The origin of dark matter, matter-anti-matter asymmetry, and inflation
aa r X i v : . [ h e p - ph ] J un The origin of dark matter, matter-anti-matter asymmetry, and inflation
Anupam Mazumdar
1, 2 Lancaster University, Physics Department, Lancaster LA1 4YB, UK Niels Bohr Institute, Blegdamsvej-17, DK-2100, Denmark.
A rapid phase of accelerated expansion in the early universe, known as inflation, dilutes all matterexcept the vacuum induced quantum fluctuations. These are responsible for seeding the initialperturbations in the baryonic matter, the non-baryonic dark matter and the observed tempera-ture anisotropy in the cosmic microwave background (CMB) radiation. To explain the universeobserved today, the end of inflation must also excite a thermal bath filled with baryons, an amountof baryon asymmetry, and dark matter. We review the current understanding of inflation, darkmatter, mechanisms for generating matter-anti-matter asymmetry, and the prospects for testingthem at ground and space based experiments.
Contents
I. Introduction II. Particle physics origin of inflation T R within MSSM 17 III. Matter-anti-matter asymmetry
IV. Dark matter
V. Acknowledgements References I. INTRODUCTION
This review aims at building a consistent picture of theearly universe where the three pillars of modern cosmol-ogy: inflation, baryogenesis and the synthesis of darkmatter can be understood in a testable framework ofphysics beyond the Standard Model (SM).Inflation (Guth, 1981), which is a rapid phase of ac-celerated expansion of space, is the leading model thatexplains the origin of matter; during this phase, primor-dial density perturbations are also stretched from sub-Hubble to super-Hubble length scales (Mukhanov et al. ,1992). A strong support for such an inflationary sce-nario comes from the precision measurement of these per-turbations in the cosmic microwave background (CMB)radiation, e.g. by the Cosmic Background Explorer(COBE) (Smoot et al. , 1992) and the Wilkinson Mi-crowave Anisotropy Probe (WMAP) (Komatsu et al. ,2011) satellites. However, one of the most seriouschallenges faced by inflationary models is that only afew of them provide clear predictions for crucial ques-tions regarding the nature of the matter created afterinflation and the mode of exiting inflation in a vac-uum that can excite the SM degrees of freedom ( d.o.f )(Mazumdar and Rocher, 2011).From observations we know that the current universecontains 4 .
6% atoms, 23% non-relativistic, non-luminousdark matter, and the rest in the form of dark energy.While some 13 . et al. , 2011). Therefore, it is manda-tory that the inflationary vacuum must excite these SMbaryons, and create the right abundance of dark mat-ter. Since the success of Big Bang Nucleosynthesis(BBN) (Iocco et al. , 2009) requires an asymmetry be-tween the baryons and anti-baryons of order one part in10 , it is necessary that the baryonic asymmetry musthave been created dynamically in the early universe be-fore the BBN (Sakharov, 1967).The prime question is what sort of visible sector be-yond the SM would accomplish all these goals – infla-tion, matter creation, and seed perturbations for theCMB. Beyond the scale of electroweak SM (at energiesabove ≥ − et al. , 2009). However the lowscale supersymmetry (SUSY) provides an excellent plat-form, which have been built on the success of the elec-troweak physics (Chung et al. , 2005; Haber and Kane,1985; Martin, 1997; Nilles, 1984). The minimal super-symmetric extension of the SM, known as MSSM, orits minimal extensions, provides many testable imprintsat the collider experiment (Nath et al. , 2010). In par-ticular, the lightest SUSY particle (LSP) can be elec-trically neutral, and will be an ideal candidate for theweakly interacting massive particle (WIMP) as a darkmatter (Ellis et al. , 1984a; Goldberg, 1983), whose abun-dance can now be calculated from the direct decay of theinflaton, or from the decay products of the inflaton, asshown in Fig. 1.If such a visible sector, with the known gauge inter-actions, can also provide us with an inflationary poten-tial capable of matching the current CMB data, thenwe would be able to identify the origin of the inflaton,its mass and couplings, and the vacuum energy den-sity within a testable theory, such as the MSSM. Theinflaton’s gauge invariant couplings would enable us toascertain the post-inflationary dynamics, and the exactmechanism for particle creation from the inflaton’s coher-ent oscillations, known as (p)reheating (Allahverdi et al. ,2010a). We would be able to precisely determine thelargest reheat temperature, T R , of the post-inflationaryuniverse, during which all the MSSM d.o.f come in chemi-cal and in kinetic equilibrium for the first time ever. Oncethe relevant d.o.f are created it would be possible to builda coherent picture where we will be able to understandthe origin of baryogenesis and the dark matter in a con-sistent framework as illustrated in Fig. 1.This review is divided into three parts. In the firstpart we will discuss the origin of inflation, and how toconnect the models of inflation to the current CMB ob-servations. We will keep our discussions general and pro-vide some examples of non-SUSY models of inflation. Wewill mainly focus on SUSY based models and its gener-alization to supergravity (SUGRA). We will discuss theepoch of reheating, preheating and thermalization for anMSSM based models of inflation. In the second partof the review, we will focus on baryogenesis. We will FIG. 1: An illustration of a visible sector model for theearly universe. EW stands for the electroweak.state the conditions for generating baryogenesis. We willdiscuss electroweak baryogenesis, baryogenesis inducedby lepton asymmetry, known as leptogenesis, and theMSSM based Afflck-Dine baryogenesis which can createnon-topological solitons, known as Q-balls. In the thirdpart, we will consider general properties of dark matter,various mechanisms for creating them, some well moti-vated candidates, and link the origin of dark matter tothe origin of inflation within SUSY. We will briefly dis-cuss the ongoing searches of WIMP as a dark mattercandidate. II. PARTICLE PHYSICS ORIGIN OF INFLATION
There are two classes of models of inflation, whichhave been discussed extensively in the literature, e.g. seereviews (Lyth and Riotto, 1999; Mazumdar and Rocher,2011). In the first class, the inflaton field belongs to thehidden sector (not charged under the SM gauge group).The direction along which the inflaton field rolls belongsto an absolute gauge singlet, whose couplings to the visi-ble sector – such as that of the SM or MSSM fields are notdetermined a-priori. A singlet inflaton would couple tothe visible and hidden sectors without any biased – suchas the case of gravity which is a true singlet, and a colorand flavor blind. In the second class, the inflaton can-didate distinctly belongs to the visible sector, where theinflaton is charged under the SM or its minimal extensionbeyond the SM gauge group. This has many advantages,which we will discuss in some details.Any inflationary models are required to be tested bythe amplitude of the density perturbations for the ob-served large scale structures (Mukhanov et al. , 1992).Therefore the predictions for the CMB fluctuations arethe most important ones to judge the merits of the mod-els, which would contain information about the powerspectrum, the tilt in the spectrum, running in the tilt,and tensor to scalar ratio. These observable quantitiescan be recast in terms of the properties of the poten-tial which we will discuss below. From the particle ori-gin point of view, one of the successful criteria is to endinflation in the right vacuum - where the SM baryonsare excited naturally for a successful baryogenesis beforeBBN (Mazumdar and Rocher, 2011).
A. Properties of inflation
The inflaton direction which leads to a graceful exitneeds to be flat with a non-negligible slope providedby a potential V ( φ ) which dominates the energy den-sity of the universe. A completely flat potential, ora false vacuum with a very tiny tunneling rate to alower vacuum, would render inflation future eternal, butnot past (Borde et al. , 2003; Borde and Vilenkin, 1994;Linde, 1983, 1986; Linde et al. , 1994, 1996). A past eter-nal inflation is possible only if the null geodesics are pastcomplete (Biswas et al. , 2010, 2006). The slow-roll infla-tion assumes that the potential dominates over the ki-netic energy of the inflaton ˙ φ ≪ V ( φ ), and ¨ φ ≪ V ′ ( φ ),therefore the Friedmann and the Klein-Gordon equationsare approximated as: H ≈ V ( φ ) / M , (1)3 H ˙ φ ≈ − V ′ ( φ ) , (2)where prime denotes derivative with respect to φ . Thereexists slow-roll conditions, which constrain the shape ofthe potential, are give by: ǫ ( φ ) ≡ M V ′ /V ) ≪ , (3) | η ( φ ) | ≡ M | V ′′ /V | ≪ . (4)These conditions are necessary but not sufficient for in-flation. The slow-roll conditions are violated when ǫ ∼ η ∼
1, which marks the end of inflation.However, there are certain models where thisneed not be true, for instance in hybrid inflationmodels (Linde, 1994), where inflation comes to anend via a phase transition, or in oscillatory mod-els of inflation where slow-roll conditions are satis-fied only on average (Damour and Mukhanov, 1998;Liddle and Mazumdar, 1998), or inflation happens inoscillations (Biswas and Mazumdar, 2009), or in fastroll inflation where the slow-roll conditions are nevermet (Linde, 2001). The K-inflation where only the ki-netic term dominates where there is no potential atall (Armendariz-Picon et al. , 1999).One of the salient features of the slow-roll inflationis that there exists a late time attractor behavior, suchthat the evolution of a scalar field after sufficient e-foldings become independent of the initial conditions(Salopek and Bond, 1990).The number of e-foldings between, t , and the end ofinflation, t end , is defined by: N ≡ ln a ( t end ) a ( t ) = Z t end t Hdt ≈ M Z φφ end VV ′ dφ , (5) where φ end is defined by ǫ ( φ end ) ∼
1, provided infla-tion comes to an end via a violation of the slow-rollconditions. The number of e-foldings can be relatedto the Hubble crossing mode k = a k H k by comparingwith the present Hubble length a H . The final result is(Liddle and Leach, 2003) N ( k ) = 62 − ln ka H − ln 10 GeV V / k + ln V / k V / end −
13 ln V / end ρ / R , (6)where the subscripts end (R) refer to the end of inflation(end of reheating). Today’s Hubble length would cor-respond to N Q ≡ N ( k = a H ) number of e-foldings,whose actual value would depend on the equation ofstate, i.e. ω = p/ρ ( p denotes the pressure, ρ denotesthe energy density), from the end of inflation to radi-ation and matter dominated epochs. A high scale in-flation with a prompt reheating with relativistic specieswould yield approximately, N Q ≈ −
60. A significantmodification can take place if the scale of inflation islow (Arkani-Hamed et al. , 2000; Green and Mazumdar,2002; Lyth and Stewart, 1995, 1996; Mazumdar, 1999;Mazumdar and Perez-Lorenzana, 2001).
B. Density Perturbations
1. Scalar perturbations
Small inhomogeneities in the scalar field, φ ( ~x, t ) = φ ( t )+ δφ ( ~x, t ), such that δφ ≪ φ , induce perturbations inthe background metric, but the separation between thebackground metric and a perturbed one is not unique.One needs to choose a gauge. A simple choice would be tofix a gauge where the non-relativistic limit of the full per-turbed Einstein equation can be recast as a Poisson equa-tion with a Newtonian gravitational potential, Φ. Theinduced metric can be written as, e.g. (Mukhanov et al. ,1992): ds = a ( τ ) (cid:2) (1 + 2Φ) dτ − (1 − δ ik dx i dx k (cid:3) , (7)Only in the presence of Einstein gravity and when thespatial part of the energy momentum tensor is diagonal,i.e. δT ij = δ ij , it follows that Φ = Ψ.During inflation the massless inflaton (with masssquared: m ∼ V ′′ ≪ H ) perturbations, δφ , arestretched outside the Hubble patch. One can track theirperturbations from a sub-Hubble to that of a super-Hubble length scales. Right at the time when the wavenumbers are crossing the Hubble patch, one finds a solu-tion for δφ as h| δφ k | i = ( H ( t ∗ ) / k ) , (8)where t ∗ denotes the instance of Hubble crossing. Onecan define a power spectrum for the perturbed scalar field P φ ( k ) = k π h| δφ k | i = (cid:20) H ( t ∗ )2 π (cid:21) ≡ (cid:20) H π (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = aH . (9)Note that the phase of δφ k can be arbitrary, and there-fore, inflation has generated a Gaussian perturbation.Now, one has to calculate the power spectrum for themetric perturbations. For a critical density universe δ k ≡ δρρ (cid:12)(cid:12)(cid:12)(cid:12) k = − (cid:18) kaH (cid:19) Φ k , (10)where Φ k ( t ) ≈ (3 / H ( δφ k / ˙ φ ) | k = aH . Therefore, one ob-tains: δ k ≡ P Φ ( k ) = 49 925 (cid:18) H ˙ φ (cid:19) (cid:18) H π (cid:19) , (11)where the right hand side can be evaluated at the time ofhorizon exit k = aH . The temperature anisotropy seenby the observer in the matter dominated epoch is propor-tional to the Newtonian potential, ∆ T k /T = − (1 / k .Besides tracking the perturbations in the longitudinalgauge with the help of Newtonian potential, there ex-ists another useful gauge known as the comoving gauge.By definition, this choice of gauge requires a comov-ing hypersurface on which the energy flux vanishes,and the relevant perturbation amplitude is known asthe comoving curvature perturbation, ζ k (Lukash, 1980;Mukhanov et al. , 1992). For the super-Hubble modes, k →
0, the comoving curvature perturbation, ζ k is aconserved quantity, and it is proportional to the New-tonian potential, ζ k = − (5 / k . Therefore, δ k canalso be expressed in terms of curvature perturbations(Liddle and Lyth, 1993, 2000) δ k = 25 (cid:18) kaH (cid:19) ζ k , (12)and the corresponding power spectrum δ k =(4 / P ζ ( k ) = (4 / H/ ˙ φ ) ( H/ π ) . With thehelp of the slow-roll equation 3 H ˙ φ = − V ′ , and thecritical density formula 3 H M = V , one obtains δ k ≈ π M V V ′ = 1150 π M Vǫ P ζ ( k ) = 124 π M Vǫ , (13)where we have used the slow-roll parameter ǫ ≡ ( M / V ′ /V ) . The COBE satellite measured theCMB anisotropy and fixes the normalization of P ζ ( k ) onvery large scales. If we assume that the primordial spec-trum can be approximated by a power law (ignoring thegravitational waves and the k − dependence of the power n s ) (Komatsu et al. , 2009) P ζ ( k ) ≃ (2 . ± . × − (cid:18) kk (cid:19) n s − , (14)where n s is called the spectral index (or spectral tilt),the reference scale is: k = 7 . a H ∼ .
002 Mpc − , andthe error bar on the normalization is given at 1 σ , and n s ( k ) = 0 . ± .
13 (15) It is important to stress that these central values anderror bars vary significantly when other parameters areintroduced to fit the data, in part because of degenera-cies between parameters (in particular n s with Ω b h , theoptical depth τ , its running, the tensor-to-scalar ratio, r ,and the fraction of cosmic strings). The spectral index n ( k ) is defined as n ( k ) − ≡ d ln P ζ d ln k . (16)This definition is equivalent to the power law behaviorif n ( k ) is close to a constant quantity over a range of k of interest. One particular value of interest is n s ≡ n ( k ). If n s = 1, the spectrum is flat and known asHarrison-Zeldovich spectrum (Harrison, 1970; Zeldovich,1970). For n s = 1, the spectrum is tilted, and n s > n s <
1) is known as a blue (red) spectrum. In the slow-roll approximation, this tilt can be expressed in terms ofthe slow-roll parameters and at first order: n s − − ǫ + 2 η + O ( ǫ , η , ǫη, ξ ) , (17)where ξ ≡ M V ′ (d V / d φ ) V , σ ≡ M V ′ (d V / d φ ) V . (18)The running of these parameters are givenby (Salopek and Bond, 1990). Since the slow-rollinflation requires that ǫ ≪ , | η | ≪
1, therefore naturallypredicts small variation in the spectral index within∆ ln k ≈ n ( k )d ln k = − ǫη + 24 ǫ + 2 ξ . (19)It is possible to extend the calculation of metric pertur-bation beyond the slow-roll approximations based on aformalism similar to that developed in Refs. (Kolb et al. ,1995; Mukhanov, 1985, 1989; Sasaki, 1986).
2. Multi-field perturbations
Inflation can proceed along many flat directions withmany light fields. Their perturbations can be trackedconveniently in a comoving gauge, on large scales ζ = − Hδφ/ ˙ φ remains a good conserved quantity, providedeach field follow slow-roll condition. The comoving cur-vature perturbations can be related to the number of e-foldings, N , given by (Salopek, 1995; Sasaki and Stewart,1996) ζ = δN = ( ∂N /∂φ a ) δφ a , (20)where N is measured by a comoving observer while pass-ing from flat hypersurface (which defines δφ ) to thecomoving hypersurface (which determines ζ ). The re-peated indices are summed over and the subscript a denotes a component of the inflaton (Lyth and Liddle,2009; Lyth and Riotto, 1999). If the random fluctuationsalong δφ a have an amplitude ( H/ π ) , one obtains: δ k = 425 P ζ = V π M ∂N∂φ a ∂N∂φ a . (21)For a single component ∂N/∂φ ≡ ( M − V /V ′ ), and thenEq. (21) reduces to Eq. (13). By using slow-roll equationswe can again define the spectral index n − − M V ,a V ,a V − M N ,a N ,a + 2 M N ,a N ,b V ,ab V N ,c N ,c , (22)where V ,a ≡ ∂V /∂φ a , and similarly N ,a ≡ ∂N/∂φ a . Fora single component we recover Eq. (17) from Eq. (22).In the case of multi-fields, one has to distinguish adi-abatic from isocurvature perturbations. Present CMBdata rules out pure isocurvature perturbation spec-trum (Beltran et al. , 2004; Komatsu et al. , 2009), al-though a mixture of adiabatic and isocurvature pertur-bations remains a possibility.
3. Gravitational waves
During inflation stochastic gravitational waves are ex-pected to be produced similar to the scalar perturbations(Allen, 1988; Grishchuk, 1975; Grishchuk and Sidorov,1989; Sahni, 1990). For reviews on gravitational waves,see (Maggiore, 2000; Mukhanov et al. , 1992). The grav-itational wave perturbations are described by a line ele-ment ds + δds , where ds = a ( τ )(d τ − d x i d x i ) , δds = − a ( τ ) h ij d x i d x j . (23)The gauge invariant and conformally invariant 3-tensor h ij is symmetric, traceless δ ij h ij = 0, and divergenceless ∇ i h ij = 0 ( ∇ i is a covariant derivative). Massless spin 2gravitons have two transverse degrees of freedom ( d.o.f )For the Einstein gravity, the gravitational wave equa-tion of motion follows that of a massless Klein Gordonequation (Mukhanov et al. , 1992). Especially, for a flatuniverse ¨ h ij + 3 H ˙ h ij + (cid:0) k /a (cid:1) h ij = 0 , (24)As any massless field, the gravitational waves also feelthe quantum fluctuations in an expanding background.The spectrum mimics that of Eq. (9) P grav ( k ) = 2 M (cid:18) H π (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k = aH . (25)Note that the spectrum has a Planck mass suppression,which suggests that the amplitude of the gravitationalwaves is smaller compared to that of the scalar perturba-tions. Therefore it is usually assumed that their contribu-tion to the CMB anisotropy is small. The corresponding spectral index can be expanded in terms of the slow-rollparameters at first order as r ≡ P grav P ζ = 16 ǫ , n t = d ln P grav ( k )d ln k ≃ − ǫ, . (26)Note that the tensor spectral index is negative. It isexpected that PLANCK could detect gravity waves if r > ∼ .
1, however the spectral index will be hard to mea-sure in forthcoming experiments. The primordial gravitywaves can be generated for large field value inflationarymodels. Using the definition of the number of e-foldingsit is possible to derive the range of ∆ φ (Hotchkiss et al. ,2008; Lyth, 1997; Lyth and Liddle, 2009))16 ǫ = r < .
003 (50 /N ) (∆ φ/M P ) . (27)Note that it is possible to get sizable, r , for∆ φ ≪ M P in assisted inflation (discussed be-low), and in inflection point inflation discussed inRef. (Ben-Dayan and Brustein, 2010). If the tensor-to-scalar ratio r and/or a running α s are introduced,the best fit for n s and error bars (at 1 σ ) n s =1 . +0 . − . , α s = − . ± . n s = 0 . ± . , r < .
22 (at 2 σ ), n s = 1 . +0 . − . , r < .
55 (at 2 σ ) , α s = − . ± .
028 (Komatsu et al. ,2009). These data therefore suggest that a red spectrumis favored ( n s = 1 excluded at 2 . σ from WMAP andat 3 . σ when other data sets are included) if there is norunning. C. Generic models of inflation
1. High scale models of inflation
The most general form for the potential of a gaugesinglet scalar field φ contains an infinite number of terms, V = V + ∞ X α =2 λ α M α − φ α . (28)The renormalizable terms allows to prevent all terms with α ≥
4. By imposing the parity Z , under which φ → − φ ,allows to prevent all terms with α odd. Most phenomeno-logical models of inflation proposed initially assume thatone or two terms in Eq. (28) dominate over the others,though some do contain an infinite number of terms. a. Power-law chaotic inflation:
The simplest inflationmodel by the number of free parameters is perhaps thechaotic inflation (Linde, 1983) with the potential domi-nated by only one of the terms in the above series V = λ α M α − φ α , (29)with α a positive integer. The first two slow-roll param-eters are given by ǫ = α M φ , η = α ( α − M φ . (30)Inflation ends when ǫ = 1, reached for φ e = αM P / √ φ Q = p N Q αM P , which is super Planckian; thisis the first challenge for this class of models. The spectralindex for the scalar and tensor to scalar ratio read: n s = 1 − α N Q + α/ , r = 4 αN Q + α/ . (31)The amplitude of the density perturbations, if normal-ized at the COBE scale, yields to extremely small cou-pling constants; λ α ≪ λ ≃ . × − ).The smallness of the coupling, λ α /M α − , is often con-sidered as an unnatural fine-tuning. Even when dimen-sion full, for example if α = 2, the generation (andthe stability) of a mass scale, √ λ M P ≃ GeV,is a challenge in theories beyond the SM, as they re-quire unnatural cancellations. These class of mod-els have an interesting behavior for initial conditionswith a large phase space distribution where there ex-ists a late attractor trajectory leading to an end of in-flation when the slow-roll conditions are violated closeto the Planck scale (Brandenberger and Kung, 1990;Kofman et al. , 2002; Linde, 1983, 1985).Note that the above mentioned monomial potentialcan be a good approximation to describe in a certainfield range for various models of inflation proposed andmotivated from particle physics; natural inflation whenthe inflaton is a pseudo-Goldstone boson (Freese et al. ,1990), or the Landau-Ginzburg potential when the in-flaton is a Higgs-type field (Bezrukov and Shaposhnikov,2008). The necessity of super Planckian VEVs representsthough a challenge to such embedding in particle physicsand supergravity (SUGRA). b. Exponential potential:
An exponential potentialalso belongs to the large field models: V ( φ ) = V exp (cid:18) − r p φM P (cid:19) . (32)It would give rise to a power law expansion a ( t ) ∝ t p , sothat inflation occurs when p >
1. The case p = 2 cor-responds to the exactly de Sitter evolution and a neverending accelerated expansion. Even for p = 2, viola-tion of slow-roll never takes place, since ǫ ( φ ) = 1 /p andinflation has to be ended by a phase transition or grav-itational production of particles (Copeland et al. , 2001;Lyth and Riotto, 1999).The confrontation to the CMB data yields: n s = 1 − /p and r = 16 /p ; the model predicts a hight tensor toscalar ratio and it is within the one sigma contour-plotof WMAP (with non-negligible r ) for p ∈ [73 −
2. Assisted inflation
Many heavy fields could collectively assist inflation byincreasing the effective Hubble friction term for all theindividual fields (Liddle et al. , 1998). This idea can be illustrated with the help of ′ m ′ identical scalar fields withan exponential potentials, see Eq. (32), where now φ −→ φ i , where i = 1 , , · · · , m . For a particular solution;where all the scalar fields are equal: φ = φ = · · · = φ m . H = 13 M m [ V ( φ ) + ˙ φ / , (33)¨ φ = − H ˙ φ − dV ( φ ) /dφ . (34)These can be mapped to the equations of a modelwith a single scalar field ˜ φ by the redefinitions ˜ φ = m φ ; ˜ V = m V ; ˜ p = mp , so the expansion rateis a ∝ t ˜ p , provided that ˜ p > /
3. The expansion becomesquicker the more scalar fields there are. In particular, po-tentials with p <
1, which for a single field are unable tosupport inflation, can do so as long as there are enoughscalar fields to make mp > N = − R H dt , we have P i ∂N∂φ i ˙ φ i = − H ,we yield: P ζ = ( H/ π ) (1 /m )( H / ˙ φ ).Note that this last expression only contains one ofthe scalar fields, chosen arbitrarily to be φ . Theestimation for the spectral tilt is given by : n − − /mp , which matches that produced by a sin-gle scalar field with ˜ p = mp . The more scalar fieldsthere are, the closer to scale-invariance is the spec-trum that they produce. The above calculation canbe repeated for arbitrary slopes, p i . In which case thespectral tilt would have been given by n = 1 − / ˜ p ,where ˜ p = P p i . The above scenario has been gen-eralized to study arbitrary exponential potentials withcouplings, V = P n z s exp( P m α sj φ j ) (Copeland et al. ,1999; Green and Lidsey, 2000). a. Assisted chaotic inflation:
Multi-scalarfields of chaotic type has interesting proper-ties (Jokinen and Mazumdar, 2004): V ∼ X i f (cid:0) φ ni /M n − (cid:1) (35)(for n ≥ φ i ≪ M P , below the Planckscale (Kanti and Olive, 1999a,b). The effective slow-roll parameters are given by: ǫ eff = ǫ/m ≪ | η eff | = | η | /m ≪
1, where ǫ, η are the slow-roll parame-ters for the individual fields. Inflation can now occur forfield VEVs (Jokinen and Mazumdar, 2004):∆ φM P ∼ (cid:18) m (cid:19) (cid:18) N Q (cid:19) (cid:16) ǫ eff (cid:17) / ≪ , (36)where N Q is the number of e-foldings. Obviously, all theproperties of chaotic inflation can be preserved at VEVs ≪ M P , including the prediction for the tensor to scalarration for the stochastic gravity waves, i.e. r = 16 ǫ eff .For ǫ eff ∼ .
01 and m ∼ n s − − ǫ eff + 2 η eff , and large tensor toscalar ratio, i.e. r = 0 . b. N-flation:
Amongst various realizations of assistedinflation, N-flation is perhaps the most interesting one.The idea is to have N ∼ M P /f ) ∼ number ofaxions, where f is the axion decay constant, of order f ∼ . M − drive inflation simultaneously with a leadingorder potential (Dimopoulos et al. , 2005): V = V + X i Λ i cos( φ i /f i ) + ... (37)where φ i are axion fields correspond to the partners ofK¨ahler moduli. The ellipses contain higher order con-tributions. In a certain Type-IIB compactification, it isassumed that all the moduli are heavy and thus stabi-lized by prior dynamics, including that of the volumemodulus. Only the axions of T i = φ i /f i + iM s R i arelight (Dimopoulos et al. , 2005). The assumption of de-coupling the dynamics of K¨ahler modulus from the axionsis still a debatable issue, see (Kallosh, 2008). After rear-ranging the potential for the axions, and expanding themaround their minima for a canonical choice of the kineticterms, the Lagrangian simplifies to the lowest order inexpansion: L = 12 ∂ µ φ i ∂ µ φ j − X i m i φ i + · · · . (38)The exact calculation of m i is hard, assuming all ofthe mass terms to be the same m i ∼ GeV, and
N > ( M P /f ) , it is possible to match the current ob-servations with a tilt in the spectrum, n ∼ .
97, and large tensor to scalar ratio: r ∼ /N Q ∼ .
13 for N Q ∼
60. There are also realizations of assisted inflationvia branes (Becker et al. , 2005; Cline and Stoica, 2005;Mazumdar et al. , 2001).
3. Hybrid inflation
The end of inflation can happen via a waterfall trig-gered by a Higgs (not necessarily the SM Higgs) field cou-pled to the inflaton, first discussed in (Copeland et al. ,1994; Linde, 1991, 1994). The model is based on thepotential given by (Linde, 1991, 1994) V ( φ, ψ ) = 12 m φ + λ (cid:0) ψ − M (cid:1) + λ ′ φ ψ , (39)where φ is the inflaton and ψ is the Higgs-type field. λ and λ ′ are two positive coupling constants, m and M are two mass parameters. It is the most general form(omitting a quartic term λ ′′ φ ) of renormalizable poten-tial satisfying the symmetries: ψ ↔ − ψ and φ ↔ − φ .Inflation takes place along the ψ = 0 valley and endswith a tachyonic instability for the Higgs-type field. Thecritical point of instability occurs at: φ c = M p λ/λ ′ . (40)The system then evolves toward its true minimum at V =0, h φ i = 0, and h ψ i = ± M . The inflationary valley, for h ψ i = 0, where the last 50 −
60 e-foldings of inflation is assumed to take. This raisesthe issue of initial conditions for ( φ, ψ ) fields and the finetuning required to initiate inflation (Clesse and Rocher,2008; Lazarides and Vlachos, 1997; Mendes and Liddle,2000; Panagiotakopoulos and Tetradis, 1999; Tetradis,1998). In Ref. (Clesse and Rocher, 2008) it was foundthat when the initial VEV of the inflaton, φ ≪ M P , asubdominant but non-negligible part of the initial condi-tions for the phase space leads to a successful inflation,i.e. around less than 15% depending on the model pa-rameters. Initial conditions with super-Planckian VEVsfor φ ≫ M P automatically leads to a successful inflationsimilarly to chaotic inflation. In the inflationary valley, h ψ i = 0, the effective potential is given by: V eff ( φ ) ≃ λM m φ , (41)The model predicts a blue tilt in the spectrum, i.e. n s >
1, in the small field regime, φ Q < M P , which is slightlydisfavored by the current data.Two variations of the hybrid inflation idea were pro-posed assuming that the term φ is negligible. The two-field scalar potentials are of the form: V pq ( φ, ψ ) = M (cid:20) − (cid:18) φ ∗ φ (cid:19) p (cid:21) + λφ ψ q . (42)They share the common feature of having an infla-tionary trajectory during which h ψ i is varying andnot vanishing. For ( p, q ) = (1 , Mutated hybrid (Stewart, 1995b), and( p, q ) = (4 ,
6) corresponds to
Smooth hybrid inflation(Lazarides and Panagiotakopoulos, 1995). The latter in-volves non-renormalizable terms of order M − to keepthe potential bounded from below.The potential is valid in the large field limit φ ≫ φ ∗ ,since in the small field limit, the potential is not boundedfrom below and should be completed. For mutated, themodel predicts a red spectral index and negligible ten-sor to scalar ratio, n s − ≃ − / (8 N Q ) ≃ .
97, and r ≃ m/ (2 λN / Q ) ≪ / (8 N Q ) ∼ − , if we assume N Q ≃
60. For smooth, the end of slow-roll inflation hap-pens by a violation of the conditions; ǫ, η ≪
1, since nowaterfall transition takes place. This allows the predic-tions for the spectral index to be n s − ≃ − / (3 N Q ) ≃ .
97 (Lazarides and Panagiotakopoulos, 1995), and theratio for tensor to scalar is found to be negligible.
