Recent developments in string model-building and cosmology
aa r X i v : . [ h e p - t h ] A p r April 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 1 Recent developments in string model-building and cosmology
Michele Cicoli
Dipartimento di Fisica ed Astronomia, Universit`a di Bologna,via Irnerio 46, 40126 Bologna, Italy ∗ E-mail: [email protected], Sezione di Bologna, ItalyAbdus Salam ICTP, Strada Costiera 11, Trieste 34014, Italy
In this talk I discuss recent developments in moduli stabilisation, SUSY breaking andchiral D-brane models together with several interesting features of cosmological modelsbuilt in the framework of type IIB string compactifications. I show that a non-trivialpre-inflationary dynamics can give rise to a power loss at large angular scales for whichthere have been mounting observational hints from both WMAP and Planck. I thendescribe different stringy embeddings of inflationary models which yield large or smalltensor modes. I finally argue that reheating is generically driven by the decay of thelightest modulus which can produce, together with Standard Model particles, also non-thermal dark matter and light hidden sector degrees of freedom that behave as darkradiation.
Keywords : String compactifications; Moduli phenomenology
1. Introduction
During this talk I will focus on type IIB on CY orientifolds with D3/D7-branes andO3/O7-planes since: • D-branes provide non-Abelian gauge symmetries and chiral matter. Theycan therefore be used to realise MSSM- or GUT-like models via either mag-netised D7-branes wrapped around 4-cycles or D3-branes at singularities; • Most of the moduli can be fixed with control over moduli space by turningon background fluxes H , F which lead to a small back-reaction on theinternal geometry; • Type IIB compactifications allow to realise a brane-world scenario wheregauge interactions are localised. Thus model-building, being a local issue,decouples (at leading order) from moduli stabilisation which is a global issue.The table below shows different local (brane) and global (bulk) issues. The localones are more model-dependent since they involve the details of particular braneset-ups, while global issues are more model-independent since they are affected bythe properties of the bulk of the extra dimensions. Interestingly, some issues likereheating, dark radiation and dark matter are both local and global since theyinvolve the coupling of closed string modes to open string degrees of freedom. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 2
Recently there has been a lot of progress in trying to combine global with localissues in compact Calabi-Yau models with explicit brane set-up, tadpole cancellationand dS moduli stabilisation compatible with chirality. Let us summarise the mainresults within the framework of type IIB Large Volume Scenarios which we willdiscuss in this talk: • Construction of explicit compact Calabi-Yau orientifolds via toric geometry; • Presence of an explicit set-up with D3/D7-branes, O3/O7-planes and fluxes(both background and gauge); • Global consistency of the underlying construction due to D3-, D5- and D7-tadpole and Freed-Witten anomaly cancellation; • Explicit fixing of dilaton and complex structure moduli by reducing theeffective number of moduli due to symmetries in the moduli space identifiedusing the Greene-Plesser construction; • Stabilisation of the K¨ahler moduli in a way compatible with chirality withinregime of validity of the effective field theory; • Two different realisations of the visible sector:(1) D7-branes in geometric regime can lead to chiral SU (5)- or MSSM-likemodels, (2) Fractional D3-branes and flavour D7-branes at del Pezzo singularitiescan accommodate SU (3) , Pati-Salam or MSSM-like models, • dS vacua without anti-branes via two fully supersymmetric methods:(1) Non-zero F-terms of charged hidden matter fields induced by D-termstabilisation, (2) Non-perturbative effects at singularities, • Spontaneous SUSY breaking by F-terms of K¨ahler moduli; • TeV-scale soft-terms via gravity mediation; • The K¨ahler moduli are promising inflaton candidates since: (1) The η -problem can be solved by the extended no-scale structure of theK¨ahler potential,(2) Explicit stringy realisations of α -attractors with inflationary potentialsof the schematic and generic form V ≃ V (cid:0) − k e − kφ (cid:1) , pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 3 (3) Possible power loss at large angular scales due to a non slow-roll pre-inflationary evolution, • Reheating is driven by the decay of the lightest modulus; • Generic production of non-thermal neutralino dark matter, and axionicdark radiation; • Prediction of the existence of a cosmic axion background with O (200 eV)energies; • Possible explanation of the observed soft X-ray excess, and 3 .
2. Moduli stabilisation and SUSY breaking
In this section we give a very brief review of the main aspects of dS closed stringmoduli stabilisation and SUSY breaking.
