Thermal Inflation with a Thermal Waterfall Scalar Field Coupled to a Light Spectator Scalar Field
aa r X i v : . [ a s t r o - ph . C O ] O c t Thermal Inflation with aThermal Waterfall Scalar FieldCoupled to a Light SpectatorScalar Field
Arron Rumsey
September 2016
This thesis is submitted in partial fulfillment of the requirements for thedegree of Master of Philosophy (MPhil) at Lancaster University. No part ofthis thesis has been previously submitted for the award of a higher degree. bstract
This thesis begins with an introduction to the state of the art of modernCosmology. The field of Particle Cosmology is then introduced andexplored, in particular with regard to the study of cosmological inflation.We then introduce a new model of Thermal Inflation, in which the mass ofthe thermal waterfall field responsible for the inflation is dependent on alight spectator scalar field. The model contains a variety of free parameters,two of which control the power of the coupling term and thenon-renormalizable term. We use the δN formalism to investigate the “endof inflation” and modulated decay scenarios in turn to see whether they areable to produce the dominant contribution to the primordial curvatureperturbation ζ . We constrain the model and then explore the parameterspace. We explore key observational signatures, such as non-Gaussianity,the scalar spectral index and the running of the scalar spectral index. Wefind that for some regions of the parameter space, the ability of the modelto produce the dominant contribution to ζ is excluded. However, for otherregions of the parameter space, we find that the model yields a sharpprediction for a variety of parameters within the model. cknowledgements As is extremely common for large pieces of academic work, this thesis wouldnot exist if it were not for many people other than myself. Also, this thesiswas completed in part whilst receiving an STFC PhD studentship.I would like to start by thanking Anupam Mazumdar for several helpfuldiscussions regarding thermalization and thermal interaction rates.I have had many helpful and insightful conversations with my academicpeers and friends Phil Stephens, Frankie Doddato, Ernest Pukartas, LingfeiWang, Mindaugas Karciauskas and Jacques Wagstaff.I would like to thank Jessica Brooks.The majority of the work on the new Thermal Inflation model that isintroduced in this thesis was done in collaboration with David Lyth. A spe-cial mention needs to be given to David, however, as he was not merely anacademic collaborator, but for a large part was also my acting supervisor. Ihave had many fruitful discussions on Particle Cosmology with David. I feelprivileged to have worked with such a pillar of modern Cosmology.I would like to thank my very dear friends and family Linda Rumsey, RobBishop, Paul Evans, Matt Eve, Guy Rusha, Kelly Hodder, Tristan Reeves,Chelle Nevill and Alise Kirtley for the support and encouragement that youiave provided to me.It would be improper if the final acknowledgment did not go to my super-visor, Kostas Dimopoulos. The work that we have done together has beenexciting, interesting and challenging. During some particular bleak timesduring the course of this work, he has been compassionate and accommodat-ing. I cherish the knowledge, wisdom and conversations that we have shared.I sincerely thank you for everything that you have done for me.ii ontents
List of Figures viList of Tables viii1 Introduction 12 Concordance Model of Cosmology — Λ CDM 3 δN Formalism . . . . . . . . . . . . . . . . . . . . . . . 333.2.2 Scalar Fields . . . . . . . . . . . . . . . . . . . . . . . . 353.2.3 Particle Production . . . . . . . . . . . . . . . . . . . . 37 φ Decay Rate, Spectral Index and Tensor Fraction . . . . . . . 514.3.1 φ Decay Rate . . . . . . . . . . . . . . . . . . . . . . . 514.3.2 Spectral Index — n s and n ′ s . . . . . . . . . . . . . . . 524.3.2.1 The Case Γ ϕ & H TI . . . . . . . . . . . . . . . 574.3.2.2 The Case Γ ϕ ≪ H TI . . . . . . . . . . . . . . . 594.3.3 Tensor Fraction r . . . . . . . . . . . . . . . . . . . . . 604.4 “End of Inflation” Mechanism . . . . . . . . . . . . . . . . . . 604.4.1 Generating ζ . . . . . . . . . . . . . . . . . . . . . . . 614.4.1.1 The Case Γ ϕ & H TI . . . . . . . . . . . . . . . 614.4.1.2 The Case Γ ϕ ≪ H TI . . . . . . . . . . . . . . . 624.4.2 Non-Gaussianity . . . . . . . . . . . . . . . . . . . . . 634.4.3 Constraining the Free Parameters . . . . . . . . . . . . 664.4.3.1 Primordial Inflation Energy Scale . . . . . . . 664.4.3.2 Thermal Inflation Dynamics . . . . . . . . . . 664.4.3.3 Lack of Observation of φ Particles . . . . . . 674.4.3.4 Light ψ . . . . . . . . . . . . . . . . . . . . . 684.4.3.5 The Field Value ψ ∗ . . . . . . . . . . . . . . . 70iv.4.3.6 Thermal Fluctuation of φ . . . . . . . . . . . 714.4.3.7 Thermalization of φ . . . . . . . . . . . . . . 744.4.3.8 The Field Value φ ∗ . . . . . . . . . . . . . . . 764.4.3.9 Energy Density of the Thermal Waterfall Field 794.4.3.10 Time of Transition from Thermal Inflation toThermal Waterfall Field Oscillation . . . . . . 824.4.3.11 Energy Density of the Oscillating SpectatorField . . . . . . . . . . . . . . . . . . . . . . . 854.4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.4.4.1 The Case α = 1 . . . . . . . . . . . . . . . . . 874.4.4.2 The Case α = 1 . . . . . . . . . . . . . . . . . 984.5 Modulated Decay Rate . . . . . . . . . . . . . . . . . . . . . . 1004.5.1 Non-Gaussianity . . . . . . . . . . . . . . . . . . . . . 1024.5.2 Constraining the Free Parameters . . . . . . . . . . . . 1044.5.2.1 Time for φ Decay . . . . . . . . . . . . . . . . 1044.5.2.2 Light ψ . . . . . . . . . . . . . . . . . . . . . 1054.5.2.3 Energy Density of the Spectator Field . . . . 1064.5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 φ φ and a Thermal Bath 121Bibliography 125 v ist of Figures ∆ TT of CMB in spherical harmonics, as observedby Planck spacecraft . . . . . . . . . . . . . . . . . . . . . . . 234.1 Potentials in Thermal Inflation scenario . . . . . . . . . . . . . 454.2 Potential given by Eq. (4.5) . . . . . . . . . . . . . . . . . . . 494.3 Allowed parameter space for ψ ∗ . . . . . . . . . . . . . . . . . 894.4 Allowed parameter space for δψ ∗ ψ ∗ . . . . . . . . . . . . . . . . . 894.5 Allowed parameter space for H T I . . . . . . . . . . . . . . . . 904.6 Allowed parameter space for m ψ . . . . . . . . . . . . . . . . . 904.7 Allowed parameter space for h . . . . . . . . . . . . . . . . . . 914.8 Prediction for f NL . . . . . . . . . . . . . . . . . . . . . . . . . 924.9 Prediction for g NL . . . . . . . . . . . . . . . . . . . . . . . . . 934.10 Prediction for N TI . . . . . . . . . . . . . . . . . . . . . . . . . 964.11 Prediction for N ∗ . . . . . . . . . . . . . . . . . . . . . . . . . 974.12 Prediction for n s for φ ∗ Case A: Chaotic Inflation . . . . . . . 984.13 Prediction for n ′ s for φ ∗ Case A: Chaotic Inflation . . . . . . . 99vi.14 Prediction for n s for φ ∗ Case B: Chaotic Inflation . . . . . . . 1004.15 Prediction for n ′ s for φ ∗ Case B: Chaotic Inflation . . . . . . . 1014.16 Prediction for n s for φ ∗ Case C: Chaotic Inflation . . . . . . . 1024.17 Prediction for n ′ s for φ ∗ Case C: Chaotic Inflation . . . . . . . 103vii ist of Tables α = 1 and Γ ϕ ≪ H TI . . . . . . . . . . . . . . . . . . . . . . . 884.3 Values of quantities in the model for α = 1, Γ ϕ ≪ H TI , m ∼ GeV and the parameter values of Table 4.2. . . . . . . . . 914.4 Prediction for non-Gaussianity parameters of the model, with α = 1, Γ ϕ ≪ H TI , m ∼ GeV and h and H ∗ values fromTable 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5 Prediction for N TI and N ∗ of the model, with Γ ϕ ≪ H TI , m ∼ GeV and n , g , H ∗ , Γ ϕ and λ values from Table 4.2. . . . . 964.6 Prediction for n s and n ′ s of the model with primordial inflationbeing Chaotic Inflation, with α = 1, Γ ϕ ≪ H TI , m ψ = 10 − GeV, m ∼ GeV and the parameter values from Table 4.2. . . . . 97viii hapter 1Introduction
Cosmological Inflation is a leading candidate for the solution of the threemain problems of the standard Big Bang scenario: the Horizon, Flatness andRelic problems. It also has the ability to seed the initial conditions required toexplain the observed large-scale structure of the Universe. (For a textbook onthis topic, see [1].) In the simplest scenario, quantum fluctuations of a scalarfield are converted to classical perturbations around the time of horizon exit,after which they become frozen. This gives rise to the primordial curvatureperturbation, ζ , which grows under the influence of gravity to give rise toall of the large-scale structure in the Universe. The observed value of thespectrum of the primordial curvature perturbation is P ζ ( k ) ∼ − .Moving away from this simplest scenario, there has been much work doneon trying to generate the observed ζ in other scenarios, such as the curvaton[2–23], inhomogeneous reheating [4, 19–22, 24–30], “end of inflation” [13, 30–36] (also see [37]) and inhomogeneous phase transition [38]. (Also see [39].)This thesis is structured as follows. In Chapter 2 we talk about the1standard model of Cosmology. We then go on to talk about the field ofParticle Cosmology in Chapter 3. In Chapter 4 we give a detailed account of anew Thermal Inflation model that we have created, where we give expressionsfor key observational quantities that are predicted by the model. We finishwith Chapter 5, in which we conclude. hapter 2Concordance Model ofCosmology — Λ CDM
At the beginning of the 20 th century, the field of Cosmology was devoid ofGeneral Relativity, proof of the existence of other galaxies and that the Uni-verse was expanding, evidence of the Cosmic Microwave Background, here-after referred to as the CMB, as well as the Big Bang Theory, or indeed anytheory of the genesis of the Universe that was scientifically based. By the end of the 20 th century however, a consistent and hugely successful model ofthe entire history of the Universe (save for the very first moments after thecreation of spacetime, if indeed even such a creation occurred) was firmlyestablished.Several hundred years ago, the Polish astronomer Nicolaus Copernicusproposed an alternative to the Ptolemaic view, which stated that the Earthwas at the centre of the Universe. Copernicus stated that it was the Sun The Heliocentric system was originally proposed by Aristarchus of Samos in the 3 rd century BC, whom Copernicus was aware of and cited. .1 General Relativity Copernican Principle .Analysis of observations of the large-scale structure of the Universe andgeneralizing the Copernican Principle allows us to state that, on large scales,the Universe is statistically both homogeneous and isotropic . Homogene-ity states that observations made at any point in the Universe will be sta-tistically representative of those made at any other point and hence there isno preferred place in the Universe. Isotropy states that the Universe looksstatistically the same in all directions. These two aspects of our Universe,when taken together, define what is termed the
Cosmological Principle ,which is a foundational pillar in the current standard model of Cosmology.
General Theory of Rel-ativity , being the 100 th anniversary of its first presentation to the worldby Einstein. After 100 years, it has survived the test of time and scientificrigor to remain the principle theory that humankind possesses regarding thebehavior of gravity on large (cosmological) scales.Let us start with a reminder of the basic mathematics of Special Relativ- .1 General Relativity x µ ≡ x x x x ≡ ctxyz = ct x in Minkowski coordinates (in a Minkowski space). This defines a threading of spacetime, which can be represented by a series of lines, corresponding toa fixed x i ( i = 1 , , slicing of spacetime into hypersurfaces,corresponding to a fixed x .The line element between two spacetime points is given by ds = − dt + dx + dy + dz (2.1)where we use natural units for which c = ~ = k B = 1. By defining a symmetric Minkowski Metric Tensor η µν ≡ − we can denote the line element as ds = η µν dx µ dx ν (2.2) .1 General Relativity Einstein summation convention.Going from a flat (Minkowski) manifold to the generically curved oneof spacetime, which is an example of a pseudo-Riemannian manifold, wehave the general symmetric metric tensor g µν (as opposed to the flat η µν ).The line element now becomes ds = g µν dx µ dx ν (2.3)The essence of General Relativity, hereafter referred to as GR, is reflected inthe Einstein Field Equation , which is R µν − g µν R = 8 πGT µν (2.4)where R µν is the Ricci Tensor , given by R µν ≡ R λµλν (2.5)where R µνβα is the Riemann Tensor , given by R µνβα ≡ ∂ α Γ µνβ − ∂ β Γ µνα + Γ µσα Γ σνβ − Γ µσβ Γ σνα (2.6)where Γ µνβ is the Christoffel Symbol , given byΓ µνβ = 12 g αµ ( ∂ β g αν + ∂ ν g αβ − ∂ α g νβ ) (2.7) .1 General Relativity R is the Ricci Scalar , given by R ≡ g µν R µν (2.8)with g µν being the inverse metric. In Eq. (2.4), G is Newton’s Gravita-tional Constant and T µν is the Energy-Momentum Tensor , also knownas the Stress-Energy Tensor, which is defined as T µν ≡ T T T T T T T T T T T T T T T T where the T component in red is the energy density, the T i components in orange are the momentum density, the T i components in orange are theenergy flux, the components in blue are the shear stress and the T ii com-ponents in green are the pressure. The T ij components are the momentumflux. Sometimes the LHS of Eq. (2.4) is combined into a single tensor, knownas the Einstein Tensor G µν ≡ R µν − g µν R (2.9)The Einstein Field Equation relates the curvature of a region of spacetime,i.e. the strength of gravity, to the amount of energy, momentum and stressthat is present within that spacetime region. .1 General Relativity During the 1920s and 1930s, four scientists worked independently on prob-lems concerning the geometry and evolution of a homogeneous and isotropicUniverse. These were Alexander Friedmann, Georges Lemaître, Howard P.Robertson and Arthur Geoffrey Walker. The most general form of the lineelement in polar coordinates that satisfies homogeneity and isotropy, as wellas allowing for uniform expansion is ds = − dt + a ( t ) " dr − Kr + r (cid:16) dθ + sin ( θ ) dφ (cid:17) (2.10)where a ( t ) is the scale factor and K is the spatial intrinsic curvature. Thevalue K = 0 corresponds to a spatially flat (Euclidean) Universe. A valueof K <
K >
FLRW metricline element , after the authors mentioned above. Analysis of the CMBshows that our Universe is spatially flat to a very high degree of precision.For a K = 0 spatially flat Universe, the line element in Cartesian coordinatesis ds = − dt + a ( t ) (cid:16) dx + dy + dz (cid:17) (2.11)where the metric tensor is g µν = − a ( t ) 0 00 0 a ( t ) 00 0 0 a ( t ) .1 General Relativity Conformal Time variable dη = dta (2.12)The line element now becomes ds = a ( η ) (cid:16) − dη + dx + dy + dz (cid:17) (2.13)Let us now assume that the content of the Universe is analogous to a perfect fluid , which is a fluid that has no shear stress, no viscosity andwhich does not conduct heat. It can be described entirely by its energydensity ρ and its isotropic pressure P . In the local rest frame, the Energy-Momentum tensor becomes simply T µν = ρ P P
00 0 0 P We have not yet employed GR in the discussion regarding our FLRWUniverse. All we have assumed so far is a uniformly expanding/contractinghomogeneous and isotropic Universe, which has a content that can be de-scribed as a perfect fluid. We now take the “00” (“ tt ”) component of the .1 General Relativity R − g R = 8 πGT (2.14) R + 12 R = 8 πGρ (2.15)where the Ricci tensor is R ≡ R λ λ ≡ ∂ Γ λ λ − ∂ λ Γ λ + Γ λα Γ α λ − Γ λαλ Γ α (2.16)We calculate the Christoffel symbols from Eq. (2.7). We also calculate theRicci scalar in Eq. (2.15) from Eq. (2.8). After this work, we obtain what isknown as the Friedmann Equation H = ρ M P − Ka (2.17)where M P = 2 . × GeV is the
Reduced Planck Mass , for which8 πG ≡ M − P in natural units and H ≡ ˙ aa is the Hubble Parameter , with thedot denoting derivative with respect to the cosmic time t . The Hubble pa-rameter is the rate of expansion/contraction of the Universe. For a spatiallyflat Universe, K = 0 and Eq. (2.17) is simply ρ = 3 M P H (2.18) .1 General Relativity ∇ µ T µν = 0) we obtain the Continuity Equation ˙ ρ = − H ( ρ + P ) (2.19)If we differentiate Eq. (2.18) with respect to t and employ Eq. (2.19) weobtain the (Friedmann) Acceleration Equation ¨ aa = − ρ + 3 P M P (2.20)In order to solve the Friedmann equation for the time-evolution of thescale factor, we first need an equation of state giving the relationship betweenthe energy density and the pressure of the perfect cosmic fluid. The equationof state is barotropic ( P = P ( ρ )) and is parameterized as w = Pρ (2.21)With c = 1, if the cosmic fluid has velocity v rms ≪
1, it is called matter (ornon-relativistic matter), whereas if it has v rms ≈
1, then it is called radiation(or relativistic matter). For the case of matter, we have w = 0, as ρ ≫ P .The continuity equation then gives ρ mat ∝ a − (2.22)For the case of radiation, we have w = , as ρ = 3 P . The continuity equationthen gives ρ rad ∝ a − (2.23) .1 General Relativity a ” factor compared to the case ofmatter is that the frequency of the radiation is red-shifted as the Universeexpands, hence it loses energy.In the case of K = 0 (i.e. our Universe), the Friedmann equation yieldsthe solution for the evolution of the scale factor as a ( t ) ∝ t w +1) (2.24)for w > − a ( t ) ∝ t (2.25)and for radiation domination we have a ( t ) ∝ t (2.26)In the case of a cosmological constant (see Section 2.3) (and also forinflation (see Section 3.2.2)), which is causing the expansion of the Universeto accelerate, beginning from around the time of the current epoch, theequation of state is w = −
1. For this situation, Eq. (2.24) is not valid.Instead, Eqs. (2.19) and (2.21) give that ˙ ρ = 0 (c.f. Eq. (2.18)). H ≡ ˙ aa isconstant and the evolution of the scale factor is a ( t ) ∝ e Ht (2.27) .1 General Relativity Density Parameter Ω( t ) ≡ ρ ( t ) ρ crit ( t ) (2.28)with the Critical Energy Density being defined as ρ crit ( t ) ≡ M P H ( t ).The critical energy density is the total density that a spatially flat Universewould have for a given value of the Hubble parameter. We also defineΩ − ≡ K ( aH ) ≡ K ˙ a (2.29)Precise measurements of the geometry and energy density of the Universemade by the Planck spacecraft, when combined with other data [40], yieldthe current value Ω = 1 . ± .