4. Inflection point inflation
One of the challenges for inflation is to realize inflationat low scales, preferably below M P , with the right tilt andthe amplitude of the power spectrum. Inflection pointinflation admits a large amount of flexibility in the fieldspace – similar to the analogy of a ball rolling on anelastic surface following the least action principle. Withthe help of two independent parameters, A and B , it ispossible to obtain a large range of tilt in the spectrum,while keeping the amplitude of the perturbations intact.Let us consider a simple realization of such a potential: V ( φ ) = Aφ − Cφ + Bφ , (43)where C = f ( A, B ) in order to obtain a point of inflec-tion suitable for inflation. The VEV at which inflationoccurs is intimately related to the two independent pa-rameters and can happen at wide ranging scales below M P , and for wide ranging values of ( A, B ).Here we will generalize this potential to any genericpotential V which can be written in the followingform (here ′ denotes differentiation with respect to φ ) (Enqvist et al. , 2010a; Hotchkiss et al. , 2011): V = V + a ( φ − φ ) + b φ − φ ) + c φ − φ ) + ·· (44)where V ≡ V ( φ ) , a ≡ V ′ ( φ ) , b ≡ V ′′ ( φ ) , c ≡ V ′′′ ( φ ), which is the Taylor expansion, truncated at n = 3, around a reference point φ , which we choose to bethe point of inflection where V ′′ ( φ ) = 0, or saddle pointwhere V ′′ ( φ ) = V ′ ( φ ) = 0. The higher order terms inEq. (44) can be neglected during inflation, provided that | V ′′′ | ≫ (cid:12)(cid:12)(cid:12)(cid:12) d m Vdφ m ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) | φ e − φ | m − , m ≥ , (45)where φ e corresponds to the field value at the end ofinflation. Assuming that the slow-roll parameters aresmall in the vicinity of the inflection point φ , and thatthe velocity ˙ φ is negligible, the potential energy V givesrise to a period of inflation.Inflation ends at the point φ e where | η | ∼
1. By solv-ing the equation of motion, the number of e-foldings ofinflation during the slow-roll motion of the inflaton from φ to φ e , where φ − ( φ − φ e ) < φ < φ + ( φ − φ e ), isfound to be (Enqvist et al. , 2010a) N = V M r ac [ F ( φ e ) − F ( φ )] ,F ( z ) = arccot (cid:18)r c a ( z − φ ) (cid:19) . (46)It useful to define the parameters X = aM P √ V and Y = p ca N M P X . Note that X is the square root of the slow-roll parameter ǫ at the point of inflection. The slow-rollparameters can then be recast in the following form: ǫ = 2 V c M N (cid:18) YS (cid:19) , (47) η = − N YS (cid:16) √ − X cos Y − √ X sin Y (cid:17) , (48) ξ = 2 N (cid:18) YS (cid:19) (49) where S = √ − X sin Y + √ X cos Y . One can solveEqs. (47-49), for X , Y and N in terms of the slow-roll. The equations are non-linear and in general can-not be solved analytically. However, since ǫ ≪ | η | , ξ ,one can find a closed form solution provided that V / ≤ GeV and X ≤ √ ǫ ≪ et al. ,2007c; Bueno Sanchez et al. , 2007; Enqvist et al. , 2010a;Hotchkiss et al. , 2011): P / ζ ≡ √ π V / ǫ / M = V / π √ M X sin Y , (50) n s ≡ η − ǫ = 1 − N Q Y cot Y , (51) α = − N Q (cid:18) Y sin Y (cid:19) . (52)One can derive the properties of a saddle point inflationprovided Y / sin Y →
1, and Y cot Y →
1. The modelfavors the current observations by matching the COBEnormalization and the spectral tilt ranging from n s ∈ [0 . , . N Q ∼ D. Supersymmetric models
One of the most compelling virtues of SUSY is thatit can protect the quadratically divergent contributionsto the scalar mass, which arise in one-loop computationfrom the fermion contribution and quartic self interac-tion of the scalar field. Such corrections generically spoilthe flatness of the inflaton potential. The quadratic di-vergence is independent of the mass of the scalar fieldand cancel, exactly if λ s = λ f , where λ f is the fermionYukawa and λ s is the quartic scalar coupling. Howeverthis procedure fails at 2-loops and one requires fine tun-ing of the couplings order by order in perturbation the-ory. In the case of the SM Higgs, a precision of roughlyone part in 10 is required in couplings to maintain theHiggs potential, often known as the hierarchy problem orthe naturalness problem . The electroweak symmetry isstill broken by the Higgs mechanism, but the quadraticdivergences in the scalar sector are absent. In the SUSYlimit the fermion and scalar masses are degenerate, butthe SUSY has to be broken softly at the TeV scale in sucha way that it does not spoil the solution to the hierarchyproblem, see (Chung et al. , 2005; Haber and Kane, 1985;Martin, 1997; Nilles, 1984).The matter fields for N = 1 SUSY are chiral super-fields Φ = φ + √ θψ + θθF , which describe a scalar φ ,a fermion ψ and a scalar auxiliary field F . The SUSYscalar potential V is the sum of the F - and D -terms: V = X i | F i | + 12 X a g a D a D a ,F i ≡ ∂W∂φ i , D a = φ † T a φ , (53)where W is the superpotential, and φ i transforms undera gauge group G with the generators of the Lie algebragiven by T a . Note that all the kinetic energy terms areincluded in the D -terms. For inflation, the effects of su-pergravity (SUGRA) becomes important. At tree level, N = 1 SUGRA potential is given by the sum of F and D -terms, see (Nilles, 1984) V = e K ( φi,φ ∗ i ) M (cid:20)(cid:0) K − (cid:1) ji F i F j − | W | M (cid:21) + g f − ab ˆ D a ˆ D b , (54) F i = W i + K i WM , ˆ D a = − K i ( T a ) ji φ j + ξ a . (55)where we have added the Fayet-Iliopoulos contribution ξ a to the D -term, and ˆ D a = D a /g a , where g a is gaugecoupling. Here K ( φ i , φ ∗ i ) is the K¨ahler potential, whichis a function of the fields φ i , and K i ≡ ∂K/∂φ i . Inthe simplest case, at tree-level K = φ ∗ i φ i (and K ji =( K − ) ji = δ ji ). In general the K¨ahler potential can beexpanded as: K = φ i φ ∗ i + ( k ijk φ i φ j φ ∗ k + c.c. ) /M P +( k ijkl φ i φ j φ ∗ k φ ∗ l φ ∗ k φ ∗ l + k ijkl φ i φ j φ k φ ∗ l + c.c. ) /M + · · · ).The kinetic terms for the scalars take the form: ∂ K∂φ i ∂φ ∗ j D µ φ i D µ φ ∗ j . (56)The real part of the gauge kinetic function matrix is givenby Re f ab . In general, f ab = δ ab (1 /g a + f ia φ i /M P + · · · ).The gauginos masses are typically given by m λ a =Re[ f ia ] h F i i / M P . For a universal gaugino masses, f ia arethe same for all the three gauge groups of MSSM. In thesimplest case, it is just a constant, f ab = δ ab /g a , and thekinetic terms for the gauge potentials, A aµ , are given by:14 (Re f ab ) F aµν F µνa . (57)SUGRA will be broken if one or more of the F i obtain aVEV. The gravitino, spin ± / h V i = 0, as a constraintto obtain the zero cosmological constant, one obtains m / = h K ij F i F ∗ j i M = e h K i /M |h W i| M . (58)
1. F-term inflation
The most well-known model of SUSY inflation drivenby F -terms is of the hybrid type and based on the su-perpotential (Copeland et al. , 1994; Dvali et al. , 1994;Linde and Riotto, 1997) W = κS (ΦΦ − M ) . (59)where, S is an absolute gauge singlet, while Φ and Φ aretwo distinct superfields belonging to complex conjugate representation, and κ is an arbitrary constant fixed by theCMB observations. It is desirable to obtain an effectivesinglet S superfield arising from a higher gauge theorysuch as GUT (Langacker, 1981), however to our knowl-edge it has not been possible to implement this idea, seethe discussion in (Mazumdar and Rocher, 2011). Typ-ically S would have other (self)couplings which wouldeffectively ruin the flatness required for hybrid inflation.This form of potential is protected from additionaldestabilizing contributions with higher power of S , if S ,Φ and Φ carrying respectively the charges +2, α and − α under R-parity. Then W carries a charge +2 so that theaction S = R d θ W + . . . is invariant.The tree level scalar potential derived from Eq. (59)reads V tree ( S, φ, φ ) = κ | M − φφ | + κ | S | ( | φ | + | φ | ) , (60)where we have denoted by S, φ, φ the scalar componentsof S, Φ , Φ. Note the similarity between Eq. (39) andEq. (60), where m = 0, and both λ and λ ′ are replaced byonly κ . We will also assume φ ∗ = φ along this direction,and the kinetic terms for the superfields are minimal, i.e.with a k¨ahler potential: K = | S | + | Φ | + | Φ | .Let us define two effective real scalar fields canonicallynormalized, σ ≡ √ S ), and ψ ≡ V tree ( σ, ψ ) = κ (cid:18) M − ψ (cid:19) + κ σ ψ . (61)The global minimum of the potential is located at S = 0, φφ = M . At large VEVs, S > S c ≡ M , the potentialalso possesses a local valley of minima (at h ψ i = 0) inwhich the field σ , now rolls on with V tree = V ≡ κ M .This non-vanishing value of the potential both sustainthe inflationary dynamics and induces a SUSY breaking.This induces a splitting in the mass of the fermionicand bosonic components of the superfields Φ and Φ,with m B ( S ) = κ | S | ± κ M and m F = κ | S | .Note that this description is valid only as long as S is sufficiently slow-rolling such that κ | S | | Φ | can beconsidered as a mass term. Therefore radiative cor-rections do not exactly cancel out (Dvali et al. , 1994;Lazarides, 2000), and provide a one-loop potential whichcan be calculated by using the Coleman-Weinberg for-mula (Coleman and Weinberg, 1973), V − loop ( φ ) = V inf ( φ ) + ∆ V ∆ V = 164 π X i ( − ) F i M i ( φ ) ln M i ( φ ) Λ( φ ) , (62)where V inf is now the renormalized potential, Λ( φ ) isthe renormalization mass scale. The sum extends overall helicity states i , F i is the fermion number, and M ( φ )is the mass of the i-th state. One obtains: V − loop ( S ) = κ N M π (cid:20) s κ Λ + ( z + 1) ln(1 + z − )+( z − ln(1 − z − ) (cid:3) , (63)0where z = | S | /M ≡ x , Λ represents a non-physicalenergy scale of renormalization and N denotes the di-mensionality. Note that the perturbative approach ofColeman and Weinberg breaks down when close to theinflection point at z ≃
1. For small coupling κ , the slow-roll conditions (for η ) are violated infinitely close to thecritical point, z = 1, which ends inflation.The normalization to COBE allows to fix the scale M as a function of κ . If the breaking of G does not pro-duce cosmic strings, the contribution to the quadrupoleanisotropy simply comes from the inflationary contribu-tion (see Eq. (13)) and the observed value can be ob-tained even with a coupling κ close to unity (Dvali et al. ,1994). Small values of κ can render the scale of infla-tion very low, as low as the TeV scale (Bastero-Gil et al. ,2003; Bastero-Gil and King, 1998; Randall et al. , 1996;Randall and Thomas, 1995).However it has been shown that the formationof cosmic strings at the end of F -term inflation ishighly probable when the model is embedded in SUSYGUTs (Jeannerot et al. , 2003). In this case, the normal-ization to COBE receives two contributions, one frominflation and other from cosmic strings (Jeannerot, 1997;Rocher and Sakellariadou, 2005a), which affects the rela-tion M ( κ ) at large κ , and imposes new stringent boundson M < ∼ × GeV, and (Jeannerot and Postma, 2005;Rocher and Sakellariadou, 2005b) κ < ∼ × − (126 /N Q ) , (64)by demanding that the cosmic strings cam at bestcontribute less than < ∼
10% of isocurvature fluctua-tions (Bevis et al. , 2008). Once M is fixed, the spec-tral index n s can be computed as the range is foundto be: n s ∈ [0 . ,
1] whether cosmic strings formor not (Jeannerot and Postma, 2005; Senoguz and Shafi,2003), and by including the soft-SUSY breaking termswithin minimal kinetic terms in the K¨ahler potential, thespectral index can be brought down to 0 . ≤ n s ≤ .
008 (Rehman et al. , 2009).
2. SUGRA corrections and solutions
For inflaton VEVs below the Planck scale, the SUGRAeffects can become important and may ruin the flatness ofthe potential. The N = 1 SUGRA potential is now givenby Eq. (54), where the F -terms containing an additionalexponential factor. Various cross terms between theK¨ahler and the superpotential leads to the soft breakingmass term for the light scalar fields (Bertolami and Ross,1987; Copeland et al. , 1994; Dine et al. , 1984, 1995b,1996b; Linde and Riotto, 1997) m SUGRA ∼ m susy + V M ∼ O (1) H , (65)where m susy ∼ O (100) GeV contains soft-SUSY breakingmass term for the low scale SUSY breaking scenarios. Once the inflaton gets a mass ∼ H , the contribution tothe second slow-roll parameter η becomes order unityand the slow roll inflation ends, i.e. | η | ≡ M V ′′ /V ∼ m SUGRA /H ∼ O (1). This is known as the SUGRA- η problem.When there are more than one chiral superfields, asin the F -term hybrid model, it can be possible to cancelthe dominant O (1) H correction to the inflaton mass bychoosing an appropriate K¨ahler term (Copeland et al. ,1994; Stewart, 1995a). For non-minimal K¨ahler poten-tials, such as K = | S | + | Φ | + | Φ | + κ S | S | /M + . . . , (66)the kinetic terms K ij ∂ µ Φ i ∂ µ Φ ∗ j are non-minimal because K ij = δ ij . One obtains: ( ∂ SS ∗ K ) − ∼ − κ S | S | /M + . . . One again obtains a problematic contribution to theinflaton mass, i.e. κ S ×O (1) H . Several mechanisms havebeen proposed to tackle this η -problem. One can impose, κ S ∼ − , which is sufficient to keep the slow roll infla-tion safe, but without much physical motivation. For ageneric inflationary model it is not possible to compute κ s at all. a. Shift and Heisenberg symmetry:
Safe non-minimal K¨ahler potentials have also been pro-posed (Antusch et al. , 2009b; Bastero-Gil and King,1999; Brax and Martin, 2005; Pallis, 2009) making useof the shift symmetry. Under this symmetry, a superfield S → S + iC , where C is a constant. (Kawasaki et al. ,2000, 2001) to protect the K¨ahler potential of theform K (Φ , ¯Φ) → K (Φ + ¯Φ). This symmetry generatesan exactly flat direction for an inflaton field and anon-invariance of the superpotential induces some slopeto its potential to allow slow-roll at the loop level.Another symmetry - the Heisenberg symmetry - hasalso been invoked to protect the form of the K¨ahlerpotential (Antusch et al. , 2009a), where the effectiveK¨ahler is a no-scale type of the form K = ln(Φ i ).This solves the SUGRA- η -problem by canceling theexponential term exp( K ). However note that K¨ahlerpotentials generically obtains quantum correctionsunlike the non-renormalization theorem which can onlyprotect the superpotential terms (Grisaru et al. , 1979).Such corrections are hard to compute without knowingthe ultra-violet completion, and the exact matter sectorfor the inflationary model (Berg et al. , 2005a,b, 2006).Note that none of these papers considered MSSM mattersector. b. Inflection point inflation:
For any smooth poten-tial, it is possible to drive inflation near the saddlepoint, V ′ = 0 , V ′′ = 0 , V ′′′ = 0, or near the pointof inflection, V ′ = 0 , V ′′ = 0 , V ′′′ = 0. Theseare special points where the effective mass term ofthe inflaton vanishes and the potential does not suf-fer through SUGRA- η problem (Allahverdi et al. , 2007c,2006; Mazumdar et al. , 2011). In the saddle point case1FIG. 2: Regions of parameter space (green) for thepotential in Eq. (67) that satisfy the WMAP 7-yearconstraints on ( δ H , n s ). Contour lines of | β CMB | areshown in black, for the values of | β CMB | indicated for n = 6 case. For O (1) H correction to the inflaton massthe value of β ∼ − (Mazumdar et al. , 2011).the potential can be made so flat that inflation can bedriven eternally (Allahverdi et al. , 2006, 2007d).From the low-energy perspective, the most generic anddangerous SUGRA corrections to the inflaton potential( with minimal and non-minimal K¨ahler potentials for φ ) would have a large vacuum energy contribution. Tocomplicate further, one may even assume that the flat-ness of φ is lifted by non-renormalizable contribution tothe potential (Mazumdar et al. , 2011): V ( φ ) = V c + c H H | φ | − a H HnM n − P φ n + | φ | n − M n − P , (67)where V c = 3 H M . As mentioned above the interestingobservation is that, in fact, there always exists a rangeof field values, ∆ φ , for which a full potential admits a point of inflection with all known sources of correctionstaken into account. Now, all the uncertainties in thecorrections to the K¨ahler potential can be absorbed in thefull potential, such that the flat region admits a slow rollinflation with ∆ φ ≫ φ (Mazumdar et al. , 2011). Thecondition for this inflection point is a H ≈ n − c H ,where we characterize the fine-tuning by β defined as: a H n − c H = 1 − ( n − β . (68)When | β | is small, a point of inflection φ exists such that V ′′ ( φ ) = 0, with φ = (cid:18)r c H n − HM n − (cid:19) n − . (69)We can Taylor expand the potential about φ as discussedin section II.C.4, and analyze the CMB constraints asshown in Fig. 2.
3. D-term inflation
In Refs. (Bin´etruy and Dvali, 1996; Halyo, 1996;Stewart, 1995a), it was noticed that a perfectly flat in- flaton potentials can be constructed using a constantcontribution coming from the D -term. In addition, theSUGRA- η problem arising in F -term models does not ap-pear for D -terms driven inflation because the D -sector ofthe potential does not receive exponential contributionsfrom non-minimal SUGRA. The model however requiresthe presence of a Fayet-Iliopoulos (FI) term ξ , and there-fore a U (1) ξ symmetry which generates it. For a K¨ahlerpotential K (Φ m , Φ n ), the D -terms D a = − g a [ D a = φ i ( T a ) ij K j + ξ a ](where K m ≡ ∂K/∂ Φ m ) give rise to a scalar potential: V ( φ, φ ∗ ) = 12 [Re f ( φ )] − X D a D a + F − terms (70)where g a and T a are respectively the gauge coupling con-stants and the generators of each factors of the symmetryof the action, ′ a ′ running over all factors of the symmetry,and f ( φ ) is the gauge kinetic function. If this symme-try contains a factor U (1) ξ , the most general action thenallows for the presence of a constant contribution ξ .The simplest realization of D -term inflation repro-duces the hybrid potential with three chiral superfields, S , φ + , and φ − with non-anomalous U (1) ξ (an abeliantheory is said to be anomalous if the trace of thegenerator is non-vanishing P q n = 0) charges q n =0 , +1 , − W D = λSφ + φ − . (71)In what follows, we assume the minimal structure for f (Φ i ) (i.e., f (Φ i )=1) and take the minimal K¨ahler po-tential, i.e. K = | φ − | + | φ + | + | S | .. Then the scalarpotential reads V D − SUGRAtree = λ exp (cid:18) | φ − | + | φ + | + | S | M (cid:19)(cid:20) | φ + φ − | (cid:18) | S | M (cid:19) + | φ + S | (cid:18) | φ − | M (cid:19) + | φ − S | (cid:18) | φ + | M (cid:19) + 3 | φ − φ + S | M (cid:21) + g ξ (cid:0) | φ + | − | φ − | + ξ (cid:1) , (72)where g ξ is the gauge coupling of U (1) ξ . The globalminimum of the potential is obtained for h S i = 0 and h Φ − i = √ ξ , which is SUSY preserving but induces thebreaking of U (1) ξ . For S > S inst ≡ g ξ √ ξ/λ the potentialis minimized for | φ + | = | φ − | = 0 and therefore, at thetree level, the potential exhibits a flat inflationary valley,with vacuum energy V = g ξ ξ / + and Φ − , because of the tran-sient D -term SUSY breaking. The radiative cor-2rections are given by the Coleman-Weinberg expres-sion (Coleman and Weinberg, 1973) and the full poten-tial inside the inflationary valley reads V D − SUGRAeff = g ξ ξ ( g ξ π " λ | S | Λ exp (cid:18) | S | M (cid:19) +( z + 1) ln(1 + z − ) + ( z − ln(1 − z − ) , (73)with z = ( λ | S | /g ξ ξ ) e | S | /M . Inflation ends when theslow-roll conditions break down, that is for z end ≃
1, andthe predictions for the inflationary parameters are verysimilar to the previous discussion on F -term inflation. E. MSSM flat direction inflation
So far we have discussed inflationary models where theinflaton sector belongs to the hidden sector (not chargedunder the SM gauge group), such models will have atleast one SM gauge singlet component, whose couplingsto other fields and mass are chosen just to match theCMB observations. These models are simple but lackproper embedding within MSSM or its minimal exten-sions.In order to construct a predictable hidden sector modelof inflation, one must know all the inflaton couplings tothe hidden and visible matter. One such unique modelhas been constructed within type IIB string theory, whereit was found that all the inflaton energy is transferredto exciting the hidden matter (Cicoli and Mazumdar,2010a,b), and the universe could be prematurely dom-inated by the hidden sector dark matter. Such obstaclesdo not arise if the last phase of inflation occurs withinMSSM.
1. Introducing MSSM and its flat directions
In addition to the usual quark and lepton superfields,MSSM has two Higgs fields, H u and H d . Two Hig-gses are needed because H † is forbidden in the super-potential. The superpotential for the MSSM is given by,see (Chung et al. , 2005; Haber and Kane, 1985; Martin,1997; Nilles, 1984) W MSSM = λ u QH u u + λ d QH d d + λ e LH d e + µH u H d , (74)where H u , H d , Q, L, u, d, e in Eq. (74) are chiral super-fields, and the dimensionless Yukawa couplings λ u , λ d , λ e are 3 × H u , H d , Q, L fields are SU (2) doublets, while u, d, e are SU (2) sin-glets. The last term is the µ term, which is a SUSY ver-sion of the SM Higgs boson mass. Terms proportional to H ∗ u H u or H ∗ d H d are forbidden in the superpotential, since W MSSM must be analytic in the chiral fields. H u and H d are required not only because they give masses to all thequarks and leptons, but also for the cancellation of gaugeanomalies. The Yukawa matrices determine the massesand CKM mixing angles of the ordinary quarks and lep-tons through the neutral components of H u = ( H + u , H u )and H d = ( H d H − d ). Since the top quark, bottom quarkand tau lepton are the heaviest fermions in the SM, weassume that only the third family, (3 ,
3) element of thematrices λ u , λ d , λ e are important.The µ term provides masses to the Higgsinos L ⊃ − µ ( ˜ H + u ˜ H − d − ˜ H u ˜ H d ) + c.c , (75)and contributes to the Higgs ( mass ) terms in the scalarpotential through − L ⊃ V ⊃ | µ | ( | H u | + | H + u | + | H d | + | H − d | ) . (76)Note that Eq. (76) is positive definite. Therefore, it can-not lead to electroweak symmetry breaking without in-cluding SUSY breaking ( mass ) soft terms for the Higgsfields, which can be negative. Hence, | µ | should almostcancel the negative soft ( mass ) term in order to allowfor a Higgs VEV of order ∼
174 GeV. That the twodifferent sources of masses should be precisely of sameorder is a puzzle for which many solutions has been sug-gested (Casas and Munoz, 1993; Giudice and Masiero,1988; Kim and Nilles, 1984).Within MSSM one can construct gauge invariant D -and F -flat directions, for the list of MSSM flat direc-tions see (Dine et al. , 1996b; Gherghetta et al. , 1996). Aflat direction can be represented by a composite gaugeinvariant operator, X m , formed from the product of k chiral superfields Φ i making up the flat direction: X m = Φ Φ · · · Φ m . The scalar component of the su-perfield X m is related to the order parameter φ through X m = cφ m (Dine et al. , 1996b).An example of a D -and F -flat direction is providedby (Dine and Kusenko, 2004; Enqvist and Mazumdar,2003) H u = 1 √ (cid:18) φ (cid:19) , L = 1 √ (cid:18) φ (cid:19) , (77)where φ is a complex field parameterizing the flat di-rection, or the order parameter, or the AD field. Allthe other fields are set to zero. In terms of the com-posite gauge invariant operators, we would write X m = LH u ( m = 2). Note that a flat direction necessarily car-ries a global U (1) quantum number, which correspondsto an invariance of the effective Lagrangian for the or-der parameter φ under phase rotation φ → e iθ φ . In theMSSM the global U (1) symmetry is B − L . For example,the LH u -direction has B − L = − F ∗ H u = λ u Qu + µH d = F ∗ L = λ d H d e ≡ φ . However there ex-ists a non-zero F-component given by F ∗ H d = µH u . Since µ can not be much larger than the electroweak scale3 M W ∼ O (1) TeV, this contribution is of the same orderas the soft SUSY breaking masses, which are going to liftthe degeneracy. Therefore, following (Dine et al. , 1996b),one may nevertheless consider LH u to correspond to a F-flat direction. The relevant D -terms read D aSU (2) = H † u τ H u + L † τ L = 12 | φ | − | φ | ≡ . (78)Therefore the LH u direction is also D -flat.
2. Gauge invariant inflatons of MSSM
A simple observation was first made in(Allahverdi et al. , 2007c, 2006, 2007e), where theinflaton properties are directly related to the soft SUSYbreaking mass term and the A-term of the MSSM.Within MSSM, it is possible to lift the flatness of thegauge invariant combinations of squarks and sleptonsaway from the point of enhanced gauge symmetry by the F -term, while maintaining the D -flatness. a. Squarks and sleptons driven inflation:
Letus consider a non-renormalizable superpotentialterm (Dine and Kusenko, 2004; Enqvist and Mazumdar,2003): W non = X n> λ n n Φ n M n − , (79)Where Φ = φ exp[ iθ ], while θ is the phase term is a gauge invariant superfield which contains the flat direc-tion. Within MSSM (with conserved R -parity) all theflat directions are lifted by the non-renormalizable op-erators with 4 ≤ n ≤ et al. , 1996). Twodistinct directions are: udd and LLe , up to an overallphase factor they are parameterized by: u αi = 1 √ φ , d βj = 1 √ φ , d γk = 1 √ φ . (80) L ai = 1 √ (cid:18) φ (cid:19) , L bj = 1 √ (cid:18) φ (cid:19) , e k = 1 √ φ , (81)where 1 ≤ α, β, γ ≤ α = β = γ ) are color indices, and1 ≤ i, j, k ≤ j = k ) denote the quark families for udd ,and 1 ≤ a, b ≤ a = b ) are the weak isospin indicesand 1 ≤ i, j, k ≤ i = j = k ) denote the lepton familiesfor LLe . Both these directions are lifted by n = 6 non-renormalizable operators (Gherghetta et al. , 1996), W ⊃ M ( LLe )( LLe ) , W ⊃ M ( udd )( udd ) . (82) Rest of the directions within MSSM are lifted by hybridoperators of type, W ∼ (1 /M n − )ΨΦ n − , which does notlead to cosmologically flat potential viable for slow-rollinflation (Allahverdi et al. , 2008, 2006, 2007e).The scalar potential along these directions includessoftly broken SUSY mass term for φ and an A -term gives rise to a specific potential (Allahverdi et al. , 2008, 2006,2007e) V ( φ ) = 12 m φ | φ | − A λφ M + λ | φ | M , (83)The A -term is a positive quantity with dimension ofmass. Note that the first and third terms in Eq. (83)are positive definite, while the A -term leads to a nega-tive contribution along the directions whenever cos( nθ + θ A ) <
0. The above potential is similar to Eq.(44). Itis possible to find a point of inflection , φ , provided that A / m φ ≡ α , where α ≪
1, and at the lowestorders in O ( α ), we obtain: V ( φ ) = 415 m φ φ + · · · , V ′ ( φ ) = 4 α m φ φ + · · · ,V ′′ ( φ ) = 0 , V ′′′ ( φ ) = 32 m φ /φ + · · · .φ = (cid:16) m φ M /λ √ (cid:17) / + O ( α ) . (84)In the case of gravity-mediated SUSY breaking scenarios, m φ ∼ A ∼ m / ∼ (100 GeV − A ∼ m φ can indeed be satisfied. Inflationoccurs within an interval | φ − φ | ∼ φ / M ≪ M P , inthe vicinity of the point of inflection, φ ∼ O (10 GeV).Within which the slow-roll parameters, ǫ, η ≪
1. TheHubble expansion rate during inflation is given by H MSSM ≃ √ m φ φ M P ∼ (100 MeV − . (85)The amplitude of density perturbations δ H (seeEqs. (13, 50,51) and the scalar spectral index n s are given by (Allahverdi et al. , 2006, 2007e;Allahverdi and Mazumdar, 2006a; Bueno Sanchez et al. ,2007): δ H = 8 √ π m φ M P φ sin [ N Q √ ∆ ] (86) n s = 1 − √ ∆ cot[ N Q √ ∆ ] , (87)where 2 × − ≤ ∆ ≡ α N − Q ( M P /φ ) ≤ . × − , and N Q ∼
50. Running in the tilt is very small. Inthis case the universe thermalizes in to MSSM radiationinstantly in less than one Hubble time after the end ofinflation (Allahverdi et al. , 2011b), see the discussion inSect. II.F.3.For φ ∼ GeV, there is an apparent fine-tuning inthe parameters
A/m φ = α ∼ − , which may look un-pleasant. However note that this fine tuning between thetwo MSSM parameters in the ratio is energy dependentand valid only at the scale of inflation at 10 GeV, butat the TeV scale where the soft masses would be mea-sured at the collider there is no apparent fine tuning inthe parameters (Allahverdi et al. , 2010c).As shown in Sect. II.D.2.b, see Fig. 2, the SUGRAcorrections will ameliorate the tuning down to β ≡ -4 n s d H m f = G e V m f = G e V m f = G e V m f = G e V FIG. 3: ( δ H , n s ) is plotted for different values of m φ and λ = 1. The 2 σ region for δ H is shown by the bluehorizontal band and the 2 σ allowed region of n s isshown by the vertical green band. The 1 σ allowedregion of n s is within the solid verticallines (Allahverdi et al. , 2008). α ∼ − , virtually addressing any fine tuning re-quired for the success of MSSM inflation. It wasshown in Refs. (Allahverdi et al. , 2008, 2007d), that the inflection point for the MSSM inflaton is an attrac-tor solution, provided there exists a phase of inflationprior of that of the MSSM with N ≥ e-foldings.Such large e-foldings can be generated within stringtheory landscape (Allahverdi et al. , 2007d), or withinMSSM (Allahverdi et al. , 2008). b. Renormalizable superpotential:
The left handedneutrinos can be of Dirac type with an appropriateYukawa coupling. The simplest way to obtain this wouldbe to augment the SM symmetry by, SU (3) C × SU (2) L × U (1) Y × U (1) B − L , where U (1) B − L is gauged. The rele-vant superpotential term is W ⊃ hN H u L. (88)Here N , L and H u are superfields containing the RHneutrinos, left-handed (LH) leptons and the Higgs whichgives mass to the up-type quarks, respectively. Note thatthe N H u L monomial represents a D -flat direction underthe U (1) B − L , as well as the SM gauge group.The value of h needs to be small, i.e. h ≤ − , inorder to explain the light neutrino mass, ∼ O (0 . et al. ,2007b,e), V ( | φ | ) = m φ | φ | + h | φ | − Ah √ | φ | . (89)For A ≈ m φ , there exists an inflection point for which V ′ ( φ ) = 0 , V ′′ ( φ ) = 0, where inflation takes place φ = √ m φ h = 6 × m φ (cid:16) .
05 eV m ν (cid:17) ,V ( φ ) = m φ h = 3 × m φ (cid:16) .
05 eV m ν (cid:17) . (90)The amplitude of density perturbations follows fromEqs. (13,50,51) (Allahverdi et al. , 2007b,e). δ H ≃ π H inf ˙ φ ≃ . × − (cid:16) m ν .