Tree-level stabilisation
The 4D type IIB tree-level K¨ahler potential K and superpotential W read: K tree = − V ( T i + ¯ T i ) − ln( S + ¯ S ) − ln (cid:18) i Z CY Ω( U ) ∧ ¯Ω (cid:19) W tree = Z CY G ∧ Ω( U )leading to a scalar potential V = e K h K i ¯ j D i W D ¯ j ¯ W − | W | i of the form: V = e K X S,U K α ¯ β D α W D ¯ β ¯ W + e K "X T K i ¯ j K i K ¯ j − | W | ≥ P T K i ¯ j K i K ¯ j = 3. The dilaton S and the complexstructure moduli U can be fixed supersymmetrically at D S W = D U W = 0 setting W = h W tree i . These are n = 2 h , + 2 real non-linear equations in n unknownswith 2 n parameters, the flux quanta, whose values are constrained by D3 tadpolecancellation. There is therefore enough freedom to find solutions whose number canbe estimated as follows. If each flux quanta can take for example 10 different valueswe have: N sol ∼ n = 10 h , +1) ∼ for h , ∼ O (100)This leads to the flux landscape. At this level of approximation the vacuum isMinkowski and SUSY is broken since F T = 0. However the T -moduli are still flat.The gravitino mass turns out to be: m / = e K/ | W | ≃ W V M p (1)Notice that natural values of the underlying parameters lead to W ∼ O (1) while W ≪ pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 4 K¨ahler moduli stabilisation K and W get corrected beyond tree-level as follows: W = W tree + W np K ≃ K tree + K p with K p = K α ′ + K g s where: W np ∼ e − τ ≪ K p ∼ K α ′ ∼ V ∼ τ / for τ ≫ V = V tree + V p + V np where V tree = 0 due to the no-scale cancellation.On the other hand the scaling behavior of V p and V np is: V p ∼ e K W K p V np ∼ e K ( W W np + W )In the natural case with W ∼ O (1) we have: V p V np ∼ K p W np ∼ e τ τ / ≫ ⇒ V p ≫ V np Thus non-perturbative effects can be neglected and moduli stabilisation has to takeplace at perturbative level. However this can be done only via tuning since V g s ≪ V α ′ due to the extended no-scale structure. On the other hand, if we tune W ∼O ( W np ) we have: V p V np ∼ K p ∼ τ / ≪ ⇒ V p ≪ V np In this case we have therefore to perform a pure non-perturbative KKLT-like sta-bilisation. However a full non-perturbative fixing has some shortcomings: • W has to be tuned; • W np gets definitely generated for rigid cycles while non-perturbative effects fornon-rigid cycles are not guaranteed; • There is a tension between moduli stabilisation and chirality which can beschematically summarised as follows: (1) If the visible sector wraps the 4-cycle τ with gauge flux F , τ gets a U (1)charge(2) If τ is also wrapped by an E3-instanton, the contribution W np ∼ e − τ wouldnot be gauge invariant(3) Chiral intersections between the E3 and the visible sector make W np ∼ Q i φ i e − τ gauge invariant(4) In order to preserve visible sector gauge symmetries at high energies weneed h φ i i = 0 ∀ i . This in turn gives W np = 0, implying that the 4-cyclesupporting the visible sector cannot be fixed by non-perturbative effects • There is a tension also between moduli stabilisation and Freed-Witten anomalycancellation which we briefly summarise as:(1) In the simplest case in order to generate a non-zero W np , an E3 has towrap a transversally invariant cycle with F = F − B = 0 pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 5 (2) In the case of non-spin cycles, Freed-Witten anomaly cancellation induces F FW = ˆ D = 0(3) One can therefore choose the B -field to cancel F FW as B = F FW . Howeverthen it becomes hard to cancel F − B simultaneously for two or moreintersecting cycles • Standard KKLT-like stabilisation procedures lead to AdS vacua where the dSuplifting is performed with anti D3-branes. The consistency of these construc-tions is presently under debate.