005 (2.30)which is consistent with the time-independent value of Ω = 1. We thereforehave that the energy density of the Universe is very close to the criticaldensity ( ρ = ρ crit ) and that the geometry of the Universe is very close tospatially flat ( K = 0). The observed energy density of the Universe is made-up of three components: Baryonic Matter (5%), Dark Matter (26%) and DarkEnergy (69%), the percentages indicating the approximate relative amountof each component. .2 The Big Bang In 1929, Edwin Hubble discovered what is known as
Hubble’s Law , whichstates that there exists a linear relationship between the distance of a galaxyfrom us and its recession velocity, with the constant of proportionality being
Hubble’s Constant , H , which has been observed using a variety of sources,with the combined data value [40] being H = 67 . ± .
46 km s − Mpc − (2.31)Therefore, the further away a galaxy is from our Milky Way, the faster it istraveling away from us. As a result of the Cosmological Principle, Hubble’sLaw implies that, over large scales, each galaxy is moving away from everyother galaxy. If we consider this fact in reverse time, it is clear that galaxieswill get ever closer to each other. At some point in the past, the Universewill be sufficiently dense, hot and energetic that the equations of GR willbreak down and leave us with a spacetime singularity. It is this point thatwe label as the beginning of our Universe, which occurred around 13 . Hot Big Bang , hereafter referred to as HBB, to begin atthe time when the reheating process from inflation (discussed in Section 3.2.3)is complete. This process must leave us with a radiation-dominated Universe .2 The Big Bang T ∼ ≫ H , so that the process has the “time” tooccur, before the expansion of the Universe stifles the process.For a collection of particles in thermal equilibrium, the distribution func-tion is f ( E ) = 1 e ( E − µT ) ± − for fermions and where E is the particle energyand µ is the Chemical Potential . For the case where the temperature T is much larger than the mass and chemical potential of the particle species,which was applicable in the very early Universe, the distribution functionsimplifies to f ( p ) ≈ e pT ± E ∼ T ≃ p , with p being the momentum of the particle. This distri-bution function yields the blackbody distribution of photons. The numberdensity is given by n = g π Z ∞ f ( p ) p d p (2.34)= A ζ (3) gπ T (2.35) .2 The Big Bang A = 1 for bosons and A = for fermions and g is the number of spinstates of the particle (relativistic degrees of freedom). The energy density isgiven by ρ = g π Z ∞ f ( p ) p d p (2.36)= B π g T (2.37)where B = 1 for bosons and B = for fermions. For the Universe, the energydensity is given by the weighted sum of all the particles as ρ = π g ∗ T (2.38)with g ∗ ≡ bosons X i g i + 78 fermions X j g j (2.39)being the total number of spin states of all of the constituent particles (theeffective number of relativistic degrees of freedom).As the temperature of the Universe falls due to the expansion, Γ for aparticular particle species will fall below H at some time. Different particlespecies will start to fall out of thermal equilibrium at different times and thus decouple from each other.One of the big successes of the HBB is the agreement between analyti-cal/numerical calculations and observation of Big Bang Nucleosynthesis ,hereafter referred to as BBN. After an excess of H (over anti- H ) had been .2 The Big Bang He . H and He wereproduced in smaller quantities and a very small amount of Li was also pro-duced. The abundance of these nuclei depend strongly on the baryon numberto photon number ratio η ≡ n B n γ (2.40)which has a value of η ∼ − .Another great success of the HBB concerns the observed large-scale struc-ture, LSS , of the Universe. The theory predicts an expanding Universe,which we observe as the
Hubble Flow , with structure forming under theinfluence of gravity according to the laws of GR. On cosmologically smallscales, matter (baryonic and dark matter) is bound gravitationally into galax-ies, with there existing a hierarchy of structure, consisting of galaxies, galaxygroups, galaxy clusters, galaxy superclusters and galaxy filaments. The lastof these structures form the boundaries with the voids of the Universe. Theentire collection of components of the LSS is referred to as the
Cosmic Web .Further evidence for a HBB model concerns the CMB. Due to the hot,dense early stages of the Universe, the HBB predicts that there should be aremnant blackbody radiation that was emitted when the Universe was youngthat should still be observable today. For a long time after BBN, photonswere being continuously and rapidly scattered off of free electrons, due toThomson scattering. As the temperature of the Universe continued to falldue to the expansion, there came a time when the free electrons becamebound with the nuclei that was present, in a process known as
Recombi- .3 Λ CDM nation . At this point, the photons fell out of thermal equilibrium with theelectrons (the latter are now bound inside neutral atoms) and thus becamedecoupled, which allowed them to travel freely, following a geodesic. There-fore, there should exist a last scattering surface , corresponding to the timeof recombination. (In reality, recombination did not occur instantaneouslyand so the last scattering surface actually has a thickness to it.)The CMB radiation was discovered in 1964 by Arno Penzias and RobertWilson, a discovery which earned them the 1978 Nobel Prize for Physics.When combined with other data, the data obtained from the Planck space-craft [40] yields values of the corresponding blackbody temperature and red-shift of the last scattering surface of T = 2 . ± .
021 K (2.41) z ls = 1089 . ± .
23 (2.42)The redshift of the last scattering surface corresponds to a time of ≈ Λ CDM
All of what has been talked about so far forms part of the current standardmodel of Cosmology. However, to complete the model, we need to considerthe effects of reionization.
Reionization amounts to the partial ionization .3 Λ CDM optical depth τ ( t ) ≡ σ T Z t t n e ( t ) d t (2.43)where n e is the number density of free electrons and σ T is the Thomsonscattering cross-section. With this definition, the probability that a photonthat is observed now that was emitted between the time of recombinationand reionization, at a time t , has traveled freely is e − τ ( t ) , with the valuebeing practically constant for emission times between recombination and .3 Λ CDM τ = 0 . ± .
012 (2.44) z reion = 8 . +1 . − . (2.45)The final piece of the standard model concerns the small perturbations inthe cosmic fluid density that existed in the early Universe. On large scales,the Universe is homogeneous and isotropic. However, on smaller scales, it isclear that this is violated, as the Universe contains planets, galaxies, emptyspace etc. Therefore, there must have existed some small differences in thedensity of the cosmic fluid at very early times, which then grew under theinfluence of gravity and the Hubble flow. These tiny perturbations presentthemselves in the CMB, as anisotropies in the average temperature of themicrowave radiation. The Planck spacecraft made precise all-sky measure-ments of the CMB, which is displayed in Fig. 2.2.A crucial concept in early Universe Cosmology is that of the PrimordialCurvature Perturbation , labeled as ζ . This will be discussed more fullyin Section 3.2.1. However, we will briefly discuss two aspects of it here. The spectrum , P ζ ( k ), of the curvature perturbation conveys how much poweris in the perturbation as a function of scale k . We also have the spectralindex , n s , which tells us how the spectrum varies with scale k . (The sub-script s denotes that this is for a scalar perturbation.) The spectral index is .3 Λ CDM ∆ TT ∼ − . Taken from [42].defined as n s − ≡ d ln ( P ζ ( k ))d ln ( k ) (2.46)with n s − tilt of the spectrum. For constantspectral index, P ζ ( k ) ∝ k n s − . P ζ is scale invariant for n s = 1. In general, n s = n s ( k ), with cosmic inflation predicting n s close but not exactly equal tounity, so that the spectrum is approximately (but not quite) scale invariant.The Planck spacecraft made measurements of the spectrum at what is knownas the pivot scale , which is k ≡ .
002 Mpc − . When combined with otherdata, the data obtained from the Planck spacecraft [40] yields values for thespectrum and spectral index of P ζ ( k ) = (2 . ± . × − (2.47) .3 Λ CDM n s = 0 . ± . CDM , C old D ark M atter. A CDM particle is cold in that it hasnegligible (meaning non-relativistic) random motion. It has negligible inter-action with other particles and also negligible self-interaction, hence must benon-baryonic, with its only real presence being observed via its gravitationaleffect on galactic dynamics. Regarding dark energy, the simplest realizationis a Cosmological Constant , denoted by Λ. This is a spatially and tem-porally constant term that is added to Einstein’s Field Equation, having theeffect of negative pressure.Taking all of our discussion so far into account, we are presented withthe
ΛCDM model as our
Concordance Model of Cosmology, which con-tains just six independent parameters: H , Ω B , Ω CDM , τ , P ζ ( k ) and n s .Fig. 2.3 shows a plot of the agreement between the ΛCDM model and thedata obtained from the Planck spacecraft. .3 Λ CDM ∆ TT of the CMB in spherical harmonics, as observedby the Planck spacecraft. Red Points: Planck Data. Green Curve: BestFit of ΛCDM Model to Planck Data. Light Blue Shading: Predictions ofall Variations of ΛCDM Model that Best Agree with Planck Data. Takenfrom [43]. hapter 3Particle Cosmology So far, we have been concentrating on the ΛCDM model of the Universe. Weare now going to discuss
Particle Cosmology , which is the field concernedwith a particle physics description of the (early) Universe. The temperatureand energy scales that were dominant during the very early Universe weresuch that Quantum Field Theory is required for a complete description ofthat period.
As already discussed, the Big Bang Theory is hugely successful in explain-ing many of the properties and features that we observe in our Universe.However, it will become clear that it is not a sufficient theory on its ownto explain everything that we observe. There are five main problems withthe Big Bang and each of these will be discussed now. We will then see inSection 3.2 how the theory of Inflation can solve these problems.24 .1 Problems of Big Bang Cosmology With the definition of the Hubble parameter being H ≡ ˙ aa , we define the hori-zon as the distance that light (information) can travel within one HubbleTime H − .We define the comoving Particle Horizon as x PH ( t ) ≡ Z t a ( t ) d t (3.1)= η ( t ) − η (0) (3.2) x PH ( a ) = Z a a H d a (3.3)with the third line coming from the fact that we take ˙ a ( t ) > t, a = 0. Any two eventsthat are separated by a distance of more than twice the particle horizon areout of casual contact. The Particle Horizon in physical coordinates is givenby a ( t ) x PH ( t ) ≡ a ( t ) Z t a ( t ′ ) d t ′ (3.4) a ( t ) x PH ( a ) = a ( t ) Z a a H d a (3.5) .1 Problems of Big Bang Cosmology Event Horizon , as x EH ( t ) ≡ Z ∞ t a ( t ) d t (3.6)= η ( ∞ ) − η ( t ) (3.7) x EH ( a ) = Z ∞ a a H d a (3.8)again with the third line coming from the fact that we take ˙ a ( t ) >
0. Thisis the maximum distance that light (information) can travel in the (infinite)future. An event that occurs at a spacetime point cannot influence an eventat a future spacetime point if the latter is outside of the former’s eventhorizon. The Event Horizon in physical coordinates is given by a ( t ) x EH ( t ) ≡ a ( t ) Z ∞ t a ( t ′ ) d t ′ (3.9) a ( t ) x EH ( a ) = a ( t ) Z ∞ a a H d a (3.10)In reality, Eqs. (3.5) and (3.10) are definitions that use unrealistic bound-ary conditions, as a = 0 , ∞ is unphysical. Therefore, we usually calculateparticle and event horizons between two well-defined limits a ( t ) x PH ( a , a ) = a ( t ) Z aa a H d a a ( t ) x EH ( a, a ) = a ( t ) Z a a a H d a (3.11) .1 Problems of Big Bang Cosmology a ≫ a or a ≪ a .The Horizon Problem arises when we consider how the horizon of variousplaces in the observable Universe has varied over the lifetime of the Universeand then compare this with observation. Our observable Universe is statisti-cally homogeneous and isotropic on cosmological scales. One manifestationof this is that of the large isotropy that exists in the CMB. Over the entiresky, the temperature of the CMB is the same to within 1 part in ∼ .Therefore, it is very safe to assume that all parts of the CMB were in ther-mal equilibrium with each other at the time of last scattering and thus werein casual contact with each other. However, when we consider two parts ofthe CMB that are separated by more than a few degrees across the sky, wefind that the particle horizon’s of these two parts do not overlap. Therefore,they have never been in casual contact with each other, as there has not beenenough time for light (information) to travel between the two parts given theage of the Universe. The Horizon Problem is thus to explain how each partof the observable Universe is so extremely statistically similar to every otherpart, given that the vast majority of parts have never been in casual contactwith each other. From looking at Eq. (2.29) regarding the Density Parameter, we see that astime evolves from the birth of the Universe, i.e. as ˙ a decreases rapidly, Ωgrows rapidly, away from a value of 1. We observe for the Density Parameterthe current value Ω = 1 . ± .
005 (Eq. (2.30)), which is very close to 1, i.e. .1 Problems of Big Bang Cosmology
28a spatially flat Universe. Therefore, in the past, the value must have beeneven more precisely close to 1, with it having had a value of 1 to a precisionof at least ∼ − around the Planck time. This Flatness Problem is thus afine-tuning problem for the initial conditions of the Universe.The Flatness Problem, when looked at from a slightly different viewpoint,can also be regarded as an Age Problem. If the value of Ω had been onlyslightly larger than it appears to have been, then the Universe would havebeen of a sufficient density that it would have collapsed at a time muchsooner than the age of our observable Universe. If, on the other hand, thevalue of Ω had been only slightly smaller than it appears to have been, thenthe Universe would have expanded too quickly for stars and galaxies to haveformed the LSS that we observe today. Therefore, the fine-tuning problemcan be regarded as an Age Problem, in that how has our observable Universebecome as old as it is? In contrast to the two problems already mentioned, the Relic Problem isnot specific to a Hot Big Bang model of the Universe, but rather somethingthat has to be considered within the total picture of Particle Cosmology.There exist a large variety of particle physics models and applications ofthese to Cosmology that have the ability to produce many types of relic;particles and other components that contradict established theory and/orobservation, such as gravitinos coming from SUGRA and moduli coming fromstring theory. Another example, which was the subject of the original work .1 Problems of Big Bang Cosmology
It is clear that there exist many more particles than anti-particles in theUniverse. However, there is no explicit mechanism in the Hot Big Bangmodel itself that can account for this
Baryon Asymmetry and accordingto the model, identically equal amounts of matter and anti-matter shouldhave been created in the early Universe.