05 eV (cid:17) (cid:16) M P m φ (cid:17) N Q . (91)Here m φ denotes the loop-corrected value of the infla-ton mass at the scale φ in Eqs. (90,91). The spectraltilt as usual has a range of values 0 . ≤ n s ≤ . et al. , 2007b,e). c. MSSM Higgses as inflaton:
The MSSM Higgses areanother fine example of a visible sector inflaton providedsome restrictions are met (Chatterjee and Mazumdar,2011). The required superpotential is given by W = µH u .H d + X k λ k k ( H u .H d ) k M k − , (92)This is the µ -term which were considered anideal candidate to generate the density perturba-tions (Enqvist et al. , 2004a,b), but now they can alsoprovide the required vacuum energy to inflate theuniverse (Chatterjee and Mazumdar, 2011). The scalarpotential along the H u H d D -flat direction is given by, V ( ϕ, θ ) = 12 m ( θ ) ϕ + ( − ( k − λ ′ k µ cos((2 k − θ ) ϕ k + 2 λ ′ k ϕ k − , (93)where ϕ = √ | φ | e iθ , and H u = (1 / √ φ, T , H d =(1 / √ − (0 , φ ) T , and m ( θ ) = 12 ( m H u + m H d + 2 µ − Bµ cos 2 θ ) , (94) λ ′ k = λ k (2 k − M k − . (95)For simplicity, we may assume µ and B to be real.This choice is compatible with the experimental con-straints, mainly from the Electron Dipole Moment mea-surements (Pospelov and Ritz, 2005). The inflectionpoint can be obtained for θ = 0 , ± π/
2, for simplicitylet us consider the case for θ = 0, and when the fol-lowing condition is satisfied, m = k µ / (2 k −
1) + ˜ λ ,where ˜ λ is the tuning required to maintain the flatnessof the potential. Although, this tuning could be harshat the inflationary scale, ϕ ∼ GeV, but the ra-tio evolves to m /µ ∼ O (1) at the electroweak scaleby virtue of running of the renormalization group equa-tions (Chatterjee and Mazumdar, 2011). The amplitude5of the CMB perturbations can be obtained for λ ∼ − and λ ∼ O (1), it is possible to obtain a similar plot likeFig. (3) for Higgs mass m ( θ = 0) ∼ −
250 GeV, whichyields the spectral tilt in the range 0 . ≤ n s ≤ . F. Preheating, reheating, thermalization
Reheating at the end of inflation is an importantaspect of inflationary cosmology. Without reheatingthe universe would be empty of matter, for a re-view see (Allahverdi et al. , 2010a). Reheating occursthrough coupling of the inflaton field φ , to the SMmatter. Such couplings must be present at least viagravitational interactions. In particular, if the infla-ton is a SM gauge singlet, the relevant couplings toSM are: λM φ ( H ¯ q l ) q R , λM φF µν F µν , g φ ¯ HH , where M is the scale below which all these effective opera-tors are valid, λ, g ∼ O (1), H is the SM Higgs dou-blet, and q l , q R are the left and the right handed SMfermions (Allahverdi and Mazumdar, 2007b).Similar couplings would arise if φ is replaced by righthanded sneutrinos, axions, moduli, or any other hiddensector field. Being a SM singlet, φ can as well coupleto other hidden sectors, moduli, axions, etc. Since thehidden sectors are largely unknown, it becomes a chal-lenge for a singlet inflaton to decay solely into the SM d.o.f (Cicoli and Mazumdar, 2010a,b).After the end of inflation, the inflaton starts coher-ent oscillations around its minimum. The frequencyof oscillations are determined by the frequency of os-cillations, ω ∼ m eff ≥ H inf . During this epochthe inflaton can decay perturbatively (Albrecht et al. ,1982; Dolgov and Kirilova, 1990; Kolb and Turner, 1988;Turner, 1983). Averaging over many oscillations re-sults in a pressureless equation of state where h p i = h ˙ φ / − V ( φ ) i vanishes, so that the energy density startsevolving like a matter domination (in a quadratic poten-tial) with ρ φ = ρ i ( a i /a ) (subscript i denotes the quanti-ties right after the end of inflation). For λφ potential thecoherent oscillations yield an effective equation of statesimilar to that of a radiation epoch. If Γ φ represents the total decay width of the inflaton to pairs of fermions. Thisreleases the energy into the thermal bath of relativisticparticles when H ( a ) = p (1 / M ) ρ i ( a i /a ) / ≈ Γ φ . Theenergy density of the thermal bath is determined by thereheat temperature T R , given by: T R = (cid:18) π g ∗ (cid:19) / p Γ φ M P = 0 . (cid:18) g ∗ (cid:19) / p Γ φ M P , (96)where g ∗ denotes the effective relativistic d.o.f in theplasma. However the inflaton decay products need tothermalize, which requires acquiring kinetic and chemi-cal equilibrium.
1. Non-perturbative particle creation
If the inflaton coupling to the matter field is large,a completely new channel of reheating opens up dueto the coherent nature of the inflaton field, proposedby (Kofman et al. , 1994, 1997; Shtanov et al. , 1995;Traschen and Brandenberger, 1990), known as preheat-ing . Let us first consider a simple toy model with inter-action Lagrangian L int = − g χ φ , (97)where χ is another scalar field, in a realistic set-up χ could take the role of the SM Higgs. We can neglectthe effect of expansion provided that the time period ofpreheating is small compared to the Hubble expansiontime H − , this is reasonable in many cases.The quantum theory of χ particle production in theexternal classical inflaton background begins by expand-ing the quantum field ˆ χ into creation and annihilationoperators ˆ a k and ˆ a † k as:ˆ χ ( t, x ) = 1(2 π ) / Z d k (cid:16) χ ∗ k ( t )ˆ a k e i kx + χ k ( t )ˆ a † k e − i kx (cid:17) , (98)where k is the momentum. Since the equation of motionfor χ is linear it can be studied simply mode by mode inFourier space. The mode functions then satisfy:¨ χ k + (cid:0) k + m χ + g Φ sin ( m φ t ) (cid:1) χ k = 0 , (99)where Φ is the amplitude of oscillation in φ . This is theMathieu equation which is written in the form χ ′′ k + ( A k − q cos 2 z ) χ k = 0 , (100)where the dimensionless time variable is z = m φ t anda prime now denotes the derivative with respect to z .Comparing the coefficients, we find A k = k + m χ m + 2 q q = g Φ m φ (101)The growth of the mode function corresponds to particleproduction (Birrell and Davies, 1982). It is well knownthat the above Mathieu equation Eq. (100) has instabil-ities for certain ranges of k : χ k ∝ exp( µ k z ) , (102)where µ k is called the Floquet exponent. For small val-ues of q , i.e. q ≪
1, resonance occurs in a narrow in-stability band about k = m , known as a “narrow reso-nance” band (Traschen and Brandenberger, 1990). Theresonance is much more efficient if q ≫ et al. ,1994, 1997). In this case, resonance occurs in broadbands, i.e. the bands include all long wavelength modes k →
0, known as broad resonance . This can be un-derstood by studying the condition for particle produc-tion in the WKB approximation for the evolution of χ χ k ∝ e ± i R ω k dt , which is valid as long as the adiabaticitycondition dω k /dt ≤ ω k (103)is satisfied. In the above, the effective frequency ω k isgiven by ω k = q k + m χ + g Φ( t ) sin ( m φ t ) , (104)By inserting the effective frequency Eq. (104) into thecondition Eq. (103) and following some algebra, the adi-abaticity condition is violated for momenta k ≤ √ gm φ Φ − m χ . (105)For modes with these values of k , the adiabaticity condi-tion breaks down in each oscillation period when φ is closeto zero. The particle number does not increase smoothly,but rather in “bursts” (Kofman et al. , 1994, 1997).The above analysis of neglecting the expansion of theuniverse is self-consistent. However, as discussed in detailin (Kofman et al. , 1997), the expansion of space can beincluded. The equation of motion for χ becomes¨ χ k + 3 H ˙ χ k + (cid:18) k a + m χ + g Φ( t ) sin ( m φ t ) (cid:19) χ k = 0 . (106)The adiabaticity condition is now violated for momentasatisfying: k /a ≤ (2 / √ gm Φ( t ) − m χ . (107)Note that the expansion of space makes broad resonancemore effective since more k modes are red-shifted into theinstability band as time proceeds. The detailed analysisyields the same expression for the resonance band exceptfor the exact value of the numerical coefficient of thefirst term on the r.h.s.. Broad parametric resonance endswhen q ≤ / et al. ,1998; Giudice et al. , 1999a; Greene and Kofman, 1999,2000; Peloso and Sorbo, 2000), and higher spin ± / et al. , 1999b; Kallosh et al. , 2000a,b;Maroto and Mazumdar, 2000; Nilles et al. , 2001b,c). a. Tachyonic prehetaing:
It is possible that effectivefrequency of certain mode can be negative. For examplein a symmetry breaking potential: V ( φ ) = λ ( φ − η ) ,for small field values, the effective mass of the fluctua-tions of φ is negative and hence a “tachyonic” resonancewill occur, as studied in (Felder et al. , 2001). For smallfield values, the equation for the fluctuations φ k of φ is¨ φ k + (cid:0) k − m φ (cid:1) φ k = 0 . (108) The modes with k < m grow with an exponent whichapproaches µ k = 1 in the limit k →
0. Given initialvacuum amplitudes for the modes φ k at the intial time t = 0 of the resonance, the field dispersion at a later time t will be given by h δφ i = Z m kdk π e t √ m φ − k . (109)The growth of the fluctuations modes terminates once thedispersion becomes comparable to the symmetry break-ing scale.Tachyonic preheating also occurs in hybrid inflationmodels, see Eq. (39). In this case, it is the fluctuationsof ψ which have tachyonic form and which grow exponen-tially (Felder et al. , 2001). Preheating in hybrid inflationwas first studied in (Garcia-Bellido and Linde, 1998) us-ing the tools of broad parametric resonance. b. End of preheating:
In the above analysis we haveneglected the back-reaction of the produced χ and φ particles on the dynamics. The presence of χ parti-cles changes the effective mass of the inflaton oscilla-tions. This back-reaction effect is negligible as long asthe change ∆ m φ in the square mass of the inflaton issmaller than m φ . In the Hartree approximation, thechange in the inflaton mass due to χ particles is givenby ∆ m φ = g h χ i (Kofman et al. , 1997). Another im-portant condition is that the energy in the χ particlesshould be sub-dominant. Therefore, ρ χ ∼ h ( ∇ χ ) i ≃ k h χ i ≪ m φ h φ i , It was found that ρ χ is smaller thanthe potential energy of the inflaton field at the time t as long as the value q at the time t is larger than 1,i.e. q ( t ) >
1. This is roughly speaking the same asthe condition for the effectiveness of broad resonance(Kofman et al. , 1997).
2. Thermalization
Neither the perturbative decay of the inflaton nor thepreheating mechanism produce a thermal spectrum of de-cay products. In a full thermal equilibrium the energydensity ρ and the number density n of relativistic parti-cles scale as: ρ ∼ T and n ∼ T , where T is the tem-perature of the thermal bath. Thus, in full equilibriumthe average particle energy is given by: h E i eq = ( ρ/n ),which obeys the scaling, h E i eq ∼ ρ / ∼ T . a. Perturbative reheating and thermalization:
If theinflaton decays perturbatively, then right after the infla-ton has decayed completely, the energy density of the uni-verse is given by: ρ ≈ φ M P ) , h E i ≈ m φ ≫ ρ / .From conservation of energy, the number density of de-cayed particles is: n ≈ ( ρ/m φ ) ≪ ρ / . Hence, pertur-bative decay results in a dilute plasma that contains asmall number of very energetic particles. A local thermalequilibrium requires re-distribution of the energy among7different particles, kinetic equilibrium , as well as increas-ing the total number of particles, chemical equilibrium .Therefore both number-conserving and number-violatingreactions must be involved.The most important processes for kinetic equili-bration are 2 → t -channel. Due to an infraredsingularity, these scatterings are very efficient evenin a dilute plasma (Allahverdi and Mazumdar, 2006b;Davidson and Sarkar, 2000). Chemical equilibrium isachieved by changing the number of particles in the re-heat plasma. In order to reach full equilibrium the totalnumber of particles must increase by a factor of n eq /n ,where n ≈ ρ/m and the equilibrium value is: n eq ∼ ρ / . This can be a very large number, i.e. n eq /n ∼O (10 ). It was recognized in (Allahverdi and Drees,2002; Davidson and Sarkar, 2000), see also (Allahverdi,2000; Jaikumar and Mazumdar, 2004) that the most rel-evant processes are 2 → t − channel. When these scatter-ing become efficient, the number of particles increasesvery rapidly, and full thermal equilibrium is establishedshortly after that (Enqvist and Sirkka, 1993). b. Non-perturbative preheating and thermalization:
In this case the occupation numbers of the excited quantaare typically very high after the initial stages of preheat-ing, Once the occupation numbers of the resonant modesbecome sufficiently large, re-scattering of the fluctua-tions begins (Khlebnikov and Tkachev, 1996, 1997a,b;Micha and Tkachev, 2003, 2004) which terminates thephase of exponential growth of the occupation num-bers. The evolution of the field fluctuations evolves toa regime of turbulent scaling which is characterized bythe spectrum n ( k ) ∼ k − / (Micha and Tkachev, 2003,2004), which is non-thermal (for a thermal distributionwe would expect n ( k ) ∼ k − ). The phase of turbulenceends once most of the energy has been drained from theinflaton field. At this time quantum processes take overand lead to the thermalization of the spectrum.
3. Calculation of T R within MSSM In the case of MSSM inflation, the inflaton couplingsto MSSM d.o.f are known (Allahverdi et al. , 2011b). Itis therefore possible to track the thermal history of theuniverse from the end of inflation. When the MSSMinflaton passes through minimum, i.e. φ = 0, the entiregauge symmetry gets restored and all the d.o.f associatedwith the MSSM gauge group become massless, which isknown as the point of enhanced gauge symmetry .These are the massless modes which couple to the in-flaton directly, for instance the d.o.f corresponding to SU (2) W × U (1) Y , or that of SU (3) c × U (1) Y . At VEVsaway from the minimum, the same modes become heavyand therefore it is kinematically unfavorable to excitethem. The actual process of excitation depends on how strongly the adiabaticity condition for the time depen-dent vacuum is violated for the inflaton zero mode. a. Couplings for LLe inflaton:
Let us illustrate thiswith L L e flat direction as an inflaton. The infla-ton non-zero VEV completely breaks the SU (2) W × U (1) Y symmetry. This results in four massive realscalars, whose masses are obtained from the D -terms (Allahverdi et al. , 2011b) V ⊃ g W φ ( χ + χ + χ ) + 14 g Y φ χ . (110)Here g W , g Y are the SU (2) W and U (1) Y gauge couplingsrespectively, and φ denote the inflaton, see Eq. (81). The χ ’s are Goldstone bosons from breakdown of SU (2) W × U (1) Y . They are eaten by the Higgs mechanism andgive rise to longitudinal components of the electroweakgauge fields. In the unitary gauge, they are completelyremoved from the spectrum. The χ particles decay tosquarks, the Higgs particles, and the ˜ L , ˜ e , ˜ e sleptonswith the decay rates given by: Γ χ = Γ χ = Γ χ =3 g W φ/ π √ , Γ χ = (9 g Y φ/ π √ χ fields. Couplings of the inflatonto the gauge fields are obtained from the flat directionkinetic terms (Allahverdi et al. , 2011b) L ⊃ g W φ (2 W + ,µ W − µ + W µ W ,µ )+ g Y φ B µ B µ , (111)where W + = ( W − iW ) / √ , W − = ( W + iW ) / √ W i,µ and B µ are the SU (2) W and U (1) Y gaugefields respectively. The gauge fields decay to (s)quarks,Higgs and Higgsino particles, and L , e , e (s)leptonswith the total decay widths: Γ W + = Γ W − = Γ W =(3 g W φ/ π √ , Γ B = (9 g Y φ/ π √ et al. , 2011b). b. Instant preheating and thermalization:
The fieldsthat are coupled to the inflaton acquire a VEV-dependentmass that varies in time due to the inflaton oscillations.For illustration, we first focus on the χ scalar, which areproduced every time the inflaton goes through zero. TheFourier eigenmodes of χ have the corresponding energy ω k = q k + m χ + g W φ ( t ) / m χ is the bare mass of the χ field. Thegrowth of the occupation number of mode k can becomputed exactly for the first zero-crossing, n k,χ =exp h − π √ k + m χ ) / ( g W ˙ φ ) i , where the inflaton nearthe zero-crossing is given by ˙ φ = (2 V ( ˆ φ )) / , from theconservation of energy, where ˆ φ is the amplitude of theinflaton oscillations, ˆ φ ≃ φ / √ ∼ GeV, where φ is the inflection point for inflation, Eq. (84). Note thatafter a few oscillations, ˙ φ ≃ m φ ˆ φ , since the expansion8rate during the inflaton oscillations is negligible by virtueof m φ ∼
100 GeV and H ( t ) ≤ H inf ∼ n χ = Z ∞ d k (2 π ) n k,χ = m φ q / √ π exp − πm χ m φ √ q ! . (113)where q ≡ ( g W ˙ φ / m φ ) ≫
1. This expression corre-sponds to the asymptotic value and assumes there is noperturbative decay of the produced χ particles. How-ever, immediately after adiabaticity is restored, τ > τ ∗ = √ q − / , χ particles decay into lighter particles (i.e.those particles that have no gauge coupling to the in-flaton). In the case of L L e inflaton these are the(s)quarks, H u Higgs(ino), and L , e , e (s)leptons.Thus the fraction that is transferred from the inflatonto χ ’s, and through their prompt decay into relativisticsquarks, at every inflaton zero-crossing, can be computedanalytically, and they are given by, ρ χ rel ρ φ ∼ . g W e − πm χ m φ √ qW + 0 . g Y e − πm χ m φ √ qY . (114)The total number of d.o.f coupled to the L L e inflatonis 32 (4 from scalars, 4 × × g Y ∼ g W ∼ . et al. ,2011b): ρ rel /ρ φ ∼ .
6% (per zero − crossing) . (115)Note that this fraction is independent from the ampli-tude of oscillations. The draining the inflaton energy isquite efficient, nearly 10% of the inflaton energy densitygets transferred to the relativistic species – but not allthe SM d.o.f are in thermal equilibrium after one oscilla-tion. It takes near about 120 oscillations to reach the full chemical and kinetic equilibrium via processes requiring2 ↔ ↔ H inf ∼ − m φ , this happens within asingle Hubble time after the end of inflation. One canestimate the final reheat temperature (Allahverdi et al. ,2011b) T R = (cid:0) /π g ∗ (cid:1) / ρ / ≃ × GeV , (116)where g ∗ = 228 .
75 and ρ = (4 / m φ φ , see Eq. (84). III. MATTER-ANTI-MATTER ASYMMETRY
If (p)reheating can provide a thermal bath where all the SM quarks and leptons are excited, it is then an im-portant question to ask – why the present day galax-ies and intergalactic medium is primarily made up ofbaryons rather than anti-baryons? The baryon abundance in the universe is denoted byΩ b ≡ ρ b /ρ c , which defines the fractional baryon density ρ b with respect to the critical energy density of the uni-verse: ρ c = 1 . h × − g cm − . The observationaluncertainties in the present value of the Hubble constant; H = 100 h km · s − · Mpc − ≈ ( h/ − are en-coded in h = 0 .
73 (Kessler et al. , 2009). It is useful towrite in terms of the baryon and photon number densities η ≡ n b − n ¯ b n γ = 2 . × − Ω b h , (117)where n b is the baryon number density and n ¯ b is foranti-baryons. The photon number density is given by n γ ≡ (2 ζ (3) /π ) T . The best present estimation ofthe baryon density comes from BBN, which is based onSM physics with 3 neutrino species (Cyburt et al. , 2008;Fields and Sarkar, 2006)0 . ≤ Ω b h ≤ .
024 (95% CL ) , (118)5 . × − ≤ η ≤ . × − (95% CL ) . (119)The observational data on D and He are consistent witheach other and the expectations from the BBN analysis,but both prefer slightly higher value compared to the Li abundance Li/H | P = (1 . ± . ± . × − , whichis smaller than D and He by at least ∼ . σ . The Li abundance is primarily measured in the stellar systemssuch as globular clusters.From the acoustic peaks of the CMB the baryonfraction can be deduced. The WMAP data implyΩ b h = 0 . ± . η = 6 . ± . × − (Komatsu et al. , 2011). The WMAP data relieson priors and the choice of number of parameters, it ispossible to yield baryon abundance as low as Ω b h =0 . ± . Li abundance, stellar Li/H mea-surements are inconsistent with both WMAP and BBNdata, and this could be an useful probe of new physics atBBN, see for a review (Jedamzik and Pospelov, 2009).Often in the literature the baryon asymmetry is givenin relation to the entropy density s = 1 . g ∗ n γ , where g ∗ measures the effective number of relativistic specieswhich itself a function of temperature. At the presenttime g ∗ ≈ .
36, while during BBN g ∗ ≈ .
11, rising upto 106 .
75 at T ≫
100 GeV. In the presence of super-symmetry at T ≫
100 GeV, the number of effective rela-tivistic species are nearly doubled to 228 .
75. The baryonasymmetry defined as the difference of baryon and anti-baryon number densities relative to the entropy density,is bounded by58 . . × − ≤ n b − n ¯ b s ≤ . . × − , (120)at (95% CL ), where the numbers areCMB (Komatsu et al. , 2011), and BBN9bounds (Cyburt et al. , 2008; Fields and Sarkar, 2006),respectively.If at the beginning η = 0, then the origin of thissmall number can not be understood in a CPT invariantuniverse by a mere thermal decoupling of nucleons andanti-nucleons at T ∼
20 MeV. The resulting asymmetrywould be too small by at least nine orders of magnitude,see (Kolb and Turner, 1988). Therefore it is importantto seek mechanisms for generating baryon asymmetry,for reviews see (Dine and Kusenko, 2004; Riotto, 1998;Rubakov and Shaposhnikov, 1996).
A. Requirements for baryogenesis
As pointed out first by Sakharov (Sakharov, 1967),baryogenesis requires three ingredients: (1) baryon num-ber non-conservation, (2) C and CP violation, and (3)out-of-equilibrium condition. All these conditions are be-lieved to be met in the very early universe. c. Baryon number non-conservation:
In the SM,baryon number B is violated by non-perturbative instan-ton processes (’t Hooft, 1976a,b). Due to chiral anoma-lies both baryon number J µB and lepton number J µL cur-rents are not conserved (Adler, 1969; Bell and Jackiw,1969). However the anomalous divergences come withan equal amplitude and an opposite sign. Therefore B − L remains conserved, while B + L may changevia processes which interpolate between the multiplenon-Abelian vacua of SU (2). The probability for the B + L violating transition is however exponentially sup-pressed (’t Hooft, 1976a,b), but at finite temperatureswhen T ≫ M W , baryon violating transitions are in factcopious (Manton, 1983).The B violation also leads to proton decay inGUTs (Langacker, 1981). The dimension 6 operator( QQQL ) / Λ generates observable proton decay unlessΛ ≥ GeV. In the MSSM the bound is Λ ≥ GeVbecause the decay can take place via a dimension 5 op-erator. In the MSSM superpotential there are also termswhich can lead to ∆ L = 1 and ∆ B = 1. Similarly thereare other processes such as neutron-anti-neutron oscilla-tions in SM and in SUSY theories which lead to ∆ B = 2and ∆ B = 1 transitions. These operators are constrainedby the measurements of the proton lifetime, which yieldthe bound τ p ≥ years (Nakamura et al. , 2010). d. C and CP violation: The maximum C violation oc-curs in weak interactions while neutral Kaon is an ex-ample of CP violation in the quark sector which has arelative strength ∼ − (Nakamura et al. , 2010). CP violation is also expected to be found in the neutrinosector. Beyond the SM there are many sources for CP violation. An example is the axion proposed for solv-ing the strong CP problem in QCD (Peccei and Quinn,1977a,b). e. Departure from thermal equilibrium: If B -violating processes are in thermal equilibrium, the in-verse processes will wash out the pre-existing asymmetry(∆ n b ) (Weinberg, 1979). This is a consequence of S -matrix unitarity and CP T -theorem. However there areseveral ways of obtaining an out-of-equilibrium processin the early universe. Departure from a thermal equilib-rium cannot be achieved by mere particle physics consid-erations but is coupled to the dynamical evolution of theuniverse.
1. Out-of-equilibrium decay or scattering:
Thecondition for out-of-equilibrium decay or scattering isthat the rate of interaction must be smaller than the ex-pansion rate of the universe, i.e. Γ < H . The universe ina thermal equilibrium can not produce any asymmetry,rather it tries to equilibrate any pre-existing asymmetry.
2. Phase transitions:
They are ubiquitous in theearly universe. The transition could be of first , orof second (or of still higher) order. First order tran-sitions proceed by barrier penetration and subsequentbubble nucleation resulting in a temporary departurefrom equilibrium. The QCD and possibly electroweakphase transitions are examples of first order phasetransitions. The nature and details of QCD phasetransition is still an open debate (Karsch et al. , 2001;Rajagopal and Wilczek, 1993). Second order phase tran-sitions have no barrier between the symmetric and thebroken phase. They are continuous and equilibrium ismaintained throughout the transition.
3. Non-adiabatic motion of a scalar field:
Anycomplex scalar field carries C and CP , but the symme-tries can be broken by terms in the Lagrangian. This canlead to a non-trivial trajectory of a complex scalar fieldin the phase space. If a coherent scalar field is trappedin a local minimum of the potential and if the shape ofthe potential changes to become a maximum, then thefield may not have enough time to readjust with the po-tential and may experience completely non-adiabatic mo-tion. This is similar to a second order phase transitionbut it is the non-adiabatic classical motion which prevailsover the quantum fluctuations, and therefore, departurefrom equilibrium can be achieved. If the field condensatecarries a global charge such as the baryon number, themotion can charge up the condensate. This is the ba-sis for the Affleck-Dine baryogenesis (Affleck and Dine,1985). B. Sphalerons
At finite temperatures B + L violation in the SMcan be large due to sphaleron transitions between de-generate gauge vacua with different Chern-Simons num-bers (Klinkhamer and Manton, 1984; Manton, 1983).Thermal scattering produces sphalerons which in effectdecay in B + L non-conserving ways below 10 GeV(Bochkarev and Shaposhnikov, 1987), and thus can ex-ponentially wash away B + L asymmetry. The three im-0portant ingredients which play important role are follow-ing. f. Chiral anomalies:
In the SM there is classical con-servation of the baryon and lepton number currents J µB and J µL , but because of chiral anomaly (at the quan-tum level) the currents are not conserved (Adler, 1969;Bell and Jackiw, 1969). Instead (’t Hooft, 1976b), ∂ µ J µB = − α π N g W µνi ˜ W iµν + α π N g F αβ ˜ F αβ ,∂ µ J µL = − α π N g W µνi ˜ W iµν + α π N g F αβ ˜ F αβ , (121)where N g is the number of generations, α and α ( W iµν and F µν ) are respectively the SU (2) and U (1) gauge cou-plings (field strengths). Note that at the quantum level B + L = 0 is violated, but B − L = 0 is still conserved. g. Gauge theory vacua: in the SU (2) gauge group,the vacua are classified by their homotopy class { Ω n (r) } ,characterized by the winding number n which labels theso called θ -vacua (’t Hooft, 1976a; Polyakov, 1977). Agauge invariant quantity is the difference in the windingnumber (Chern-Simons number) N CS ≡ n + − n − = α π Z d xW µνa ˜ W aµν . (122)In the electroweak sector the field density W ˜ W is relatedto the divergence of B + L current. Therefore a changein B + L reflects a change in the vacuum configurationdetermined by the difference in winding number∆( B + L ) = − α π N g Z d xW µνa ˜ W aµν = − N g N CS . (123)For three generations of SM leptons and quarks the min-imal violation is ∆( B + L ) = 6. Note that the protondecay p → e + π requires ∆( B + L ) = 2, so that despite B -violation, proton decay is completely forbidden in theSM. The probability amplitude for tunneling from an n vacuum at t → −∞ to an n + N CS vacuum at t → + ∞ can be estimated by the WKB method (’t Hooft, 1976a) P ( N CS ) B + L ∼ exp (cid:18) − πN CS α ( M Z ) (cid:19) ∼ − N CS . (124)The baryon number violation rate is negligible at zerotemperature, but as argued at finite temperatures thesituation is completely different (Kuzmin et al. , 1985;Manton, 1983). h. Thermal tunneling: below the critical temperatureof the electroweak phase transition, the sphaleron rate isexponentially suppressed (Carson et al. , 1990):Γ ∼ . × κT (cid:16) α π (cid:17) (cid:18) E sph ( T ) B ( λ /g ) (cid:19) e − E sph /T . (125)where κ is the functional determinant which can take thevalues 10 − ≤ κ ≤ − (Dine et al. , 1992). Above the critical temperature the rate is however unsuppressed.Since the Chern-Simons number changes at most by∆ N CS ∼
1, one can estimate from Eq. (122) that∆ N CS ∼ g l sph W i ∼ → W i ∼ (1 /g l sph ). There-fore a typical energy of the sphaleron configuration isgiven by E sph ∼ l sph ( ∂W i ) ∼ (1 /g l sph ). At tem-peratures greater than the critical temperature there isno Boltzmann suppression, so that the thermal energy ∝ T ≥ E sph . This determines the size of the sphaleron: l sph ≥ /g T Based on this coherence length scale onecan estimate the baryon number violation per volume ∼ l sph , and per unit time ∼ l sph . On dimensional groundsthe transition probability would then be given byΓ sph ∼ (1 /l sph t ) ∼ κ ( α T ) . (126)where κ is a constant which incorporates various un-certainties. However, the process is inherently non-perturbative, and it has been argued that damping of themagnetic field in a plasma suppresses the sphaleron rateby an extra power of α (Arnold et al. , 1997), with theconsequence that Γ sph ∼ α T . Lattice simulations withhard thermal loops also give Γ sph ∼ O (10) α T (Moore,1999). i. Washing out B + L : Assuming that in the early uni-verse the SM d.o.f are in equilibrium, the transitions∆ N CS = +1 and ∆ N CS = − sph + / Γ sph − = exp( − ∆ F/T ), where ∆ F is the free en-ergy difference between the two vacua. Because of a finite B + L density, there is a net chemical potential µ B + L .Therefore one obtains (Bochkarev and Shaposhnikov,1987) dn B + L dt = Γ sph + − Γ sph − ∼ N g Γ sph T n B + L . (127)It then follows that an exponential depletion of n B + L dueto sphaleron transitions remains active as long asΓ sph T ≥ H ⇒ T ≤ α M P g / ∗ ∼ GeV . (128)This result imply that below T = 10 GeV, thesphaleron transitions can wash out any B + L asymmetrybeing produced earlier in a time scale τ ∼ ( T /N g Γ sph ).This seems to wreck GUT baryogenesis based on B − L conserving groups such as the minimal SU (5). C. Mechanisms for baryogenesis
There are several scenarios for baryogenesis, the maincontenders being GUT baryogenesis, electroweak baryo-genesis, leptogenesis, and baryogenesis through the decayof a field condensate, or Affleck-Dine baryogenesis. Herewe give a brief description of these various alternatives.1
1. GUT-baryogenesis
This model relied on out-of-equilibrium decays ofheavy GUT gauge bosons
X, Y → qq , and X, Y → ¯ q ¯ l ,for reviews see (Dolgov, 1992; Kolb and Turner, 1988).The decay rate of the gauge boson goes as Γ X ∼ α X M X ,where M X is the mass of the gauge boson and α / X isthe GUT gauge coupling. Assuming that the universewas in thermal equilibrium at the GUT scale, the decaytemperature is given by T D ≈ g − / ∗ α / X ( M X M P ) / , (129)which is smaller than the gauge boson mass. Thus, at T ≈ T D , one expects n X ≈ n ¯ X ≈ n γ , and hence thenet baryon density is proportional to the photon num-ber density n B = ∆ Bn γ . However below T D the gaugeboson abundances decrease and eventually they go out-of-equilibrium. The net entropy generated due to theirdecay heats up the universe with a temperature which wedenote here by T R . Let us naively assume that the en-ergy density of the universe at T D is dominated by the X bosons with ρ X ≈ M X n X , and their decay products leadto radiation with an energy density ρ = ( π / g ∗ T R ,where g ∗ ∼ O (100) for T ≥ M GUT . Equating the expres-sions for the two energy densities one obtains n X ≈ π g ∗ T R M X . (130)Therefore the net baryon number comes out to be B ≡ n B s ≈ ∆ Bn X g ∗ n γ ≈ T R M X ∆ B . (131) T rh is determined from the relation Γ X ≈ H ( T D ) ∼ ( π / g ∗ T R /M . Thus, B ≈ g − / ∗ α X M P M X ! / ∆ B . (132)Uncertainties in C and CP violation are now hidden in∆ B , but can be tuned to yield total B ∼ − in manymodels.Above we have assumed that the universe is in thermalequilibrium when T ≥ M X . This might not be true, sincefor 2 ↔ ∼ α T , which becomes smaller than H at sufficiently hightemperatures. Elastic 2 → ∼ GeV, while chemical equilibrium is lost already at T ∼ GeV (Elmfors et al. , 1994; Enqvist and Eskola,1990).