Large Volume Scenario
A very elegant way-out to the previous problems can be found if h , ≥
2. In factin this case one can fix the moduli without tuning since for W ∼ O (1) we have: V p V np ∼ K p W np ∼ e τ s τ b / ∼ O (1) if 1 ≪ τ s ≪ τ b This is obtained dynamically if τ s is a diagonal blow-up mode and the internalvolume is of the form V = τ / b − τ / s ≃ τ / b . This is a very promising situationsince:(1) τ s is a local effect, and so it naturally reproduces the required hierarchy τ s ≪ τ b (2) τ s is a rigid cycle, and so W np gets easily generated.This leads to the Large Volume Scenario where the K¨ahler potential and the super-potential look like ( ξ is an O (1) topological quantity controlling α ′ effects): K = − V − ξg / s V W = W + A s e − a s T s The scalar potential after axion minimisation becomes (the λ ’s are O (1) constants): V = λ √ τ s e − a s τ s V − λ τ s W e − a s τ s V + λ W g / s V (2)admitting an AdS minimum at: τ s ∼ g − s ∼ O (10) for g s ≃ . V ≃ W e a s τ s ∼ e /g s ≫ m / ≃ W V M p ≃ M p e − /g s ≪ M p Hence this scenario can yield low-energy SUSY naturally. Moreover SUSY is spon-taneously broken since at the minimum: F T b ∼ M p V / = 0 F T s ∼ M p V 6 = 0 pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 6 The Large Volume Scenario also does not feature any conflict between non-perturbative effects and chirality and Freed-Witten anomaly cancellation since:(1) τ s does not intersect other cycles, and so there are no chiral intersections be-tween the visible sector and instantons on τ s , resulting in a non-zero W np . Thevisible sector cycle has instead to be fixed by either D-terms or g s effects;(2) In order to fix the moduli one needs only non-perturbative effects for diagonalblow-up modes. In this case one can choose the B -field to cancel all Freed-Witten fluxes since non-perturbative effects are supported on non-intersectingcycles.In the Large Volume Scenario the minimal number of K¨ahler moduli to berealistic is h , ≥
4. In fact, there are two possible ways to realise the visible sector: • In models with D7-branes in the geometric regime, the internal volume looksschematically as: V = τ / b − τ / s − ( τ vs + τ vs ) / At leading order D-terms fix τ vs ∼ τ vs leaving a flat direction which we call τ vs . Subdominant non-perturbative and α ′ effects fix τ b and τ s at τ / b ∼ e τ s and τ s ∼ g − s . Finally g s effects stabilise τ vs . • In models with D3-branes at singularities, the Calabi-Yau volume should takeinstead the form: V = τ / b − τ / s − τ / − τ / The two 4-cycles τ vs and τ vs are exchanged by the orientifold involution inorder to obtain unitary groups for the visible sector. The shrinking of τ vs and τ vs to zero size is induced by D-term fixing. At subleading order non-perturbative and α ′ effects fix τ b and τ s at τ / b ∼ e τ s and τ s ∼ g − s .Let us now analyse separately the phenomenological implications of these two dif-ferent realisations of the visible sector. Unsequestered models
When the visible sector is built with D7-branes in geometric regime, the F-term of τ vs is non-zero: F vs ∼ m / M p = 0. The soft-terms and the mass of the volumemode scale as: m V ∼ M p V / ≪ M soft ∼ m / ∼ M p V One can therefore set either M soft ∼ O (1) TeV to solve the hierarchy problem or m V > O (50) TeV to avoid any cosmological moduli problem. These two differentchoices require two different values of the internal volume which in turn set all theother relevant energy scales. The table below shows the values of all these energyscales for V ∼ and V ∼ respectively. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 7 V ∼ Energy scales for
V ∼ M p ∼ GeV M p ∼ GeV M s ∼ m τ vs1 ∼ m a vs1 ∼ M p V − / ∼ GeV M s ∼ m τ vs1 ∼ m a vs1 ∼ GeV M KK ∼ M p V − / ∼ GeV M KK ∼ GeV m τ s ∼ m a s ∼ M p V − ln V ∼
100 TeV m τ s ∼ m a s ∼ · GeV m / ∼ M p V − ∼
10 TeV m / ∼ GeV M soft ∼ m τ vs2 ∼ M p V − (ln V ) − ∼ M soft ∼ m τ vs2 ∼ GeV m τ b ∼ M p V − / ∼ m τ b ∼ GeV m a vs2 ∼ Λ f − a vs2 ∼ f a vs2 ∼ M s m a vs2 ∼ f a vs2 ∼ M s m a b ∼ M p e −V / ∼ m a b ∼ Sequestered models
When the visible sector lives on D3-branes at singularities, the F-term of τ vs van-ishes: F vs ∝ ξ FI ∝ τ vs →