Lastly, there is the issue of the creation of the LSS of the observable Uni-verse. Assuming just an expanding FLRW Universe, there arises the questionof how the observable structure in the Universe came into existence, as thisexpanding spacetime is exactly homogeneous and isotropic. Although ourobservable Universe is highly homogeneous and isotropic on cosmological .2 Cosmic Inflation
We now discuss one of the key areas of Particle Cosmology in the 21 st centuryand indeed the main topic of work in this thesis: Cosmic Inflation. Firstly,let us define inflation. With regard to the scale factor a ( t ), inflation is definedas any period of spacetime expansion in which we have¨ a > t H − a < − ˙ HH < .2 Cosmic Inflation H becomes more and more constantduring the period of inflation, with the expansion becoming more and moreexponential, i.e. a ∝ e Ht . It should be noted that Eqs. (3.12)–(3.14) are allequivalent.If we now include GR in our discussion, we have one final definition ofinflation. Using the (Friedmann) Acceleration Equation, Eq. (2.20), we have ρ + 3 P <
P < − ρ , in order to achieve a period of inflation.Now we will discuss how a period of inflation can solve the five problemsof an isolated Big Bang Cosmology mentioned above. Firstly, the HorizonProblem. The solution to this problem is to say that the entire observableUniverse used to be inside the horizon, prior to the end of inflation. Dur-ing inflation however, different scales of relevance to our observable Universewere “stretched” to outside of the horizon at different times, rememberingthat one of the definitions of inflation is that of a decreasing comoving Hubblelength, Eq. (3.13). At later times, after inflation, these scales then startedto re-enter the horizon, again at different times for different scales.Now, the Flatness Problem. From looking at Eq. (2.29) regarding theDensity Parameter, we can see that the RHS will tend towards 0 duringinflation. The reason for this, is that during inflation ¨ a > a will grow rapidly, which will have the affect of rapidly drivingthe value of Ω towards 1 throughout the entire period of inflation. However, .2 Cosmic Inflation aH ≪ H , a long time before cosmologicalscales exit the horizon during inflation. We now discuss the solution to the Relic Problem. Any relics that existbefore or during inflation will be rapidly diluted away by the inflation. Theywill quickly become non-interacting as their number density decreases, due tothe rapid expansion, as the interaction rates quickly fall below H . However,we may still have relics that are produced thermally after the end of inflation.This will depend mainly on the reheat temperature of the Universe after theend of inflation. Therefore, to avoid specific types of relics that may be anissue, for example in how they spoil BBN, any particular model of inflationmust have a reheat temperature low enough so as to not produce an unac-ceptable amount of such relics. Alternatively, a second period of inflation,known as Thermal Inflation (see Chapter 4), can affectively eradicate them.Let us now discuss the Baryon Asymmetry Problem. Inflation can ac-commodate Baryogenesis mechanisms that produce the asymmetry betweenmatter and anti-matter. In order to achieve Baryogenesis, we require thefollowing conditions, known as the Sakharov Conditions
A) Violation of B (Baryon Number) conservationB) Violation of CP symmetryC) Absence of thermal equilibriumIt is the last of these conditions that can be achieved during a period ofinflation. We assume the current value of a being a = 1. .2 Cosmic Inflation δN Formalism
We now go on to define a crucial quantity known as the Primordial CurvaturePerturbation, ζ . We will then see how we can use the so-called δN formalismto calculate this. Let us consider a coordinate gauge in which the spatialthreads are comoving and the temporal slices are such that each one has auniform energy density. The spatial part of the spacetime metric is g ij = a ( x , t ) γ ij ( x ) (3.16)where we have a ( x , t ) ≡ a ( t ) e ζ ( x ,t ) (3.17)and γ ij ( x ) ≡ (cid:16) I e h ( x ) (cid:17) ij (3.18)where h ( x ) is the primordial tensor perturbation. We therefore have as ourdefinition of ζ ζ ( x , t ) ≡ ln a ( x , t ) a ( t ) ! (3.19) .2 Cosmic Inflation g ij = ˜ a ( x , t )˜ γ ij ( x ) (3.20)where ˜ a ( x , t ) ≡ a ( t ) e ψ ( x ,t ) (3.21)and ˜ γ ij ( x ) ≡ (cid:16) I e h ( x ) (cid:17) ij (3.22)At any given value of t , the scale factors a ( x , t ) and ˜ a ( x , t ) differ only becauseof the difference in the time coordinate of the two spacetime slices. Therefore,in order to maintain generality, we will drop the ~ on the scale factor. Wecan define a quantity called the e-folding number as N ≡ ln (cid:18) a a (cid:19) (3.23)which is the number of exponential expansions of the Universe between whenthe scale factor is a and when it is a . As H ≡ ˙ aa , this can also be expressedas N = Z t t H d t (3.24)and is thus sometimes called the number of Hubble times. The difference inthe number of e-foldings between any two generic spacetime slices is given .2 Cosmic Inflation δN ( x ) = δ Z t t a ( x , t ) d a ( x , t )d t d t (3.25) = ψ ( x , t ) − ψ ( x , t ) (3.26)We will define what we will call a flat spacetime slice as the one where ψ ( x , t ) = 0 (3.27)We define the term δN ( x , t ) to denote the number of e-foldings between theflat slice and a slice of uniform energy density at time t . Therefore, we reachwhat is called the δN formalism ζ ( x , t ) = δN ( x , t ) (3.28)which thus allows us to calculate the primordial curvature perturbation fromcalculating the difference in the number of e-foldings of expansion betweena flat slice and a latter uniform energy density slice, which is extremelyuseful when considering the perturbation that is produced from a particularinflation model. Inflation models most often assume that the content of the Universe duringand immediately after inflation is dominated by the presence of one or more .2 Cosmic Inflation φ n . A scalar field is homogeneous (being as it is homogenizedby inflation) and will behave like a perfect fluid, as its stress is isotropic.Let us consider the contents of the Universe to simply be a single scalarfield, φ . The action that governs this scenario is S = Z √− g (cid:18) M P R + L (cid:19) d x (3.29)where g is the determinant of the metric tensor g µν , R is the Ricci scalar and L is the Lagrangian density of the scalar field, which is L = − ∂ µ φ∂ µ φ − V ( φ ) (3.30)where the first term is called the kinetic term and the second term is calledthe scalar field potential. By using the action principle, δS = 0 , we can obtainthe Energy-Momentum tensor for the scalar field, which is T µν = − ∂ L ∂g µν + g µν L (3.31)Substituting Eq. (3.30) into here gives T µν = ∂ µ φ∂ ν φ − δ µν (cid:18) ∂ α φ∂ α φ + V ( φ ) (cid:19) (3.32)The “ ” (“ tt ”) component of this Energy-Momentum tensor gives the energydensity for a homogeneous scalar field, which is T = ρ = 12 ˙ φ + V ( φ ) (3.33) .2 Cosmic Inflation T = T = T components give the pressure for a homogeneous scalarfield, which is P = 12 ˙ φ − V ( φ ) (3.34)For simplicity, let us continue to assume that the cosmological fluid duringand immediately after inflation consists principally of just the one scalar field, φ . If the kinetic energy density term of this field, ˙ φ , is small, i.e. if thefield varies slowly, or not at all, then we will have a situation in which P ≈ − ρ (3.35)Therefore, the equation of state will be w = Pρ ≈ − (3.36)and we will therefore have a period of (quasi-de Sitter) inflation, in whichthe scale factor goes (nearly) like a ( t ) ∝ e Ht (see Section 2.1.1). As φ is thecomponent in the cosmological fluid that is responsible for driving inflation,we call the field, as well as its associated particle within the context of QFT,the Inflaton . We now briefly discuss the method by which we actually obtain an energydensity perturbation from a scalar field. During inflation, the inflaton field φ will naturally acquire quantum fluctuations, δφ , as a direct result of theuncertainty principle. We assume that φ is in a vacuum state and so we have .2 Cosmic Inflation
380 particles as the eigenvalue of the number operator. The field equation ofthe first-order perturbation δφ is ¨ δφ k + 3 H ˙ δφ k + ka ! δφ k + V ′′ δφ k = 0 (3.37)where φ k is the Fourier transform of φ ( x ) φ k = 1(2 π ) Z φ ( x ) e − i k · x d x (3.38)We concern ourselves only with a light field. Given this, we have V ′′ ≪ ka ! (3.39)Therefore, we have the field equation ¨ δφ k + 3 H ˙ δφ k + ka ! δφ k ≃ (3.40)As the key interest in this discussion is the time around horizon exit, let usconcentrate on this and so let us set H = H ∗ (3.41)where H ∗ is the scale-independent constant value of H at around the timeof horizon exit. The use of the constant value H ∗ as opposed to the scale-dependent value H k , for when the scale k exits the horizon, is an approxi-mation, used to simplify the derivation here. The approximation is valid, as .2 Cosmic Inflation H ≈ constant , i.e. H varies extremelyslowly with scale. We now transform from cosmic time t to conformal time η = − aH ∗ (3.42)and also consider instead the comoving field perturbation ϕ ≡ aδφ (3.43)Eq. (3.40) now becomes d ϕ k ( η )d η + ω k ( η ) ϕ k ( η ) = 0 (3.44)where ω k ( η ) = k − η (3.45)where we have assumed a ( t ) ∝ e Ht . We have thus obtained a harmonic oscil-lator scenario. Following the usual procedure of QFT, we will now promotevariables to operators and quantize this harmonic oscillator. We express thecomoving perturbation in terms of Fourier components as ˆ ϕ k ( η ) = ϕ k ( η )ˆ a ( k ) + ϕ ∗ k ( η )ˆ a † ( − k ) (3.46)where ˆ a † and ˆ a are creation and annihilation operators respectively, thatsatisfy h ˆ a ( k ) , ˆ a † ( k ′ ) i = (2 π ) δ ( k − k ′ ) (3.47) .2 Cosmic Inflation [ˆ a ( k ) , ˆ a ( k ′ )] = 0 (3.48)We assume that the vacuum state is that of the Bunch-Davies vacuum. Thisvacuum state is the ground state of the system within a curved spacetimebackground. For very early times, as η → −∞ , the Bunch-Davies vacuumgives the initial condition ϕ k ( η ) = 1 √ k e − ikη (3.49)As this is for very early times, it also corresponds to very small wavelengthsand so corresponds to the Minkowski vacuum, i.e. the vacuum of the systemwithin a flat spacetime background. The solution for ϕ k ( η ) is ϕ k ( η ) = ( kη − i ) kη e − ikη √ k (3.50)which is the mode function for the Bunch-Davies vacuum. The spectrum isgiven by h ˆ ϕ k ˆ ϕ k ′ i = 14 πk P ϕ ( k ) δ ( k + k ′ ) (3.51)Substituting Eq. (3.46) and the commutation relations Eqs. (3.47) and (3.48)into Eq. (3.51) yields P ϕ ( k, η ) = k π | ϕ k ( η ) | (3.52)Substituting Eq. (3.50) into this, dividing by a (to return back to the δφ perturbations) and evaluating it a few Hubble times after horizon exit gives .2 Cosmic Inflation P δφ ( k ) = (cid:18) H ∗ π (cid:19) (3.53)which is the Hawking Temperature for de Sitter spacetime. See [45] for theoriginal derivation of this result.Now let us consider a time well after horizon exit. The solution for ϕ k ( η ) ,Eq. (3.50), tends to ϕ k ( η ) → − ikη √ k (3.54)i.e. a purely imaginary solution. Eq. (3.46) now becomes ˆ ϕ k ( t ) = ϕ k ( t ) (cid:16) ˆ a ( k ) − ˆ a † ( − k ) (cid:17) (3.55)Therefore, we can see that, before horizon exit, the perturbation ϕ k of thescalar field φ is a quantum object. However, well after horizon exit, the per-turbation has become an almost scale-invariant classical perturbation. Theclassical perturbation is conserved whilst outside the horizon [46].After the end of inflation, there must exist a mechanism for transferringthe energy density of φ into the components that will initiate the Hot BigBang. This mechanism is called Reheating . At around the time of horizonentry, the classical perturbation starts to oscillate and thus we have a particleinterpretation for φ within the context of QFT. Reheating can be sudden ortake some cosmic time to complete and is complete when we have a cosmicfluid whose components are radiation (i.e. relativistic), which are all in ther-mal equilibrium with each other and that this fluid is the initiation of the .2 Cosmic Inflation H has fallen to the same order as the decay rate of the φ field H ∼ Γ φ (3.56)The temperature at the point where reheating is complete is known as the Reheat Temperature and is given by T reh ∼ q M P Γ φ (3.57)There also exists the possibility of having a period of Preheating . Thisis where most of the energy density of the inflaton decays immediately (explo-sively) into radiation, due to non-perturbative effects. However, preheatingis typically incomplete. Therefore, the final stages of inflaton decay are per-turbative. If Γ φ ≪ H ∗ , then preheating products are irrelevant, as the energydensity of the Universe becomes dominated by the oscillating inflaton again.However, if Γ φ . H ∗ , then the Hot Big Bang will begin after preheating. hapter 4A New Thermal InflationModel Thermal Inflation [47–50] is a brief period of inflation (lasting about 10 e-folds) that could have occurred after a period of prior primordial inflation.It occurs due to finite-temperature effects arising from a coupling between athermal waterfall field and the thermal bath created from the partial or com-plete reheating from the prior inflation. If we start with a zero-temperaturescalar field theory, we can calculate the affect that placing the system in athermal bath at temperature T has on the theory by introducing an inter-action term in the form of a 1-loop correction. After calculating the appro-priate variables within the context of thermal field theory, we can take thehigh- T approximation of the correction, which gives a thermal contribution V T ≃ g T φ to the effective potential V eff , where g is the coupling constant43 .1 Thermal Inflation φ and the thermal bath. This results in a thermalcorrection to the effective mass of m ≃ g T . Within the context of statisti-cal mechanics, the interpretation of V eff is that of the free energy of φ whenthe field is in thermal equilibrium with the thermal bath at temperature T ,with the minima in V eff defining the equilibrium states, with h φ i representingthe thermal average, as opposed to the vacuum expectation value (VEV).Let us take the following potential V ( φ, T ) = V + (cid:18) g T − m (cid:19) φ + λ φ M P (4.1)Initially, the temperature of the Universe will be sufficiently high that thetemperature term in the brackets will be greater than the mass term in thebrackets. This will have the affect of holding the φ field at φ = 0 . Whenthe energy density of the Universe falls below the value of V in the thermalinflation potential, the V term will come to dominate the energy densityof the Universe and thermal inflation will begin. It will continue until veryshortly after the point when the temperature term has become smaller thanthe mass term, at which point spontaneous symmetry breaking will occurand so φ will start to roll down the potential towards either the positive ornegative VEV. The shape of the potential in this scenario is displayed inFig. 4.1.This scenario is quite general and would not be particularly unexpectedin the early Universe. However, Thermal Inflation was originally proposed asa solution to the moduli problem [47, 48]. Moduli are scalar fields that arise For ease of visualization and calculation, it is usually assumed that a scalar field rollsdown to the positive VEV. .1 Thermal Inflation -< > < > V Fig. 4.1: Blue:
T > m √ g Cyan:
T < m √ g Purple: T = 0 in string theory. They are flaton fields, which are flat directions in SUSY.These have no tree-level terms in the potential from SUSY and they get amass term from SUSY breaking (they do not have a quartic (self-interaction)term). Flaton fields have nearly flat potentials (with √ V ′′ being the relevantquantity, with a prime indicating the (partial) derivative of V with respect tothe flaton field) and large VEVs, ∼ M P . The problem is that when inflationends and a modulus starts to oscillate around its large VEV, the oscillationswill also be very large and the energy density of the moduli will start todominate the energy density of the Universe. This has the affect of creatingan abundance of moduli particles that are long-lived and do not decay priorto BBN, thus creating unwanted relics. Thermal Inflation alleviates thisproblem by diluting away the moduli during the period of inflation. They .2 The Model It is possible for the mass of a certain scalar field to be dependent on anotherscalar field [13, 22, 25–28, 31–34, 38, 39]. More specifically, the mass of a ther-mal waterfall field that is responsible for a bout of thermal inflation couldbe dependent on another scalar field. If the latter is light during primordialinflation, quantum fluctuations of the field are converted to almost scale-invariant classical field perturbations at around the time of horizon exit. Ifthe scalar field remains light all the way up to the end of thermal inflation,then thermal inflation will end at different times in different parts of theUniverse, because the value of the spectator field determines the mass of thethermal waterfall field, which in turn determines the end of thermal inflation.This is the “end of inflation” mechanism [31] and it will generate a contribu-tion to the primordial curvature perturbation ζ . In addition to this, if thescalar field remains light up until the decay of the thermal waterfall field, thedecay rate of the thermal waterfall field will be modulated, due to the massof the thermal waterfall field (which controls the decay rate) being dependenton the light scalar field. The decay of the thermal waterfall field will generatea second contribution to ζ . The motivation of this work is to explore thesetwo scenarios to see if either of them can produce the dominant contributionto the primordial curvature perturbation with characteristic observational .2 The Model c = ~ = k B = 1 and the re-duced Planck Mass is M P = 2 . × GeV.The potential that is considered in this model is V ( φ, T, ψ ) = V + g T − m + h ψ α M α − P ! φ + λ φ n +4 M nP + 12 m ψ ψ (4.2)where φ is the thermal waterfall scalar field, ψ is a light spectator scalarfield, T is the temperature of the thermal bath, g , h and λ are dimensionlesscoupling constants, α ≥ and n ≥ are integers and the − m and m ψ terms come from soft SUSY breaking. We do not include a φ term, becausethe thermal waterfall field is a flaton, whose potential is stabilised by the h has a factorial term absorbed into it. Also, we are absorbing the n +4)! factor into λ . .2 The Model m ≡ m − h ψ α M α − P (4.3)i.e. we combine the bare mass and coupling term into a new mass quantity.The variation of m , which is due only to the variation of ψ , is δm = − αh ψ α − mM α − P δψ (4.4)We only consider the case where the mass of φ is coupled to one field. Ifthe mass were coupled to several similar fields, the results are just multipliedby the number of fields. If the multiple fields are different, then there willbe only a small number that dominate the contribution to the mass pertur-bation. Therefore we consider only one for simplicity.Using Eq. (4.3), our redefined mass quantity, the potential becomes V ( φ, T, ψ ) = V + (cid:18) g T − m (cid:19) φ + λ φ n +4 M nP + 12 m ψ ψ (4.5)This potential is shown in Fig. 4.2. It would appear from the potential thatdomain walls will be produced, due to the fact that in some parts of theUniverse φ will roll down to + h φ i while in others parts it will roll down to − h φ i . However, this does not occur, as we can interpret φ as being the realpart of a complex field whose potential contains only one continuous VEV. A complex φ may result in the copious appearance of cosmic strings after the end ofthermal inflation. However, we assume that their energy scale is very low and so they willnot have any serious affect on the CMB observables. Moreover, depending on the overall .2 The Model V ( φ, , ψ ) = V − m φ + λ φ n +4 M nP + 12 m ψ ψ (4.6)To obtain the VEV of φ , we find the minimum of the zero temperaturepotential. The VEV is h φ i = mM nP q (2 n + 4) λ n +1 (4.7) background theory, such cosmic strings may well be unstable. Thus, we ignore them. .2 The Model V = 0 at the VEV. V is obtained by inserting the VEV into the zerotemperature potential and then looking along the ψ = 0 direction. We obtain V ∼ m n +40 M nP λ ! n +1 (4.8)We use the Friedmann equation M P H ∼ V (4.9)giving the energy density of the Universe during thermal inflation to obtainthe Hubble parameter during thermal inflation as H TI ∼ m n +20 √ λ M P ! n +1 (4.10)Within this model, we will consider two cases regarding the decay rateof the inflaton, Γ ϕ , with ϕ , the inflaton, being the field driving the period ofprimordial inflation prior to thermal inflation. Firstly, the case that Γ ϕ & H TI ,i.e. that reheating from primordial inflation occurs before or around the timeof the start of thermal inflation. Secondly, we will consider the case that Γ ϕ ≪ H TI , i.e. that reheating from primordial inflation occurs at some timeafter the end of thermal inflation. In the case of Γ ϕ & H TI , thermal inflationwill begin at a temperature T ∼ V (4.11) We are ignoring a cosmological constant as it is negligible. Considering it would give V = 0 at the VEV. .3 φ Decay Rate, Spectral Index and Tensor Fraction T corresponds to the temperature when the potential energy density becomescomparable with the energy density of the thermal bath, for which the densityis ρ γ ∼ T . In the case of Γ ϕ ≪ H TI , thermal inflation will begin at atemperature T ∼ (cid:16) M P H TI Γ ϕ (cid:17) (4.12)In both cases, thermal inflation ends at a temperature T = m √ g (4.13) T corresponds to the temperature when the tachyonic mass term of thethermal waterfall field becomes equal to the thermally-induced mass term inEq. (4.5). φ Decay Rate, Spectral Index and TensorFraction φ Decay Rate
The decay rate of φ is given by Γ ∼ max ( g m φ, osc , m φ, osc M P ) (4.14) .3 φ Decay Rate, Spectral Index and Tensor Fraction m φ, osc is the effective mass of φ during the time of φ ’s oscillationsaround its VEV after the end of thermal inflation. This is calculated as m φ, osc ∼ m (4.15)Therefore we obtain Γ ∼ max ( g m , m M P ) (4.16)The first expression is for decay into the thermal bath via direct interactionsand the second is for gravitational decay. We will only consider the case inwhich the direct decay is the dominant channel ( g is not taken to be verysmall). This is the case when m ≪ gM P (4.17)Therefore we have just Γ ∼ g m (4.18) n s and n ′ s Thermal Inflation has the effect of changing the number of e-folds beforethe end of primordial inflation at which cosmological scales exit the horizon.This affects the value of the spectral index n s of the curvature perturbation ζ , assuming ζ is generated due to the perturbations of the spectator scalarfield. The spectral index is given by [1] n s ≃ − ǫ H + 2 η ψψ (4.19) .3 φ Decay Rate, Spectral Index and Tensor Fraction ǫ H and η ψψ are slow-roll parameters, defined as ǫ H ≡ M P H ′ ( ϕ ) H ( ϕ ) ! (4.20)and η ψψ ≡ V ψψ H (4.21)where H ′ ( ϕ ) is the derivative of the Hubble parameter with respect to theinflaton field ϕ and V ψψ ≡ ∂ V∂ψ . ǫ H and η ψψ are to be evaluated at the pointwhere cosmological scales exit the horizon during primordial inflation. In thelimit of slow-roll inflation, which we consider to be the case for our primordialinflation period, ǫ H → ǫ , which is defined as ǫ ≡ M P V ′ ( ϕ ) V ( ϕ ) ! (4.22)where V ( ϕ ) is the inflaton potential and V ′ ( ϕ ) is the derivative of that po-tential with respect to the inflaton field ϕ . ǫ is to be evaluated at the pointwhere cosmological scales exit the horizon during primordial inflation. Thespectral index now becomes n s ≃ − ǫ + 2 η ψψ (4.23)Regarding the various scalar fields involved in this model, the reason why ǫ depends only on ϕ is because this slow-roll parameter captures the infla-tionary dynamics of primordial inflation, which is governed only by ϕ in ourmodel (we are assuming that both ψ and φ have settled to a constant value .3 φ Decay Rate, Spectral Index and Tensor Fraction η depends only on ψ is because this parametercaptures the dependance on the spectral index of the field(s) whose pertur-bations contribute to the observed primordial curvature perturbation ζ . Inour case, this is only the ψ field.The definition of the running of the spectral index is [1] n ′ s ≡ d n s d ln ( k ) (4.24) ≃ − d n s d N (4.25)the second line coming from d ln ( k ) = d ln ( aH ) ≃ H d t ≡ − d N , where k = aH .From Eq. (4.23) we have n ′ s ≃ ǫ d N − η ψψ d N (4.26) ≃ ǫ d ln ( ǫ )d N − η ψψ d N (4.27)By differentiating the natural log of ǫ with respect to N , we obtain [1] d ln ( ǫ )d N ≃ − ǫ + 2 η (4.28) .3 φ Decay Rate, Spectral Index and Tensor Fraction η is a slow-roll parameter given by η ≡ M P V ′′ ( ϕ ) V ( ϕ ) (4.29)where V ′′ ( ϕ ) is the second derivative of the inflaton potential with respectto the inflaton field ϕ . η is to be evaluated at the point where cosmologicalscales exit the horizon during primordial inflation. By differentiating η ψψ with respect to N using the quotient rule, we obtain d η ψψ d N = 3 H V ψψ d N − V ψψ d H d N H (4.30) = − V ψψ d H d N H (4.31) = − η ψψ H d H d N (4.32) = − η ψψ d ln ( H )d N (4.33)with the second line coming from the fact that we have V ψψ not dependingon N , as we are assuming that both ψ and φ have settled to a constant value(Sections 4.4.3.5 and 4.4.3.8 respectively) by the time cosmological scales exitthe horizon during primordial inflation. By differentiating the natural log of H with respect to N , we obtain [1] d ln ( H )d N ≃ ǫ (4.34) .3 φ Decay Rate, Spectral Index and Tensor Fraction d η ψψ d N ≃ − ǫη ψψ (4.35)Therefore, the final result for the running of the spectral index is n ′ s ≃ − ǫ + 4 ǫη + 4 ǫη ψψ (4.36)From now on we assume that H has the constant value H ∗ by the timecosmological scales exit the horizon up until the end of primordial inflation.In order to obtain ǫ and η , we require N ∗ , the number of e-folds before theend of primordial inflation at which cosmological scales exit the horizon. Weconsider the period between when the pivot scale, k ≡ . Mpc − , exitsthe horizon during primordial inflation and when it reenters the horizon longafter the end of thermal inflation. We have R ∗ = H − ∗ k − = H − (4.37)where R ∗ is a length scale when the pivot scale exits the horizon duringprimordial inflation. Therefore H − ∗ = a ∗ a piv H − (4.38)where a ∗ is the scale factor at the time when the pivot scale exits the horizonduring primordial inflation and a piv is the scale factor at the time when thepivot scale reenters the horizon. .3 φ Decay Rate, Spectral Index and Tensor Fraction Γ ϕ & H TI We have H − ∗ = a ∗ a end,inf a end,inf a reh,inf a reh,inf a start,TI a start,TI a end,TI a end,TI a reh,TI a reh,TI a eq a eq a piv H − (4.39) = e − N ∗ a end,inf a reh,inf a reh,inf a start,TI e − N TI a end,TI a reh,TI a reh,TI a eq a eq a piv H − (4.40)where N TI is the number of e-folds of thermal inflation and the scale factorsare the following: a end,inf is at the end of primordial inflation, a reh,inf is at pri-mordial inflation reheating, a start,TI is at the start of thermal inflation, a end,TI is at the end of thermal inflation, a reh,TI is at thermal inflation reheating and a eq is at the time of matter-radiation equality. For the period between theend of primordial/thermal inflation and primordial/thermal inflation reheat-ing, a goes as T − . The proof is as follows. During this time, T ∼ ( M P H Γ) .As H goes as t − we have T ∝ t − . During the field oscillations, the Universeis matter dominated and so we have a ∝ t . Therefore t ∝ a . Putting thisall together we find T ∝ t − ∝ a − and therefore a ∝ T − . For all other times, a goes as T − and so we have e N ∗ = H ∗ k T reh,inf T end,inf ! T start,TI T reh,inf T reh,TI T end,TI ! T piv T reh,TI e − N TI (4.41)We need to calculate T piv . We consider the period between when the pivotscale reenters the horizon and the present. Throughout this period the Uni- .3 φ Decay Rate, Spectral Index and Tensor Fraction ρ ∝ a − (4.42) ∝ T (4.43)Therefore, from the Friedmann equation, we have M P H ∝ T (4.44)This gives H H = T T (4.45) T piv = . Mpc − T H ! (4.46) = 9 . × − GeV (4 s.f.) (4.47)We now obtain N ∗ as N ∗ ≈ ln (1 . × GeV − ) H ∗ . ! + 23 ln ϕ π H ∗ ! + 14 ln π V M P H ∗ ! + 23 ln π H TI ! + 14 ln π (9 . × − GeV ) M P Γ ! − N TI (4.48)where we have used g ∗ ≈ as the number of spin states (effective relativisticdegrees of freedom) of all of the particles in the thermal bath, at the time of .3 φ Decay Rate, Spectral Index and Tensor Fraction Γ ϕ ≪ H TI We have H − ∗ = a ∗ a end,inf a end,inf a start,TI a start,TI a end,TI a end,TI a reh,TI a reh,TI a eq a eq a piv H − (4.49) = e − N ∗ a end,inf a start,TI e − N TI a end,TI a reh,TI a reh,TI a eq a eq a piv H − (4.50)Using a ∝ T − for the period between the end of primordial inflation andthe start of thermal inflation, as well as for the period between the end ofthermal inflation and thermal inflation reheating and using a ∝ T − for allother times, we have e N ∗ = H ∗ k T start,TI T end,inf ! T reh,TI T end,TI ! T piv T reh,TI e − N TI (4.51)Using T piv = 9 . × − GeV (Eq. (4.47)), we obtain N ∗ as N ∗ ≈ ln (1 . × GeV − ) H ∗ . ! + 23 ln π H ∗ ! + 14 ln π (9 . × − GeV ) M P Γ ! − N TI (4.52)where we have used g ∗ ≈ as the number of spin states (effective relativisticdegrees of freedom) of all of the particles in the thermal bath at the time .4 “End of Inflation” Mechanism r The definition of the Tensor Fraction, r [1], is r ≡ P h P ζ (4.53)where P h and P ζ are the spectrums of the primordial tensor and curvatureperturbations respectively. The spectrum P h is given by P h ( k ) = 8 M P (cid:18) H k π (cid:19) (4.54)for a given wavenumber k . Using this, together with ρ ∗ = 3 M P H ∗ , given thatwe are saying H k = H ∗ for our current case, as well as the observed value P ζ ( k ) = 2 . × − , we obtain r = ρ ∗ . × GeV (4.55) In this section, we investigate the “end of inflation” mechanism. We aim toobtain a number of constraints on the model parameters and the initial con-ditions for the fields. Considering these constraints, we intend to determinethe available parameter space (if any). In this parameter space we will cal- .4 “End of Inflation” Mechanism ζ As φ is coupled to ψ , the “end of inflation” mechanism will generate a con-tribution to the primordial curvature perturbation ζ [31]. We will use the δN formalism to calculate this contribution. In this formalism, the finalcoordinate slice is the transition slice, going from thermal inflation to fieldoscillation. The δN formalism allows us to calculate the primordial curvatureperturbation as ζ = δN TI = d N TI d m δm + 12! d N TI d m δm + 13! d N TI d m δm + ... (4.56)The number of e-folds between the start and end of thermal inflation is givenby N TI = ln (cid:18) a a (cid:19) = ln (cid:18) T T (cid:19) (4.57)where a = a start,TI and a = a end,TI . Γ ϕ & H TI Substituting T and T , Eqs. (4.11) and (4.13) respectively, into Eq. (4.57)gives N TI ≃ ln √ gV m (4.58) .4 “End of Inflation” Mechanism δN formalism to third order gives ζ = δN TI = − δmm + 12 δm m − δm m (4.59)By substituting our mass definition and its differential, Eqs. (4.3) and (4.4),into Eq. (4.59) we obtain the power spectrum of the primordial curvatureperturbation, which to first order is P ζ = αh H ∗ ψ α − πm M α − P (4.60)It must be noted that although there will be perturbations in ψ that aregenerated during thermal inflation that will become classical due to the in-flation, the scales to which these correspond are much smaller than cosmo-logical scales, as thermal inflation lasts for only about 10 e-folds. Thereforewe do not consider them here.A required condition for the perturbative expansion in Eq. (4.59) to besuitable is that each term is much smaller than the preceding one. Thisrequirement gives h H ∗ ψ α − m M α − P ≪ (4.61) Γ ϕ ≪ H TI Substituting T and T , Eqs. (4.12) and (4.13) respectively, into Eq. (4.57)gives N TI ≃