2. Electroweak baryogenesis
A popular baryogenesis candidate is based on the elec-troweak phase transition, during which one can in prin-ciple meet all the Sakharov conditions. There is the sphaleron-induced baryon number violation above thecritical temperature, various sources of CP violation, andan out-of-equilibrium environment if the phase transi-tion is of the first order. In that case bubbles of bro-ken SU (2) × U (1) Y are nucleated into a symmetric back-ground with a Higgs field profile that changes throughthe bubble wall (Kuzmin et al. , 1987, 1985).There are two possible mechanisms which work in dif-ferent regimes: local and non-local baryogenesis. In thelocal case both CP violation and baryon number vio-lation takes place near the bubble wall. This requiresthe velocity of the bubble wall to be greater than thespeed of the sound in the plasma (Ambjorn et al. , 1990;Turok and Zadrozny, 1990, 1991), and the electroweakphase transition to be strongly first order with thin bub-ble walls.In the non-local case the bubble wall velocity speed issmall compared to the sound speed in the plasma. In thismechanism the fermions, mainly the top quark and thetau-lepton, undergo CP violating interactions with thebubble wall, which results in a difference in the reflectionand the transmission probabilities for the left and rightchiral fermions. The net outcome is an overall chiral fluxinto the unbroken phase from the broken phase. The fluxis then converted into baryons via sphaleron transitionsinside the unbroken phase. The interactions are takingplace in a thermal equilibrium except for the sphalerontransitions, the rate of which is slower than the rate atwhich the bubble sweeps the space (Cohen et al. , 1993;Nelson et al. , 1992).For a constant velocity profile of the bubble, v w , thenet baryon asymmetry is generated by: n B ≃ − Γ sph T Z dt µ B , (133)where µ B is the chemical potential, which determines thetilt in the free energy of the sphaleron transitions, andnumerically it is equivalent to: µ B ≡ ρ ( z − v w t ) / [(2 N +5 / T ]. Here ρ determines the profile of the bubble, and N denotes the number of Higgs doublets. The net baryonasymmetry can be calculated by following Eq. (126): n B s ≃ κα (cid:18) π g ∗ (cid:19) (cid:18) F z v w T (cid:19) τ T , (134)where τ is the transport time of the scattered fermionsoff the bubble wall, and F z ≡ R ∞ dzρ ( z ). For the max-imum wall velocity v w ∼ / √ τ T ∼ − n B /s ≃ − F z / ( v w T ). The func-tion F z also takes into account the CP phase, in thefavorable scenario one would expect F z / ( v w T ) ∼ − .The details of the transport equations can be found inRefs. (Kainulainen et al. , 2001; Nelson et al. , 1992).One great challenge for the electroweak baryogenesis isthe smallness of CP violation in the SM at finite temper-atures. It has been pointed out that an additional Higgsdoublet (McLerran et al. , 1991; Turok and Zadrozny,21991) would provide an extra source for CP violationin the Higgs sector. However, the situation is much im-proved in the MSSM where there are two Higgs dou-blets H u and H d , and two important sources of CP violation (Ellis et al. , 1982). The Higgses couple tothe charginos and neutralinos at one loop level leadingto a CP violating contribution. There is also a newsource of CP violation in the mass matrix of the topsquarks which can give rise to considerable CP violation(Huet and Nelson, 1996).Bubble nucleation depends on the thermal tunnelingrate, and the expansion rate of the universe. The tunnel-ing rate has to overcome the expansion rate in order tohave a successful phase transition via bubble nucleationat a given critical temperature T c > T t > T . The effec-tive potential for the Higgs at finite temperatures can becomputed, which takes the form: V eff ( φ, T ) = ( − µ + αT ) φ − γT φ + ( λ/ φ (135)The order parameter is given by the ratio of h φ ( T c ) /T c i ∼ γ/λ , which has to be larger than one for first order phasetransition. For T c ∼
100 GeV, one obtains the conditionfor the sphaleron energy (Rubakov and Shaposhnikov,1996; Shaposhnikov, 1987) E sph ( T c ) T c ≥ (cid:20) E sph ( T c ) T c (cid:21) +9 log(10)+log( κ ) . (136)which implies (Bochkarev et al. , 1991) E sph ( T c ) T c ≥
45 for κ = 10 − , (137)In terms of the Higgs field value at T c , φ ( T c ) T c = g πB ( λ/g ) E sph ( T c ) T c ∼ E sph ( T c ) T c , (138)where g is gauge coupling of SU (2) L , and B ∼ . φ ( T c ) /T c ≥ . , (139)which implies that the phase transition should bestrongly first order in order that sphalerons do not washaway all the produced baryon asymmetry. This result isthe main constraint on electroweak baryogenesis.Lattice studies suggest that in the SM the phase tran-sition is strongly first order only below Higgs mass m H ∼
72 GeV (Kajantie et al. , 1996; Rummukainen et al. ,1998). Above this scale the transition is just a cross-over.Such a Higgs mass is clearly excluded by the LEP mea-surements (Nakamura et al. , 2010), thus excluding elec-troweak baryogenesis within the SM. However, this opensup a possibility to include new physics beyond the SM. a. Electroweak baryogenesis induced by new physics: it was pointed out that by modifying the SMHiggs self-interactions, especially the cubic term, it is possible to enhance the first order phase tran-sition (Anderson and Hall, 1992). One such ex-ample has been considered in (Grojean et al. , 2005;Mohapatra and Zhang, 1992) where non-renormalizablecontribution to the Higgs potential has been consideredof type: V (Φ) = λ (cid:18) Φ † Φ − v (cid:19) + 1Λ (cid:18) Φ † Φ − v (cid:19) (140)where Φ is the SM electroweak Higgs doublet and Λ is thescale of new physics which induces the corrections belowthe energy scale of Λ ∼ O (1) TeV. At zero temperaturethe CP-even scalar state can be expanded in terms of itszero-temperature VEV, h φ i = v ≃
246 GeV, and thephysical Higgs boson H : Φ = φ/ √ H + v ) / √ V ( φ, T ) = cT φ / V ( φ,
0) (141)where c is given in the high-temperature expansion of theone-loop thermal potential: c = 116 (cid:18) y t + 3 g + g ′ + 4 m H v − v Λ (cid:19) , (142)where g and g ′ are the SU (2) L and U (1) Y gauge cou-plings, and y t is the top Yukawa coupling. Note thatthere is no trilinear term in the effective potential.The critical temperature T c at which the minima φ = 0and φ = 0 are degenerate is given by T c = Λ m H + 2Λ m H v − v c Λ v . (143)The VEV of the Higgs field at the critical temperaturein terms of m H , Λ and v is h φ ( T c ) i = v c = 32 v − m H Λ v . (144)From Eqs. (143) and (144), one finds that for any given m H there is an upper bound on Λ to make sure thatthe phase transition is always first order ( v c > T =0 minimum at φ = 0 is a global minimum ( T c > v m H , √ v p m H + 2 m c ! < Λ < √ v m H (145)where m c = v p (4 y t + 3 g + g ′ ) / ≈
200 GeV. In or-der to ensure that the thermal mass correction is positive: c > → Λ > √ v / p m H + 2 m c . For these ranges ofΛ the ratio v c /T c >
1, ensuring a successful sphalerontransition for the Higgs mass m H ≥
115 GeV. One niceaspect of this model is that the non-renormalizable scaleΛ can also be constrained from the precision electroweakobservable, which can be tested in near future by theLHC (Grojean et al. , 2005).3
3. Electroweak baryogenesis in MSSM
In the MSSM the ratio Φ( T c ) /T c can be increasedby virtue of the scalar loops which can make the cu-bic term in the temperature dependent Higgs potentiallarge; V eff ( ϕ, T ) = ( − µ + αT ) ϕ − γT ϕ + ( λ/ ϕ .In particular the right handed stop e t R coupling to theHiggs with a large Yukawa coupling. This leads toa strong first order phase transition – as the ratio ofΦ( T c ) /T c ∼ γ/λ ≥
1, where γ determines the orderparameter (Carena et al. , 1996; Cline et al. , 1998, 2000;Laine, 1996; Laine and Rummukainen, 1998).The finite temperature cubic term is given by: γT ϕ ≃ ( T / π )[ m e t R ( ϕ, T )] / , where the lightest right handedstop mass m e t R ≈ m U + ξT + 0 . M Z cos(2 β ) + m t − e A t m Q ! , (146)where e A t = A t − µ/ tan( β ) is the stop mixing parameter, A t is the trilinear term in the MSSM superpotential, and µ is the soft-SUSY breaking mass parameter for the right-handed stop. The coefficient γ of the cubic term γT ϕ in the effective potential reads γ MSSM ≈ γ SM + h t sin ( β )4 √ π − ˜ A t m Q ! / , (147)and can be at least one order of magnitude larger than γ SM . The implications for the particle spectrum are: • A light right-handed stop: 120 GeV ≤ m e t ≤
170 GeV ≤ m t . • A heavy left-handed stop: m Q ≥ • A light SM-like Higgs: m H ≤
120 GeV, for 5 < tan β < CP -evenHiggs mass is m H ≥
115 GeV (Nakamura et al. , 2010).Note that within MSSM, the lightest Higgs mass isbounded by: m H ≤ M Z cos β . Hence, even an MSSM-based electroweak baryogenesis may be at the verge ofbeing ruled out.MSSM also provides new CP violating complex phasesin the Higgsino sector, i.e. arg ( µM , ) ≥ − , with µ, M , ≤
400 GeV. The CP -violating phases arealso constrained by the electric dipole moments. Tomatch the observational limit on | d e | < . × − e cm (Regan et al. , 2002), one requires first and secondgeneration sfermion masses greater than 10 TeV. whilethe 2-loop electron dipole moment contribution comesout to be: | d e | ≥ × − e cm.The definitive test of the MSSM based electroweakbaryogenesis will obviously come from the Higgs andthe stop searches at the LHC (Carena et al. , 2003;Chung et al. , 2009).
4. Electroweak baryogenesis beyond MSSM
Some of these problems of MSSM can be resolvedin nMSSM (next-to minimal SUSY SM), with the helpof introducing an extra singlet in the MSSM superpo-tential: W = m S + λSH u H d + W MSSM . The S field gets a VEV to explain the µ ≡ λ h S i -term, butit also generates a singlet tadpole – its contribution tothe vacuum energy, δV = t s S ∼ (1 / π ) n ( S/M P ) F s ,can be suppressed with the help of discrete symmetries, Z R or Z R , where F s ∼ m soft M P (Abel et al. , 1995;Panagiotakopoulos and Tamvakis, 1999). As a result thesoft-SUSY breaking Higgs potential becomes: V soft = t s ( S + h.c. )+ m s | S | + a λ ( SH u H d + h.c. )+ V MSSM , (148)Note that the trilinear, a λ SH u H d now contributes to the γ -term at the tree level, indicating potentially strongerfirst order phase transition even without a light stop andfor m H >
120 GeV. The CP phases are distributed ingaugino masses as well as in the singlet, but not in thetree level of a λ .One can similarly proceed with 4 SM singlets, and theHiggs doublet as in the case of U (1) ′ electroweak baryo-genesis discussed in Ref. (Kang et al. , 2005), for a re-view see (Kang et al. , 2009), where the superpotentialcontains: W = hSH u H d + λS S S + W MSSM . (149)It is assumed that the U (1) ′ is broken at higher VEVs,such as 1 − S , S , S haveVEVs greater than those of S and, H u and H d . Themass of Z ′ bosons are M Z ′ ∼ O (1) TeV. The tree levelHiggs potential can now contain CP violating contribu-tions from the phases β , β (Kang et al. , 2009, 2005): V soft = V MSSM + m s | S | + X i =1 m S i | S i | − A h h | S || H u || H d | cos β − A λ λ | S || S || S | cos β − m SS | S || S | cos β − m SS | S || S | cos β − | m S S || S || S | cos( − β + β + γ ) (150)The potential can yield strong first order phase transitionwithout large stop masses, and the new contributionsto electron dipole moments can be tamed by tuning theYukawa sector (Kang et al. , 2009, 2005).
5. Thermal Leptogenesis
At temperatures 10 GeV ≥ T ≥
100 GeV, the B + L is completely erased by the sphaleron transitions, a netbaryon asymmetry in the universe can still be generatedfrom a non-vanishing B − L (Fukugita and Yanagida,1986; Harvey and Kolb, 1981; Luty, 1992), even if there4were no baryon number violating interactions. The lep-ton number violating interactions can produce baryonasymmetry, a process which is known as leptogene-sis, for recent reviews, see (Buchmuller et al. , 2005a,b;Davidson et al. , 2008).The lepton number violation requires physics beyondthe SM. The most attractive mechanism arises in SO (10)which is left-right symmetric (for details, see (Langacker,1981)), and has a natural foundation for the see-sawmechanism (Gell-Mann and Slansky, 1980; Minkowski,1977; Mohapatra and Senjanovic, 1980; Yanagida, 1979)as it incorporates a singlet right-handed Majorana neu-trino N R with a mass M R . A lepton number violationappears when the Majorana right handed neutrino de-cays into the SM lepton doublet and Higgs doublet, andtheir CP conjugate state through N R → H + l , N R → ¯ H + ¯ l , (151)where ( H ) l is the SM (Higgs) lepton. The relevant L violating interaction is then given by L ⊃
12 ( M N ) ii N i N i + y ij N i ¯ ℓ j iτ H ∗ + h . c . , (152)where i, j = 1 , ,
3. The above interac-tion is also responsible for generating the observedneutrino masses via the canonical seesaw mecha-nism (Akhmedov et al. , 2003; Buchmuller et al. , 2002;Buchmuller and Plumacher, 1998, 2000), as required bythe neutrino oscillation data (Gonzalez-Garcia et al. ,2010). This mass turns out to be m ν ≈ | y | v /M N with v = 174 GeV, what implies right-handed neutrino massscale of M N ∼ O (10 ) GeV for | y | ∼ m ν ∼ . M ≪ M , M (correspond-ing to N , N , N ). The CP asymmetry can be es-timated from the N decay, the asymmetry is gener-ated through the interference between tree level andone-loop diagrams, which is given by (Covi et al. , 1996;Flanz et al. , 1995; Fukugita and Yanagida, 1986; Luty,1992; Plumacher, 1998) ǫ = Γ( N → lH ) − Γ( N → ¯ l ¯ H )Γ( N → lH ) + Γ( N → lH ) , (153)= 18 π yy † X i =1 , , Im[( yy † ) i ] f ( M i /M ) , (154)where f is a function which represents radiative cor-rections. In the case of SM, f ( x ) = √ x [( x − / ( x −
1) + ( x −
1) ln(1 + 1 /x )], and in the case of MSSM, f ( x ) = √ x [2 / ( x −
1) + ln(1 + 1 /x )].Let us take an example of the SM where the CP phasecan be labeled by, | ǫ | = 3 M / (16 πv ) p ∆ m sin δ ,where ∆ m is the atmospheric mass scale of light neu-trinos (Gonzalez-Garcia et al. , 2010) and δ is the effective CP violating phase. The total lepton asymmetry is thengiven by η L = | ǫ | Y N κ (155) where Y N is the abundance of the right handed Majo-rana neutrino N and κ is a thermal wash-out factor,which takes into account that the scatterings such as¯ ℓH ↔ ℓ ¯ H tend to wash out any lepton asymmetry beingcreated.In order to process the total lepton asymmetryinto baryons, we need to know the chemical poten-tials (Khlebnikov and Shaposhnikov, 1988) B = X i (2 µ qi + µ u R i + µ d R i ) , L = X i (2 µ li + µ e R i ) , (156)where i denotes three leptonic generations. The Yukawainteractions establish an equilibrium between the differ-ent generations ( µ li = µ l and µ qi = µ q , etc.), and oneobtains expressions for B and L in terms of the numberof colors N = 3, and the number of charged Higgs fields N H B = − N µ l , L = 14 N + 9 N N H N + 3 N H µ l , (157)together with a relationship between B and B − L (Khlebnikov and Shaposhnikov, 1988) B = (cid:18) N + 4 N H N + 13 N H (cid:19) ( B − L ) . (158)The final asymmetry is then given by B = (28 / B − L ) in the case of SM and B = (8 / B − L ) for theMSSM (Khlebnikov and Shaposhnikov, 1988)The baryon asymmetry based on the decays of righthanded neutrinos in a thermal bath has been com-puted within MSSM (Buchmuller et al. , 2002, 2005a;Giudice et al. , 2004), where besides the right handedneutrinos the right handed (s)neutrinos also participatein the interactions. The decay of a RH (s)neutrinowith mass M i results in a lepton asymmetry via one-loop self-energy and vertex corrections, see Eq. (153).If the asymmetry is mainly produced from the decay ofthe lightest right handed states, and assuming hierar-chical right handed (s)neutrinos M ≪ M , M , we willhave (Davidson and Ibarra, 2002) η ≃ × − κ (cid:18) m − m .
05 eV (cid:19) (cid:18) M GeV (cid:19) , (159)for O (1) CP -violating phases ( m ν < m ν < m ν are the masses of light mostly light handed neutrinos).Here κ is the efficiency factor accounting for the de-cay, inverse decay and scattering processes involvingthe right handed states (Buchmuller et al. , 2002, 2003,2005a; Giudice et al. , 2004).A decay parameter K can be defined as K ≡ Γ H ( T = M ) , (160)where Γ is the decay width of the lightest right handed(s)neutrino. If K <
1, the decay of right handed5states will be out of equilibrium at all times. In thiscase the right handed states, which are mainly producedvia scatterings of the left handed (s)leptons off the top(s)quarks and electroweak gauge/gaugino fields, neverreach thermal equilibrium. The cross-section for produc-ing the right handed (s)neutrinos is ∝ T − ( M ), when T > M ( < M ), and hence most of them are producedwhen T ∼ M . The efficiency factor reaches its maxi-mum value for κ ≃ . m ν = 10 − eV. For largervalues of m ν it drops again, because the inverse decaysbecome important and suppress the generated asymme-try. Producing sufficient asymmetry then sets a lowerbound, M ≥ GeV (Buchmuller et al. , 2002, 2003,2005a; Giudice et al. , 2004). Successful thermal leptoge-nesis therefore requires that T R ≥ GeV. a. Resonant leptogenesis:
If the mass splitting be-tween, say M , M , is comparable to their decaywidths, the CP asymmetry resonantly gets enhanced,see Eq. (153). For example, let us consider N and N . The dominant contribution to the CP asymme-try arises in the mixing of N and N , and it is givenby (Pilaftsis and Underwood, 2004, 2005; Plumacher,1998) ǫ = Im( y † y ) π ( y † y ) ( M − M ) M M ( M − M ) + ( M Γ − M Γ ) . (161)Now, assuming M ∼ M and M − M ∼ Γ − Γ There-fore, a large L asymmetry can be produced even if theinitial abundance of N and N is small. b. Flavored leptogensis:
So far we have assumed thatall the leptonic flavors, i.e. τ, µ, e , behave alike in athermal bath. Especially in a non-SUSY case, where wecan imagine a thermal bath of SM d.o.f with a tempera-ture ≤ GeV, the τ -Yukawa interactions are in ther-mal equilibrium, while temperatures below 10 GeV themuon Yukawa interactions are faster than the expansionrate of the universe and the leptogenesis rate. Since thewash out factor κ is inversely proportional to the leptonviolating interaction rates, so each of the flavored symme-tries is subject to its own wash out effect (Abada et al. ,2006a,b). The above Eq. (155) gets modified to: η = Y N n f X i = τ,... κ i ǫ i , (162)where n f corresponds to the number of active flavorsparticipating in thermal interactions. For temperaturesranging 10 GeV ≤ T ≤ GeV the flavored leptoge-nesis can enhance the net baryon asymmetry by a factor2 or 3 (Davidson et al. , 2008). c. Dirac leptogenesis:
It is possible that the decay of aheavy particle accompanied by the CP distributed thelepton number equally between left handed and right handed particles with the net lepton number zero. A spe-cific example will be when the decay gives rise to a neg-ative lepton number in left-handed neutrinos, and a pos-itive lepton number of equal magnitude in right-handedneutrinos. If the observed neutrinos are Dirac in naturewith a small Yukawa couplings h ∼ − , then the leftand right handed neutrinos will not come to thermal equi-librium before the electroweak scale, H ∼ Γ ⇒ T /M P ∼ h T . Since the sphalerons interact with the left-handedneutrinos, violating B + L and conserving B − L , part ofthe lepton number in left handed neutrinos get convertedinto baryon number. Ar lower temperatures, the universecontains a total positive baryon number, total positivelepton number, and B − L = 0 (Dick et al. , 2000). d. Leptogenesis via scattering:
If there exists ashadow world similar to the SM sector, but hidden, andthe only mediator is the heavy singlet neutrinos N , thenit is possible to realize leaking the lepton number fromthe hidden to the visible sector (Bastero-Gil et al. , 2002;Bento and Berezhiani, 2001). Let us consider a simpleinteraction between hidden (lepton doublet l ′ and Higgs φ ′ ) and visible sector fields (SM lepton doublet l , and theSM Higgs φ ) via h ia l i N a φ + h ′ ka ℓ ′ k N a φ ′ + 12 M ab N a N b + H . C . (163)where the Yukawa interactions are given by h ia and h ka .After integrating out the heavy neutrinos N with a mass M a = g a M , where M being the overall mass scale and g a are order one real constants, we obtain an effectivedimensional 5 operators A ij M l i l j φφ + D ik M l i l ′ k φφ ′ + A ′ kn M ℓ ′ k ℓ ′ n φ ′ φ ′ + H . C . , (164)with coupling constant matrices of the form A = hg − h T , A ′ = h ′ g − h ′ T and D = hg − h ′ T . Let us supposethat the reheat temperatures in both hidden, T ′ R , andvisible sector, T R , are below M . The only way thetwo sectors can interact via the lepton number violatingscatterings mediated by the heavy neutrinos N whichstay out of equilibrium, since T R ≪ M . The CP phase can be obtained in lφ ↔ ℓ ′ φ ′ and ℓφ ↔ ¯ ℓ ′ ¯ φ ′ .the net asymmetry is given by ∆ σ = 3 J S/ π M ,where J = Im Tr[( h ′ † h ′ ) g − ( h † h ) g − ( h † h ) ∗ g − ] is the CP -violation parameter and S is the c.m. of energysquare. The final B − L asymmetry of the universe isgiven by (Bento and Berezhiani, 2001) B − L = n B − L s = " ∆ σ n Hs R , ≈ − J (cid:18) GeV M (cid:19) (cid:18) T R GeV (cid:19) (165)where s is the entropy density, and for Yukawa constantsspreading in the range 0 . − e. Non-thermal leptogenesis:
There existvarious scenarios of non-thermal leptogene-sis (Allahverdi and Mazumdar, 2003; Asaka et al. ,1999, 2000b; Giudice et al. , 1999a; Lazarides and Shafi,1991; Murayama et al. , 1993) which can work for T R ≤ M N . One classic example is when the righthanded sneutrino, a scalar field, with mass M N , domi-nates the energy density of the universe and decays intothe SM leptons and Higgs to reheat the universe andsimultaneously creating the lepton asymmetry. The CP asymmetry can be created again from the interferencebetween a tree level and one-loop quantum corrections,which yields the net asymmetry: η ∼ n L s ∼ ǫ ρ N sM N ∼ ǫ T R M N . (166)A similar expression can be used if any sneutrino con-densate decays after inflation (Berezhiani et al. , 2001;Mazumdar, 2004a,b; Mazumdar and Perez-Lorenzana,2004a,b; Postma and Mazumdar, 2004), in which case T R is replaced by the decay temperature of the sneu-trino condensate, i.e. T D . The right handed neutri-nos and sneutrinos could also be excited non-thermallyduring preheating if they couple to the inflaton, whichwould generate non-thermal leptogenesis (Giudice et al. ,1999a). f. Soft leptogenesis:
In a perfect SUSY preservinglimit the mass and the width of the right-handed neu-trino and sneutrino would be the same. Let us considera single generation, where the mass is M N , and theirwidth is given by Γ = Y M N π = mM N / πv , m ≡ Y v /M N , where v ∼
174 GeV is the Higgs VEV, and Y N is the Yukawa coupling. However, in a realistic scenariowe would expect soft SUSY breaking terms which wouldbe relevant for soft-leptogenesis (Allahverdi et al. , 2003;D’Ambrosio et al. , 2003; Grossman et al. , 2003, 2005): L soft = BM N e N e N + AY e L e NH + h.c. (167)This model has one physical CP violating phase given by: φ = arg( AB ∗ ). The soft SUSY breaking terms introducemixing between the sneutrino e N and the anti-sneutrino e N † in a similar fashion to the B − ¯ B and K − ¯ K sys-tems. The mass and width difference of the two sneutrinomass eigenstates are given by∆ m = | B | , ∆Γ = 2 | A | Γ M N . (168)The CP violation in the mixing is responsible for gener-ating the lepton-number asymmetry in the final states ofthe e N decay. This lepton asymmetry is converted into thebaryon asymmetry through the sphaleron process. Thebaryon to entropy ratio is given by (D’Ambrosio et al. ,2003): n B s = − − α (cid:20) | B | | B | + Γ (cid:21) | A | M N sin φ , (169) where the efficiency parameter α depends on the mech-anism that produces the right-handed sneutrinos. In athermal production, the largest conceivable value couldbe of order α ∼ . m ν ∼ − eV (D’Ambrosio et al. , 2003). It may be slightlychallenging to fix the parameters to obtain the right lep-ton asymmetry either making | B | / Γ or Γ / | B | small. Theabove requirement gives a non-trivial constraint on theparameters (D’Ambrosio et al. , 2003; Grossman et al. ,2003): A ∼ GeV , M N ≤ GeV , B ≤ , φ ∼ . (170)Small value of | B | cannot be obtained in gravity me-diated SUSY breaking scenarios, but it might be pos-sible to arrange within gauge mediated SUSY break-ing (Grossman et al. , 2004).
6. Affleck-Dine Baryogenesis
As we discussed already, within MSSM there ex-ists cosmologically flat directions (Dine et al. , 1996b;Gherghetta et al. , 1996). Field fluctuations alongsuch flat directions are smoothed out by infla-tion (Enqvist and Mazumdar, 2003), which effectivelystretches out any gradients, and only the zero mode ofthe scalar condensate remains. Baryogenesis can thenbe achieved by the perturbative decay of a conden-sate (Allahverdi and Mazumdar, 2007a, 2008) that car-ries baryonic charge, as was first pointed out by Affleckand Dine (AD) (Affleck and Dine, 1985). As we willdiscuss, the flat direction condensate can get dynami-cally charged with a large B and/or L by virtue of CP -violating self-couplings.In the original version (Affleck and Dine, 1985)baryons were produced by a direct decay of the con-densate. It was however pointed out that in the caseof gauge mediated SUSY breaking (Kusenko, 1997b;Kusenko and Shaposhnikov, 1998), and in the case ofgravity mediated SUSY breaking (Enqvist et al. , 2000;Enqvist and McDonald, 1998, 1999, 2000), that the ADflat direction condensate in most cases is not stablebut fragments and eventually forms non-topological soli-tons called Q -balls (Coleman, 1985). In gauge me-diated SUSY breaking scenarios these Q-balls can bemade a long lived dark matter candidate (Kusenko et al. ,1998; Kusenko and Shoemaker, 2009). For reviewssee (Dine and Kusenko, 2004; Enqvist and Mazumdar,2003).Since, SUSY is broken by the finite energy density ofthe inflaton, the AD condensate receives corrections inthe case of F-term inflation. Let us consider a genericsuperpotential for the AD field given by Eq. (79), thenthe effective potential for the AD field will be given by7 −0.5 0 0.5 1 1.5 2 2.5−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.100.10.2 f f (a) FIG. 4: The dynamical motion of AD condensate φ = φ + iφ for a gravity mediated case with d = 4(solid) and d = 6 (dashed) with the initial condition θ i = − π/ et al. , 1995b, 1996b) V ( φ ) = − C I H I | φ | + aλ d H φ d dM d − + h . c . ! + m φ | φ | , + A φ λ d φ d dM d − + h . c . ! + | λ | | φ | d − M d − . (171)The first and the third terms are the Hubble-induced andlow-energy soft mass terms, respectively, while the secondand the fourth terms are the Hubble-induced and low-energy A terms. The last term is the contribution fromthe non-renormalizable superpotential. The coefficients | C I | , a, λ d ∼ O (1), and the coupling λ ≈ / ( d − A φ term has a mass dimension. The a, A -terms in Eq. (171) violate the global U (1) symmetrycarried by φ . If | a | is O (1), the phase θ of h φ i is related tothe phase of a through nθ + θ a = π ; otherwise θ will takesome random value, which will generally be of O (1). Thisis the initial CP -violation which is required for baryoge-nesis/leptogenesis. The AD baryogenesis is quite robustand can occur even in presence of positively large Hubble-induced corrections (Kasuya and Kawasaki, 2006).At large VEVs the first term dictates the dynamics ofthe AD field. If C I < φ = 0 and during inflation the condensate willevolve to its global minimum in one Hubble time. On theother hand if C I >
0, the absolute value of the AD fieldsettles during inflation to the minimum given by | φ | ≃ ( H I M d − ) /d − . After the end of inflation the minimumof the condensate evolves from its initial large VEV toits global minimum φ = 0, note that the dynamics isnon-trivial when the condensate starts oscillating when H ( t ) ∼ m φ ∼ O (100) GeV. The dynamics of the ADcondensate is non-trivial as shown in Fig. (4).If inflation is driven by D-term, one does not get theHubble induced mass correction to the flat direction sothat C I , a = 0. Also the Hubble induced a -term is ab-sent. However the Hubble induced mass correction even-tually dominates once D-term induced inflation comes to an end.The baryon/lepton number density is related to thedynamics of the AD field by n B,L = βi ( ˙ φ † φ − φ † ˙ φ ) , (172)where β is corresponding baryon and/or lepton charge ofthe AD field.The equation of motion for the AD field isgiven by ¨ φ + 3 H ˙ φ + ∂V ( φ ) /∂φ ∗ = 0 . (173)The above two equations give rise to˙ n B,L + 3 Hn B,L = 2 β I m (cid:20) ∂V ( φ ) ∂φ ∗ φ (cid:21) , = 2 β m φ dM d − I m( aφ d ) . (174)The net baryon and/or lepton number can be obtainedby integrating the above equation a ( t ) n B,L ( t ) = 2 β | a | m φ M d − Z t a ( t ′ ) | φ ( t ′ ) | d sin( θ ) dt ′ , (175)Note that ′′ a ′′ introduces an extra CP phase which can beparameterized by sin( δ ). Note that the asymmetry is notgoverned by the Hubble induced A term, the amplitude ofthe oscillations will be damped and so the A -term, whichis proportional to a large power of φ will become gradu-ally negligible. The net baryon and/or lepton asymmetryis given by (Dine et al. , 1996b) n B,L ( t osc ) = β d − d − m φ φ sin 2 θ sin δ , ≈ β d − d − m φ (cid:0) m φ M d − (cid:1) / ( d − , (176)where sin δ ∼ sin 2 θ ≈ O (1). When the inflaton de-cay products have completely thermalized with a reheattemperature T R , the baryon and/or lepton asymmetry isgiven by n B,L s = 14 T R M H ( t osc ) n B,L ( t osc ) , = d − d − β T R M m φ (cid:0) m φ M d − (cid:1) / ( d − , (177)where we have used H ( t osc ) ≈ m φ , and s is the entropydensity of the universe at the time of reheating. For d = 4, the baryon-to-entropy ratio is n B,L s ≈ × − × β (cid:18) m φ (cid:19) (cid:18) T R GeV (cid:19) , (178)and for d = 6 n B,L s ≈ − × β (cid:18) m φ (cid:19) / (cid:18) T R
100 GeV (cid:19) , (179)8where we have taken the net CP phase to be ∼ O (1).The asymmetry remains frozen unless there is additionalentropy production afterwards.The lepton asymmetry calculated above inEqs. (178,179) can be transformed into baryon numberasymmetry via sphalerons n B /s = (8 / n L /s . ADleptogenesis has important implications in neutrinophysics also, because in the MSSM, the LH u directionis lifted by the d = 4 non-renormalizable operator whichalso gives rise to neutrino masses (Asaka et al. , 2000a;Dine et al. , 1996b): W = 12 M i ( L i H u ) = m ν i h H u i ( L i H u ) , (180)where we have assumed the see-saw relation m ν i = h H u i /M i with diagonal entries for the neutrinos ν i , i =1 , ,
3. The final n B /s can be related to the lightest neu-trino mass since the flat direction moves furthest alongthe eigenvector of L i L j which corresponds to the small-est eigenvalue of the neutrino mass matrix (Asaka et al. ,2000a; Dine et al. , 1996b). n L s ≈ × − × β (cid:18) m / m φ (cid:19) (cid:18) T R GeV (cid:19) (cid:18) − eV m νl (cid:19) , (181)where m ν l denotes the lightest neutrino. Similarlythe d = 6 case corresponds to the flat direction udd (Enqvist and McDonald, 2000; McDonald, 1997).