0. This cancellation induces a sequestering of the visiblesector from the sources of SUSY breaking which are the F-terms of the bulk moduli.Thus gaugino masses turn out to be suppressed with respect to m / : M / ∼ M p V ≪ m / ∼ M p V Depending on the exact moduli-dependence of the matter K¨ahler metric and themechanism responsible to achieve a dS minimum, scalar masses instead scale as: m ∼ M p V / ∼ m V or m ∼ M p V ∼ M / Setting
V ∼ , one can obtain M / ∼ O (1) TeV. All the other main energy scalesare listed in the table below. Energy scales M p ∼ GeV M GUT ∼ M s V / ∼ GeV M s ∼ m τ vs1 ∼ m a vs1 ∼ m τ vs2 ∼ m a vs2 ∼ GeV M KK ∼ GeV m τ s ∼ m a s ∼ GeV m / ∼ GeV m τ b ∼ GeV M / ∼ m a open ∼ f a open ∼ M s √ τ vs ≪ M s m a b ∼ Scalar masses can be either m ∼ M / ∼ m ∼ m τ b ∼ GeV as in split SUSY-like scenarios. This scenario cantherefore allow for: low-energy SUSY, a promising framework to embed standardGUT theories, a right inflationary scale, no cosmological moduli problem for τ b , aviable QCD axion from open string modes, reheating driven by the decay of τ b with T rh ∼ −
10 GeV, non-thermal dark matter and axionic dark radiation producedfrom the decay of τ b . pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 8 dS from hidden F-terms Let us now briefly present a general mechanism which can lead to dS vacua. Inglobally consistent models τ b is wrapped by a hidden stack of D7-branes becauseof D7-tadpole cancellation. Moreover Freed-Witten anomaly cancellation inducesa non-zero gauge flux on τ b . This modulus therefore acquires a U (1)-charge andappears in the Fayet-Iliopoulos term of the D-term potential: V bulk D = 1 τ b X i q D i | φ i | − ξ D ! with ξ D = 3(2 V ) / The total scalar potential reads: V tot = V bulk D + V F = 1 τ b (cid:0) q D | φ dS | − ξ D (cid:1) + m / | φ dS | + V O ( V − ) where is V O ( V − ) the moduli potential (2). The minimum for φ dS lies at q D | φ dS | = ξ D − m / τ b q D Substituting this result in V tot we obtain: V tot = V bulk D, + V F = m / τ b q D + m / ξ D q D + V O ( V − ) (3)The first term on the RHS of (3) is negligible since it scales as V − / while thesecond term on the RHS behaves as V − / and can play the rˆole of an upliftingterm. Minimising with respect to τ s and V we obtain h V tot i = 3 W a / s V " δ V / − s ln (cid:18) V W (cid:19) with δ ≃ . a / s q D ! Clearly W can be tuned to get h V tot i = 0. In particular, W ∼ O (1) gives riseto solutions around V ∼ -10 which are the values needed to get TeV-scaleSUSY. This uplifting mechanism has an interesting higher dimensional understandingin terms of T-branes. In fact, the effective field theory has to be expanded aroundthe correct background. For a hidden D7-stack this is parameterised by an adjointcomplex scalar Φ. The non-zero gauge flux breaks SO (8) to U (4) (focusing on thecase of 4 D7s on top of an O7), and so Φ decomposes as → ⊕ +2 ⊕ − . Adeformation of Φ can be written as δ Φ = (cid:18) φ φ +2 φ − − φ T (cid:19) The 8D BPS equation of motion for a hidden D7-brane is J ∧F D + (cid:2) Φ , ¯Φ (cid:3) d vol = 0,implying that if J ∧ F D = 0 for F D = 0, (cid:2) Φ , ¯Φ (cid:3) = 0. Thus Φ cannot be in theCartan and has to take the simple form: h Φ i = (cid:18) φ +2 (cid:19) pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 9 This is a T-brane background. The gauge group is broken to SO (4). Given thatthere is no U (1) left, one should not see any D-term contribution if the effectivefield theory is expanded around the correct background. However by expanding thebrane action around this vacuum expectation value in the presence of backgroundfluxes (soft SUSY-breaking scalar masses) one finds the same uplifting term in (3).
3. Inflation3.1.
Slow-roll inflation
The emerging picture from COBE, WMAP, Planck and BICEP is a striking sim-plicity since:(1) The scalar fluctuations are Gaussian;(2) The spectral index is almost scale-invariant: n s ≃ . ± . (3) There is no evidence for tensor modes: r < .
11 at 95% CL. This picture can be elegantly described by an early epoch of accelerated expansiondriven by a scalar field. The most popular scenario is slow-roll inflation which isrealised when: ǫ ≡ M p (cid:18) V ′ V (cid:19) ≪ η ≡ M p V ′′ V ≃ (cid:18) m inf H inf (cid:19) ≪ V ≃ H M p . The duration of inflation tosolve the flatness and horizon problems has to be: N e = 1 M p Z φ in φ end √ ǫ dφ & P S ( k ) ≃ A S k n s − A S ≃ H inf π √ M p √ ǫ ≃ · − n s − η − ǫ ≃ − . r ≡ A T /A S = 16 ǫ < .