83 ln √ g ( M P H TI Γ ϕ ) m (4.62) .4 “End of Inflation” Mechanism δN formalism to third order gives ζ = δN TI = − δmm + 43 δm m − δm m (4.63)By substituting our mass definition and its differential, Eqs. (4.3) and (4.4),into Eq. (4.63) we obtain the power spectrum of the primordial curvatureperturbation, which to first order is P ζ = 83 αh H ∗ ψ α − πm M α − P (4.64)The condition for the perturbative expansion in Eq. (4.63) to be suitable,i.e. that each term is much smaller than the preceding one, yields the sameconstraint as in Eq. (4.61). One of the distinct observational signatures that we hope to generate throughthis model is the production of characteristic and observable non-Gaussianityin the curvature perturbation. Non-Gaussianity refers to the departure thatthe distribution of, in this particular case, the curvature perturbation is frompurely Gaussian, i.e. of the familiar bell-shaped distribution. For a purelyGaussian distribution, there is no correlation between different modes of theperturbation. For a non-Gaussian distribution however, there is correlation,with the 3-point correlator for the curvature perturbation being [1] h ζ k ζ k ζ k i = (2 π ) δ k + k + k B ζ ( k , k , k ) (4.65) .4 “End of Inflation” Mechanism B ζ ( k , k , k ) is a function called the Bispectrum , being given by B ζ ( k , k , k ) = 65 f NL ( k , k , k ) [ P ζ ( k ) P ζ ( k ) + P ζ ( k ) P ζ ( k ) + P ζ ( k ) P ζ ( k )] (4.66)where f NL ( k , k , k ) effectively parameterises the bispectrum (it is thevalue of the reduced bispectrum) and P ζ ( k ) is the spectrum of the curvatureperturbation (the spectrum that is being used in this thesis is that definedby P ζ ( k ) ≡ k π P ζ ( k ) ).The 4-point (connected) correlator for the curvature perturbation is h ζ k ζ k ζ k ζ k i = (2 π ) δ k + k + k + k T ζ (4.67)where T ζ is a function called the Trispectrum , being given by [51] T ζ ( k , k , k , k ) = τ NL [ P ζ ( k ) P ζ ( k ) P ζ ( k ) + 11 perms. ]+ 5425 g NL [ P ζ ( k ) P ζ ( k ) P ζ ( k ) + 3 perms. ] (4.68)where k ≡ | k + k | and τ NL and g NL effectively parameterise the trispec-trum.We will consider what is termed local non-Gaussianity, which for thebispectrum corresponds to the “squeezed” configuration of the momentatriangle, in that the magnitude of one of the momentum vectors is muchsmaller than the other two, which are of similar magnitude to each other,e.g. k ≪ k , k and k ≈ k . Within the framework of the δN formalism, the .4 “End of Inflation” Mechanism f NL is obtained as [51] f NL = 56 N ′′ N ′ (4.69)where the prime denotes the derivative with respect to ψ . By substituting N TI from Eq. (4.58) or Eq. (4.62) into Eq. (4.69) we obtain f NL ≃ A − mm ′′ m ′ ! (4.70)where A = for Γ ϕ & H TI (Eq. (4.58)) or A = for Γ ϕ ≪ H TI (Eq. (4.62)).Then, from our mass definition m , Eq. (4.3), we obtain f NL ≃ A " α − α m M α − P h ψ α − ! (4.71)The non-Gaussianity parameter g NL is obtained as [51] g NL = 2554 N ′′′ N ′ (4.72)By substituting N TI from Eq. (4.58) or Eq. (4.62) into Eq. (4.72) we obtain g NL ≃ B − mm ′′ m ′ + m m ′′′ m ′ ! (4.73) .4 “End of Inflation” Mechanism B = for Γ ϕ & H TI (Eq. (4.58)) or B = for Γ ϕ ≪ H TI (Eq. (4.62)).Then, our m from Eq. (4.3) gives g NL ≃ B " α − α m M α − P h ψ α − ! + (2 α − α ) ( α − α m M α − P h ψ α − m M α − P h ψ α + 1 ! (4.74)In the parameter space available (if any), we will investigate the range ofvalues for f NL and g NL . We want the energy scale of primordial inflation to be V . GeV (4.75)so that the inflaton contribution to the curvature perturbation is negligible.Therefore, from the Friedmann equation M P H ∗ = V we require H ∗ . GeV (4.76)
We will consider only the case in which the inflationary trajectory is 1-dimensional, in that only the φ field is involved in determining the trajectoryof thermal inflation in field space. We do this only to work with the simplest .4 “End of Inflation” Mechanism ψ field does not affect the inflationary trajectory during thermalinflation, we require from our m mass definition, Eq. (4.3), m ≫ h ψ α M α − P (4.77)Therefore we have m ≃ m (4.78)From our potential, Eq. (4.2), Eq. (4.77) gives m < g T (4.79)For Γ ϕ & H TI , substituting T from Eq. (4.11) into Eq. (4.79) gives m < g n +2 √ λ ! n M P (4.80)and for Γ ϕ ≪ H TI , substituting T from Eq. (4.12) into Eq. (4.79) gives m < "(cid:16) g Γ ϕ (cid:17) n +1 M n +1 P √ λ n +2 (4.81) φ Particles
Given that we have not observed any φ particles, the most liberal constrainton the present value of the effective mass of φ is m φ, now & TeV (4.82) .4 “End of Inflation” Mechanism m φ, now ∼ − m + (2 n + 4)(2 n + 3) λ h φ i n +2 M nP (4.83)Substituting the VEV of φ , Eq. (4.7), into here gives m φ, now ∼ m (4.84)for all reasonable values of n . Therefore, we require m & TeV (4.85) ψ In order that ψ acquires classical perturbations during primordial inflation,we require the effective mass of ψ to be light during this time, i.e. | m ψ, eff | ≪ H ∗ (4.86)where we are using notation such that | m ψ, eff | ≡ r(cid:12)(cid:12)(cid:12) m ψ, eff (cid:12)(cid:12)(cid:12) . We have m ψ, eff = m ψ + (cid:16) α − α (cid:17) h φ ψM P ! α − (4.87)Therefore we require m ψ ≪ H ∗ (4.88) .4 “End of Inflation” Mechanism hφ ∗ ψ ∗ M P ! α − ≪ H ∗ (4.89)where φ ∗ and ψ ∗ are the values of φ and ψ during primordial inflation re-spectively.We require that ψ remains at ψ ∗ , the value during primordial inflation,all the way up to the end of thermal inflation. The reason for this is that if ψ starts to move, then its perturbation will decrease. This is because ψ un-freezes when the Hubble parameter becomes less than ψ ’s mass, i.e. H < m ψ .In this case, the perturbation of ψ also unfreezes, because it has the samemass as ψ . The density of the oscillating ψ field decreases as matter, so m ψ ψ ∝ a − . Therefore, ψ ∝ a − . The same is true for the perturbation,i.e. δψ ∝ a − . This means that the perturbation decreases exponentially (as a ∝ e Ht ) and so the whole effect of perturbing the end of thermal inflation isdiminished. Requiring that ψ is light at all times up until the end of thermalinflation is sufficient to ensure that the field and its perturbation remain at ψ ∗ and δψ ∗ respectively. Therefore we require m ψ ≪ H TI (4.90)which is of course stronger than requiring just m ψ ≪ H ∗ , Eq. (4.88).Given that we have not observed any ψ particles, the most liberal con-straint on the present value of the effective mass of ψ is m ψ, now & TeV (4.91) .4 “End of Inflation” Mechanism ψ ∗ Substituting the observed spectrum value P ζ ( k ) = 2 . × − into bothEq. (4.60) and Eq. (4.64), with α ∼ , gives the same constraint, which is ψ ∗ ∼ − m M α − P h H ∗ ! α − (4.92)This constraint automatically satisfies the requirement of a suitable perturba-tive expansion, Eq. (4.61). Substituting Eq. (4.92) into Eq. (4.77), regardingthe dynamics of thermal inflation, gives h ≫ − H ∗ ! α m M α − P (4.93)Rearranging this for m gives the constraint m ≪ (cid:16) H ∗ (cid:17) α hM α − P (4.94)We require the field value of ψ to be much larger than its perturbation,i.e. ψ ∗ ≫ δψ ∗ , so that the perturbative approach is valid. Therefore weobtain, with δψ ∗ ∼ H ∗ , ψ ∗ ≫ H ∗ (4.95)and δψ ∗ ψ ∗ ≪ (4.96) .4 “End of Inflation” Mechanism ψ ∗ , Eq. (4.92), with ψ ∗ ≫ δψ ∗ and δψ α − ∗ ∼ H α − ∗ gives m ≫ h H α ∗ M α − P (4.97) φ The effective mass of φ at the end of primordial inflation is m φ, end,inf ∼ g T − m (4.98)We have gT end,inf ≫ m , which therefore gives m φ, end,inf ∼ g T end,inf (4.99)The proof is as follows. Instead of considering the time of the end of primor-dial inflation, we consider the time at the start of thermal inflation, which isof course a time at a lower temperature. For the case Γ ϕ & H TI , we would besaying that gT ≫ m (4.100) gV ≫ m (4.101)Substituting V , Eq. (4.8), into Eq. (4.101) gives m ≪ g n +2 √ λ ! n M P (4.102) .4 “End of Inflation” Mechanism Γ ϕ ≪ H TI , we would be saying that gT ≫ m (4.103) g (cid:16) M P H TI Γ ϕ (cid:17) ≫ m (4.104)Substituting H TI , Eq. (4.10), into Eq. (4.104) gives m ≪ "(cid:16) g Γ ϕ (cid:17) n +1 M n +1 P √ λ n +2 (4.105)which is identical to Eq. (4.81) except for the difference in the limit.As we are dealing with the thermal fluctuation of φ about φ = 0 , we have h δφ i T = h φ i T . The thermal fluctuation of φ is q h φ i T ∼ T (4.106)and we require g ≪ (4.107)A detailed derivation of this is given in Appendix A.In order to keep m ψ, eff light, we require, from Section 4.4.3.4, hT ψ ∗ M P ! α − ≪ H TI (4.108)During the time between the end of primordial inflation and primordial in-flation reheating, T ∝ a − and H ∝ a − and during radiation domination, .4 “End of Inflation” Mechanism T ∝ a − and H ∝ a − . Therefore, if Eq. (4.108) is satisfied, then equivalentconstraints for higher T and H are guaranteed to be satisfied as well. In thecase of Γ ϕ & H TI , by substituting Eqs. (4.8), (4.10), (4.11) and (4.92) intoEq. (4.108) we obtain the constraint h ≪ λ α − n +4 (cid:18) m M P (cid:19) (2 α − n +2)2 n +2 √ H ∗ M P m ! α − (4.109)Rearranging this for m gives the following. For α = 1 and any value of n , orfor α = 2 and n = 1 we have m ≫ (10 H ∗ ) (2 α − n +1) h n +2 λ α − M α + nP αn − n − (4.110)For all other α and n combinations, the above inequality is reversed, giving m ≪ (10 H ∗ ) (2 α − n +1) h n +2 λ α − M α + nP αn − n − (4.111)In the case of Γ ϕ ≪ H TI , by substituting Eqs. (4.10), (4.12) and (4.92) intoEq. (4.108) we obtain the constraint h ≪ m n +20 √ λ M P ! α − n +4 (cid:16) M P q Γ ϕ (cid:17) α − √ H ∗ M P m ! α − (4.112)Rearranging this for m gives the following. For the values of n and α givenin Table 4.1 we have m ≫ (10 H ∗ M P ) (4 α − n +1) h n +4 (cid:16) √ λ M P (cid:17) α − ( M P Γ ϕ ) (2 α − n +1) αn − α − n − (4.113) .4 “End of Inflation” Mechanism n and α values, the above inequality is reversed, giving n α – ∞ All –
84 1 –
55 1 – –
13 1 – – ∞ , Table 4.1: Values for which Eq. (4.113) applies. m ≪ (10 H ∗ M P ) (4 α − n +1) h n +4 (cid:16) √ λ M P (cid:17) α − ( M P Γ ϕ ) (2 α − n +1) αn − α − n − (4.114) φ In order that φ interacts with the thermal bath and therefore that we actuallyhave the g T φ term in our potential, Eq. (4.2), we require Γ therm > H (4.115)where Γ therm is the thermalization rate of φ , which is given by Γ therm = n h σv i (4.116) ∼ σ T (4.117)where n ∼ T is the number density of particles in the thermal bath, σ isthe scattering cross-section for the interaction of φ and the particles in the .4 “End of Inflation” Mechanism v is the relative velocity between a φ particle and a thermalbath particle (which in our case is ≈ c = 1 ) and h i denotes a thermal average.The scattering cross-section σ is given by σ ∼ g E (4.118)where E c.m. is the centre-of-mass energy, which is E c.m. ∼ T (4.119)Substituting Eq. (4.119) into Eq. (4.118) gives σ ∼ g T (4.120)This scattering cross-section is the total cross-section for all types of scat-tering (e.g. elastic) that can take place between φ and the particles in thethermal bath. For a complete Field Theory derivation of the elastic scat-tering cross-section between φ and the thermal bath, see Appendix B. Thethermalization rate now becomes Γ therm ∼ g T (4.121)During the time between the end of primordial inflation and primordial in-flation reheating, T ∝ a − and H ∝ a − and during radiation domination T ∝ a − and H ∝ a − . Therefore, if the constraint Γ therm > H is satisfied atthe time of the end of primordial inflation, then it is satisfied all the way up .4 “End of Inflation” Mechanism Γ therm > H ∗ (4.122)Taking Eq. (4.121) with T ∼ ( M P H ∗ Γ ϕ ) gives Γ ϕ > H ∗ g M P (4.123)We also require Γ therm > H to be satisfied throughout the whole of thermalinflation. Therefore, we have the constraint g T > H TI (4.124)Substituting H TI and T , Eqs. (4.10) and (4.13) respectively, into the abovegives m < g √ ! n +1 √ λ M P (4.125) φ ∗ We consider three possible cases for the value of the thermal waterfall field φ during primordial inflation, with m φ, inf being the effective mass of φ duringprimordial inflation:A) φ heavy, i.e. | m φ, inf | ≫ H ∗ , in which φ rolls down to its VEV.B) φ light, i.e. | m φ, inf | ≪ H ∗ , in which φ is at the Bunch-Davies value (tobe explained below).C) φ light, in which a SUGRA correction to the potential is appreciable, .4 “End of Inflation” Mechanism φ rolling down to φ = 0 . Case A
Substituting h φ i and ψ ∗ , Eqs. (4.7) and (4.92) respectively, into Eq. (4.89)gives h ≪ H α − ∗ q (2 n + 4) λm M nP α − n +1 M P m ! α − (4.126)Rearranging this for m gives m ≪ q (2 n + 4) λM nP α − (10 M P ) α − H α − ∗ h ! n +1 αn +4 α − n − (4.127) Case B
We consider φ to be at the Bunch-Davies value φ BD ∼ M nP H ∗ √ λ ! n +2 (4.128)corresponding to the Bunch-Davies vacuum [45], which is the unique quan-tum state that corresponds to the vacuum, i.e. no particle quanta, in theinfinite past in conformal time in a de Sitter spacetime. φ BD is of this form as λ φ n +4 M nP ∼ H ∗ , this being because the probability of this Bunch-Davies stateis proportional to the factor e − VH . Substituting ψ ∗ and φ BD , Eqs. (4.92)and (4.128) respectively, into Eq. (4.89) gives h ≪ H α − ∗ √ M P m ! α − √ λM nP H ∗ ! α − n +2 (4.129) .4 “End of Inflation” Mechanism m gives m ≪ q M P H α − ∗ h √ λM nP H ∗ ! α − n +2 α − (4.130) Case C If φ is light during primordial inflation, i.e. | m φ, inf | ≪ H ∗ , our potential,Eq. (4.2), can receive an appreciable SUGRA correction during primordialinflation of [52–54] ∆ V ∼ c ∗ H ∗ φ (4.131)where c ∗ is a coupling constant. The SUGRA correction is appreciable onlyfrom the time of primordial inflation up until primordial inflation reheating,as it is suppressed at all times after this [54]. As the scale factor a grows(almost) exponentially during inflation, φ is driven rapidly to , i.e. we have φ ∗ = 0 . Therefore the effective mass of φ during primordial inflation is m φ, inf ∼ − m + c ∗ H ∗ (4.132)In order to keep this light we therefore need m ≪ H ∗ (4.133)and c ∗ < (4.134) .4 “End of Inflation” Mechanism We require the energy density of φ to be subdominant at all times, in orderthat it does not cause any inflation by itself. During the period betweenthe end of primordial inflation and the start of thermal inflation, the energydensity of φ is ρ φ ∼ g T φ (4.135) ∼ g T (4.136)the second line coming from the thermal fluctuation of φ , which is ∼ T .Therefore, considering the Friedmann equation, we require g T ≪ M P H TI (4.137)During the time between the end of primordial inflation and primordial in-flation reheating, T ∝ a − and H ∝ a − and during radiation domination T ∝ a − and H ∝ a − . Therefore, if Eq. (4.137) is satisfied, then equiva-lent constraints for higher T and H are guaranteed to be satisfied as well.For the case Γ ϕ & H TI , by substituting H TI and T , Eqs. (4.10) and (4.11)respectively, into Eq. (4.137) we obtain g ≪ (4.138)which is the same constraint as Eq. (4.107). For the case Γ ϕ ≪ H TI , bysubstituting H TI and T , Eqs. (4.10) and (4.12) respectively, into Eq. (4.137) .4 “End of Inflation” Mechanism m ≫ (cid:20)(cid:16) g Γ ϕ (cid:17) n +1 √ λ M P (cid:21) n +2 (4.139) φ ∗ Case A