7. Baryogenesis below the electroweak scale
At temperatures below the electroweak scale, thesphaleron transitions are rather inactive. If the uni-verse reheats below the electroweak scale then it is achallenge to generate the required baryon asymmetry.The leptogenesis based scenarios are hard to imple-ment below the electroweak scale. However within SMthere exists a possibility of realizing cold electroweakbaryogenesis as discussed in (Cornwall et al. , 2001;Cornwall and Kusenko, 2000; Enqvist et al. , 2010b;Garcia-Bellido et al. , 1999; Krauss and Trodden, 1999;Tranberg et al. , 2010; Tranberg and Smit, 2003, 2006).SUSY further opens a door to realize baryon asymmetryat temperatures even close to the BBN via R-parity vio-lating interactions (Cline and Raby, 1991; Kitano et al. ,2008; Kohri et al. , 2009; Scherrer et al. , 1991). Here wewill discuss both the scenarios. a. Cold electroweak baryogenesis:
There are mecha-nisms to obtain cold electroweak baryogenesis where itis assumed that the SM d.o.f are not in thermal equilib-rium. Moreover in a cold environment the CP-violationin SM is much larger than at the electroweak tempera-tures, and most of the baryon asymmetry is produced atthe initial quench when the Higgs field is rolling downthe potential. Baryon production essentially stops afterthe first few oscillations, after which the coherent Higgs field will start decaying, thereby reheating the universe.However, the hurdle is to obtain this fast quench withoutthe presence of strong first order phase transition.There are couple of possibilities of realizing cold ini-tial condition and out of equilibrium condition. Therecould be a very low scale of inflation which might notbe responsible for generating the seed perturbations,or the universe could be simply trapped in a vacuumwhere the SM d.o.f are not even excited. The out-of-equilibrium condition can be obtained during the co-herent oscillations of the scalar fields. In order to re-alize this idea, we would require a scalar field coupledto the SM Higgs, σ H . During the coherent oscilla-tions, it is possible to have the baryon number violat-ing sphaleron transitions (Cornwall and Kusenko, 2000;Garcia-Bellido et al. , 1999; Tranberg and Smit, 2003).This can happen since the Higgs oscillations can ex-cite the electroweak gauge bosons from the time de-pendent vacuum fluctuations with a very large occu-pation number, similar to case of preheating. Theselong wavelength fluctuations of the gauge fields are re-sponsible for overcoming the sphaleron barriers whichleads to the baryon number violation. Furthermore,there could be extra sources of CP-violations duringthe oscillations as pointed out in (Cornwall et al. , 2001;Cornwall and Kusenko, 2000; Tranberg et al. , 2010). b. R-parity violation and baryogenesis:
The currentlimits on some of the R -parity violating interactions arepoorly understood. Let us now consider a scenario whereB and L are violated within MSSM, with a superpoten-tial: W = µ ′ i L i H u + λ ijk L i L j e k + λ ′ ijk L i Q j d k + λ ′′ ijk u i d j d k , (182)where L i = ( ν i , e i ), Q i = ( u i , d i ), H u = ( h + u , h u ) T , H d = ( h d , h − d ) T , etc are SU (2) L doublets and u ci , d ci are SU (2) L singlet quarks. In Eq.(182), the first threeterms violate lepton number by one unit (∆ L = 1),while the last term violates baryon number by one unit(∆ B = 1). For the stability of proton we assume that λ ijk = λ ′ ijk = 0. This can be accomplished if there existsany conservation of lepton number, which then forces µ ′ i to be zero. However, the electric dipole moment of neu-tron gives (Barbier et al. , 2005)Im ( λ ′′ λ ′′ ) < . (cid:18) . V td (cid:19) ˜ M TeV ! (183)and the non-observation of n − ¯ n oscillation gives an upperbound on λ ′′ k to be (Barbier et al. , 2005) | λ ′′ k | < (cid:0) − − − (cid:1) s τ osc ˜ M TeV ! / (184)While λ ′′ is hardly constrained and can be taken to beas large as O (1). Let us consider that a scalar field φ decays to MSSM d.o.f right before BBN primarily into9gauge bosons and gauginos via R -parity violating cou-plings λ ′′ ijk .Let us assume that the gauginos are heavier than thequarks and squarks. As a result their decay to a pairof quark and squark through one loop quantum correc-tion gives rise to a net CP violation. The magnitude ofCP violation in the decay: ˜ g → t ˜ t c can be estimatedas (Cline and Raby, 1991): ǫ = Γ (cid:0) ˜ g → t ˜ t c (cid:1) − Γ (cid:0) ˜ g → ¯ t ˜ t (cid:1) Γ tot˜ g ≈ λ ′′ π Im ( A ∗ m ˜ g ) | m ˜ g | (185)where A is the trilinear SUSY breaking term andwe also assume a maximal CP violation. As a resultthe decay of gauginos produce more squarks (antisqarks)than antisquarks (squarks). The baryon number violat-ing (∆ B = 1) decay, induced by λ ′′ of squarks (anti-squarks) to quarks (antiquarks) then gives rise to a netbaryon asymmetry. Note that the decay of squarks (anti-squarks) are much faster than any other processes thatwould erase the produced baryon asymmetry. Hence theB-asymmetry can simply be given by: η B ∼ B ˜ g ǫ n φ s ∼ B ˜ g ǫ T R m φ , (186)where B ˜ g ∼ . φ to ˜ g ˜ g , and in the above equation s is the entropy densityresulted through the decay of φ . For T R /m φ ∼ − and m φ ∼ GeV. Therefore a reasonable CP violation oforder ǫ ∼ . − .
001 could accommodate the desiredbaryon asymmetry of O (10 − ) close to the temperatureof T ∼ − et al. ,2009). IV. DARK MATTER
There is a conclusive evidence that a considerable frac-tion of the current energy density is in the form of a non-baryonic dark matter. The dynamical motions of astro-nomical objects such as rotation curves for spiral galax-ies (Begeman et al. , 1991), velocity dispersion of individ-ual galaxies in galaxy clusters, large x-ray temperaturesof clusters (Flores et al. , 2007), bulk flows and the pecu-liar motion of our own local group (Dressler et al. , 1987),all implies the presence of a dark matter. The massof galaxy clusters inferred by their gravitational lensingof background images is also consistent with the largedark-to-visible mass ratios (Bolton et al. , 2006). Per-haps the most compelling evidence, at a statistical sig-nificance of 8 σ comes from the two colliding clusters ofgalaxies, known as the Bullet cluster (Clowe et al. , 2006;Markevitch et al. , 2004). It was found that the spatialoffset of the center of the total mass from the center of thebaryonic mass peaks cannot be explained with an alter-ation of gravitational force law. Furthermore, the largescale structure formation from the initial seed perturba-tions from inflation requires a significant non-baryonic dark matter component (Abazajian et al. , 2009). Interms of the critical density, ρ c = 3 H M / π =1 . × − g cm − and with Hubble constant H ≡ h km sec − Mpc − , the dark matter density inferredfrom WMAP and large scale structure data is Ω DM ≡ ρ DM /ρ c ∼ .
22 (Komatsu et al. , 2011).The dark matter is assumed to be a weakly in-teracting massive particle (WIMP), yet undiscovered.There are many well motivated particle physics candi-dates, e.g. (Bertone et al. , 2005; Jungman et al. , 1996;Kusenko, 2009; Taoso et al. , 2008), all of which arisefrom beyond the SM physics. The dark matter is as-sumed to be stable on the scale of cosmological structureformation. By virtue of new symmetries, for exampleR-parity conservation in SUSY allows the lightest SUSYparticle (LSP) to be absolutely stable (Ellis et al. , 1984a;Goldberg, 1983), or in the case of extra dimensions, theKaluza-Klein (KK) parity leaves the lightest KK particle(LKP) stable (Servant and Tait, 2003).In many cases some of these symmetries which pro-tect the dark matter particle from decaying are brokenby sufficiently suppressed higher-dimensional operators,such that the dark matter might as well have a finitelife time comparable to the age of the universe. In thecontext of SUSY grand unification, operators with massdimension 6 are expected to make SUSY dark matterunstable, with a time-scale τ ∼ π (cid:18) M m X (cid:19) ∼ sec (cid:18) TeV m X (cid:19) (cid:18) M GUT × GeV (cid:19) (187)where M GUT ∼ GeV, and m X is the dark matterparticle. The lower dimensional operators would yieldmuch shorter time scale, as it would lead to dark mat-ter decay long before the structure formation. WithinSUSY one compelling candidate could be the gravitinowith R-parity weakly broken in the hadronic sector,yielding the required baryon asymmetry also in the pro-cess (Kohri et al. , 2009).The widely accepted lore is that after radiation-matterequality, when the universe becomes matter dominated,the density perturbations in the dark matter begin togrow, and drive the oscillations of the photon-baryonicfluid around the dark matter gravitational potentialwells. Immediately after the epoch of recombinationthe baryons kinematically decouple from photons, whichthen free-stream through the universe; the baryons onthe other hand slowly fall into the potential wells createdby the dark matter particles, eventually becoming lightemitting galaxy, see for more details (Dodelson, 2003;Kolb and Turner, 1988; Peebles, 1994). There are threebroad categories of dark matter which have been centralto our discussion.0 A. Types of dark matter
1. Hot Dark Matter
If the dark matter particle is collisionless, then theycan damp the fluctuations from higher to lower densityregions above the free-streaming scale. This hot darkmatter consists of particles which are relativistic at thetime of structure formation and therefore lead to largedamping scales (Bond and Szalay, 1983).The SM neutrinos are the simplest examples of hotdark matter. In the early universe they can be decoupledfrom a relativistic bath at T ∼ ν h = P i m ν i
90 eV . (188)Various observational constraints combining Ly- α for-est, CMB, SuperNovae and Galaxy Clusters data leadsto (Fogli et al. , 2008; Seljak et al. , 2006): P m ν < .
17 eV (95 % CL). Similar limits can be applied toany generic hot dark matter candidate, such as ax-ions (Hannestad et al. , 2010) or to hot sterile neutri-nos (Dodelson et al. , 2006; Kusenko, 2009). The free-streaming length for neutrinos is (Kolb and Turner,1988): λ F S ∼ (cid:18)
30 eV m ν (cid:19) Mpc . (189)For instance, the universe dominated by the eV neutri-nos would lead to suppressed structures at 600 Mpc scale,roughly the size of supercluster. Furthermore, hot darkmatter would predict a top-down hierarchy in the forma-tion of structures, with small structures forming by frag-mentation of larger ones, while observations show thatlarger galaxies have formed from the mergers of the ini-tially small galaxies.
2. Cold Dark Matter
The standard theory of structure formation requirescold dark matter (CDM), whose free-streaming length issuch that only fluctuations roughly below the Earth massscale are suppressed (Bertschinger, 2006; Green et al. ,2004, 2005; Hofmann et al. , 2001; Loeb and Zaldarriaga,2005). The CDM candidates are heavy and non-relativistic at the time of their freeze-out from thermalplasma. The current paradigm of ΛCDM is falsifiablewhose predictive power can be used to probe the struc-tures at various cosmological scales, such as the abun-dance of clusters at z ≤ ρ ( r ) = ρ ( r/R s ) γ [1 + ( r/R s ) α ] ( β − γ ) /α , (190)where ρ and the radius R s vary from halo to halo. theparameters α, β and γ vary slightly from one profile toother. The four most popular ones are: • Navarro, Frenk and White (NFW) pro-file (Navarro et al. , 1997), where α = 1 , β =3 , γ = 1, and R s = 20 Kpc. • Moore profile (Moore et al. , 1999), where α =1 . , β = 3 , γ = 1 .
5, and R s = 28 Kpc. • Kra profile (Kravtsov et al. , 1998), where α =2 , β = 3 , γ = 0 .
4, and R s = 10 Kpc. • Modified Isothermal profile (Bergstrom et al. ,1998), where α = 2 , β = 3 , γ = 0, and R s =3 . γ , in the inner most regions ispart of the numerical uncertainties and still under debate,as all four simulations provide different numbers. Thesimulations hint towards a cuspy profile, as the density inthe inner regions becomes large, while from the rotationcurves of low surface brightness (LSB) galaxies point to-wards uniform dark matter density profile with constantdensity cores (Gentile et al. , 2004). In our own galaxythe situation is even more murky, as the observationsof the velocity dispersion of stars near the core suggestsa supermassive black hole at the center of our Galaxy,with a mass M SMBH ≈ . × M ⊙ (Ghez et al. , 1998).Many galaxies have been found to host supermassiveblackholes of 10 − M ⊙ . It has been argued thatif supermassive blackhole exists at the galactic center,the accretion of dark matter by the blackhole wouldenhance the dark matter density (Peebles, 1994). Toalleviate some of these problems, dark matter with astrong elastic scattering cross section (Dave et al. , 2001;Spergel and Steinhardt, 2000), or large annihilation crosssections (Kaplinghat et al. , 2000) have been proposed.There are further discrepancies between observationsand numerical simulations. The number of satellite ha-los as predicted by simulations exceeds the number ofobserved Dwarf galaxies in a typical galaxy like Milky-Way (Klypin et al. , 1999; Moore et al. , 1999). Howeverrecent hydrodynamical simulations with ΛCDM, includ-ing the supernovae induced outflows suggest a fall in thedark-matter density to less than half of what it wouldotherwise be within the central Kpc.1
3. Warm Dark Matter
Besides hot and cold dark matter, the early uni-verse can also provide warm dark matter (WDM) can-didates whose velocity dispersion lies between that ofhot and CDM. The presence of WDM reduces thepower at small scales due to larger free-streaminglength compared to that of a CDM (Bode et al. , 2001;Sommer-Larsen and Dolgov, 2001).The origin of WDM can be found within ster-ile states. For instance, the see-saw mechanismfor the active neutrino masses from the SM singletstates (Gell-Mann and Slansky, 1980; Minkowski, 1977;Mohapatra and Senjanovic, 1981; Yanagida, 1979) wouldnaturally generate masses to the active m ( ν , , ) ∼ y h H i /M N , and sterile neutrinos m ( ν a ) ∼ M N ( a > i, j = 1 , · · · n + 3. The typi-cal mixing angles in this case are: θ ai ∼ y ai h H i /M N .In order to explain the neutrino masses from atmo-spheric and solar neutrino data, n = 2 is sufficient, how-ever for pulsar kicks (Kusenko, 2006; Kusenko and Segre,1996, 1999), supernovae explosion (Fryer and Kusenko,2006; Hidaka and Fuller, 2006, 2007), as well as sterileneutrino as a dark matter candidate (Abazajian et al. ,2001; Asaka et al. , 2005; Dodelson and Widrow, 1994;Dolgov and Hansen, 2002; Petraki and Kusenko, 2008;Shi and Fuller, 1999), we require at least n = 3, so intotal 6 sterile Majorana states, for a review on all theseeffects, see (Kusenko, 2009). The presence of such extrasterile neutrinos is also supported by ¯ ν µ → ¯ ν e oscillationsobserved at LSND (Aguilar et al. , 2001), and the recentresults by MiniBoone (Aguilar-Arevalo et al. , 2010).A sterile neutrino with a KeV mass can be an idealWDM candidate which can be produced in the earlyuniverse by oscillation/conversion of thermal active neu-trinos, with a momentum distribution significantly sup-pressed from a thermal spectrum (Abazajian et al. , 2001;Dodelson and Widrow, 1994). A typical free-streamingscale is given by, see (Abazajian and Koushiappas, 2006) λ F S ≈
840 Kpc h − (cid:18) m s (cid:19) (cid:18) < p/T > . (cid:19) , (191)where m s is the mass of the sterile flavor eigenstate, 0 . ≥h p/T i / . ≥ ∼ . m s ≥ −
20 KeV (95 % CL) ( m W DM ≥ − B. WIMP production
1. Thermal relics
At early times it is assumed that the dark matter parti-cle, denoted by X is in chemical and kinetic equilibrium,i.e. in local thermodynamic equilibrium. The dark mat-ter will be in equilibrium as long as reactions can keep X in chemical equilibrium and the reaction rate can pro-ceed rapidly enough as compared to the expansion rateof the universe, H ( t ). When the reaction rate becomessmaller than the expansion rate, then the particle X canno longer be in its equilibrium, and thereafter its abun-dance with respect to the entropy density becomes con-stant. When this occurs the dark matter particle is saidto be “frozen out.”The equilibrium abundance of X relative to the en-tropy density depends upon the ratio of the mass ofthe particle to the temperature. Let us define the vari-able Y ≡ n X /s , where n X is the number density of X with mass m X , and s = 2 π g ∗ T /
45 is the en-tropy density, where g ∗ counts the number of relativistic d.o.f . The equilibrium value of Y , Y EQ ∝ exp( − x ) for x = m X /T ≫
1, while Y EQ ∼ constant for x ≪ Y EQ can be computed exactlyby solving the Boltzmann equation (Kolb and Turner,1988): ˙ n X + 3 Hn X = −h σv i ( n X − ( n eqX ) ) , (192)where dot denotes time derivative, σ is the total annihila-tion cross section, v is the velocity, bracket denotes ther-mally averaged quantities, and n eq is the number densityof X in thermal equilibrium: n eq = g ( mT / π ) / e − m X /T , (193)where T is the temperature. In terms of Y = n X /s and x = m X /T , and using the conservation of entropy percomoving volume ( sa = constant), we rewrite Eq. (192)as: dYdx = − h σv i sHx (cid:0) Y − ( Y eq ) (cid:1) . (194)In the case of heavy X , the cross section can be expandedwith respect to the velocity in powers of v , h σv i = a + b h v i + O ( h v i ) + ... ≈ a + 6 b/x , where x = m X /T and a, b are expressed in GeV − . Typically a = 0 for s-waveannihilation, and a = 0 for p-wave annihilation. Wecan rewrite Eq. (194) in terms of a new variable: ∆ = Y − Y eq , ∆ ′ = − Y eq ′ − f ( x )∆(2 Y eq + ∆) , (195)where prime denotes d/dx , and f ( x ) = πg ∗ m X M P ( a + 6 b/x ) x − . (196)2One can find a simple analytic solution for Eq. (195) fortwo extreme regimes∆ = − Y eq ′ f ( x )(2 Y eq + ∆) , x ≪ m X T f , ∆ ′ ≪ Y eq ′ (197)∆ − ∆ ′ = − f ( x ) , x ≫ m X T f , ∆ ′ ≫ Y eq ′ . (198)Integrating the last equation for ( x f , ∞ ), and using∆( x f ) ≫ ∆ ∞ , we find∆ − ∞ ≈ Y − ∞ = r πg ∗ M P (cid:18) m X x f (cid:19) (cid:18) a + 3 bx f (cid:19) . (199)In terms of the present energy density, ρ X = m X n X = m X s Y ∞ , where s = 2889 . − is the present entropydensity, the relic abundance of dark matter particle interns of the critical energy density is given by:Ω X h ≈ . × M P x f √ g ∗ ( a + 3 b/x f ) GeV − , (200)where the freeze-out temperature is defined by solvingthis equation ∆( x f ) = cY eq ( x f ) iteratively for early andlate time solutions, for c ∼ O (1) x f = ln " c ( c + 2) √ g ast π / m X M P ( a + 6 b/x f ) g / ∗ √ x f . (201)An approximate order of magnitude estimation of theabundance can be written as:Ω X h ≈ × − cm s − h σv i ∼ . h σv i . (202)For a WIMP interacting with a heavy gauge boson, wouldnaturally yield an upper bound on m X . From the aboveEq. (202), on dimensional grounds, Ω ∼ / h σv i ∼ /m X for h σv i ≃ α ( m X /M ) , where M is the mass of thenew gauge boson. For m X ∼ M ∼ X h ∼ O (1), for α ∼ .
1. Actually, the unitaritybound limits the dark matter mass to be below m X ≤
300 TeV (Griest and Kamionkowski, 1990). For a real-istic scenario α ∼ .
01, the unitarity bound would yield m X ≤ h σv i ≈ a + 6 b/x may not hold universally. When amass of second particle becomes nearly degenerate withthe dark matter particle X as in the case of coannihila-tion (Binetruy et al. , 1984; Griest and Seckel, 1991), orthe cross section is strongly varying function of the cen-ter of mass energy as in the case of a resonant anni-hilation (Griest and Seckel, 1991). In the latter case, σ ,gets a boost by resonant annihilation when m X ≈ m A / X annihilates with an exchange of particle, A , witha mass m A .
2. Coannihilating WIMPs
If there are N particles, X i ( i = 1 , . . . , N ) with thelightest one, X , which have nearly degenerated masses m i , such that m ≤ m ≤ · · · ≤ m N − ≤ m N ,and internal d.o.f (statistical weights) g i . The next tolightest dark matter particle will be N . In this casethe above calculation of relic density, Eq. (192), getsmodified (Binetruy et al. , 1984; Griest and Seckel, 1991;Servant and Tait, 2003). dndt = − Hn − N X i,j =1 h σ ij v ij i (cid:0) n i n j − n eq i n eq j (cid:1) , (203)where n = P Ni =1 n i is the number density of the relicparticle, since all other particles decay much before thelong-lived X . The total annihilation rate for X i − X j into a SM particle is given by: σ ij = X X σ ( X i X j → X SM ) , and (204) v ij = q ( p i · p j ) − m i m j /E i E j , (205)is the relative particle velocity, with p i and E i are thefour-momentum and energy of particle i . One requiresto define a thermal averaged h σ ij v ij i , which is definedby: h σ ij v ij i = R d p i d p j f i f j σ ij v ij R d p i d p j f i f j , (206)where f i are distribution functions in the Maxwell-Boltzmann approximation.Typically, when the scattering rate of particles off SMparticles in a thermal background is much faster thantheir annihilation rate, then in the above Eq. (192), h σv i is replaced by: h σ eff v i = N X ij σ ij g i g j g eff (1 + ∆ i ) / (1 + ∆ j ) / e − x (∆ i +∆ j ) . (207)where ∆ i = ( m i − m ) /m , and g eff = P Ni g i (1 +∆ i ) / exp( − x ∆ i ). In the case of co-annihilation, thefreeze-out temperature is determined by x f = ln " c ( c + 2) √ g eff π / m X M P ( a eff + 6 b eff /x f ) g / ∗ √ x f . (208)where a eff and b eff are the coefficients of the Taylorexpansion of σ eff . The relic abundance for N is nowgiven byΩ N h ≈ × − cm s − g / ∗ x − f ( I a + 3 I b /x f ) , where (209) I a = x f Z ∞ x f a eff x − dx , I b = 2 x f Z ∞ x f b eff x − dx . X . They alleventually decay into X , and at the time of freeze-out the density of all heavy particles is exponentiallysuppressed except when there is mass degeneracy oc-curs between heavy particles and the X . The detailsof coannihilation has been studied extensively withinSUSY (Edsjo and Gondolo, 1997), and publicly availablenumerical codes include coannihilations with all SUSYparticles (Gondolo et al. , 2004).
3. Non-thermal relics
The dark matter particle, X , can also be created inan out of equilibrium condition, i.e. X must not havebeen equilibrium when it froze out. A sufficient condi-tion for non-equilibrium is that the annihilation rate (perparticle) must be smaller than the expansion rate of theuniverse: n X h σv i < H .Let us assume that X were non-relativistic at the timeof production and they were never in local thermody-namical equilibrium. The largest dark matter densitywill thus be determined by the largest freeze out temper-ature, which can be attainable in the universe. Assumingthis to be the reheat temperature, T R and the universefollows a radiation domination, then the ratios of energydensities will be given by (Kolb et al. , 1998): ρ X ( t ) ρ r ( t ) = ρ X ( t R ) ρ r ( t R ) (cid:18) T R T (cid:19) , (210)where T is the present temperature and t correspondsto the present time, ρ γ is the energy density in radiation,and ρ X = m X n X denotes the energy density in the darkmatter with the number density n X . If we further as-sume that X particles were created at time t = t ∗ < t R ,sometime during the coherent oscillations of the inflatonand before the completion of reheating, then both the X particle energy density and the inflaton energy den-sity would redshift approximately at the same rate untilreheating is completed. Therefore, ρ X ( t R ) ρ r ( t R ) ≈ ρ X ( t ∗ )3 M H ( t ∗ ) , (211)assuming that the inflaton energy density dominated theuniverse. Since, Ω X = ρ X ( t ) /ρ c ( t ), where ρ c ( t ) =3 H M and H = 100 h km sec − Mpc − , then usingEq. (210), one obtains (Kolb et al. , 1998):Ω X h ≈ Ω r h (cid:18) T R T (cid:19) (cid:18) M X M P (cid:19) n X ( t ∗ )3 M P H ( t ∗ ) . (212) ∼ (cid:18) T R GeV (cid:19) ρ X ( t ∗ ) ρ inf ( t ∗ ) , (213)where Ω r h ≈ . × − is the fraction of critical energydensity in radiation today, and T ∼ . × − GeV.The above expression tells us that a non-thermal creation would require a very small fraction of the inflaton energydensity to be transferred to the dark matter particle X ,otherwise the universe would be dominated by the darkmatter particles.For a singlet hidden sector dark matter, it is really achallenge not to overproduce them directly from the decayof the inflaton. If the inflaton sector belongs to the hid-den sector, then it is natural to have inflaton couplings tosuch hidden sector dark matter field. There are three pos-sible ways to obtain a small fraction of ρ X ( t ∗ ) /ρ inf ( t ∗ )in order to match the current observations. (a) Gravitational production The dark matter can be created from the transitionof the equation of state of the universe from inflation tomatter domination or radiation domination, due to non-adiabatic evolution of the vacuum (Chung et al. , 2001,1999b, 2000; Kolb et al. , 1998). The underlying mech-anism is similar to the metric fluctuations which seedthe structure formation, except now the excitations cancreate massive particles. The gravitational production ofdark matter is universal, and it can occur even if the darkmatter coupling to the inflaton is vanishingly small.Let us consider a simple action for X field with a metric ds = dt − a ( t ) d x = a ( η ) (cid:2) dη − d x (cid:3) , where η is aconformal time. S = Z dt Z d x a (cid:18) ˙ X − ( ∇ X ) a − m X X − ξRX (cid:19) , (214)where R is the Ricci scalar. Let us expand the X field interms of creation and annihilation operators which obey:[ a k , a † k ] = δ (3) ( k − k ), and X = Z d k (2 π ) / a ( η ) h a k u k ( η ) e i k · x + a † k u ∗ k ( η ) e − i k · x i , (215)where the mode functions obey the identity u k u ′ ∗ k − u ′ k u ∗ k = i , and prime denotes derivative w.r.t. η . Themode equation is given by: u ′′ k ( η ) + [ k + m X a + (6 ξ − a ′′ /a ] u k ( η ) = 0 , (216)The parameter ξ = 1 / , for conformal and ξ = 0 forminimal coupling. Here we will consider the conformalcoupling for simplicity. The number density of X parti-cles can be estimated by a Bogoliubov transformation: u η k ( η ) = α k u η k ( η ) + β k u ∗ η k ( η ) , (217)where η = −∞ , and η = + ∞ . The energy density ofproduced particles is given by (Chung et al. , 1999b): ρ X ( η ) = m X H inf (˜ a ( η )) − Z ∞ d ˜ k π ˜ k | β ˜ k | , (218)where the number operator is defined at η . Assum-ing that the transition from inflation-radiation or matter4domination is smooth, the largest energy density can beobtained if m X /H inf ∼
1. If 0 . ≤ m X /H inf ≤
2. If H inf ∼ m φ ∼ GeV and m φ is the mass of the infla-ton, then X particles produced gravitationally can matchthe density today of the order of the critical density pro-vided they are long lived. Such super heavy massive darkmatter particle X is known as Wimpzillas! (b) Direct decay of the inflaton The dark matter can also be created from direct infla-ton decay if m X < m φ /
2, with a rate Γ X ∼ h X m φ / π ,where h X is the interaction strength. The total inflatondecay rate is given by Γ d ∼ p / T R /M P , while the in-flaton number density at the time of decay is given by n φ ∼ T /m φ . This constrains the overall coupling to h X ≤ π r T R M P − m X , (219)where m X is in units of GeV. This is required due to thefact that the produced X must not overclose the universewhich, for Ω X ≤ .
22 and H = 70 km sec − Mpc − ,reads n X /n γ ≤ × − m − X , (220)when m X is expressed in units of GeV (Allahverdi et al. ,2002). It is evident from the overclosure bound that h X needs to be very small. (c) Creation during reheating If the process of reheating is slow and not instanta-neous, then it is possible to create WIMP from the am-bient plasma which is in the process of acquiring ther-malization via scatterings. The Boltzmann equationsfor inflaton energy density, ρ φ , radiation energy den-sity, ρ r , and dark matter energy density ρ X are givenby (Chung et al. , 1999a; Kolb and Turner, 1988):˙ ρ φ + 3 Hρ φ + Γ φ ρ φ = 0˙ ρ R + 4 Hρ R − Γ φ ρ φ − h σ | v |i m X h ρ X − ( ρ eqX ) i = 0˙ ρ X + 3 Hρ X + h σ | v |i m X h ρ X − ( ρ eqX ) i = 0 , (221)where dot denotes time derivative, and thermal averagedcross section is given by: h σ | v |i . The equilibrium energydensity for the X particles, ρ eqX , is determined by theradiation temperature, T = (30 ρ R /π g ∗ ) / . Following(Chung et al. , 1999a; Kolb and Turner, 1988), it is usefulto introduce two dimensionless constants, α φ and α X ,defined in terms of Γ φ = α φ m φ , h σ | v |i = α X m − X , andΦ ≡ ρ φ m − φ a ; R ≡ ρ r a ; X ≡ ρ X m − X a . With these parameters the Boltzmann equations are:Φ ′ = − c x √ Φ x + R Φ R ′ = c x √ Φ x + R Φ + c x − √ Φ x + R (cid:0) X − X eq (cid:1) X ′ = − c x − √ Φ x + R (cid:0) X − X eq (cid:1) . (222)where x = am φ , prime denotes d/dx , and the constants c , c , and c are given by c = r π M P m φ α φ , c = c m φ m X α X α φ , c = c m φ m X . The equilibrium value of X is given in terms of thetemperature T and Eq. (193): X eq = n X m X =( m X /m φ )(1 / π ) / x ( T /M X ) / exp( − M X /T ).It is straightforward to solve the system of equa-tions in Eq. (222) with initial conditions at x = x I , R ( x I ) = X ( x I ) = 0, and Φ( x I ) = Φ I =(3 / π )( M P /m φ )( H I /m φ ) x I . At early time solution for R can be easily obtained: R ≃ . c (cid:16) x / − x / I (cid:17) Φ / I ( H ≫ Γ φ ) . (223)By maximizing the above equation, which is obtainedat x/x I = (8 / / = 1 .
48, the largest temperature ofthe ambient plasma can be even larger than the reheattemperature (Kolb and Turner, 1988) T MAX /T R = 0 . (cid:0) / π g ∗ (cid:1) / (cid:0) H inf M P /T R (cid:1) / . (224)From Eq. (223) when x/x I > T scales as a − / , whichimplies that entropy is created in the early-time regime.For the choices of m φ , α φ , g ∗ , and α X , and Ω X h =0 .