The energy density of φ during primordial inflation is ρ φ, inf = − m + h ψ α ∗ M α − P ! h φ i + λ h φ i n +4 M nP (4.140) ∼ − m h φ i + λ h φ i n +4 M nP (4.141)with the second line coming from Eq. (4.77) regarding the dynamics of ther-mal inflation. Therefore, with the energy density of the Universe being ∼ M P H ∗ , we require m h φ i ≪ M P H ∗ (4.142)and √ λ h φ i n +2 ≪ M n +1 P H ∗ (4.143)Substituting h φ i , Eq. (4.7), into Eq. (4.142) gives the same constraint as fromsubstituting h φ i into Eq. (4.143). This constraint is m ≪ (cid:16) √ λ M P H n +1 ∗ (cid:17) n +2 (4.144)However, for all viable parameter values in our model, this constraint is neverthe dominant constraint when we consider it alongside all of the other con- .4 “End of Inflation” Mechanism φ ∗ Case B
The energy density of φ during primordial inflation is ρ φ, inf = − m + h ψ α ∗ M α − P ! φ + λ φ n +4BD M nP (4.145) ∼ − m φ + λ φ n +4BD M nP (4.146)with the second line coming from Eq. (4.77) regarding the dynamics of ther-mal inflation. Therefore, with the energy density of the Universe being ∼ M P H ∗ , we require m φ BD ≪ M P H ∗ (4.147)and √ λ φ n +2BD ≪ M n +1 P H ∗ (4.148)Substituting φ BD , Eq. (4.128), into Eq. (4.147) gives m ≪ (cid:16) √ λ M P H n ∗ (cid:17) n +2 (4.149)and substituting φ BD into Eq. (4.148) gives just H ∗ ≪ M P (4.150) .4 “End of Inflation” Mechanism φ ∗ Case C
We do not have any additional constraints here for this case, as φ = 0 . In order for the equations of the δN formalism that are derived within thecontext of the “end of inflation” mechanism to be valid, we require the tran-sition from thermal inflation to thermal waterfall field oscillation to be suf-ficiently fast [34]. More specifically, we require ∆ t ≪ δt → (4.151)where ∆ t ≡ t − t is the time taken for the transition to occur and δt → is the proper time between a uniform energy density spacetime slice justbefore the transition at t and one just after the transition at t when φ starts to oscillate around its VEV. Qualitatively, we require the thicknessof the transition slice to be much smaller than its warping. The primordialcurvature perturbation that is generated by the “end of inflation” mechanismis given by ζ = H TI δt → (4.152) .4 “End of Inflation” Mechanism ζ ≫ H TI ∆ t (4.153)To calculate φ and φ , the value of φ at times t and t respectively,we use the fact that the process is so rapid that it takes place in less thana Hubble time, so that the Universe expansion can be ignored. Then theequation of motion is ¨ φ + ∂V∂φ = 0 (4.154)At the end of thermal inflation, φ is not centered on the origin, but has startedto roll down the potential slightly. At this time, g T is much smaller than m . Therefore we have ∂V∂φ ∼ − m φ (4.155)So we have the equation of motion ¨ φ ∼ m φ (4.156)The solution is φ ∼ Ae m t (4.157) .4 “End of Inflation” Mechanism A is a constant and we are keeping only the growing mode. Thereforewe have ln φ φ ! ∼ m ( t − t ) (4.158) ∼ m ∆ t (4.159)We know that φ ∼ T ∼ m (4.160)and φ ∼ h φ i (4.161)Therefore we have ln q (2 n + 4) λ (cid:18) M P m (cid:19) n n +1 ∼ m ∆ t (4.162)For all values of n , λ and m , we have ∆ t ≥ m − . Therefore, from Eq. (4.153)we have ζ ≫ H TI m (4.163)Given that ζ ∼ − , we require H TI ≪ − m (4.164) .4 “End of Inflation” Mechanism m ψ ≪ H TI , Eq. (4.90). This gives m ψ ≪ − m (4.165)A further constraint is obtained by substituting H T I , Eq. (4.10), intoEq. (4.164). We obtain m ≪ − n − √ λ M P (4.166) As ψ has acquired perturbations from primordial inflation, we require it notto dominant the energy density of the Universe after the end of thermal infla-tion when it is oscillating, at which time the effective mass of ψ is increasedsignificantly due to the coupling of ψ to φ . This is so as not to allow ψ toact as a curvaton, i.e. not to allow ψ ’s perturbations to generate a dominantcontribution to the primordial curvature perturbation when ψ decays. Thereason for this is just so that we do not have a curvaton inflation scenario, asthe perturbations that are generated via the modulated mass in our modelthat could give the dominant contribution to ζ would be negligible.The energy density of the oscillating ψ field after the end of thermal .4 “End of Inflation” Mechanism ρ ψ, osc = h ψ α M α − P φ + 12 m ψ ψ (4.167) ∼ h ψ α ∗ M α − P h φ i + 12 m ψ ψ ∗ (4.168)For simplicity, we assume that ψ decays around the same time as φ , i.e. that H does not change much between the time when φ decays and the time when ψ decays. Therefore, the energy density of the Universe at the time when ψ decays is ∼ M P Γ . We therefore require ρ ψ, osc ≪ M P Γ (4.169) h ψ α ∗ M α − P h φ i + 12 m ψ ψ ∗ ≪ M P Γ (4.170)Therefore we require m ψ ≪ M P Γ ψ ∗ (4.171)and h h φ i ψ α ∗ ≪ M αP Γ (4.172)Substituting h φ i , Γ and ψ ∗ , Eqs. (4.7), (4.18) and (4.92) respectively, intoEq. (4.172) gives the constraint h ≫ g m ) α − m M nP q (2 n + 4) λ α − n +1 m √ H ∗ M P ! α (4.173) .4 “End of Inflation” Mechanism In this section, we combine the above constraints to find out the allowedparameter space (if any). α = 1 Even with α set to the value α = 1 , we still have six free parameters inthe model. Therefore, the parameter space is very multi-dimensional andthe number of allowed regions is potentially very vast. Given this, we showresults only for one such allowed region of parameter space that we have ex-plored. Additionally, we also only show results for the case where Γ ϕ ≪ H TI ,in that reheating from primordial inflation occurs at some time after the endof thermal inflation, as this scenario was found to yield more parameter spacethan the case where Γ ϕ & H TI .From Eq. (4.107) we require g ≪ . We also require the constraint givenby Eq. (4.123) to be satisfied, where g is present as g − . Therefore, thislatter constraint will start to become very strong very quickly as we decrease g . We find that a value of g = 0 . yields allowed parameter space, for reason-able values of H ∗ and Γ ϕ . The parameter space that we find here however,when all constraints are considered together and regardless of the φ ∗ case, isactually a sharp prediction of single values for all but one of the free param-eters and the other quantities in the model, to within an order of magnitude,rather than a range of parameter space. The values of the free parametersare displayed in Table 4.2.The allowed range of values for ψ ∗ , the contrast δψ ∗ ψ ∗ and H T I are dis- .4 “End of Inflation” Mechanism n g . H ∗ GeV Γ ϕ × − GeV λ − h − Table 4.2: Values of the free parameters for which parameter space exists,for α = 1 and Γ ϕ ≪ H TI .played in Figs. 4.3–4.5 respectively. Fig. 4.6 shows the allowed parameterspace for m ψ . Within the range m ∼ – GeV, the mass m ψ can spanmany orders of magnitude, with only an upper limit of ∼ − – − GeV.Within the model, there is no effective lower bound on m ψ . Finally, Fig. 4.7shows the allowed parameter space for h , for a value of h = 10 − . Fromlooking at this plot, it may initially seem as if there is no allowed parameterspace available for our given value of h . However, as we are working withinan order of magnitude for each value, we can see that all of the visible con-straints do allow for our given value of h , for a thermal waterfall field massvalue of m ∼ GeV.Values of other quantities in the model for a mass value of m ∼ GeVand the parameter values of Table 4.2 are shown in Table 4.3. In this tablewe include the tensor fraction, which for a value of H ∗ ∼ GeV yields thenegligible value r ∼ − .Figs. 4.8 and 4.9 show the prediction of the model for the non-Gaussianityparameters f NL and g NL respectively, with h and H ∗ values from Table 4.2,together with the central value and range for the parameters as obtained by .4 “End of Inflation” Mechanism m ( GeV ) * ( GeV ) Fig. 4.3: The allowed parameter space for ψ ∗ , with α = 1 , Γ ϕ ≪ H T I and theparameter values of Table 4.2. The constraints on m that are shown are thefollowing: Green: Eq. (4.166) Blue: Eq. (4.94) Purple: Eq. (4.114) m ( GeV ) - - - δψ * ψ * Fig. 4.4: The allowed parameter space for δψ ∗ ψ ∗ , with α = 1 , Γ ϕ ≪ H T I and theparameter values of Table 4.2. The constraints on m that are shown are thefollowing: Green: Eq. (4.166) Blue: Eq. (4.94) Purple: Eq. (4.114)the Planck spacecraft [55]. From looking at Fig. 4.9, it may initially seemas if our predicted value for g NL for a thermal waterfall field mass value of .4 “End of Inflation” Mechanism m ( GeV ) - - - H TI ( GeV ) Fig. 4.5: The allowed parameter space for H T I , with α = 1 , Γ ϕ ≪ H T I andthe parameter values of Table 4.2. The constraints on m that are shown arethe following: Green: Eq. (4.166) Blue: Eq. (4.94) Purple: Eq. (4.114) m ( GeV ) - - - - - m ψ ( GeV ) Fig. 4.6: The allowed parameter space for m ψ , with α = 1 , Γ ϕ ≪ H T I andthe parameter values of Table 4.2. The upper bound given by the Red lineis that of Eq. (4.90) and the upper bound given by the Brown line is that ofEq. (4.165). (We choose to display only down to − GeV, i.e. this value isnot a lower bound.) The constraints on m that are shown are the following:Green: Eq. (4.166) Blue: Eq. (4.94) Purple: Eq. (4.114) .4 “End of Inflation” Mechanism m ( GeV ) - - - - (cid:0) h Fig. 4.7: The allowed parameter space for h , with α = 1 , Γ ϕ ≪ H T I and theparameter values of Table 4.2. The Black line is h = 10 − , the Red line isthe lower bound given by Eq. (4.93) and the Brown line is the upper boundgiven by Eq. (4.112). The constraints on m that are shown are the following:Green: Eq. (4.166) Blue: Eq. (4.94) Purple: Eq. (4.114)Quantity Order of Magnitude Value ψ ∗ GeV δψ ∗ ψ ∗ − H T I − GeV h φ i GeV V GeV T GeV T GeV
Γ 10 GeV r − Table 4.3: Values of quantities in the model for α = 1 , Γ ϕ ≪ H TI , m ∼ GeVand the parameter values of Table 4.2. m ∼ GeV is ruled-out. However, as we are working within an order ofmagnitude for each value, we can see that our predicted value, i.e. the Black .4 “End of Inflation” Mechanism m ∼ GeV. Thevalues of f NL and g NL for a thermal waterfall field mass of m ∼ GeVare shown in Table 4.4, with them both being within current observationalbounds [55]. m ( GeV ) - - f NL Fig. 4.8: Prediction of the model for the non-Gaussianity parameter f NL ,with α = 1 , Γ ϕ ≪ H T I and h and H ∗ values from Table 4.2. (A plot ofEq. (4.71), with ψ = ψ ∗ .) The Blue and Red lines are the central value andupper bound of f NL respectively as obtained by the Planck spacecraft [55],with the lower bound being outside the displayed range of f NL .Parameter Order of Magnitude Value f NL − g NL − Table 4.4: Prediction for non-Gaussianity parameters of the model, with α = 1 , Γ ϕ ≪ H TI , m ∼ GeV and h and H ∗ values from Table 4.2. n s and n ′ s : Chaotic Inflation We provide results for the spectral index and its running when the period of .4 “End of Inflation” Mechanism m ( GeV ) - - - g NL Fig. 4.9: Prediction of the model for the non-Gaussianity parameter g NL , with α = 1 , Γ ϕ ≪ H T I and h and H ∗ values from Table 4.2. (A plot of Eq. (4.74),with ψ = ψ ∗ .) The Blue and Red lines are the central value and lower/upperbounds of g NL respectively as obtained by the Planck spacecraft [55].primordial inflation is that of slow-roll Chaotic Inflation, with the potential V ( ϕ ) = 12 m ϕ ϕ (4.174)From Section 4.3.2, the spectral index n s is given by n s ≃ − ǫ + 2 η ψψ (4.175)with ǫ being given by Eq. (4.22) and η ψψ being given by Eq. (4.21) andwhere both are to be evaluated at the point where cosmological scales exitthe horizon during primordial inflation. The potential of Eq. (4.174) gives ǫ = 2 M P ϕ ∗ (4.176) .4 “End of Inflation” Mechanism ϕ ∗ in terms of N ∗ by using the equation N ∗ ≈ M P Z ϕ ∗ ϕ end V ( ϕ ) V ′ ( ϕ ) d ϕ (4.177)We define the end of primordial inflation to be when ǫ = 1 . This gives ϕ end = √ M P (4.178)Therefore we have N ∗ ≈ M P Z ϕ ∗ √ M P V ( ϕ ) V ′ ( ϕ ) d ϕ (4.179) ≈ M P Z ϕ ∗ √ M P ϕ d ϕ (4.180) ϕ ∗ ≈ q N ∗ + 2 M P (4.181)Substituting Eq. (4.181) into Eq. (4.176) gives ǫ ≈ N ∗ + 1 (4.182)We also need to calculate η ψψ . Using our potential, Eq. (4.2), we obtain V ψψ at the time cosmological scales exit the horizon as V ψψ | ∗ = m ψ + (cid:16) α − α (cid:17) h φ ∗ ψ ∗ M P ! α − (4.183) .4 “End of Inflation” Mechanism η ψψ as η ψψ = 13 H ∗ m ψ + (cid:16) α − α (cid:17) h φ ∗ ψ ∗ M P ! α − (4.184)Our final result for the spectral index is therefore n s ≈ − N ∗ + 1 + 23 H ∗ m ψ + (cid:16) α − α (cid:17) h φ ∗ ψ ∗ M P ! α − (4.185)From Section 4.3.2, the running of the spectral index n ′ s is given by n ′ s ≃ − ǫ + 4 ǫη + 4 ǫη ψψ (4.186)with η being given by Eq. (4.29), which is to be evaluated at the pointwhere cosmological scales exit the horizon during primordial inflation. Thepotential of Eq. (4.174) gives η = 2 M P ϕ ∗ (4.187)which is identical to the value of ǫ , Eq. (4.176). Substituting Eq. (4.181) intoEq. (4.187) gives η ≈ N ∗ + 1 (4.188)Our final result for the running of the spectral index is therefore n ′ s ≈ − N ∗ + 1) + 4(6 N ∗ + 3) H ∗ m ψ + (cid:16) α − α (cid:17) h φ ∗ ψ ∗ M P ! α − (4.189) .4 “End of Inflation” Mechanism n s and n ′ s , we first need to obtain N ∗ . The predictionof the model for N TI and N ∗ are shown in Figs. 4.10 and 4.11 respectively,with n , g , H ∗ , Γ ϕ and λ values from Table 4.2. The kink that is visible in theplot of N ∗ at around m ≈ GeV is a result of the fact that for m valueslarger than this, we do not have any period of thermal inflation, as can beseen in the plot of N TI . The values of N TI and N ∗ for a thermal waterfallfield mass of m ∼ GeV are shown in Table 4.5. (cid:1) (cid:2) m ( GeV ) N TI Fig. 4.10: Prediction of the model for N TI , with Γ ϕ ≪ H T I and n , g , Γ ϕ and λ values from Table 4.2. (A plot of Eq. (4.62), with m = m .)Parameter Value N TI ≈ N ∗ ≈ Table 4.5: Prediction for N TI and N ∗ of the model, with Γ ϕ ≪ H TI , m ∼ GeV and n , g , H ∗ , Γ ϕ and λ values from Table 4.2.The prediction of the model for n s and n ′ s for each φ ∗ case and for aspectator field mass at the upper bound of m ψ = 10 − GeV are shown in .4 “End of Inflation” Mechanism (cid:3) (cid:4) m ( GeV ) N * Fig. 4.11: Prediction of the model for N ∗ , with Γ ϕ ≪ H T I and n , g , H ∗ , Γ ϕ and λ values from Table 4.2. (A plot of Eq. (4.52), with m = m and Γ = g m .)Figs. 4.12–4.17, with the parameter values of Table 4.2. The predicted valuesof n s and n ′ s of the model for a thermal waterfall field mass of m ∼ GeVfor all three φ ∗ Cases are the same to within at least four significant figures.They are also both insensitive to the value of m ψ within its allowed range. n s and n ′ s are shown in Table 4.6, with them both being within currentobservational bounds [40].Quantity Value n s ≈ . n ′ s ≈ − . Table 4.6: Prediction for n s and n ′ s of the model with primordial inflationbeing Chaotic Inflation, with α = 1 , Γ ϕ ≪ H TI , m ψ = 10 − GeV, m ∼ GeVand the parameter values from Table 4.2. .4 “End of Inflation” Mechanism (cid:5) (cid:6) m ( GeV ) n s Fig. 4.12: Prediction of the model for n s for φ ∗ Case A with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.185), with φ ∗ = h φ i , m = m and Γ = g m .) The Blue and Red lines are the central value andlower/upper bounds of n s respectively as obtained by the Planck spacecraft[40]. α = 1 The present value of the effective mass of ψ is m ψ, now = (cid:16) α − α (cid:17) h h ψ i M P ! α − h φ i + m ψ (4.190) m ψ, now = m ψ (4.191)the second line coming from the fact that h ψ i = 0 today. Therefore, fromEq. (4.91) we have m ψ & TeV (4.192) .4 “End of Inflation” Mechanism m ( GeV ) - -- - n s Fig. 4.13: Prediction of the model for n ′ s for φ ∗ Case A with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.189), with φ ∗ = h φ i , m = m and Γ = g m .) The Blue line is the central value of n ′ s as obtainedby the Planck spacecraft [40], with the lower and upper bounds being outsidethe displayed range of n ′ s .In order for this and Eq. (4.165) to both be satisfied, we require the lowerbound in Eq. (4.192) to be much smaller than the upper bound in Eq. (4.165).This gives m ≫ GeV (4.193)We now require the lower bound here to be much smaller than the upperbound in Eq. (4.166). This gives λ ≫ (2 n + 4)! 10 n +26 GeV M P (4.194)where we have explicitly factored out the n +4)! term from our definitionof λ . Given that n ≥ , this constraint is in conflict with the requirementthat λ . . Therefore, for α = 1 , we find that the “end of inflation” mecha- .5 Modulated Decay Rate m ( GeV ) n s Fig. 4.14: Prediction of the model for n s for φ ∗ Case B with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.185), with φ ∗ = φ BD , m = m and Γ = g m .) The Blue and Red lines are the central value andlower/upper bounds of n s respectively as obtained by the Planck spacecraft[40].nism cannot produce the dominant contribution to the observed primordialcurvature perturbation within this Thermal Inflation model as it currentlystands. Now we investigate the modulated decay scenario to see if it can producethe dominant contribution to the primordial curvature perturbation ζ . As inSection 4.4, we aim to obtain a number of constraints on the model param-eters and the initial conditions for the fields. Considering these constraints,we intend to determine the available parameter space (if any). In this pa-rameter space we will calculate distinct observational signatures (such as .5 Modulated Decay Rate m ( GeV ) - -- - n s Fig. 4.15: Prediction of the model for n ′ s for φ ∗ Case B with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.189), with φ ∗ = φ BD , m = m and Γ = g m .) The Blue line is the central value of n ′ s as obtainedby the Planck spacecraft [40], with the lower and upper bounds being outsidethe displayed range of n ′ s .non-Gaussianity) that may test this scenario in the near future.From Section 4.3.1, the decay rate of the φ field is given by Γ ∼ g m (4.195)The primordial curvature perturbation that is produced by a varying decayrate [24] is given to first order by ζ = δN = − δ ΓΓ (4.196)Differentiating Γ ∼ g m with respect to m gives δ Γ ∼ g δm (4.197) .5 Modulated Decay Rate m ( GeV ) n s Fig. 4.16: Prediction of the model for n s for φ ∗ Case C with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.185), with φ ∗ = 0 , m = m and Γ = g m .) The Blue and Red lines are the central value andlower/upper bounds of n s respectively as obtained by the Planck spacecraft[40].Therefore we obtain the primordial curvature perturbation as ζ = δN ∼ − δmm (4.198)This is of the same order of magnitude as the primordial curvature pertur-bation that is produced by the “end of inflation” mechanism, Eqs. (4.59)and (4.63). As with the “end of inflation” mechanism scenario, we will consider what istermed local non-Gaussianity, which for the bispectrum corresponds to the“squeezed” configuration of the momenta triangle, in that the magnitude of .5 Modulated Decay Rate m ( GeV ) - -- - n s Fig. 4.17: Prediction of the model for n ′ s for φ ∗ Case C with primordialinflation being Chaotic Inflation, with α = 1 , Γ ϕ ≪ H T I , m ψ = 10 − GeV andthe parameter values from Table 4.2. (A plot of Eq. (4.189), with φ ∗ = 0 , m = m and Γ = g m .) The Blue line is the central value of n ′ s as obtainedby the Planck spacecraft [40], with the lower and upper bounds being outsidethe displayed range of n ′ s .one of the momentum vectors is much smaller than the other two, which areof similar magnitude to each other, e.g. k ≪ k , k and k ≈ k . Withinthe framework of the δN formalism, the non-Gaussianity parameter f NL isobtained as [1] f NL = 5 ΓΓ ′′ Γ ′ − ! (4.199)where the prime denotes the derivative with respect to ψ . By substituting Γ from Eq. (4.195) into Eq. (4.199) we obtain f NL ∼ mm ′′ m ′ − ! (4.200) .5 Modulated Decay Rate m , Eq. (4.3), we obtain f NL ∼ − " α − α m M α − P h ψ α ∗ − ! (4.201) Regarding the constraints that appear in Section 4.4.3, all are relevant to thismodulated decay scenario except for those that appear in Section 4.4.3.10,as we do not require that the transition from thermal inflation to thermalwaterfall field oscillation be sufficiently fast here, as well as those appearing inSection 4.4.3.11, as the spectator field will not be oscillating in this scenario(see Section 4.5.2.2). φ Decay
After the end of thermal inflation, we require there to exist an amount oftime, i.e. an amount of Universe expansion, prior to the decay of the φ field.This is so that the modulated decay rate mechanism can have an effect. Ifthe φ field decayed immediately after the end of thermal inflation, the decayrate would not be effectively modulated. Therefore, we require H T I > Γ (4.202)Substituting Eqs. (4.10) and (4.195) into here gives m > √ λ g n +2 M P (4.203) .5 Modulated Decay Rate ψ For the “end of inflation” mechanism scenario, we required the effective massof ψ to be light all the way up until the end of thermal inflation. However,in this modulated decay rate scenario, we require the effective mass to belight for longer, all the way up until φ decays, in order that ψ does not startoscillating, so that the perturbations in ψ remain at decay time and thushave the effect of perturbing the decay rate. Therefore we require | m ψ, eff | ≪ H (4.204)The φ field decays when H falls to ∼ Γ . Therefore we require | m ψ, eff | ≪ Γ (4.205)We have m ψ, eff = m ψ + (cid:16) α − α (cid:17) h φ ψM P ! α − (4.206)Therefore we require m ψ ≪ Γ (4.207)and hφ ψ ∗ M P ! α − ≪ Γ (4.208) h h φ i ψ ∗ M P ! α − ≪ Γ (4.209) .5 Modulated Decay Rate h φ i , ψ ∗ and Γ , Eqs. (4.7), (4.92) and (4.195) respectively, intoEq. (4.209) gives the constraint h ≪ (cid:16) g m M α − P (cid:17) α − q (2 n + 4) λm M nP α − n +1 √ H ∗ m M α − P ! α − (4.210) We require the energy density of ψ to be subdominant after thermal inflationup until it decays, in order that it does not cause any inflation by itself. Theenergy density of ψ after thermal inflation is ρ ψ = h ψ α ∗ M α − P φ + 12 m ψ ψ ∗ (4.211) ∼ h ψ α ∗ M α − P h φ i + 12 m ψ ψ ∗ (4.212)For simplicity, we assume that ψ decays around the same time as φ , i.e. that H does not change much between the time when φ decays and the time when ψ decays. Therefore, the energy density of the Universe at the time when ψ decays is ∼ M P Γ . We therefore require ρ ψ ≪ M P Γ (4.213) h ψ α ∗ M α − P h φ i + 12 m ψ ψ ∗ ≪ M P Γ (4.214) .5 Modulated Decay Rate m ψ ≪ M P Γ ψ ∗ (4.215)and h h φ i ψ α ∗ ≪ M αP Γ (4.216)These two constraints are identical to Eqs. (4.171) and (4.172) that appearin Section 4.4.3.11. This is easy to understand, in that the only differencebetween the derivation of the two constraints in Section 4.4.3.11 and thosethat appear here, is that, in the latter, we start with ψ taking its frozenvalue ψ ∗ , whereas in the former, we start with ψ taking its average oscillationvalue. However, the average oscillation value is just ψ ∗ . Substituting h φ i , ψ ∗ and Γ , Eqs. (4.7), (4.92) and (4.195) respectively, into Eq. (4.216) gives theconstraint h ≫ g m ) α − m M nP q (2 n + 4) λ α − n +1 m √ H ∗ M P ! α (4.217)which is identical to Eq. (4.173). In order for Eqs. (4.210) and (4.217) to both be satisfied, we require the upperbound in Eq. (4.210) to be much larger than the lower bound in Eq. (4.217).This gives m ≪ q (2 n + 4) λ (cid:16) g √ H ∗ (cid:17) n +1 √ M P n − (4.218) .5 Modulated Decay Rate H ∗ ≫ M P (2 n + 4) n +1 (4.219)We now require this lower bound to be much smaller than the upper boundin Eq. (4.76). This gives (2 n + 4) n +1 ≫ M P GeV (4.220)For all values of n , this constraint is grossly violated. Therefore, we find thatthis modulated decay rate scenario cannot produce the dominant contribu-tion to the observed primordial curvature perturbation within this ThermalInflation model as it currently stands. hapter 5Summary and Conclusions The research activity and success that has occurred within the field of Cos-mology over the last several decades, both observationally and theoreti-cally/computationally, is vast. We have reached a point in which the amountof data being obtained observationally, as well as its precision, is of a suf-ficient level to strongly guide the direction that theoretical topics withinCosmology should take. This era of “Precision Cosmology” is, for example,having an ever-greater input into the topic of Inflation. It has the powerto rule-out with high confidence certain models of Inflation, whilst givingfurther (strong) support to others.The momentum that Cosmology research is experiencing at the momentis not just down to the field of Cosmology however. Several other fields areproviding significant input into Cosmology, the main one being, at least forour work, particle physics. The topics of QFT and, in general, the StandardModel of particle physics play key roles in the development of modern Par-ticle Cosmology. 10910This thesis explores one such contribution to the current state of ParticleCosmology. We have developed a model of Thermal Inflation in which athermal waterfall scalar field, φ , is coupled to a light spectator scalar field, ψ . If this spectator field remains light from the time of primordial inflationup until the time when the thermal waterfall field decays, then a contribu-tion to the primordial curvature perturbation ζ will be generated by twomechanisms: “end of inflation” and modulated decay. The motivation forthe creation of this new model was to determine whether it can produce thedominant contribution to the primordial curvature perturbation and then toscrutinise its predictions for various quantities that can be observationally(and theoretically) tested.Our model explores two different cases for the decay of the inflaton ϕ :Inflaton decay is complete before the start of thermal inflation and inflatondecay occurs after the end of thermal inflation. We have also explored severaldifferent cases for the initial value of φ during primordial inflation, which welabel φ ∗ Case A, B and C. With regard to the decay of the thermal waterfallfield φ , we have only considered the case that the decay is via direct inter-actions with the particles in the thermal bath that exist due to the (partial)reheating of the inflaton ϕ . Regarding the decay of the spectator field, weassume that this field, ψ , decays around the same time as the thermal wa-terfall field φ .We have used the δN formalism to study the perturbations that are gen-erated from both the “end of inflation” and modulated decay mechanismswithin this model. We find that ζ is of the same magnitude to first order forboth cases.11We have constrained the two mechanisms within the model by using alarge array of constraints, coming from both observational and theoreticalconsiderations. Some of these are commonplace constraints that appear inmany inflation models, such as the requirement that the model does not spoilthe high-degree of success of BBN. However, some of the constraints that weemploy are rarely seen elsewhere in inflation model-building. Therefore, webelieve that, in general, our model is significantly more comprehensive thanmost others and that some of our constraints should be applied to other in-flation models.We first discuss the results for the modulated decay scenario within ourmodel. After considering all constraints together, we found strong tensionbetween several constraints relating to h , m and H ∗ , as these yield the con-straint given by Eq. (4.220), which can never be satisfied. In conclusiontherefore, we find that the modulated decay rate mechanism scenario withinour model cannot produce the dominant contribution to the observed pri-mordial curvature perturbation as it currently stands.Now we discuss the results for the “end of inflation” scenario within ourmodel. We first start with the case α = 1 . Similarly with the modulateddecay scenario, after considering all constraints together, we found tensionbetween several constraints, relating to m ψ and m . These combined con-straints yield the constraint given by Eq. (4.194), which can never be satisfied,given that we require λ . . In conclusion therefore, we find that the “endof inflation” mechanism scenario for α = 1 within our model cannot producethe dominant contribution to the observed primordial curvature perturbationas it currently stands.12We now turn to the case α = 1 . We have given results only for the case Γ ϕ ≪ H TI . The reason for this is that this case yielded more liberal constraintsthan the case Γ ϕ & H TI . For reasonable values of H ∗ and Γ ϕ , a value of g = 0 . yields a sharp prediction to within an order of magnitude for all but one of thequantities in the model, regardless of the φ ∗ case. The quantity m ψ can spanmany orders of magnitude when looking within the range m ∼ – GeVand we report only an upper bound of m ψ ∼ − – − GeV.Regarding the tensor fraction, we obtain a prediction of r ∼ − for avalue of H ∗ ∼ GeV.We also report values for the non-Gaussianity parameters f NL and g NL for local-type non-Gaussianity, i.e. for when the momenta triangle is in thesqueezed configuration. We predict values of f NL ∼ − and g NL ∼ − , fora value of m ∼ GeV, which are within current observational bounds.In order to obtain predictions from our model for the scalar spectral indexand its running, we needed to choose an inflation model for the period of pri-mordial inflation. We chose to use a simple slow-roll Chaotic Inflation model,with the potential given by Eq. (4.174). We obtain values of n s ≈ . and n ′ s ≈ − . , for values of m ∼ GeV, N TI ≈ and N ∗ ≈ andindependent of the φ ∗ Case and the value of m ψ within its allowed range.These values for n s and n ′ s are within current observational bounds.As already mentioned, the analysis that we have performed on our modelhas been limited to a relatively small region of the vast parameter space.Therefore, it is possible that there exist other allowed regions, for differentvalues of some or all of the free parameters in the model. This is a potentialarea for future research. In addition, as our analysis has been purely analyti-13cal, where we have analysed most quantities to within an order of magnitude,a future area of work could be to perform numerical analysis. In particular,the constraint that was imposed in order that the inflationary trajectory was1-dimensional, in that only the φ field was involved in determining the tra-jectory of thermal inflation in field space, could be removed, as a numericalanalysis lends itself better to such multi-dimensional inflationary trajectoriesthan does an analytic one.Cosmology is still experiencing a golden era at present, with ever moredata and theoretical insight progressing the field and giving rise to break-throughs. We can hope that this situation and momentum grows, with itcontinuing to lead us on an incredibly exciting journey further down theavenue of the most fundamental questions that humankind can ask. ppendix ADerivation of ThermalFluctuation of φ Our aim is to calculate the thermal fluctuation of φ that exists when thefield is in thermal equilibrium with a thermal bath. This derivation hasbeen worked through following Ref. [56]. As we are dealing with the thermalfluctuation of φ about φ = 0 , we will simply call δφ by φ .The solution to the perturbation equation ¨ φ − ∇ φ + V ′′ φ = 0 (A.1)is φ = 1 √ π Z √ ω k (cid:16) e − iω k t + i k · x a − k + e iω k t − i k · x a + k (cid:17) d k (A.2)where ω k = r k + (cid:12)(cid:12)(cid:12) m φ, eff (cid:12)(cid:12)(cid:12) (A.3)11415Using D ˆ a + k ˆ a − k ′ E ≡ D n k (cid:12)(cid:12)(cid:12) ˆ a + k ˆ a − k ′ (cid:12)(cid:12)(cid:12) n k E h n k | n k i = n k δ ( k − k ′ ) (A.4)and D ˆ a + k ˆ a + k ′ E = D ˆ a − k ˆ a − k ′ E = 0 (A.5)where n k is the occupation number, as well as the commutation relation h ˆ a − k , ˆ a + k ′ i = δ ( k − k ′ ) (A.6)we obtain the 2-point correlator as D φ E = 116 π ZZ √ ω k ω k ′ (cid:16) e − iω k t + i k · x + iω k ′ t − i k ′ · x (cid:16) δ ( k − k ′ ) + D ˆ a + k ′ ˆ a − k E(cid:17) + e iω k t − i k · x − iω k ′ t + i k ′ · x D ˆ a + k ˆ a − k ′ E(cid:17) d k d k ′ (A.7)Working through the integral, we obtain D φ E = 18 π Z ω k (cid:18)