22 (Chung et al. , 1999a):Ω X h = α X (cid:18) g ∗ (cid:19) / (cid:18) T R m X (cid:19) (225) (d) Non-perturbative creation of dark matter The dark matter can be created non-perturbativelyduring the coherent oscillations of the inflaton. a. Superheavy dark matter during preheating:
If thedark matter couples to the inflaton directly, then it ismore efficient to excite them from the coherent oscilla-tions of the inflaton during preheating. One of the mostinteresting applications of preheating is the copious pro-duction of particles which have a mass greater than theinflaton mass m φ . Such processes are impossible in per-turbation theory and in the theory of narrow parametricresonance.Following Eq. (97), let us suppose that the dark mat-ter X is coupled to the inflaton with an interaction term:5(1 / g X φ . During the broad resonance regime, as wehave discussed in section II.F.1, superheavy X -particleswith mass m X ≫ m φ can be produced. The momentumdependent frequency ω k ( t ) violates the adiabatic condi-tion of time dependent vacuum, see Eq. (103), when k + m X < ∼ ( g φm φ ˆ φ ) / − g ˆ φ , (226)where ˆ φ is the amplitude of the inflaton oscillations.The maximal range of momenta for which particle pro-duction occurs corresponds to φ ( t ) = φ ∗ , where φ ∗ ≈ / q ( m φ ˆ φ ) /g . The maximal value of momentum forparticles produced at that epoch can be estimated by k + m χ = ( gm φ ˆ φ ) /
2. The resonance becomes effi-cient for gm φ ˆ φ > ∼ m X . Thus, the inflaton oscillationsmay lead to a copious production of superheavy particleswith m X ≫ m φ if the amplitude of the field φ is largeenough, g ˆ φ > ∼ m X /m φ (Kofman et al. , 1997). b. Dark matter from the fragmentation of a scalarcondensate:
Let us assume that coherent oscillations ofa scalar condensate in a potential U ( φ ) has a frequencywhich is large compared to the expansion rate of the uni-verse. The equation of state is obtained by averaging, p/ρ = ( | ˙ φ | /ρ ) −
1, over one oscillation cycle T . The re-sult is: p = ( γ − ρ , where γ = (2 /T ) R T (1 − U ( φ ) /ρ ) dt (Turner, 1983). For the case U ∼ m φ , one finds γ = 1,so that one effectively obtains the usual case of pressure-less, non-relativistic cold matter. When the motion ofthe condensate is not simply oscillatory, such as in thecase of a rotating trajectory with a phase, one can gener-alize the above calculation by integrating over the orbitof the condensate. Let us consider the potential U ( φ ) = 12 m φ | φ | (cid:0) φ /µ (cid:1) x , (227)one finds that γ = (1 + x ) / (1 + x/ , p = x/ (2 + x ).There arises a negative pressure whenever x <
0. This isa sign of an instability of the scalar field under arbitrarilysmall perturbations. The quantum fluctuations in thecondensate grow according to when effective mass of thescalar field is much larger than the expansion¨ δ k = − K k δ k . (228)If K = 2 x <
0, quantum fluctuations of the condensateat the scale, λ = 2 π/ | k | , will grow exponentially in time: δφ k ( t ) = δφ (0)exp (cid:0) − K k t (cid:1) . (229)In reality the onset of non-linearity sets the scale atwhich the spatial coherence of the field can no longerbe maintained and the condensate fragments. Theinitial fluctuations in the condensate owes to the in-flationary perturbations. If the condensate carries aglobal charge, due to charge conservation the energy-to-charge ratio changes as the the condensate frag-ments. This is what happens in the case of MSSM. The AD condensate which was responsible for generatingthe baryon asymmetry at the first instance could frag-ment (Kasuya and Kawasaki, 2000a,b; Kusenko et al. ,2009). The ground state of these fragmented lumps isa non-topological soliton with a fixed charge, called the Q -ball (Coleman, 1985; Kusenko, 1997b; Lee and Pang,1992). In gauge mediated SUSY breaking scenarios the Q -balls can absolutely stable and can be a candidate forCDM (Kusenko et al. , 1998; Kusenko and Shaposhnikov,1998; Kusenko and Shoemaker, 2009). The Q -balls canalso be formed from the fragmentation of the infla-ton (Enqvist et al. , 2002a,b). The slow surface evapo-ration of a Q -ball will also create SUSY LSP, which wewill discuss below. C. Candidates
In this subsection we will discuss some of the dark mat-ter candidates which are well motivated, and the chal-lenges they face.
1. Primordial blackholes
The primordial blackholes (PBH) can be created inthe early universe (Carr and Hawking, 1974; Hawking,1971), and they can survive the age of the universe witha typical lifetime of an evaporating blackhole which isgiven by: τ sec ≈ (cid:18) M grams (cid:19) , (230)If the initial mass M ≈ g, the blackhole will be evap-orating now, for heavier blackholes the Hawking evap-oration is negligible, and they can be a CDM candi-date. When M ∼ g, the blackholes would decayat the time of BBN. The PBHs are formed from thecollapse of order one perturbations, δ ≡ δρ/ρ ∼ O (1),inside the Hubble patch. The detailed numerical simu-lation suggest δ c ∼ . M H ≈ g (cid:18) GeV T (cid:19) , (231)where T is temperature of the thermal bath during radi-ation. In spite of novelty in this idea, the detailed calcu-lations suggest that it is hard to form primordial black-holes just from the collapse of sub-Hubble over densedregions - one requires more power on small scales n < . − .
30 in P ζ ( k ) = Ak n − (Carr and Lidsey, 1993;Drees and Erfani, 2011; Green and Liddle, 1997), whilethe CMB data points towards n ∼ .
96. It was shown ina hydrodynamical simulation (Jedamzik and Niemeyer,1999) that primordial blackholes can also be produced6in a first order phase transition, and during preheat-ing (Green and Malik, 2001).The abundance of PBH contains many uncertainties,as the details of the initial gravitational collapse andthe initial number density of n PBH depends on manyphysical circumstances. These uncertainties can how-ever be encoded in terms of the ratio determined bythe initial time, t i ; ρ PBH ( t i ) /ρ ( t i ) = M n
PBH ( t i ) /ρ ( t i ) =4 M n
PBH / T i s ( T i ), by assuming ρ = 3 sT / β ′ ( M ) ≈ . × − (cid:18) MM ⊙ (cid:19) / (cid:18) n PBH ( t )1 Gpc − (cid:19) . (232)where t corresponds to present time. In terms of thisfraction β ′ , the PBH abundance is given by (Carr, 1975;Green and Liddle, 1999)Ω PBH h ≈ . (cid:18) β ′ ( M )1 . × − (cid:19) (cid:18) MM ⊙ (cid:19) − / . (233)The value of β ′ ( M ) can be constrained fromΩ PBH ≤ Ω CDM , which yields β ′ ( M ) < . × − (Ω CDM / . M/ g) / , for mass M ≥ g.A tighter constrain on β ′ arises from a range of astro-physical observations, such as BBN, CMB anisotropy,and γ -ray backgrounds for M ≤ g, the boundweakens for larger mass blackholes (Carr et al. , 2010).
2. Axions
The axions were introduced to solve the strong CPproblem (Peccei and Quinn, 1977a,b) which requires anew global chiral symmetry U (1) P Q that is broken spon-taneously at the Peccei-Quinn (PQ) scale f a (for reviews,see (Kim, 1987; Raffelt, 1990; Sikivie, 2008; Turner,1990)). The corresponding pseudo-Nambu-Goldstoneboson is the axion a (Weinberg, 1978; Wilczek, 1978),which couples to the gluons L = af a /N g s π G aµν e G aµν , (234)where N is the color anomaly of the PQ symmetry de-pends on the interactions. This interaction term compen-sates the vacuum contribution in the QCD Lagrangian L Θ = Θ( g s / π ) G aµν e G aµν , in a way that Θ → ¯Θ +Arg (det M ) < − (Nakamura et al. , 2010), in order tomatch the electric dipole moment of the neutron. Thedynamical solution yields when h a i = − ¯Θ f a /N , at whichthe effective potential for the axion has its minimum.The axion can interact via heavy quark while all otherSM fields do not carry any PQ charge, in which case N = 1 (Kim, 1979; Shifman et al. , 1980). The axioncan directly couple to the SM, and at the lowest order itwill induce non-renormalizable coupling with the gluons,where N = 6 (Dine et al. , 1981; Zhitnitsky, 1980).The axion searches, various astrophysical andcosmological observations suggest that the PQ scale (Nakamura et al. , 2010; Raffelt, 2008; Sikivie,2000) must be large, f a /N > ∼ × GeV , (235)and the axion mass must be very small, m a ≤ .
01 eV.The cosmological constraints on f a > × GeV( m a ≤ . N eff =3 . +1 . − . (Iocco et al. , 2009). Another interesting boundarises from isocurvature perturbations from CMB. Atbest one can allow less than 10% of the total pertur-bations to arise from sources other than the inflatonfluctuations–the axions being massless during inflationcan account for such fluctuations, which limits f a ≥ − GeV, however, it depends on the scale ofinflation (Beltran et al. , 2007; Steffen, 2009).The axion life time depends on the axion-photoninteraction, which gives a long life time comparedto the age of the universe, i.e., τ a = Γ − a → γγ =64 π/ ( g aγγ m a ) ∼ ( f a N − / GeV) s for m u /m d ∼ .
56 (Kolb and Turner, 1988).Axion is massless for T ≥ ≥ Λ QCD andit acquires mass only through instanton effects for T ≤ Λ QCD . For f a /N ≤ × GeV (correspond-ing to m a ≥ . T f ≤
150 MeV. The axion is kept in thermal equilibrium with ππ ↔ πa . The relic thermal abundance is given byΩ a h = 0 .
077 (10 /g ∗ ( T f )) ( m a /
10 eV), where g ∗ denotesthe number of effectively massless degrees of freedom.In an opposite limit, when f a /N is very large, the ax-ions are never in thermal equilibrium, and in particularwhen T R < f a the PQ symmetry is never restored. Themain production mechanism is due to the coherent os-cillations of the axion due to the initial misalignmentangle Θ i of the axion. At T ∼ Λ QCD , the axion ob-tains a temperature dependent effective mass and oscil-late coherently around its minimum when m a ( T osc ) ≃ H ( T ). These oscillations of the axion condensate be-haves as cold dark matter (Abbott and Sikivie, 1983;Dine and Fischler, 1983; Preskill et al. , 1983) with a relicdensity that is governed by the initial misalignment angle − π < Θ i ≤ π (Beltran et al. , 2007; Sikivie, 2008):Ω a h ∼ . ξ f (Θ i ) Θ i (cid:18) f a /N GeV (cid:19) / (236)with ξ = O (1) parametrizing theoretical uncertaintiesrelated to details of the quark–hadron transition and f (Θ i ) accounting for anharmonicity in Θ i – f (Θ i ) → i →
0. For 10 GeV ≤ f a /N ≤ GeV, this“misalignment mechanism” can provide the correct darkmatter abundance.
D. SUSY WIMP
The most general gauge invariant and renormalizablesuperpotential would also include baryon number B or7lepton number L violating terms, with each violating byone unit: W ∆ L =1 = λ ijk L i L j e k + λ ′ ijk L i Q j d k + µ ′ i L i H µ and W ∆ B =1 = λ ′′ ijk u i d j d k , where i = 1 , , B = +1 / Q i , B = − / u i , d i , and B = 0 for all others. The total lepton num-ber assignments are L = +1 for L i , L = − e i , and L = 0 for all the others. Unless λ ′ and λ ′′ terms are verymuch suppressed, one would obtain rapid proton decaywhich violates both B and L by one unit.There exists a discrete Z symmetry, which can forbidbaryon and lepton number violating terms, known as R -parity (Fayet, 1979). For each particle: P R = ( − B − L )+2 s (237)with P R = +1 for the SM particles and the Higgs bosons,while P R = − s is spin of the particle. Besides for-bidding B and L violation from the renormalizable inter-actions, R -parity has interesting phenomenological andcosmological consequences. The lightest sparticle with P R = −
1, the LSP, must be absolutely stable. If elec-trically neutral, the LSP is a natural candidate for darkmatter (Dimopoulos and Hall, 1988; Ellis et al. , 1984a).The advantage here is that their cross sections are gov-erned by the SM gauge group – and therefore the darkmatter paradigm is embedded within a visible sector.However, there are some exceptions which we will dis-cuss first.
1. Gravitino
The gravitino is a spin-3 / m / ∼O (100) GeV − O (1) KeV) depending on the details ofthe SUSY breaking schemes, for instance, in gauge andgravity-mediated SUSY breaking scenarios (Dine et al. ,1996a, 1995a; Giudice and Rattazzi, 1999). Indeed, with-out considering the SUSY breaking mechanisms and theSUSY breaking scale, we can treat the gravitino mass asa free parameter. Production:
Gravitinos with both the helicities canbe produced from a thermal bath, they are never in ther-mal equilibrium. They are produced mainly throughthe scatterings–within MSSM there are many scatter-ing channels which include fermion, sfermion, gauge andgaugino quanta all of which have a cross-section ∝ /M .The thermal abundance is given by (up to a logarith-mic correction) for g ∗ = 228 .
75 as in the case of MSSM: Y ± / ≃ ( T R / GeV) × − (Ellis et al. , 1984b),and Y ± / ≃ [1 + M e g ( T R ) / m / ]( T R / GeV) × FIG. 5: The bound on reheat temperature T R withrespect to an unstable gravitino mass m / , whereneutralino is the LSP with a mass 117 GeV ( indicatedby the shaded light-orange region in which m / ≤ m ¯ χ for ( m / , m ) = (300 , A = 0, tan β = 30.The thermal relic density is given by: Ω ¯ χ h = 0 . LSP , the ¯ χ densityfrom decays of thermally produced gravitinos exceedsΩ ¯ χ h = 0 . et al. , 2008).10 − (Bolz et al. , 2001), where M e g is the gluino mass.Note that for M e g ≤ m / both the helicity states haveessentially the same abundance, while for M e g ≫ m / production of helicity ± / / h canbe approximated by the convenient expression for theuniversal gaugino masses M , , = m / at M GUT and m / ≪ M , , (Pradler and Steffen, 2008)Ω / h ≃ . (cid:16)
10 GeV m / (cid:17)(cid:16) m / (cid:17) (cid:16) T R GeV (cid:17) (238)Thermally produced gravitinos have a negligible free-streaming velocity today. However, the gravitinos cre-ated from decays can be warm or hot dark matter.Besides thermal production, gravitino can be pro-duced non-thermally from the decay of the NLSP(next-to-lightest SUSY particle). Obviously differentNLSP’s give slightly different abundances. For sneutrino,see Refs. (Arina and Fornengo, 2007; Ellis et al. , 2008;Feng et al. , 2004), for stop NLSP, see (Berger et al. ,2008; Diaz-Cruz et al. , 2007; Kang et al. , 2008). A sim-ple approximation yields (Pospelov et al. , 2008; Steffen,2006) Y dec e l ≡ n e l s ≃ . × − (cid:18) m e l (cid:19) , (239)where the total slepton number density is given by assum-ing an equal number density of positively and negativelycharged slepton’s. The NLSP’s can have a long lifetime τ e l . For a slepton NLSP, one finds in the limit m l → τ e l ≃ Γ − = 48 πm / M m e l − m / m e l ! − , (240)8which holds not only for a charged slepton NLSP butalso for the sneutrino NLSP. Similar expressions for thelifetimes for the neutralino NLSP (Feng et al. , 2004)and the stop NLSP can be found in(Diaz-Cruz et al. ,2007). If the NLSP decays into the gravitino LSPafter BBN, the SM particles emitted in addition tothe gravitino can affect the abundances of the pri-mordial light elements. Also the presence of a long-lived negatively charged particle, champ, i.e. e l − ,can lead to bound states that catalyze BBN reac-tions (De Rujula et al. , 1990; Dimopoulos et al. , 1990).The new acceptable limits on champs: 100( q X /e ) ≤ m X ≤ ( q X /e ) TeV, virtually ruled out any lowscale SUSY champs (Chuzhoy and Kolb, 2009). Itwas suggested in Ref. (Pospelov, 2007) that bound-state formation of champ with He can lead to a sub-stantial production of primordial Li , which puts theconstraint τ e l < ∼ × sec. (Bird et al. , 2008;Hamaguchi et al. , 2007; Pospelov, 2007; Pospelov et al. ,2008; Pradler and Steffen, 2008; Takayama, 2008). Uncertainties:
The main uncertainties on gravitinoabundance arise from the hidden sectors . If the in-flaton sector is embedded within a hidden sector thenthere are many more sources of gravitino production–the inflaton could decay directly into gravitino duringreheating or preheating (Frey et al. , 2006; Giudice et al. ,1999b; Kallosh et al. , 2000a,b; Maroto and Mazumdar,2000; Nilles et al. , 2001a; Nilles and Peloso, 2001), theinflaton couplings to other hidden sectors can similarlyexcite gravitinos giving rise to large uncertainties in theirtotal abundance.Ω / h = Ω MSSM / h + Ω Inflaton / h + Ω Hidden / h , (241)All these contributions can easily overproduce gravitinos,i.e. Ω / h ∼
1, especially the last two sectors are largelyunconstrained by particle physics. These uncertaintiescan be minimized if the last phase of inflation occurswithin MSSM, as discussed in Sect. ?? , then the onlypredominant source of gravitino production arises fromthe decay of the MSSM inflaton and from the MSSMthermal bath.Another solution has been put forward – for high scaleand hidden sector models of inflation – since the flat di-rections of MSSM can be displaced from their minimumduring inflation, the flat direction VEV at early timeswould generate time dependent masses to the MSSMfields which are coupled to the flat direction. As a result,the inflaton might not even decay into all the MSSM d.o.f due to kinematical blocking (Allahverdi and Mazumdar,2005, 2006b, 2007b). Furthermore, the flat directionVEV also generates masses to gauge bosons and gauginoswhich participate in scatterings, therefore delaying theactual thermalization process and lowering the reheatingtemperature, i.e. T R . Both these effects address the ther-mal gravitino overproduction problem without any needof extra assumptions (Allahverdi and Mazumdar, 2005). Unstable gravitino:
An unstable gravitino decaysto particle-sparticle pairs, and its decay rate is givenby Γ / ≃ m / / M . If m / <
50 TeV, the graviti-nos decay during or after BBN, which can ruin its suc-cessful predictions for the primordial abundance of lightelements (Cyburt et al. , 2003; Kawasaki et al. , 2005).If the gravitinos decay radiatively, the most stringentbound, (cid:0) n / /s (cid:1) ≤ − − − , arises for m / ≃
100 GeV − et al. , 2003). On the otherhand, much stronger bounds are derived if the graviti-nos mainly decay through the hadronic modes. In par-ticular, for a hadronic branching ratio ≃
1, and in thesame mass range, (cid:0) n / /s (cid:1) ≤ − − − will berequired (Kawasaki et al. , 2005, 2008). This puts con-straint on reheat temperature of the universe, i.e. T R , atwhich these unstable gravitinos are produced, see Fig. 5.An intriguing possibility arises if R-parity is broken. Thegravitino LSP with m / ∼ T R from Fig. 5. The gravitino in this case cannot decay intohadrons which is kinematically suppressed, and the three-body decay life time is typically larger than the age of theuniverse (Kohri et al. , 2009). For a GeV mass gravitino,the present day free-streaming velocity is ≤ − km/s,which corresponds to that of a cold dark matter. Detection:
The direct detection of gravitino willbe impossible at the LHC, their production will beextremely suppressed. If the NLSP is long lived(quasi-stable) as in the case of stau then they wouldpenetrate the collider detector in a way similar tomuons (Drees and Tata, 1990; Feng and Moroi, 1998;Nisati et al. , 1997). If the produced staus are slow,then from the associated highly ionizing tracks andtime of flight measurements one can determine theirmass (Ellis et al. , 2006). This might give some indirecthandle on gravitinos.
2. Axino
The axino, e a , is a superpartner of the axion, is an-other example of a gauge singlet dark matter candi-date (Kim, 1984; Kim and Nilles, 1984; Nilles and Raby,1982). It interacts extremely weakly since its couplingsare suppressed by the PQ scale f a > ∼ GeV (Raffelt,2007, 2008; Sikivie, 2008), and its mass can rangefrom eV to GeV. In the hadronic axion model (Kim,1979; Shifman et al. , 1980) in a SUSY setting, the ax-ino couples to MSSM field indirectly via loops of heavy(s)quarks. Typically e a decouples early at a tempera-ture T f ≥ GeV, below this temperature, they aremainly created from scatterings of MSSM fields in ankinetic equilibrium. The thermal abundance is givenby (Brandenburg and Steffen, 2004; Choi et al. , 2008;9 −6 −4 −2 m~a [GeV℄ T R [ G e V ℄ (cid:10)WMAPCDM h2 = 0:113+0:016(cid:0)0:018 ex ludedhot warm oldfa=N = 1011 GeV FIG. 6: The plot shows the reheating temperature T R with respect to the axion mass for f a /N = 10 GeV.The gray band indicates Ω e a h = 0 . +0 . − . . Thermallyproduced axinos can be classified as hot, warm, andcold dark matter (Covi et al. , 2001) as indicated in theplot. The plot is taken from (Brandenburg and Steffen,2004).Covi et al. , 2001)Ω e a h ≃ . g s ( T R ) ln (cid:18) . g s ( T R ) (cid:19) (cid:18) GeV f a /N (cid:19) × (cid:16) m e a . (cid:17) (cid:18) T R GeV (cid:19) (242)with the axion-model-dependent color anomaly N of thePQ symmetry breaking scale f a . Thermally produced ax-inos can be hot, warm, and cold dark matter (Covi et al. ,2001) as shown in Fig. 6.Non-thermal production of e a has many uncertainties.The e a can be created from the decay of the NLSP, directdecay from the inflaton or moduli, or any other hiddensector. The expression for the final abundance is similarto that of Eq. (241), where 3 / −→ e a . In this regard e a faces similar challenges as that of a gravitino dark matter.The e a LSP is inaccessible in direct/indirect darkmatter searches if R-parity is conserved. Their directproduction at collider is strongly suppressed. Never-theless, quasi-stable e τ ’s could appear in collider de-tectors (and neutrino telescopes (Ahlers et al. , 2006;Albuquerque et al. , 2007)) as a possible signature of the e a LSP.
3. Neutralino
In the MSSM the binos e B (superpartner of B), winos f W (superpartner of W ) and Higgsinos ( e H u and e H d )mix into 4 Majorana fermion eigenstates, called neu-tralinos with 4 mass eigenstates: e χ , e χ , e χ , e χ , or-dered with increasing mass. The LSP is thus denotedby e χ = N e B + N f W + N e H u + N e H d . The gaug-ino fraction, f G = N + N , and Higgsino fraction, m [GeV] m [ G e V ] A =0, m> b =40b → s g G e V LE P II m c ˜ > m t ˜ a m ≤ × -10 m [GeV] m [ G e V ] A =0, m> b =40b → s g G e V n s = , d H = . - n s = , d H = . - n s = . , d H = . - → m c ˜ > m t ˜ a m ≤ × -10 d a r k m a t t e r a ll o w e d FIG. 7: On the left hand panel the co-annihilation bandis shown for ( m − m / ) plane for tan β = 40. Theconstraints are shown in Figure, see (Nath et al. , 2010).On the right hand panel the overlapping contoursbetween MSSM inflation and neutralino dark matter fordifferent values of ( n s . δ H ) within 95%c.l. are shown forthe same parameter region (Allahverdi et al. , 2007a). f H = N + N , are determined by the mixing ma-trix, N , which diagonalizes the neutralino mass ma-trix (Jungman et al. , 1996; Kane et al. , 1994).The e χ ’s were in thermal equilibrium for primordialtemperatures of T > T f ∼ m e χ /
20. At T f , the annihi-lation rate of the (by then) non-relativistic e χ ’s becomessmaller than the Hubble rate so that they decouple fromthe thermal plasma, see Sect. IV.B.1.It is easy to work with a limited set of parameters, themSUGRA model is a simple model which contains onlyfive parameters: m , m / , A , tan β and sign ( µ ) . (243) m is the universal scalar soft breaking parameter, m / is the universal gaugino mass, A is the universal cu-bic soft breaking mass, measures at M GUT , and tan β = h e H u i / h e H d i at the electroweak scale.The model parameters are already constrained by dif-ferent experimental results. (i) the light Higgs massbound of M h >
114 GeV from LEP (Barate et al. ,2003), (ii) the b → sγ branching ratio bound of 1 . × − < B ( B → X s γ ) < . × − (we assume here a rela-tively broad range, since there are theoretical errors in ex-tracting the branching ratio from the data) (Alam et al. ,1995), (iii) the 2 σ bound on the dark matter relic den-sity: 0 . < Ω CDM h < .
129 (Komatsu et al. , 2009),(iv) the bound on the lightest chargino mass of M ˜ χ ± >
104 GeV from LEP (Nakamura et al. , 2010) and (v) themuon magnetic moment anomaly a µ , where one gets a3.3 σ deviation from the SM from the experimental re-sult (Bennett et al. , 2004).The allowed mSUGRA parameter space, at present,has mostly three distinct regions: (i) the stau-neutralino(˜ τ − ˜ χ ), coannihilation region where ˜ χ is the lightest0SUSY particle (LSP), (ii) the ˜ χ having a dominant Hig-gsino component (focus point) and (iii) the scalar Higgs( A , H ) annihilation funnel (2 M ˜ χ ≃ M A ,H ). Thesethree regions have been selected out by the CDM con-straint. There stills exists a bulk region where none ofthese above properties is observed, but this region is nowvery small due to the existence of other experimentalbounds. The allowed parameter space for the neutralinodark matter (blue region) for tan( β ) = 40 is shown in theleft panel of Fig. 7. Detection:
In general the observable signals for SUSYat LHC are: n leptons+ m jets + E/ T (missing transverseenergy), where either n or m could be 0. The existence ofmissing energy in the signal will tell us the possibility ofdark matter candidate. There are SM backgrounds, e.g. W and Z bosons decaying to neutrinos providing E/ T .The clean signal for SUSY would be jets + E/ T , withoutisolated leptons. One of the key analysis for mSUGRA isto measure the M eff which is the sum of the transversemomenta of the four leading jets and the missing trans-verse energy: M eff = p T, + p T, + p T, + p T, + E/ T . Onehas to further measure the masses (squarks, sleptons), A and tan β , and the mixing matrices which lead to the cal-culation of the relic density. One particularly favored pa-rameter space is the coannihilation region where the stauand the neutralino masses are close for smaller values of m . The mass difference, ∆ m , governs the relic abun-dance due to the Boltzmann suppression factor e − ∆ M/T ,see section IV.B.2. Therefore measuring ∆ M directlygives handle on measuring the relic abundance at theLHC, see for a detailed discussion in (Nath et al. , 2010). MSSM inflation and dark matter:
After consid-ering all these bounds it was found that there exists aninteresting overlap between the constraints from MSSMinflation and the e χ abundance, see the right hand panelof Fig. 7 for tan β = 40. The constraints on the parame-ter space arising from the inflation appearing to be con-sistent with the constraints arising from the dark mattercontent of the universe and other experimental results.It is also interesting to note that the allowed regionfor u d d as an MSSM inflaton with a mass m φ , re-quired by the CMB observations for λ = 1, see Fig. 3in Sect. II.E.2, lies in the stau-neutralino coannihilationregion which requires smaller values of the SUSY par-ticle masses (Allahverdi et al. , 2007a). Similar analysiswere performed in (Balazs et al. , 2005, 2004), where theauthors studied the overlap between MSSM parametersfor the electroweak baryogenesis and the e χ dark matterabundance.
4. Sneutrino
The lightest right handed (RH) sneutrino e N can be agood dark matter candidate when the SM gauge groupis augmented to SU (3) C × SU (2) L × U (1) Y × U (1) B − L , W h m N Ž H G e V L FIG. 8: Ω h vs m ˜ N . The solid lines from left to rightare for Ω h = 0.094 and 0.129 respectively. The Z ′ -inomass is equal to the Bino mass since the new U (1) gaugecoupling is the same as the hypercharge gauge coupling.The plot is taken from (Allahverdi et al. , 2007b).with a superpotential (Allahverdi et al. , 2007b, 2010b) W = W MSSM + W B − L + yN H u L . (244)The model introduces new gauge boson Z ′ , two Higgsfields H ′ , H ′ , and their superpartners, the Yukawa cou-pling is denoted by y . The spontaneous breaking of U (1) B − L will generate Majorana neutrino masses, or if y ≈ − , then Dirac neutrino masses, or a mixtureof both Dirac and Majorana (Allahverdi et al. , 2011a).If the right handed sneutrino is the LSP then it pro-vides another compelling candidate which is well moti-vated from particle theory and can be embedded withleast unknown uncertainties (Allahverdi et al. , 2007b;Arina and Fornengo, 2007; Lee et al. , 2007).Scatterings via the new U (1) gauge interactions bringthe RH sneutrino into thermal equilibrium. In order tocalculate the relic abundance of the RH sneutrino, weneed to know the masses of the additional gauge boson Z ′ and its SUSY partner ˜ Z ′ , the new Higgsino masses, HiggsVEVs which break the new U (1) B − L gauge symmetry,the RH sneutrino mass, the new gauge coupling, and thecharge assignments for the additional U (1). The primarydiagrams responsible to provide the right amount of relicdensity are mediated by ˜ Z ′ in the t -channelBy assuming that the new gauge symmetry is brokenaround TeV in Fig. 8, we show the relic density values forsmaller masses of sneutrino where the lighter stop massis ≤ et al. , 2009) allowed values of the relicdensity, i.e., 0 . − .
129 is satisfied for many points. Inthe case of a larger sneutrino mass in this model, thecorrect dark matter abundance can be obtained by anni-hilation via Z ′ pole (Allahverdi et al. , 2007b; Lee et al. ,2007). Detection:
Since the dark matter candidate, the RHsneutrino, interacts with quarks via the Z ′ boson, it ispossible to see it via the direct detection experiments.The detection cross sections are not small as the inter-action diagram involves Z ′ in the t -channel. The typical1cross section is about 2 × − pb for a Z ′ mass around2 TeV. It is possible to probe this model in the upcom-ing dark matter detection experiments. The signal forthis scenario at the LHC will contain standard jets plusmissing energy and jets plus leptons plus missing energy.The jets and the leptons will be produced from the cas-cade decays of squarks and gluinos into the final statecontaining the sneutrino.
5. Stable and evaporating Q-ball, LSP dark matter
The AD condensate fragments to form Q-balls, for reviews see (Dine and Kusenko, 2004;Enqvist and Mazumdar, 2003), and a finite size Q-ball has a minimum of energy and it is stable withrespect to decay into free quanta if U ( φ ) /φ = min, for φ > Q , the energy of a soliton is thengiven by (Coleman, 1985; Kusenko, 1997b; Lee and Pang,1992): E = | νQ | < m φ | Q | , which ensures its stabilityagainst decay into plane wave solutions with ϕ ≃ ϕ inside and ϕ ≃ φ ( t, x ) = e iωt ϕ ( x ). The value of ν was computed in (Kusenko,1997a,b). Note that the global U (1) symmetry is thusbroken inside the soliton by the VEV, however, remainsunbroken outside the soliton. The most crucial piece isthe presence of a global U (1) charge of the Q -ball whichactually prevents it from decaying and makes the solitonstable.In the context of gauge mediated SUSYbreaking the AD potential takes the form U ( ϕ ) ≈ m φ log (cid:16) | ϕ | /m φ (cid:17) (Dvali et al. ,1998; Kusenko and Shaposhnikov, 1998), where m φ ∼ O (TeV), represents the SUSY break-ing scale. The profile of the Q -ball is given by ϕ ( r ) ∼ exp( − m φ r ), where the energy, radius and theVEV of a Q -ball goes as (Enqvist and Mazumdar, 2003;Kusenko and Shaposhnikov, 1998) E ≈ √ πm φ Q / , R ≈ Q / √ m φ , ϕ ≈ m φ √ π Q / (245)This allows for the existence of some entirely sta-ble Q-balls with a large baryon number Q ∼ B (B-balls). Indeed, if the mass of a B-ball is M B ∼ (1 TeV) × B / , then the energy per baryon number( M B /B ) ∼ (1 TeV) × B − / is less than 1 GeV for B > . Such large B-balls cannot dissociate intoprotons and neutrons and are entirely stable – thanksto the conservation of energy and the baryon num-ber. If they were produced in the early universe,they would exist at present as a form of dark mat-ter (Kusenko and Shaposhnikov, 1998). There are as-trophysical and terrestrial lilmits (Kusenko et al. , 1998;Kusenko and Shoemaker, 2009), and direct searches forthe Q-balls, which places a lower limit on Q > (Arafune et al. , 2000). In the gravity mediated case the B -balls are not sta-ble, but they evaporate via surface evaporation. In thisprocess AD condensate can generate the required baryonasymmetry and also create dark mater. The appropriatecandidate will be the udd flat direction, lifted by n = 6operator, which carries the baryon number and the rightdark matter abundance, see Eq. (179).When a B -ball decays, for each unit of B produced,corresponding to the decay of 3 squarks to quarks, therewill be at least three units of R -parity produced, corre-sponding to at least 3 LSPs (depending on the nature ofthe cascade produced by the squark decay and the LSPmass, more LSP pairs could be produced). Let N LSP ≥ f B be the fraction of the total B asymmetry con-tained in B -balls. Then the baryon to dark matter ratio, r B = ρ B /ρ DM , and the dark matter abundance are givenby (Enqvist and McDonald, 1998, 1999), r B = m n N LSP f B m LSP , Ω LSP ≈ f B (cid:18) m LSP m n (cid:19) (246)where m n is the nucleon mass and m LSP is the LSP mass.It is rather natural to have r B < B -ball decays will collide withthemselves and with other weakly interacting particlesin the background and settle locally into a kinetic equi-librium. Thermal contact can be maintained until T f ∼ m LSP /
20, and a rough freeze-out condition for LSPs (ifthey were initially in thermal equilibrium) will be givenby: n LSP h σ eff v i ≈ H f m e χ /T f , where σ eff is the LSP an-nihilation cross-section and the subscript f refers to thefreeze-out values. The thermally averaged cross sectioncan be written as h σ eff v i = a + bT /m e χ , where a and b depend on the couplings and the masses of the lightfermions (Jungman et al. , 1996).Assuming an efficient LSP production, so that f B = 1,one finds for the LSP density for b ≈ Hm e χ T − f n − f ,where n f ≈ ( m e χ T f ) / exp[ − m χ /T f ]. The LSPs pro-duced in B -ball decays will spread out by a random walkwith a rate ν determined by the collision frequency di-vided by a thermal velocity v th ≈ ( T /m e χ ) / . Sincethe decay is spherically symmetric, it is very likely thatthe LSPs have a Gaussian distribution. In terms of thedensity parameter Ω e χ , the neutralino abundance can bewritten as (Fujii and Hamaguchi, 2002)Ω e χ ≃ . m χ · − GeV h σv i · T d (cid:18) g ∗ ( T d ) (cid:19) / . (247)where the decay temperature of the B-ball is given by T d ≪ (cid:16) m χ
100 GeV (cid:17) / N tot e χ ! / (cid:18) g ( T ) (cid:19) / GeV . (248)Below this temperature the annihilations of e χ are neg-ligible. Similar analysis can be performed for other LSPcandidates. For example, if the gravitino is an LSP, the2gravitino abundance from the Q-balls decay will be givenby (Shoemaker and Kusenko, 2009):Ω / h ≈ . (cid:16) m / (cid:17) (cid:18) N g f B (cid:19) (cid:18) Ω b h . (cid:19) , (249)where N g = 3 and f b ∼
1. The above expression isvalid for temperatures below 10 GeV. Similar expressioncan be derived for the Q-balls decaying into axino darkmatter (Roszkowski and Seto, 2007).