12 + n k (cid:19) d k (A.8) = 18 π Z ∞ k ω k (cid:18)
12 + n k (cid:19) d k Z π sin( θ ) d θ Z π d ϕ (A.9) = 12 π Z ∞ k ω k (cid:18)
12 + n k (cid:19) d k (A.10)16The occupation number n k is given by the Bose-Einstein distribution n ω k = 1 e ωkT − (A.11)where we are neglecting the chemical potential. Substituting this into Eq. (A.10)gives D φ E T = 12 π Z ∞ k ω k (cid:16) e ωkT − (cid:17) d k (A.12)The term that appears in Eq. (A.10) is not present in Eq. (A.12) as it canbe removed by one of the following: − Having n ω k ≫ − Subtracting the vacuum − Performing normal orderingEq. (A.12) can be expressed as D φ E T = T π J (1) − | m φ, eff | T , ! (A.13)The proof is as follows. The definition of the J term is J ( ν ) ∓ ( κ, τ ) ≡ Z ∞ κ ( x − κ ) ν e x − τ ∓ x + Z ∞ κ ( x − κ ) ν e x + τ ∓ x (A.14)where for our derivation we define κ ≡ | m φ, eff | T τ ≡ µT (A.15)17where µ is the chemical potential. Therefore we obtain (with τ = 0 as µ = 0 ) J (1) − | m φ, eff | T , ! ≡ Z ∞ | mφ, eff | T r x − (cid:16) | m φ, eff | T (cid:17) e x − x (A.16)Substituting this into Eq. (A.13) gives D φ E T = T π Z ∞ | mφ, eff | T r x − (cid:16) | m φ, eff | T (cid:17) e x − x (A.17)Now, we can obtain Eq. (A.17) from Eq. (A.12) by simply changing theintegration variable as k → x ≡ ω k T (A.18) = r k + (cid:12)(cid:12)(cid:12) m φ, eff (cid:12)(cid:12)(cid:12) T (A.19) d x = kω k T d k (A.20)We can make progress in trying to solve the integral of J (1) − as follows. J ( − − can be written as J ( − − = 2 ∞ X n =1 K ( nκ ) (A.21)where K is the modified Bessel function of the second kind. There exists arecurrence relation ∂J ( ν ) ∓ ∂κ = − νκJ ( ν − ∓ (A.22)18Therefore we have J (1) − = − Z κJ ( − − d κ (A.23) = − Z κ ∞ X n =1 K ( nκ ) d κ (A.24)The summation of the modified Bessel functions can be expressed as ∞ X n =1 K ( nκ ) = 12 (cid:20) C + ln (cid:18) κ π (cid:19)(cid:21) + π κ + π ∞ X l =1 √ κ + 4 l π − lπ ! (A.25)as given in Ref. [57], where C is Euler’s constant. Therefore we obtain J (1) − = − Z " Cκ + κ ln (cid:18) κ π (cid:19) + π + 2 πκ ∞ X l =1 √ κ + 4 l π − lπ ! d κ (A.26)In general, this integral doesn’t converge, due to the summation term. How-ever, for κ ≪ , the summation term vanishes and the integral does converge.We adopt this case, with only a mild constraint on g being introduced. Fromthe definition of κ , Eq. (A.15), κ ≪ requires T ≫ | m φ, eff | (A.27)19As we are considering the time of the end of primordial inflation, | m φ, eff | ∼ g T end,inf , as given by Eq. (4.99). Therefore we have T end,inf ≫ gT end,inf (A.28) g ≪ (A.29)So, we now have J (1) − ≃ − Cκ + 14 κ − κ ln (cid:18) κ π (cid:19) − πκ + A (A.30)where A is a constant of integration. We can obtain A by equating Eq. (A.30)with Eq. (A.14) and setting κ = τ = 0 . We obtain A = 2 Z ∞ xe x − x (A.31) = π (A.32)Given that | m φ, eff | ∼ g T end,inf , we have κ = g . Therefore Eq. (A.30) becomes J (1) − ≃ π − πg − g ln (cid:18) g π (cid:19) − C g + 14 g (A.33)For all values of g , i.e. g ≤ , we have J (1) − ∼ π (A.34)20Substituting this into Eq. (A.13) gives D φ E T ∼ T (A.35) q h φ i T ∼ T (A.36) ppendix BField Theory Derivation ofElastic Scattering Cross-sectionbetween φ and a Thermal Bath We assume the thermal bath to only consist of one scalar field, which we call χ . We define the following 4-momenta: p for a φ particle before the collision, p for a χ particle before the collision, p for a φ particle after the collisionand p for a χ particle after the collision. Working within the centre-of-massframe, the 4-momenta are: p = ( E , p ) (B.1) p = ( p, − p ) (B.2) p = ( E , p ′ ) (B.3) p = ( p ′ , − p ′ ) (B.4)12122The differential elastic cross-section is given by d σ = 1 | v − v | E p (cid:12)(cid:12)(cid:12) − ig (cid:12)(cid:12)(cid:12) d p ′ (2 π ) E d ( − p ′ )(2 π ) p ′ (2 π ) δ ( p + p − p − p ) (B.5)where v and v are the velocities of the φ and χ particles before the collisionrespectively, where v = c . As we are setting c = 1 , the first term in Eq. (B.5) ≈ . Collecting terms together gives d σ ≈ − g π E E pp ′ δ ( p + p − p − p )d p ′ d ( − p ′ ) (B.6)Integrating this gives σ ≈ − g π E E ZZ pp ′ δ ( p + p − p − p )d p ′ d ( − p ′ ) (B.7) ≈ − g π E E Z pp ′ δ ( E + p − E − p ′ )d p ′ (B.8) ≈ − g π E E Z p δ ( E + p − E − p ′ ) p ′ d p ′ Z Ω dΩ (B.9)From the 4-momenta of p and p , Eqs. (B.1) and (B.3) respectively, we have E = q p + m φ, eff (B.10)and E = q p ′ + m φ, eff (B.11)23Substituting these equations into Eq. (B.9) gives σ ≈ − g π E E Z p δ (cid:16)q p + m φ, eff + p − q p ′ + m φ, eff − p ′ (cid:17) p ′ d p ′ Z Ω dΩ (B.12)The argument of the delta function is only = 0 if p ′ = p . Therefore we have σ ≈ − g π E Z Ω dΩ (B.13)We have E = p + m φ, eff (B.14)From the 4-momenta of p , Eq. (B.2), we know that E = p . In addition,given that the energy of a χ particle is T , we therefore have E = T + m φ, eff (B.15)Let us consider the time of primordial inflation reheating. We have m φ, eff ∼ gT (B.16)Therefore E ∼ (cid:16) g (cid:17) T (B.17) ∼ T (B.18)24Substituting Eq. (B.18) into Eq. (B.13) gives σ ≈ − g π T Z Ω dΩ (B.19) ≈ − g πT (B.20) | σ | ≈ g πT (B.21) ibliography [1] D. H. Lyth and A. R. Liddle, The Primordial Density Perturbation:Cosmology, Inflation and the Origin of Structure . CambridgeUniversity Press, 2009.[2] D. H. Lyth and D. Wands,
Generating the curvature perturbationwithout an inflaton , Phys.Lett.
B524 (2002) 5–14, arXiv:hep-ph/0110002v2 [hep-ph] .[3] D. H. Lyth, C. Ungarelli and D. Wands,
The Primordial densityperturbation in the curvaton scenario , Phys.Rev.
D67 (2003) 023503, arXiv:astro-ph/0208055v3 [astro-ph] .[4] K.-Y. Choi and O. Seto,
Modulated reheating by curvaton , Phys.Rev.
D85 (2012) 123528, arXiv:1204.1419v1 [astro-ph.CO] .[5] K. Dimopoulos,
Can a vector field be responsible for the curvatureperturbation in the Universe? , Phys.Rev.
D74 (2006) 083502, arXiv:hep-ph/0607229v2 [hep-ph] .[6] K. Dimopoulos, M. Karciauskas, D. H. Lyth and Y. Rodriguez,
Statistical anisotropy of the curvature perturbation from vector field
IBLIOGRAPHY perturbations , JCAP (2009) 013, arXiv:0809.1055v5[astro-ph] .[7] K. Dimopoulos and M. Karciauskas,
Non-minimally coupled vectorcurvaton , JHEP (2008) 119, arXiv:0803.3041v2 [hep-th] .[8] M. Karciauskas,
The Primordial Curvature Perturbation from VectorFields of General non-Abelian Groups , JCAP (2012) 014, arXiv:1104.3629v2 [astro-ph.CO] .[9] K. Dimopoulos,
Statistical Anisotropy and the Vector CurvatonParadigm , Int.J.Mod.Phys.
D21 (2012) 1250023, arXiv:1107.2779v2[hep-ph] .[10] K. Dimopoulos,
Erratum: Statistical Anisotropy and the VectorCurvaton Paradigm , Int.J.Mod.Phys.
D21 (2012) 1292003, arXiv:1107.2779v2 [hep-ph] .[11] M. Karciauskas, K. Dimopoulos and D. H. Lyth,
Anisotropicnon-Gaussianity from vector field perturbations , Phys.Rev.
D80 (2009)023509, arXiv:0812.0264v2 [astro-ph] .[12] M. Karciauskas, K. Dimopoulos and D. H. Lyth,
Erratum: Anisotropicnon-Gaussianity from vector field perturbations , Phys.Rev.
D85 (2012)069905(E), arXiv:0812.0264v2 [astro-ph] .[13] S. Yokoyama and J. Soda,
Primordial statistical anisotropy generatedat the end of inflation , JCAP (2008) 005, arXiv:0805.4265v6[astro-ph] . IBLIOGRAPHY
Statistical Anisotropy fromVector Curvaton in D-brane Inflation , arXiv:1108.4424v2 [hep-th] .[15] N. Bartolo, E. Dimastrogiovanni, S. Matarrese and A. Riotto, Anisotropic Trispectrum of Curvature Perturbations Induced byPrimordial Non-Abelian Vector Fields , JCAP (2009) 028, arXiv:0909.5621v2 [astro-ph.CO] .[16] K. Dimopoulos,
Supergravity inspired Vector Curvaton , Phys.Rev.
D76 (2007) 063506, arXiv:0705.3334v1 [hep-ph] .[17] K. Dimopoulos and M. Karciauskas,
Parity Violating StatisticalAnisotropy , JHEP (2012) 040, arXiv:1203.0230v3 [hep-ph] .[18] K. Dimopoulos, G. Lazarides and J. M. Wagstaff,
Eliminating the η -problem in SUGRA Hybrid Inflation with Vector Backreaction , JCAP (2012) 018, arXiv:1111.1929v2 [astro-ph.CO] .[19] H. Assadullahi, H. Firouzjahi, M. H. Namjoo and D. Wands,
Modulated curvaton decay , arXiv:1301.3439v1 [hep-th] .[20] D. Langlois and T. Takahashi, Density Perturbations from ModulatedDecay of the Curvaton , arXiv:1301.3319v1 [astro-ph.CO] .[21] S. Enomoto, K. Kohri and T. Matsuda, Modulated decay in themulti-component Universe , arXiv:1301.3787v1 [hep-ph] .[22] K. Kohri, C.-M. Lin and T. Matsuda, Delta-N Formalism for Curvatonwith Modulated Decay , arXiv:1303.2750v1 [hep-ph] . IBLIOGRAPHY
Curvaton dynamics , Phys.Rev.
D68 (2003) 123515, arXiv:hep-ph/0308015v1 [hep-ph] .[24] G. Dvali, A. Gruzinov and M. Zaldarriaga,
New mechanism forgenerating density perturbations from inflation , Phys.Rev.
D69 (2004)023505, arXiv:astro-ph/0303591v1 [astro-ph] .[25] G. Dvali, A. Gruzinov and M. Zaldarriaga,
Cosmological perturbationsfrom inhomogeneous reheating, freezeout, and mass domination , Phys.Rev.
D69 (2004) 083505, arXiv:astro-ph/0305548v1[astro-ph] .[26] M. Postma,
Inhomogeneous reheating scenario with low scale inflationand/or MSSM flat directions , JCAP (2004) 006, arXiv:astro-ph/0311563v2 [astro-ph] .[27] F. Vernizzi,
Generating cosmological perturbations with massvariations , Nucl.Phys.Proc.Suppl. (2005) 120–127, arXiv:astro-ph/0503175v1 [astro-ph] .[28] F. Vernizzi,
Cosmological perturbations from varying masses andcouplings , Phys.Rev.
D69 (2004) 083526, arXiv:astro-ph/0311167v3[astro-ph] .[29] M. Zaldarriaga,
Non-Gaussianities in models with a varying inflatondecay rate , Phys.Rev.
D69 (2004) 043508, arXiv:astro-ph/0306006v1 [astro-ph] . IBLIOGRAPHY
Density Fluctuations inThermal Inflation and Non-Gaussianity , JCAP (2009) 012, arXiv:0910.3053v3 [hep-th] .[31] D. H. Lyth,
Generating the curvature perturbation at the end ofinflation , JCAP (2005) 006, arXiv:astro-ph/0510443v3[astro-ph] .[32] M. P. Salem,
On the generation of density perturbations at the end ofinflation , Phys.Rev.
D72 (2005) 123516, arXiv:astro-ph/0511146v5[astro-ph] .[33] L. Alabidi and D. H. Lyth,
Curvature perturbation from symmetrybreaking the end of inflation , JCAP (2006) 006, arXiv:astro-ph/0604569v3 [astro-ph] .[34] D. H. Lyth,
The hybrid inflation waterfall and the primordial curvatureperturbation , JCAP (2012) 022, arXiv:1201.4312v4[astro-ph.CO] .[35] D. H. Lyth and A. Riotto,
Generating the Curvature Perturbation atthe End of Inflation in String Theory , Phys.Rev.Lett. (2006)121301, arXiv:astro-ph/0607326v1 [astro-ph] .[36] M. Sasaki, Multi-brid inflation and non-Gaussianity , Prog.Theor.Phys. (2008) 159–174, arXiv:0805.0974v3 [astro-ph] . IBLIOGRAPHY
Modulated fluctuations fromhybrid inflation , Phys.Rev.
D70 (2004) 083004, arXiv:astro-ph/0403315v1 [astro-ph] .[38] T. Matsuda,
Cosmological perturbations from an inhomogeneous phasetransition , Class.Quant.Grav. (2009) 145011, arXiv:0902.4283v3[hep-ph] .[39] L. Alabidi, K. Malik, C. T. Byrnes and K.-Y. Choi, How the curvatonscenario, modulated reheating and an inhomogeneous end of inflationare related , JCAP (2010) 037, arXiv:1002.1700v2[astro-ph.CO] .[40]
Planck
Collaboration, P. A. R. Ade et al. , Planck 2015 results. XIII.Cosmological parameters , arXiv:1502.01589v2 [astro-ph.CO] .[41] ESA: C. Carreau, The History of Structure Formation in the Universe ,March, 2013. http://sci.esa.int/science-e-media/img/68/Planck_history_of_Universe_Crop_orig.jpg . Downloaded01/10/2016.[42] ESA: Planck Collaboration,
Cosmic Microwave Background seen byPlanck , March, 2013. http://sci.esa.int/science-e-media/img/61/Planck_CMB_Mollweide_wallpaper.jpg . Downloaded 12/10/2015.[43] ESA: Planck Collaboration,
Planck’s Power Spectrum of TemperatureFluctuations in the Cosmic Microwave Background , March, 2013. http://sci.esa.int/science-e-media/img/63/Planck_power_spectrum_orig.jpg . Downloaded 12/10/2015.
IBLIOGRAPHY
The Inflationary Universe: A Possible Solution to theHorizon and Flatness Problems , Phys. Rev.
D23 (1981) 347–356.[45] T. S. Bunch and P. C. W. Davies,
Quantum Field Theory in de SitterSpace: Renormalization by Point Splitting , Proc. Roy. Soc. Lond.
A360 (1978) 117–134.[46] D. H. Lyth, K. A. Malik and M. Sasaki,
A General proof of theconservation of the curvature perturbation , JCAP (2005) 004, arXiv:astro-ph/0411220v3 [astro-ph] .[47] D. H. Lyth and E. D. Stewart,
Cosmology with a TeV mass GUTHiggs , Phys.Rev.Lett. (1995) 201–204, arXiv:hep-ph/9502417v1[hep-ph] .[48] D. H. Lyth and E. D. Stewart, Thermal inflation and the moduliproblem , Phys.Rev.
D53 (1996) 1784–1798, arXiv:hep-ph/9510204v2[hep-ph] .[49] T. Barreiro, E. J. Copeland, D. H. Lyth and T. Prokopec,
Someaspects of thermal inflation: The Finite temperature potential andtopological defects , Phys.Rev.
D54 (1996) 1379–1392, arXiv:hep-ph/9602263v2 [hep-ph] .[50] T. Asaka and M. Kawasaki,
Cosmological moduli problem and thermalinflation models , Phys.Rev.
D60 (1999) 123509, arXiv:hep-ph/9905467v1 [hep-ph] . IBLIOGRAPHY
Primordial trispectrum frominflation , Phys. Rev.
D74 (2006) 123519, arXiv:astro-ph/0611075v2[astro-ph] .[52] M. Dine, L. Randall and S. D. Thomas,
Supersymmetry breaking in theearly universe , Phys. Rev. Lett. (1995) 398–401, arXiv:hep-ph/9503303v2 [hep-ph] .[53] M. Dine, L. Randall and S. D. Thomas, Baryogenesis from flatdirections of the supersymmetric standard model , Nucl. Phys.
B458 (1996) 291–326, arXiv:hep-ph/9507453v1 [hep-ph] .[54] D. H. Lyth and T. Moroi,
The Masses of weakly coupled scalar fields inthe early universe , JHEP (2004) 004, arXiv:hep-ph/0402174v2[hep-ph] .[55] Planck
Collaboration, P. A. R. Ade et al. , Planck 2015 results. XVII.Constraints on primordial non-Gaussianity , arXiv:1502.01592v2[astro-ph.CO] .[56] V. Mukhanov, Physical Foundations of Cosmology . CambridgeUniversity Press, 2005.[57] I. Gradshteyn and I. Ryzhik,
Table of Integrals, Series and Products ,5 thth