E. Detection of WIMP
The direct detection of WIMP is the cleanest way toseek the identity of the dark matter, their detection ispossible through elastic collision with the nuclei at terres-trial targets (Goodman and Witten, 1985). This methodis especially promising for detecting SUSY WIMP can-didates, such as neutralino or sneutrino. There are alsoways of inferring WIMP in the sky by using the galaxy it-self as a detector in the indirect dark matter searches viastudying the gamma rays, high energy neutrinos, chargedleptons, proton, anti-proton background from the decayor annihilation of the dark matter particles.
1. Direct detection of WIMPs
Important quantity is the recoil energy deposited bythe WIMP interaction with the nucleus of mass m N inan elastic collision, E r = m r v (1 − cos θ ) /m N , where m r is the WIMP nucleus reduced mass, θ is the scatteringangle in the dark matter-nucleus center-of-mass frame,and v is the velocity relative to the detector, and it is ofthe order of the galactic rotation velocity ∼
200 km/s.Typical recoil energies are E r ∼ O (1 − dRdE R = N T ρ m W Z v max v min d~v f ( ~v ) v dσdE R (250)where N T represents the number of the target nuclei, m X is the dark matter mass and ρ ∼ . / cm is thelocal WIMP density in the galactic halo, ~v and f ( ~v ) arethe WIMP velocity and velocity distribution function inthe Earth frame, which we take it to be Maxwellian, and dσ/dE R is the WIMP-nucleus differential cross section.The velocity v min = p ( m N E r / m ), and v max is theescape velocity of the WIMP in the Earth frame, v esc =544 +64 − km/s.In fact, the Earth velocity with respect to the darkmatter halo must be written as v e = v (1 . .
07 cos ωt )where 1 . v is the galactic velocity of the Sun and ω = 2 π/ et al. ,1986; Freese et al. , 1988). In the above expression, f ( v )must be replaced by f ( | ~v − ~v e | ). There also exists a forward-backward asymmetry in a directional signal asfirst pointed out in (Copi et al. , 1999; Spergel, 1988).For a given momentum transfer q , the differential crosssection depends on the nuclear form factor dσdq = σ m r v F ( q ) , (251)where F ( q ) is a dimensionless form factor such that F (0) = 1, in which case σ corresponds to the total cross-section. It is possible to estimate the parameters σ and F ( q ), for example in the case of neutralino WIMP, whichis a Majorana fermion therefore it only has axial andscalar couplings (Jungman et al. , 1996). a. Spin-dependent cross-section:
The axial part ofthe neutralino-quark interaction is mediated via Z bo-son and squark exchange L qχ ∼ ( ¯ Xγ µ γ X ) (¯ qγ µ γ q ).At the level of nutralino-nucleon interaction by con-sidering the nucleon matrix element h n | ¯ qγ µ γ | n i ,the effective Lagrangian is given by: L eff nX =2 √ G F a ( n ) ( ¯ Xγ µ γ X ) (¯ nγ µ γ n ), where G F is the Fermiconstant and a n is a dimensionless parameter. For anucleus of spin J, with h S p i and h S n i being the aver-age spins “carried” by protons and neutrons respectively,the cross-section at zero momentum transfer is givenby (Jungman et al. , 1996) dσdq ( q = 0) = 8 πv G F Λ J ( J + 1) . (252)where Λ = ( a p h S p i + a n h S n i ) /J . Additional cor-rections to the form factor is required to takeinto account of the non-zero momentum trans-fer. There are many experiments which are sensi-tive to spin-dependent cross section with pure pro-ton couplings, DAMA (Bernabei et al. , 2004), PI-CASSO (Archambault et al. , 2009), KIMS (Lee et al. ,2007). Recently COUPP (Behnke et al. , 2011) has setthe best constraint on spin-dependent cross section downto 7 × − cm for a WIMP mass ∼
30 GeV. b. Spin-independent cross-section:
The scalar part ofthe neutralino-quark interaction is mediated via Higgsesand squark exchanges: L qX = f q (¯ qq )( ¯ XX ). To expressthe neutralino-nucleon coupling one needs the nucleonmatrix element m q h n | ¯ qq | n i ≡ m ( n ) , the effective interac-tion has the form: L eff nX = f ( n ) ( ¯ XX )(¯ nn ), where f n con-tains the information about hadron physics, and typicallyit is the same for proton or neutron, i.e., f p ∼ f n . In thecase of right handed sneutrino also there exists no spin-dependent part as it has no axial-vector interactions, andthe cross-section is dominated by the spin-independentpart. One can define a single spin-independent WIMP-nucleon cross section σ ≡ σ SI , independent of the spinof the nucleon (Jungman et al. , 1996). dσdq = [ Z f p + ( A − Z ) f n ] F ( q ) πv , (253)3FIG. 9: Spin-independent elastic WIMP-nucleoncross-section σ as function of WIMP mass m χ . The newXENON100 limit at 90% CL is shown as the thick(blue) line together with the 1 σ and 2 σ sensitivity ofthis run (shaded blue band). From (Aprile et al. , 2011).In the spin-independant case, for low nuclear recoil en-ergies F ( q ) ∼
1, there is a coherence effect which booststhe WIMP-nucleus cross section by a factor A m r . Asa result this technique is better suited to detect theWIMP candidate with heavy nucleus targets. Recentlythe most stringent limits on the elastic spin-independentWIMP-nucleon cross-section has been given by num-ber of experiments, such as CDMS-II (Ahmed et al. ,2010), EDELWEISS-II (Armengaud et al. , 2011) andXENON100 (Aprile et al. , 2011). These limits areshown in Fig. 9. The shaded gray area also showsthe expected region of CMSSM for the WIMP massand the cross section are indicated at 68% and95% CL (Buchmueller et al. , 2011). The current re-sults also covers the 90% CL areas favored by Co-GeNT (green) (Aalseth et al. , 2011) and DAMA (lightred) (Savage et al. , 2009).It should be noted that CoGeNT (Aalseth et al. ,2011), and DAMA/NaI (Bernabei et al. , 2004) andDAMA/LIBRA (Bernabei et al. , 2008) collaborationshave observed an annual modulation signal. The com-bined results of the latter group stands at greater than8 σ statistical significance. The modulation signal phasematches well with the expected annual signal of WIMPS,and subsequent data (Bernabei et al. , 2010) has in-creased the statistical significance of the modulation sig-nal. However the annual modulation claim has not beenverified by any other experiments, especially the null re-sults from CDMS, XENON10, and XENON100 data.The CDMS data (Ahmed et al. , 2010) shows 2 signalevents with 0 . ± .
2. Indirect detection
The indirect dark matter detection depends on thenature of the WIMP. If the WIMP belongs to the vis-ible sector, or if it has some SM interactions, then their annihilation or decay would yield to the known parti-cle physics spectrum, for a recent review on indirectdetection, see (Porter et al. , 2011). However the spec-trum would depend on where they are produced, theirenergy deposition, and what are their final states, e.g. γ, e ± , · · · etc. The signal could be a hard spectrum witha monochromatic line if WIMPS annihilate directly intophotons (Bergstrom et al. , 1998), or a continuum spec-trum if they annihilate into a pair of intermediate parti-cles (( q = u, d, c, s, t, b ) , ¯ q, Z, g, W ± , l ± ). The formerprocess is generically suppressed compared to the lat-ter. Most of these latter particles, i.e. W, Z, g decayinto p, ¯ p , π , and a tiny fraction of deuterium or anti-deuterium D / ¯ D . The π s decay to gamma rays, whilethe π ± decays produce e ± . If the final states of decay orannihilations are e ± s or µ ± s, they dominantly producea hard e ± spectrum, with the µ ± decays into ν µ and ν e . If the final states have τ ± , they produce a softer e ± spectrum and a strong neutrino signal. The τ ± can alsodecay hadronically to pions and thus can also produce astrong γ -ray signal.The source spectrum is generically given byΦ s ( E ) = 14 π h σv i m X X f dN f dE B f,s , (254)where f denotes the annihilating final states, each withbranching fraction B f,s with E being the energy of sec-ondary particles. The production rate per annihilationof species f is given by dN f /dE . For a decaying darkmatter h σv i / m X can be replaced by Γ /M X , where Γ isthe decay rate.The flux of such final states would depend on the an-nihilation or decay rate, which in turn would depend onthe dark matter density ∝ ρ X . Therefore, the naturalsources to look at in the sky are the nearby galactic cen-ters – where there are large astrophysical uncertainties,dwarf galaxies – which have small astrophysical back-ground, and galactic centers – where the dark matterdensity is very large but distant sources would yield alocal tiny flux.Gamma rays and neutrinos are perhaps the cleanestsignals if they are produced as a result of WIMP an-nihilations or decays as they are undeflected by mag-netic fields and effectively indicating the direction to theirsource. The flux is given by the integral of the WIMPdensity-squared along the line of sight from the observerto the source, multiplied by the production spectrum φ γ ( E, ψ ) = J ( ψ ) × Φ γ ( E )Φ γ ( E, ψ ) = dN f dE h σv i πm X B f,s Z l . o . s dsρ ( r ( s, ψ )) , (255)where “ f ′′ denotes the final states and the coordinate s runs along the line of sight, E is the gamma-ray energy,and ψ is the elongation angle with respect to the centerof the source. The astrophysics related term is hidden4in (Bergstrom et al. , 1998) J ( ψ ) = 18 . . / cm ! Z l . o . s . ρ ( ℓ ) dℓ (256)where the integration is in the direction ψ along the line ℓ . The above expression is also valid for neutrinos if thesource is not far away from us.On the other hand the charged particles do nothave the directional sensitivity, they lose it in theircourse of path in a random motion due to the in-terstellar magnetic field. The motion of e ± are de-flected by the interstellar radiation field by synchro-ton radiation in presence of magnetic field. If pro-duced at energies ≥
100 GeV and if their sources arewithin few kiloparsecs, then they can reach the solar sys-tem. Furthermore, the cosmic rays from the primaryand secondary products can also generate γ rays dur-ing their course through the ISM, all these effects canbe captured numerically in the publically available codeGALPROP (Strong and Moskalenko, 1998; Strong et al. ,2000, 2004), for a review see (Strong et al. , 2007).The main challenge is to disentangle the WIMP sig-nals from astrophysical backgrounds, the powerful dis-criminator is the spectral tilt in the power spectrum.It is quite possible to have a significant fraction ofWIMP annihilation or decay into monoenergetic pho-tons, giving rise a distinctive line in the gamma-ray spec-trum (Bergstrom et al. , 1998), but the likely signal wouldbe to have a relatively hard continuum spectrum with abump or edge near the WIMP mass that is above theastrophysical background.In recent years there have been many new experimentswhich have propelled the indirect search for dark mat-ter research vigorously. For instance, the ATIC data– which shows a significant bump in the electron fluxaround 300 −
800 GeV (Chang et al. , 2008), where con-ventional astrophysical sources would have predicted adecaying power law spectrum, and the PAMELA – whichshows a positron fraction which has a rising slope above10 GeV (Adriani et al. , 2009, 2010), but the anti-protonflux and the fraction matches well with the expectationsof the astrophysical origins. The Fermi-LAT collabora-tion which has also produced an electron spectrum from7 GeV to 1 TeV (Abdo et al. , 2009; Ackermann et al. ,2010) does not confirm the rising slope of the ATIC, seeFig. 10, rather the data matches well with the HESSat higher end of the spectrum (Aharonian et al. , 2008,2009).The rise in the e + fraction in the PAMELA datamay have its roots in the astrophysical objects such asnearby pulsars, Monogem at a distance of d = 290pc, andGeminga at a distance of d = 160pc. Both are nearbyobjects to Earth which can contribute significantly to the e ± flux, and can match both the PAMELA and Fermi-LAT data sets, see (Grasso et al. , 2009). A dark mat-ter interpretation of these data sets requires an ad-hocassumption on the leptohilic nature of the WIMP with FIG. 10: Combined electron and positron spectrum asmeasured by Fermi-LAT for one year of observations,together with other measurements (Abdo et al. , 2009;Ackermann et al. , 2010). The systematic errors for themeasurement are shown by the grey band. Thesystematic uncertainty associated with the absoluteenergy scale is shown by the non-vertical arrow. Thedashed line shows the background from secondary e ± incosmic rays from GALPROP.an annihilation cross-section h σv i ∼ − cm , morethan 2 − et al. , 2009), see for a de-tailed derivation (Slatyer, 2010) in the mass range of500 −
900 GeV (Grasso et al. , 2009). It is also possible toconceive a scenario of non-thermal dark matter produc-tion in order to evade the strict bound on cross-section,or decaying dark matter scenarios, i.e. (Arvanitaki et al. ,2009; Fairbairn and Zupan, 2009).Finally, one would also expect a gamma ray signal fromthe galactic center, the typical signature for a WIMPannihilation will be a line at the WIMP mass, due tothe 2 γ s, or γZ production channels. The Fermi-LATcollaboration has released a diffused galactic and extragalactic γ -ray background (Abdo et al. , 2010b), howeverno lines were observed yet (Abdo et al. , 2010a).The high energy neutrinos are being another fron-tier for the indirect dark matter searches as the con-struction of new experiment IceCube with large volumeis underway, in which case even Earth could be usedas a detector to study the nature of dark matter, e.g.(Albuquerque et al. , 2004), or neutrino emission fromthe passage of Q-balls (Kusenko and Shoemaker, 2009).Searching for WIMPs in more than one type of experi-ments; direct, indirect, and the LHC will be necessary inorder to stamp the origin of such elusive particles.5 V. ACKNOWLEDGEMENTS
The author would like to thank A. Kusenko for manyuseful discussions and collaborating at the initial stagesof this review. He would also like to thank R. Allahverdi,R. Brandenberger, C. Boehm, A. Chatterjee, P. Dayal,B. Dutta, K. Enqvist, A. Ferrantelli, J. Garcia-Bellido,S. Hotchkiss, K. Kazunori, D. Lyth, S. Nadathur, K. Pe-traki, N. Sahu, Q. Shafi, and P. Stephens for helpful dis-cussions.
References
Aalseth, C. E., et al. (CoGeNT), 2011, Phys. Rev. Lett. ,131301.Abada, A., S. Davidson, A. Ibarra, F.-X. Josse-Michaux,M. Losada, et al. , 2006a, JHEP , 010.Abada, A., S. Davidson, F.-X. Josse-Michaux, M. Losada,and A. Riotto, 2006b, JCAP , 004.Abazajian, K., G. M. Fuller, and M. Patel, 2001, Phys. Rev.
D64 , 023501.Abazajian, K., and S. M. Koushiappas, 2006, Phys. Rev.
D74 , 023527.Abazajian, K. N., et al. (SDSS), 2009, Astrophys. J. Suppl. , 543.Abbott, L. F., and P. Sikivie, 1983, Phys. Lett.
B120 , 133.Abdo, A. A., et al. (The Fermi LAT), 2009, Phys. Rev. Lett. , 181101.Abdo, A. A., et al. , 2010a, Phys. Rev. Lett. , 091302.Abdo, A. A., et al. (The Fermi-LAT), 2010b, Phys. Rev. Lett. , 101101.Abel, S. A., S. Sarkar, and P. L. White, 1995, Nucl. Phys.
B454 , 663.Ackermann, M., et al. (Fermi LAT), 2010, Phys. Rev.
D82 ,092004.Adler, S. L., 1969, Phys.Rev. , 2426.Adriani, O., et al. (PAMELA), 2009, Nature , 607.Adriani, O., et al. (PAMELA), 2010, Phys. Rev. Lett. ,121101.Affleck, I., and M. Dine, 1985, Nucl. Phys.
B249 , 361.Aguilar, A., et al. (LSND), 2001, Phys. Rev.
D64 , 112007.Aguilar-Arevalo, A. A., et al. (The MiniBooNE), 2010, Phys.Rev. Lett. , 181801.Aharonian, F., et al. (H.E.S.S.), 2008, Phys. Rev. Lett. ,261104.Aharonian, F., et al. (H.E.S.S.), 2009, Astron. Astrophys. , 561.Ahlers, M., J. Kersten, and A. Ringwald, 2006, JCAP ,005.Ahmed, Z., et al. (The CDMS-II), 2010, Science , 1619.Akhmedov, E. K., M. Frigerio, and A. Y. Smirnov, 2003,JHEP , 021.Alam, M. S., et al. (CLEO), 1995, Phys. Rev. Lett. , 2885.Albrecht, A. J., P. J. Steinhardt, M. S. Turner, andF. Wilczek, 1982, Phys. Rev. Lett. , 1437.Albuquerque, I., G. Burdman, and Z. Chacko, 2004, Phys.Rev. Lett. , 221802.Albuquerque, I. F. M., G. Burdman, and Z. Chacko, 2007,Phys. Rev. D75 , 035006.Allahverdi, R., 2000, Phys. Rev.
D62 , 063509.Allahverdi, R., R. Brandenberger, F.-Y. Cyr-Racine, andA. Mazumdar, 2010a, eprint 1001.2600. Allahverdi, R., and M. Drees, 2002, Phys. Rev.
D66 , 063513.Allahverdi, R., B. Dutta, and A. Mazumdar, 2003, Phys. Rev.
D67 , 123515.Allahverdi, R., B. Dutta, and A. Mazumdar, 2007a, Phys.Rev.
D75 , 075018.Allahverdi, R., B. Dutta, and A. Mazumdar, 2007b, Phys.Rev. Lett. , 261301.Allahverdi, R., B. Dutta, and A. Mazumdar, 2008, Phys. Rev. D78 , 063507.Allahverdi, R., B. Dutta, and R. N. Mohapatra, 2011a, Phys.Lett.
B695 , 181.Allahverdi, R., B. Dutta, and Y. Santoso, 2010b, Phys. Lett.
B687 , 225.Allahverdi, R., B. Dutta, and Y. Santoso, 2010c, Phys.Rev.
D82 , 035012.Allahverdi, R., K. Enqvist, J. Garcia-Bellido, A. Jokinen, andA. Mazumdar, 2007c, JCAP , 019.Allahverdi, R., K. Enqvist, J. Garcia-Bellido, and A. Mazum-dar, 2006, Phys. Rev. Lett. , 191304.Allahverdi, R., K. Enqvist, and A. Mazumdar, 2002, Phys.Rev. D65 , 103519.Allahverdi, R., A. Ferrantelli, J. Garcia-Bellido, andA. Mazumdar, 2011b, eprint 1103.2123.Allahverdi, R., A. R. Frey, and A. Mazumdar, 2007d, Phys.Rev.
D76 , 026001.Allahverdi, R., A. Kusenko, and A. Mazumdar, 2007e, JCAP , 018.Allahverdi, R., and A. Mazumdar, 2003, Phys. Rev.
D67 ,023509.Allahverdi, R., and A. Mazumdar, 2005, eprint hep-ph/0505050.Allahverdi, R., and A. Mazumdar, 2006a, eprint hep-ph/0610069.Allahverdi, R., and A. Mazumdar, 2006b, JCAP , 008.Allahverdi, R., and A. Mazumdar, 2007a, JCAP , 023.Allahverdi, R., and A. Mazumdar, 2007b, Phys. Rev.
D76 ,103526.Allahverdi, R., and A. Mazumdar, 2008, Phys. Rev.
D78 ,043511.Allen, B., 1988, Phys. Rev.
D37 , 2078.Ambjorn, J., T. Askgaard, H. Porter, and M. Shaposhnikov,1990, Phys.Lett.
B244 , 479.Anderson, G. W., and L. J. Hall, 1992, Phys. Rev.
D45 , 2685.Antusch, S., M. Bastero-Gil, K. Dutta, S. F. King, and P. M.Kostka, 2009a, JCAP , 040.Antusch, S., K. Dutta, and P. M. Kostka, 2009b, Phys. Lett.
B677 , 221.Aprile, E., et al. (XENON100), 2011, eprint 1104.2549.Arafune, J., T. Yoshida, S. Nakamura, and K. Ogure, 2000,Phys.Rev.
D62 , 105013.Archambault, S., et al. , 2009, Phys. Lett.
B682 , 185.Arina, C., and N. Fornengo, 2007, JHEP , 029.Arkani-Hamed, N., S. Dimopoulos, N. Kaloper, and J. March-Russell, 2000, Nucl. Phys. B567 , 189.Arkani-Hamed, N., D. P. Finkbeiner, T. R. Slatyer, andN. Weiner, 2009, Phys. Rev.
D79 , 015014.Armendariz-Picon, C., T. Damour, and V. F. Mukhanov,1999, Phys. Lett.
B458 , 209.Armengaud, et al. (EDELWEISS), 2011, eprint 1103.4070.Arnold, P. B., D. Son, and L. G. Yaffe, 1997, Phys.Rev.
D55 ,6264.Arvanitaki, A., et al. , 2009, Phys. Rev.
D80 , 055011.Asaka, T., S. Blanchet, and M. Shaposhnikov, 2005, Phys.Lett.
B631 , 151. Asaka, T., M. Fujii, K. Hamaguchi, and T. Yanagida, 2000a,Phys.Rev.
D62 , 123514.Asaka, T., K. Hamaguchi, M. Kawasaki, and T. Yanagida,1999, Phys. Lett.
B464 , 12.Asaka, T., K. Hamaguchi, M. Kawasaki, and T. Yanagida,2000b, Phys. Rev.
D61 , 083512.Baacke, J., K. H. , and C. Patzold, 1998, Phys. Rev.
D58 ,125013.Balazs, C., M. S. Carena, A. Menon, D. E. Morrissey, andC. E. M. Wagner, 2005, Phys. Rev.
D71 , 075002.Balazs, C., M. S. Carena, and C. E. M. Wagner, 2004, Phys.Rev.
D70 , 015007.Barate, R., et al. (LEP Working Group for Higgs bosonsearches), 2003, Phys. Lett.
B565 , 61.Barbier, R., et al. , 2005, Phys. Rept. , 1.Bastero-Gil, M., E. J. Copeland, J. Gray, A. Lukas, andM. Plumacher, 2002, Phys. Rev.
D66 , 066005.Bastero-Gil, M., V. Di Clemente, and S. F. King, 2003, Phys.Rev.
D67 , 083504.Bastero-Gil, M., and S. F. King, 1998, Phys. Lett.
B423 , 27.Bastero-Gil, M., and S. F. King, 1999, Nucl. Phys.
B549 ,391.Becker, K., M. Becker, and A. Krause, 2005, Nucl. Phys.
B715 , 349.Begeman, K., A. Broeils, and R. Sanders, 1991,Mon.Not.Roy.Astron.Soc. , 523.Behnke, E., et al. , 2011, Phys. Rev. Lett. , 021303.Bell, J., and R. Jackiw, 1969, Nuovo Cim.
A60 , 47.Beltran, M., J. Garcia-Bellido, and J. Lesgourgues, 2007,Phys. Rev.
D75 , 103507.Beltran, M., J. Garcia-Bellido, J. Lesgourgues, and A. Ri-azuelo, 2004, Phys. Rev.
D70 , 103530.Ben-Dayan, I., and R. Brustein, 2010, JCAP , 007.Bennett, G. W., et al. (Muon g-2), 2004, Phys. Rev. Lett. ,161802.Bento, L., and Z. Berezhiani, 2001, Phys. Rev. Lett. ,231304.Berezhiani, Z., A. Mazumdar, and A. Perez-Lorenzana, 2001,Phys. Lett. B518 , 282.Berg, M., M. Haack, and B. Kors, 2005a, Phys. Rev.
D71 ,026005.Berg, M., M. Haack, and B. Kors, 2005b, JHEP , 030.Berg, M., M. Haack, and B. Kors, 2006, Phys. Rev. Lett. ,021601.Berger, C. F., L. Covi, S. Kraml, and F. Palorini, 2008, JCAP , 005.Bergstrom, L., P. Ullio, and J. H. Buckley, 1998, Astropart.Phys. , 137.Bernabei, R., et al. , 2004, Int. J. Mod. Phys. D13 , 2127.Bernabei, R., et al. (DAMA), 2008, Eur. Phys. J.
C56 , 333.Bernabei, R., et al. , 2010, Eur. Phys. J.
C67 , 39.Bertolami, O., and G. G. Ross, 1987, Phys. Lett.
B183 , 163.Bertone, G., D. Hooper, and J. Silk, 2005, Phys. Rept. ,279.Bertschinger, E., 2006, Phys. Rev.
D74 , 063509.Bevis, N., M. Hindmarsh, M. Kunz, and J. Urrestilla, 2008,Phys. Rev. Lett. , 021301.Bezrukov, F. L., and M. Shaposhnikov, 2008, Phys. Lett.
B659 , 703.Bin´etruy, P., and G. R. Dvali, 1996, Phys. Lett.
B388 , 241.Binetruy, P., G. Girardi, and P. Salati, 1984, Nucl.Phys.
B237 , 285.Bird, C., K. Koopmans, and M. Pospelov, 2008, Phys. Rev.
D78 , 083010. Birrell, N. D., and P. C. W. Davies, 1982, cambridge, Uk:Univ. Pr. 340p.Biswas, T., T. Koivisto, and A. Mazumdar, 2010, JCAP , 008.Biswas, T., and A. Mazumdar, 2009, eprint 0901.4930.Biswas, T., A. Mazumdar, and W. Siegel, 2006, JCAP ,009.Bochkarev, A., and M. Shaposhnikov, 1987, Mod.Phys.Lett. A2 , 417.Bochkarev, A. I., S. V. Kuzmin, and M. E. Shaposhnikov,1991, Phys. Rev. D43 , 369.Bode, P., J. P. Ostriker, and N. Turok, 2001, Astrophys. J. , 93.Bolton, A. S., S. Burles, L. V. E. Koopmans, T. Treu, andL. A. Moustakas, 2006, Astrophys. J. , 703.Bolz, M., A. Brandenburg, and W. Buchmuller, 2001, Nucl.Phys.
B606 , 518.Bond, J. R., and A. S. Szalay, 1983, Astrophys. J. , 443.Borde, A., A. H. Guth, and A. Vilenkin, 2003, Phys. Rev.Lett. , 151301.Borde, A., and A. Vilenkin, 1994, Phys. Rev. Lett. , 3305.Brandenberger, R. H., and J. H. Kung, 1990, Phys. Rev. D42 ,1008.Brandenburg, A., and F. D. Steffen, 2004, JCAP , 008.Brax, P., and J. Martin, 2005, Phys. Rev.
D72 , 023518.Buchmueller, O., et al. , 2011, Eur. Phys. J.
C71 , 1634.Buchmuller, W., P. Di Bari, and M. Plumacher, 2002,Nucl.Phys.
B643 , 367.Buchmuller, W., P. Di Bari, and M. Plumacher, 2003, Nucl.Phys.
B665 , 445.Buchmuller, W., P. Di Bari, and M. Plumacher, 2005a, Ann.Phys. , 305.Buchmuller, W., R. D. Peccei, and T. Yanagida, 2005b, Ann.Rev. Nucl. Part. Sci. , 311.Buchmuller, W., and M. Plumacher, 1998, Phys. Lett. B431 ,354.Buchmuller, W., and M. Plumacher, 2000, Int. J. Mod. Phys.
A15 , 5047.Bueno Sanchez, J. C., K. Dimopoulos, and D. H. Lyth, 2007,JCAP , 015.Bustamante, M., L. Cieri, and J. Ellis, 2009, eprint 0911.4409.Carena, M. S., M. Quiros, M. Seco, and C. Wagner, 2003,Nucl.Phys.
B650 , 24.Carena, M. S., M. Quiros, and C. Wagner, 1996, Phys.Lett.
B380 , 81.Carr, B. J., 1975, Astrophys. J. , 1.Carr, B. J., and S. W. Hawking, 1974, Mon. Not. Roy. Astron.Soc. , 399.Carr, B. J., K. Kohri, Y. Sendouda, and J. Yokoyama, 2010,Phys. Rev.
D81 , 104019.Carr, B. J., and J. E. Lidsey, 1993, Phys. Rev.
D48 , 543.Carson, L., X. Li, L. D. McLerran, and R.-T. Wang, 1990,Phys.Rev.
D42 , 2127.Casas, J. A., and C. Munoz, 1993, Phys. Lett.
B306 , 288.Chang, J., et al. , 2008, Nature , 362.Chatterjee, A., and A. Mazumdar, 2011, eprint 1103.5758.Choi, K.-Y., L. Roszkowski, and R. Ruiz de Austri, 2008,JHEP , 016.Chung, D. J. H., P. Crotty, E. W. Kolb, and A. Riotto, 2001,Phys. Rev. D64 , 043503.Chung, D. J. H., B. Garbrecht, M. J. Ramsey-Musolf, andS. Tulin, 2009, Phys. Rev. Lett. , 061301.Chung, D. J. H., E. W. Kolb, and A. Riotto, 1999a, Phys.Rev.
D60 , 063504. Chung, D. J. H., E. W. Kolb, and A. Riotto, 1999b, Phys.Rev.
D59 , 023501.Chung, D. J. H., E. W. Kolb, A. Riotto, and I. I. Tkachev,2000, Phys. Rev.
D62 , 043508.Chung, D. J. H., et al. , 2005, Phys. Rept. , 1.Chuzhoy, L., and E. W. Kolb, 2009, JCAP , 014.Cicoli, M., and A. Mazumdar, 2010a, eprint 1010.0941.Cicoli, M., and A. Mazumdar, 2010b, JCAP , 025.Clesse, S., and J. Rocher, 2008, eprint 0809.4355.Cline, J. M., M. Joyce, and K. Kainulainen, 1998, Phys.Lett.
B417 , 79.Cline, J. M., M. Joyce, and K. Kainulainen, 2000, JHEP , 018, erratum added online, Oct/2/2001.Cline, J. M., and S. Raby, 1991, Phys. Rev.
D43 , 1781.Cline, J. M., and H. Stoica, 2005, Phys. Rev.
D72 , 126004.Clowe, D., et al. , 2006, Astrophys. J. , L109.Cohen, A. G., D. Kaplan, and A. Nelson, 1993,Ann.Rev.Nucl.Part.Sci. , 27.Coleman, S. R., 1985, Nucl.Phys. B262 , 263.Coleman, S. R., and E. J. Weinberg, 1973, Phys. Rev. D7 ,1888.Copeland, E. J., A. R. Liddle, and J. E. Lidsey, 2001, Phys.Rev. D64 , 023509.Copeland, E. J., A. R. Liddle, D. H. Lyth, E. D. Stewart, andD. Wands, 1994, Phys. Rev.
D49 , 6410.Copeland, E. J., A. Mazumdar, and N. J. Nunes, 1999, Phys.Rev.
D60 , 083506.Copi, C. J., J. Heo, and L. M. Krauss, 1999, Phys. Lett.
B461 , 43.Cornwall, J. M., D. Grigoriev, and A. Kusenko, 2001, Phys.Rev.
D64 , 123518.Cornwall, J. M., and A. Kusenko, 2000, Phys. Rev.
D61 ,103510.Covi, L., H.-B. Kim, J. E. Kim, and L. Roszkowski, 2001,JHEP , 033.Covi, L., E. Roulet, and F. Vissani, 1996, Phys. Lett. B384 ,169.Cyburt, R. H., J. R. Ellis, B. D. Fields, and K. A. Olive, 2003,Phys. Rev.
D67 , 103521.Cyburt, R. H., B. D. Fields, and K. A. Olive, 2008, JCAP , 012.D’Ambrosio, G., G. F. Giudice, and M. Raidal, 2003, Phys.Lett.
B575 , 75.Damour, T., and V. F. Mukhanov, 1998, Phys. Rev. Lett. ,3440.Dave, R., D. N. Spergel, P. J. Steinhardt, and B. D. Wandelt,2001, Astrophys. J. , 574.Davidson, S., and A. Ibarra, 2002, Phys. Lett. B535 , 25.Davidson, S., E. Nardi, and Y. Nir, 2008, Phys. Rept. ,105.Davidson, S., and S. Sarkar, 2000, JHEP , 012.De Rujula, A., S. L. Glashow, and U. Sarid, 1990, Nucl. Phys. B333 , 173.Diaz-Cruz, J. L., J. R. Ellis, K. A. Olive, and Y. Santoso,2007, JHEP , 003.Dick, K., M. Lindner, M. Ratz, and D. Wright, 2000, Phys.Rev. Lett. , 4039.Dimopoulos, S., D. Eichler, R. Esmailzadeh, and G. D. Stark-man, 1990, Phys. Rev. D41 , 2388.Dimopoulos, S., and L. J. Hall, 1988, Phys. Lett.
B207 , 210.Dimopoulos, S., S. Kachru, J. McGreevy, and J. G. Wacker,2005, eprint hep-th/0507205.Dine, M., and W. Fischler, 1983, Phys. Lett.
B120 , 137.Dine, M., W. Fischler, and D. Nemeschansky, 1984, Phys. Lett.
B136 , 169.Dine, M., W. Fischler, and M. Srednicki, 1981, Phys. Lett.
B104 , 199.Dine, M., and A. Kusenko, 2004, Rev. Mod. Phys. , 1.Dine, M., R. G. Leigh, P. Huet, A. D. Linde, and D. A. Linde,1992, Phys.Lett. B283 , 319.Dine, M., A. E. Nelson, Y. Nir, and Y. Shirman, 1996a, Phys.Rev.
D53 , 2658.Dine, M., A. E. Nelson, and Y. Shirman, 1995a, Phys.Rev.
D51 , 1362.Dine, M., L. Randall, and S. D. Thomas, 1995b, Phys. Rev.Lett. , 398.Dine, M., L. Randall, and S. D. Thomas, 1996b, Nucl. Phys. B458 , 291.Dodelson, S., 2003, amsterdam, Netherlands: Academic Pr.(2003) 440 p.Dodelson, S., A. Melchiorri, and A. Slosar, 2006, Phys. Rev.Lett. , 04301.Dodelson, S., and L. M. Widrow, 1994, Phys. Rev. Lett. ,17.Dolgov, A. D., 1992, Phys. Rept. , 309.Dolgov, A. D., and S. H. Hansen, 2002, Astropart. Phys. ,339.Dolgov, A. D., and D. P. Kirilova, 1990, Sov. J. Nucl. Phys. , 172.Drees, M., and E. Erfani, 2011, eprint 1102.2340.Drees, M., and X. Tata, 1990, Phys. Lett. B252 , 695.Dressler, A., et al. , 1987, Astrophys. J. , L37.Drukier, A. K., K. Freese, and D. N. Spergel, 1986, Phys.Rev.
D33 , 3495.Dvali, G. R., A. Kusenko, and M. E. Shaposhnikov, 1998,Phys. Lett.
B417 , 99.Dvali, G. R., Q. Shafi, and R. K. Schaefer, 1994, Phys. Rev.Lett. , 1886.Edsjo, J., and P. Gondolo, 1997, Phys.Rev. D56 , 1879.Ellis, J. R., S. Ferrara, and D. V. Nanopoulos, 1982,Phys.Lett.
B114 , 231.Ellis, J. R., J. S. Hagelin, D. V. Nanopoulos, K. A. Olive, andM. Srednicki, 1984a, Nucl. Phys.
B238 , 453.Ellis, J. R., J. E. Kim, and D. V. Nanopoulos, 1984b, Phys.Lett.
B145 , 181.Ellis, J. R., K. A. Olive, and Y. Santoso, 2008, JHEP , 005.Ellis, J. R., A. R. Raklev, and O. K. Oye, 2006, JHEP ,061.Elmfors, P., K. Enqvist, and I. Vilja, 1994, Phys.Lett. B326 ,37.Enqvist, K., and K. Eskola, 1990, Mod.Phys.Lett. A5 , 1919.Enqvist, K., A. Jokinen, and J. McDonald, 2000, Phys.Lett. B483 , 191.Enqvist, K., S. Kasuya, and A. Mazumdar, 2002a, Phys. Rev.
D66 , 043505.Enqvist, K., S. Kasuya, and A. Mazumdar, 2002b, Phys. Rev.Lett. , 091301.Enqvist, K., S. Kasuya, and A. Mazumdar, 2004a, Phys. Rev.Lett. , 061301.Enqvist, K., and A. Mazumdar, 2003, Phys. Rept. , 99.Enqvist, K., A. Mazumdar, and A. Perez-Lorenzana, 2004b,Phys. Rev. D70 , 103508.Enqvist, K., A. Mazumdar, and P. Stephens, 2010a, JCAP , 020.Enqvist, K., and J. McDonald, 1998, Phys.Lett.
B425 , 309.Enqvist, K., and J. McDonald, 1999, Nucl.Phys.
B538 , 321.Enqvist, K., and J. McDonald, 2000, Nucl.Phys.
B570 , 407.Enqvist, K., and J. Sirkka, 1993, Phys. Lett.
B314 , 298. Enqvist, K., P. Stephens, O. Taanila, and A. Tranberg, 2010b,JCAP , 019.Fairbairn, M., and J. Zupan, 2009, JCAP , 001.Fayet, P., 1979, Phys. Lett.
B86 , 272.Felder, G. N., et al. , 2001, Phys. Rev. Lett. , 011601.Feng, J. L., and T. Moroi, 1998, Phys. Rev. D58 , 035001.Feng, J. L., S. Su, and F. Takayama, 2004, Phys. Rev.
D70 ,075019.Fields, B., and S. Sarkar, 2006, eprint astro-ph/0601514.Flanz, M., E. A. Paschos, and U. Sarkar, 1995, Phys. Lett.
B345 , 248.Flores, R. A., et al. , 2007, Mon. Not. Roy. Astron. Soc. ,883.Fogli, G. L., et al. , 2008, Phys. Rev.
D78 , 033010.Freese, K., J. A. Frieman, and A. Gould, 1988, Phys. Rev.
D37 , 3388.Freese, K., J. A. Frieman, and A. V. Olinto, 1990, Phys. Rev.Lett. , 3233.Frey, A. R., A. Mazumdar, and R. C. Myers, 2006, Phys. Rev. D73 , 026003.Fryer, C. L., and A. Kusenko, 2006, Astrophys. J. Suppl. ,335.Fujii, M., and K. Hamaguchi, 2002, Phys.Rev.
D66 , 083501.Fukugita, M., and T. Yanagida, 1986, Phys. Lett.
B174 , 45.Garcia-Bellido, J., D. Y. Grigoriev, A. f, and M. E. Shaposh-nikov, 1999, Phys. Rev.
D60 , 123504.Garcia-Bellido, J., and A. D. Linde, 1998, Phys. Rev.
D57 ,6075.Gell-Mann, R., M. Ramond, and S. Slansky, 1980, supergrav-ity (P. van Nieuwenhuizen et al. eds.), North Holland, Am-sterdam, p. 315.Gentile, G., P. Salucci, U. Klein, D. Vergani, and P. Kalberla,2004, Mon. Not. Roy. Astron. Soc. , 903.Gherghetta, T., C. F. Kolda, and S. P. Martin, 1996, Nucl.Phys.
B468 , 37.Ghez, A., B. Klein, M. Morris, and E. Becklin, 1998, Astro-phys.J. , 678, * Fermilab Library Only *.Giudice, G. F., and A. Masiero, 1988, Phys. Lett.
B206 , 480.Giudice, G. F., A. Notari, M. Raidal, A. Riotto, and A. Stru-mia, 2004, Nucl. Phys.
B685 , 89.Giudice, G. F., M. Peloso, A. Riotto, and I. Tkachev, 1999a,JHEP , 014.Giudice, G. F., and R. Rattazzi, 1999, Phys. Rept. , 419.Giudice, G. F., A. Riotto, and I. Tkachev, 1999b, JHEP ,036.Goldberg, H., 1983, Phys. Rev. Lett. , 1419.Gondolo, P., et al. , 2004, JCAP , 008.Gonzalez-Garcia, M. C., M. Maltoni, and J. Salvado, 2010,JHEP , 056.Goodman, M. W., and E. Witten, 1985, Phys. Rev. D31 ,3059.Grasso, D., et al. (FERMI-LAT), 2009, Astropart. Phys. ,140.Green, A. M., S. Hofmann, and D. J. Schwarz, 2004, Mon.Not. Roy. Astron. Soc. , L23.Green, A. M., S. Hofmann, and D. J. Schwarz, 2005, JCAP , 003.Green, A. M., and A. R. Liddle, 1997, Phys. Rev. D56 , 6166.Green, A. M., and A. R. Liddle, 1999, Phys. Rev.
D60 ,063509.Green, A. M., and J. E. Lidsey, 2000, Phys. Rev.
D61 ,067301.Green, A. M., and K. A. Malik, 2001, Phys. Rev.
D64 ,021301. Green, A. M., and A. Mazumdar, 2002, Phys. Rev.
D65 ,105022.Greene, P. B., and L. Kofman, 1999, Phys. Lett.
B448 , 6.Greene, P. B., and L. Kofman, 2000, Phys. Rev.
D62 , 123516.Griest, K., and M. Kamionkowski, 1990, Phys. Rev. Lett. ,615.Griest, K., and D. Seckel, 1991, Phys. Rev. D43 , 3191.Grisaru, M. T., W. Siegel, and M. Rocek, 1979, Nucl. Phys.
B159 , 429.Grishchuk, L. P., 1975, Sov. Phys. JETP , 409.Grishchuk, L. P., and Y. V. Sidorov, 1989, Class. Quant.Grav. , L161.Grojean, C., G. Servant, and J. D. Wells, 2005, Phys. Rev. D71 , 036001.Grossman, Y., T. Kashti, Y. Nir, and E. Roulet, 2003, Phys.Rev. Lett. , 251801.Grossman, Y., T. Kashti, Y. Nir, and E. Roulet, 2004, JHEP , 080.Grossman, Y., R. Kitano, and H. Murayama, 2005, JHEP ,058.Guth, A. H., 1981, Phys. Rev. D23 , 347.Haber, H. E., and G. L. Kane, 1985, Phys. Rept. , 75.Halyo, E., 1996, Phys. Lett.
B387 , 43.Hamaguchi, K., T. Hatsuda, M. Kamimura, Y. Kino, andT. T. Yanagida, 2007, Phys. Lett.
B650 , 268.Hannestad, S., A. Mirizzi, G. G. Raffelt, and Y. Y. Y. Wong,2010, JCAP , 001.Harrison, E. R., 1970, Phys. Rev. D1 , 2726.Harvey, J. A., and E. W. Kolb, 1981, Phys.Rev. D24 , 2090.Hawking, S., 1971, Mon. Not. Roy. Astron. Soc. , 75.Hidaka, J., and G. M. Fuller, 2006, Phys. Rev.
D74 , 125015.Hidaka, J., and G. M. Fuller, 2007, Phys. Rev.
D76 , 083516.Hofmann, S., D. J. Schwarz, and H. Stoecker, 2001, Phys.Rev.
D64 , 083507.’t Hooft, G., 1976a, Phys.Rev.
D14 , 3432.’t Hooft, G., 1976b, Phys.Rev.Lett. , 8.Hotchkiss, S., G. German, G. G. Ross, and S. Sarkar, 2008,JCAP , 015.Hotchkiss, S., A. Mazumdar, and S. Nadathur, 2011, eprint1101.6046.Huet, P., and A. E. Nelson, 1996, Phys.Rev. D53 , 4578.Hunt, P., and S. Sarkar, 2010, Mon. Not. Roy. Astron. Soc. , 547.Iocco, F., G. Mangano, G. Miele, O. Pisanti, and P. D. Ser-pico, 2009, Phys. Rept. , 1.Jaikumar, P., and A. Mazumdar, 2004, Nucl. Phys.
B683 ,264.Jeannerot, R., 1997, Phys. Rev.
D56 , 6205.Jeannerot, R., and M. Postma, 2005, JHEP , 071.Jeannerot, R., J. Rocher, and M. Sakellariadou, 2003, Phys.Rev. D68 , 103514.Jedamzik, K., and J. C. Niemeyer, 1999, Phys. Rev.
D59 ,124014.Jedamzik, K., and M. Pospelov, 2009, New J. Phys. ,105028.Jokinen, A., and A. Mazumdar, 2004, Phys. Lett. B597 , 222.Jungman, G., M. Kamionkowski, and K. Griest, 1996, Phys.Rept. , 195.Kainulainen, K., T. Prokopec, M. G. Schmidt, and S. Wein-stock, 2001, JHEP , 031.Kajantie, K., M. Laine, K. Rummukainen, and M. E. Sha-poshnikov, 1996, Phys.Rev.Lett. , 2887.Kallosh, R., 2008, Lect. Notes Phys. , 119.Kallosh, R., L. Kofman, A. D. Linde, and A. Van Proeyen, D61 , 103503.Kallosh, R., L. Kofman, A. D. Linde, and A. Van Proeyen,2000b, Class. Quant. Grav. , 4269.Kane, G. L., C. F. Kolda, L. Roszkowski, and J. D. Wells,1994, Phys. Rev. D49 , 6173.Kang, J., P. Langacker, T. Li, and T. Liu, 2009, eprint0911.2939.Kang, J., P. Langacker, T.-j. Li, and T. Liu, 2005, Phys. Rev.Lett. , 061801.Kang, J., M. A. Luty, and S. Nasri, 2008, JHEP , 086.Kanti, P., and K. A. Olive, 1999a, Phys. Lett. B464 , 192.Kanti, P., and K. A. Olive, 1999b, Phys. Rev.
D60 , 043502.Kaplinghat, M., L. Knox, and M. S. Turner, 2000, Phys. Rev.Lett. , 3335.Karsch, F., E. Laermann, and A. Peikert, 2001, Nucl. Phys. B605 , 579.Kasuya, S., and M. Kawasaki, 2000a, Phys. Rev.
D62 ,023512.Kasuya, S., and M. Kawasaki, 2000b, Phys. Rev.
D61 ,041301.Kasuya, S., and M. Kawasaki, 2006, Phys. Rev.
D74 , 063507.Kawasaki, M., K. Kohri, and T. Moroi, 2005, Phys. Lett.
B625 , 7.Kawasaki, M., K. Kohri, T. Moroi, and A. Yotsuyanagi, 2008,Phys. Rev.
D78 , 065011.Kawasaki, M., M. Yamaguchi, and T. Yanagida, 2000, Phys.Rev. Lett. , 3572.Kawasaki, M., M. Yamaguchi, and T. Yanagida, 2001, Phys.Rev. D63 , 103514.Kessler, R., et al. , 2009, Astrophys. J. Suppl. , 32.Khlebnikov, S. Y., and M. E. Shaposhnikov, 1988, Nucl. Phys.
B308 , 885.Khlebnikov, S. Y., and I. I. Tkachev, 1996, Phys. Rev. Lett. , 219.Khlebnikov, S. Y., and I. I. Tkachev, 1997a, Phys. Rev. Lett. , 1607.Khlebnikov, S. Y., and I. I. Tkachev, 1997b, Phys. Lett. B390 , 80.Kim, J. E., 1979, Phys. Rev. Lett. , 103.Kim, J. E., 1984, Phys. Lett. B136 , 378.Kim, J. E., 1987, Phys. Rept. , 1.Kim, J. E., and H. P. Nilles, 1984, Phys. Lett.
B138 , 150.Kitano, R., H. Murayama, and M. Ratz, 2008, Phys. Lett.
B669 , 145.Klinkhamer, F. R., and N. Manton, 1984, Phys.Rev.
D30 ,2212.Klypin, A. A., A. V. Kravtsov, O. Valenzuela, and F. Prada,1999, Astrophys. J. , 82.Kofman, L., A. D. Linde, and V. F. Mukhanov, 2002, JHEP , 057.Kofman, L., A. D. Linde, and A. A. Starobinsky, 1994, Phys.Rev. Lett. , 3195.Kofman, L., A. D. Linde, and A. A. Starobinsky, 1997, Phys.Rev. D56 , 3258.Kohri, K., A. Mazumdar, and N. Sahu, 2009, eprint0905.1625.Kolb, E. ., Edward W., and E. . Turner, Michael S., 1988,redwood city, USA: Addison-Wesley, 719 P. (Frontiers inphysics, 70).Kolb, E. W., D. J. H. Chung, and A. Riotto, 1998, eprinthep-ph/9810361.Kolb, E. W., J. E. Lidsey, M. Abney, E. J. Copeland, andA. R. Liddle, 1995, Nucl. Phys. Proc. Suppl. , 118.Komatsu, E., et al. (WMAP), 2009, Astrophys. J. Suppl. , 330.Komatsu, E., et al. (WMAP), 2011, Astrophys. J. Suppl. ,18.Kosowsky, A., and M. S. Turner, 1995, Phys. Rev. D52 , 1739.Krauss, L. M., and M. Trodden, 1999, Phys. Rev. Lett. ,1502.Kravtsov, A. V., A. A. Klypin, J. S. Bullock, and J. R. Pri-mack, 1998, Astrophys. J. , 48.Kusenko, A., 1997a, Phys.Lett. B404 , 285.Kusenko, A., 1997b, Phys.Lett.
B405 , 108.Kusenko, A., 2006, Phys. Rev. Lett. , 241301.Kusenko, A., 2009, Phys. Rept. , 1.Kusenko, A., V. Kuzmin, M. E. Shaposhnikov, andP. Tinyakov, 1998, Phys.Rev.Lett. , 3185.Kusenko, A., A. Mazumdar, and T. Multamaki, 2009, Phys.Rev. D79 , 124034.Kusenko, A., and G. Segre, 1996, Phys. Rev. Lett. , 4872.Kusenko, A., and G. Segre, 1999, Phys. Rev. D59 , 061302.Kusenko, A., and M. E. Shaposhnikov, 1998, Phys.Lett.
B418 , 46.Kusenko, A., and I. M. Shoemaker, 2009, Phys. Rev.
D80 ,027701.Kuzmin, V., V. Rubakov, and M. Shaposhnikov, 1987,Phys.Lett.
B191 , 171.Kuzmin, V. A., V. A. Rubakov, and M. E. Shaposhnikov,1985, Phys. Lett.
B155 , 36.Laine, M., 1996, Nucl.Phys.
B481 , 43.Laine, M., and K. Rummukainen, 1998, Nucl.Phys.
B535 ,423.Langacker, P., 1981, Phys. Rept. , 185.Lazarides, G., 2000, eprint hep-ph/0011130.Lazarides, G., and C. Panagiotakopoulos, 1995, Phys. Rev. D52 , 559.Lazarides, G., and Q. Shafi, 1991, Phys. Lett.
B258 , 305.Lazarides, G., and N. D. Vlachos, 1997, Phys. Rev.
D56 ,4562.Lee, H.-S., K. T. Matchev, and S. Nasri, 2007, Phys. Rev.
D76 , 041302.Lee, H. S., et al. (KIMS), 2007, Phys. Rev. Lett. , 091301.Lee, T., and Y. Pang, 1992, Phys.Rept. , 251.Lewin, J. D., and P. F. Smith, 1996, Astropart. Phys. , 87.Liddle, A. R., and S. M. Leach, 2003, Phys. Rev. D68 ,103503.Liddle, A. R., and D. H. Lyth, 1993, Phys. Rept. , 1.Liddle, A. R., and D. H. Lyth, 2000, iSBN-13-9780521828499.Liddle, A. R., and A. Mazumdar, 1998, Phys. Rev.
D58 ,083508.Liddle, A. R., A. Mazumdar, and F. E. Schunck, 1998, Phys.Rev.
D58 , 061301.Linde, A., 2001, JHEP , 052.Linde, A. D., 1983, Phys. Lett. B129 , 177.Linde, A. D., 1985, Phys. Lett.
B162 , 281.Linde, A. D., 1986, Phys. Lett.
B175 , 395.Linde, A. D., 1991, Phys. Lett.
B259 , 38.Linde, A. D., 1994, Phys. Rev.
D49 , 748.Linde, A. D., D. A. Linde, and A. Mezhlumian, 1994, Phys.Rev.
D49 , 1783.Linde, A. D., D. A. Linde, and A. Mezhlumian, 1996, Phys.Rev.
D54 , 2504.Linde, A. D., and A. Riotto, 1997, Phys. Rev.
D56 , 1841.Loeb, A., and M. Zaldarriaga, 2005, Phys. Rev.
D71 , 103520.Lukash, V. N., 1980, Sov. Phys. JETP , 807.Luty, M. A., 1992, Phys. Rev. D45 , 455.Lyth, D. H., 1997, Phys. Rev. Lett. , 1861. Lyth, D. H., and A. R. Liddle, 2009, cambridge UniversityPress.Lyth, D. H., and A. Riotto, 1999, Phys. Rept. , 1.Lyth, D. H., and E. D. Stewart, 1995, Phys. Rev. Lett. ,201.Lyth, D. H., and E. D. Stewart, 1996, Phys. Rev. D53 , 1784.Maggiore, M., 2000, Phys. Rept. , 283.Manton, N., 1983, Phys.Rev.
D28 , 2019.Markevitch, M., et al. , 2004, Astrophys. J. , 819.Maroto, A. L., and A. Mazumdar, 2000, Phys. Rev. Lett. ,1655.Martin, S. P., 1997, eprint hep-ph/9709356.Mazumdar, A., 1999, Phys. Lett. B469 , 55.Mazumdar, A., 2004a, Phys. Rev. Lett. , 241301.Mazumdar, A., 2004b, Phys. Lett. B580 , 7.Mazumdar, A., S. Nadathur, and P. Stephens, 2011, eprint1105.0430.Mazumdar, A., S. Panda, and A. Perez-Lorenzana, 2001,Nucl. Phys.
B614 , 101.Mazumdar, A., and A. Perez-Lorenzana, 2001, Phys. Lett.
B508 , 340.Mazumdar, A., and A. Perez-Lorenzana, 2004a, Phys. Rev.Lett. , 251301.Mazumdar, A., and A. Perez-Lorenzana, 2004b, Phys. Rev. D70 , 083526.Mazumdar, A., and J. Rocher, 2011, Phys. Rept. , 85.McDonald, J., 1997, Phys.Rev.
D55 , 4240.McLerran, L. D., M. E. Shaposhnikov, N. Turok, and M. B.Voloshin, 1991, Phys.Lett.
B256 , 451.Mendes, L. E., and A. R. Liddle, 2000, Phys. Rev.
D62 ,103511.Micha, R., and I. I. Tkachev, 2003, Phys. Rev. Lett. ,121301.Micha, R., and I. I. Tkachev, 2004, Phys. Rev. D70 , 043538.Minkowski, P., 1977, Phys. Lett.
B67 , 421.Mohapatra, R. N., and G. Senjanovic, 1980, Phys. Rev. Lett. , 912.Mohapatra, R. N., and G. Senjanovic, 1981, Phys. Rev. D23 ,165.Mohapatra, R. N., and X. Zhang, 1992, Phys. Rev.
D46 ,5331.Moore, B., T. R. Quinn, F. Governato, J. Stadel, and G. Lake,1999, Mon. Not. Roy. Astron. Soc. , 1147.Moore, B., et al. , 1999, Astrophys. J. , L19.Moore, G. D., 1999, Phys.Rev.
D59 , 014503.Mukhanov, V. F., 1985, JETP Lett. , 493.Mukhanov, V. F., 1989, Phys. Lett. B218 , 17.Mukhanov, V. F., H. A. Feldman, and R. H. Brandenberger,1992, Phys. Rept. , 203.Murayama, H., H. Suzuki, T. Yanagida, and J. Yokoyama,1993, Phys. Rev. Lett. , 1912.Nakamura, K., et al. (Particle Data Group), 2010, J. Phys. G37 , 075021.Nath, P., et al. , 2010, Nucl. Phys. Proc. Suppl. , 185.Navarro, J. F., C. S. Frenk, and S. D. M. White, 1997, Astro-phys. J. , 493.Nelson, A. E., D. B. Kaplan, and A. G. Cohen, 1992, Nucl.Phys.
B373 , 453.Niemeyer, J. C., and K. Jedamzik, 1998, Phys. Rev. Lett. ,5481.Niemeyer, J. C., and K. Jedamzik, 1999, Phys. Rev. D59 ,124013.Nilles, H. P., 1984, Phys. Rept. , 1.Nilles, H. P., K. A. Olive, and M. Peloso, 2001a, Phys. Lett.
B522 , 304.Nilles, H. P., and M. Peloso, 2001, eprint hep-ph/0111304.Nilles, H. P., M. Peloso, and L. Sorbo, 2001b, JHEP , 004.Nilles, H. P., M. Peloso, and L. Sorbo, 2001c, Phys. Rev. Lett. , 051302.Nilles, H. P., and S. Raby, 1982, Nucl. Phys. B198 , 102.Nisati, A., S. Petrarca, and G. Salvini, 1997, Mod. Phys. Lett.
A12 , 2213.Pallis, C., 2009, JCAP , 024.Panagiotakopoulos, C., and K. Tamvakis, 1999, Phys. Lett.
B446 , 224.Panagiotakopoulos, C., and N. Tetradis, 1999, Phys. Rev.
D59 , 083502.Peccei, R., and H. R. Quinn, 1977a, Phys.Rev.
D16 , 1791.Peccei, R., and H. R. Quinn, 1977b, Phys.Rev.Lett. , 1440.Peebles, P. J. E., 1994, princeton, USA: Univ. Pr. (1993) 718p.Peloso, M., and L. Sorbo, 2000, JHEP , 016.Petraki, K., and A. Kusenko, 2008, Phys. Rev. D77 , 065014.Pilaftsis, A., and T. E. J. Underwood, 2004, Nucl. Phys.
B692 , 303.Pilaftsis, A., and T. E. J. Underwood, 2005, Phys. Rev.
D72 ,113001.Plumacher, M., 1998, Nucl. Phys.
B530 , 207.Polyakov, A. M., 1977, Nucl.Phys.
B120 , 429.Porter, T. A., R. P. Johnson, and P. W. Graham, 2011, eprint1104.2836.Pospelov, M., 2007, Phys. Rev. Lett. , 231301.Pospelov, M., J. Pradler, and F. D. Steffen, 2008, JCAP , 020.Pospelov, M., and A. Ritz, 2005, Annals Phys. , 119.Postma, M., and A. Mazumdar, 2004, JCAP , 005.Pradler, J., and F. D. Steffen, 2008, Phys. Lett. B666 , 181.Preskill, J., M. B. Wise, and F. Wilczek, 1983, Phys. Lett.
B120 , 127.Raffelt, G. G., 1990, Phys. Rept. , 1.Raffelt, G. G., 2007, J. Phys.
A40 , 6607.Raffelt, G. G., 2008, Lect. Notes Phys. , 51.Rajagopal, K., and F. Wilczek, 1993, Nucl. Phys.
B399 , 395.Randall, L., M. Soljacic, and A. H. Guth, 1996, Nucl. Phys.
B472 , 377.Randall, L., and S. D. Thomas, 1995, Nucl. Phys.
B449 , 229.Regan, B. C., E. D. Commins, C. J. Schmidt, and D. DeMille,2002, Phys. Rev. Lett. , 071805.Rehman, M. U., Q. Shafi, and J. R. Wickman, 2009, eprint0908.3896.Riotto, A., 1998, 326, eprint hep-ph/9807454.Rocher, J., and M. Sakellariadou, 2005a, Phys. Rev. Lett. ,011303.Rocher, J., and M. Sakellariadou, 2005b, JCAP , 004.Roszkowski, L., and O. Seto, 2007, Phys. Rev. Lett. ,161304.Rubakov, V. A., and M. E. Shaposhnikov, 1996, Usp. Fiz.Nauk , 493.Rummukainen, K., M. Tsypin, K. Kajantie, M. Laine, andM. E. Shaposhnikov, 1998, Nucl.Phys. B532 , 283.Sahni, V., 1990, Phys. Rev.
D42 , 453.Sakharov, A. D., 1967, Pisma Zh. Eksp. Teor. Fiz. , 32.Salopek, D. S., 1995, Phys. Rev. D52 , 5563.Salopek, D. S., and J. R. Bond, 1990, Phys. Rev.
D42 , 3936.Sasaki, M., 1986, Prog. Theor. Phys. , 1036.Sasaki, M., and E. D. Stewart, 1996, Prog. Theor. Phys. ,71.Savage, C., G. Gelmini, P. Gondolo, and K. Freese, 2009,JCAP , 010. Scherrer, R. J., J. M. Cline, S. Raby, and D. Seckel, 1991,Phys. Rev.
D44 , 3760.Seljak, U., A. Makarov, P. McDonald, and H. Trac, 2006,Phys. Rev. Lett. , 191303.Senoguz, V. N., and Q. Shafi, 2003, Phys. Lett. B567 , 79.Servant, G., and T. M. P. Tait, 2003, Nucl. Phys.
B650 , 391.Shaposhnikov, M., 1987, Nucl.Phys.
B287 , 757.Shi, X.-D., and G. M. Fuller, 1999, Phys. Rev. Lett. , 2832.Shifman, M. A., A. I. Vainshtein, and V. I. Zakharov, 1980,Nucl. Phys. B166 , 493.Shoemaker, I. M., and A. Kusenko, 2009, Phys. Rev.
D80 ,075021.Shtanov, Y., J. H. Traschen, and R. H. Brandenberger, 1995,Phys. Rev.
D51 , 5438.Sikivie, P., 2000, Nucl. Phys. Proc. Suppl. , 41.Sikivie, P., 2008, Lect. Notes Phys. , 19.Slatyer, T. R., 2010, JCAP , 028.Smoot, G. F., et al. , 1992, Astrophys. J. , L1.Sommer-Larsen, J., and A. Dolgov, 2001, Astrophys. J. ,608.Spergel, D. N., 1988, Phys. Rev. D37 , 1353.Spergel, D. N., and P. J. Steinhardt, 2000, Phys. Rev. Lett. , 3760.Steffen, F. D., 2006, JCAP , 001.Steffen, F. D., 2009, Eur. Phys. J. C59 , 557.Stewart, E. D., 1995a, Phys. Rev.
D51 , 6847.Stewart, E. D., 1995b, Phys. Lett.
B345 , 414.Strong, A. W., and I. V. Moskalenko, 1998, Astrophys. J. , 212.Strong, A. W., I. V. Moskalenko, and V. S. Ptuskin, 2007, Ann. Rev. Nucl. Part. Sci. , 285.Strong, A. W., I. V. Moskalenko, and O. Reimer, 2000, As-trophys. J. , 763.Strong, A. W., I. V. Moskalenko, and O. Reimer, 2004, As-trophys. J. , 962.Takayama, F., 2008, Phys. Rev. D77 , 116003.Taoso, M., G. Bertone, and A. Masiero, 2008, JCAP ,022.Tetradis, N., 1998, Phys. Rev.
D57 , 5997.Tranberg, A., A. Hernandez, T. Konstandin, and M. G.Schmidt, 2010, Phys. Lett.
B690 , 207.Tranberg, A., and J. Smit, 2003, JHEP , 016.Tranberg, A., and J. Smit, 2006, JHEP , 012.Traschen, J. H., and R. H. Brandenberger, 1990, Phys. Rev. D42 , 2491.Turner, M. S., 1983, Phys. Rev.
D28 , 1243.Turner, M. S., 1990, Phys. Rept. , 67.Turok, N., and J. Zadrozny, 1990, Phys.Rev.Lett. , 2331.Turok, N., and J. Zadrozny, 1991, Nucl.Phys. B358 , 471.Weinberg, S., 1978, Phys. Rev. Lett. , 223.Weinberg, S., 1979, Phys. Rev. Lett. , 850.Wilczek, F., 1978, Phys. Rev. Lett. , 279.Yanagida, T., 1979, in Proceedings of the Workshop on theUnied Theory and the Baryon Number in the Universe (O.Sawada and A. Sugamoto, eds.), KEK, Tsukuba, Japan, p.95.Zeldovich, Y. B., 1970, Astron. Astrophys. , 84.Zhitnitsky, A. R., 1980, Sov. J. Nucl. Phys.31