11. This upper bound can be translated also into H inf < GeV and M inf = V / < · GeV ≃ M GUT . String inflation
Given that recent Planck data can very well be explained by a simple slow-rollinflationary model with a canonically normalised inflaton field, why should one tryto embed inflation in a complicated theory as string theory? Because inflation isUV-sensitive, and so one has to embed it in a complete theory of quantum gravityas string theory in order to trust any inflationary model building.The UV-sensitivity of inflation is related to the necessity to obtain abnormallyflat potentials. This is the so-called η -problem which is very similar to the hierarchy pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 10 problem for the Higgs which asks why m H ≪ M p . Similarly for the inflaton onecould ask why m inf ≪ H inf if there are no symmetries protecting the inflatonpotential and controlling Planck-suppressed operators of the form:∆ V ≃ λV ϕ M p ⇒ ∆ m ∼ λ VM p ∼ λH ⇒ ∆ η ≃ λ ∼ O (1)Moreover observable gravity waves require a trans-Planckian field motion of theinflaton due to the famous Lyth bound: ∆ ϕM p ≃ p r . which implies ∆ ϕ > M p for r > . V ( ϕ ) = V + m ϕ + ϕ ∞ X i =0 λ i (cid:18) ϕM p (cid:19) i String inflationary models mainly divide into two classes: (1) Open string inflation: the inflaton is generically a brane position modulus.There is no symmetry solving the η -problem, and so all these models involvesome degree of fine-tuning. There is also an upper bound on the inflaton rangefrom the size of the extra dimensions which leads to the prediction of unde-tectable tensor modes.(2) Closed string inflation: the inflaton is in general an axion or a volume modu-lus. There are approximate symmetries solving the η -problem and models withdetectable tensor modes.From the previous discussion, we have learned that, in order to trust inflation, theinflaton should be a pseudo Nambu-Goldstone boson with a flat potential (overtrans-Planckian distances for large r ). Moreover, all the other fields should bedecoupled from the inflationary dynamics (for example by making them heavy). Thesymmetries that can be used can be:
Abelian yielding single field inflation, or non-Abelian leading to multi-field inflationary models, which are however disfavouredby the non-observation of non-Gaussianities. In turn Abelian symmetries can beeither a compact U (1) in the case of axion inflation, or a non-compact rescaling inthe case of inflation driven by volume moduli. Compact Abelian pseudo NG bosons
In the case of compact Abelian pseudo Nambu-Goldstone bosons, the inflaton is anaxion and the symmetry is a U (1):Φ → e iα Φ Φ = ρ e iθ θ → θ + α Trading θ for ϕ via the canonical normalisation θ = ϕ/f , the periodic shift sym-metry becomes ϕ → ϕ + αf . This is broken by effects of the form V e ± iϕ/f whichgive rise to the following inflaton potential: V = V (cid:20) − cos (cid:18) ϕf (cid:19)(cid:21) ⇒ ǫ, η ∝ (cid:18) M p f (cid:19) ≪ ⇔ f > M p pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 11 It is however very hard to get a trans-Planckian f in a low-energy effective theorywhich is fully under control. Nevertheless there are complicated models with aneffective trans-Planckian f which give large tensor modes of order r > . Sometechnical control issues in these models are: (1) Renomalisation of M p due to N light species running in loops: δM p ∼ N π M p (2) Corrections to the effective field theory(3) Decoupling of all fields orthogonal to the inflationary direction by making themheavier than the inflaton Non-compact Abelian pseudo NG bosons
The most common non-compact Abelian pseudo Nambu-Goldstone bosons used asinflatons are volume moduli which enjoy a rescaling symmetry of the form: Φ → e α Φ Φ = ρ e iθ ρ → e α ρ The canonical normalisation ρ = e ϕ/f yields a non-periodic shift symmetry ϕ → ϕ + αf . Notice that the effective field theory is under control when ρ ≫ ⇔ ϕ ≫ f , implying that ϕ ≫ M p is a natural regime for f ∼ M p . Moreover thedecoupling of the fields orthogonal to the inflaton is easier because of the no-scalecancellation which gives a mass to the S and U -moduli at tree-level keeping the T -moduli massless. The symmetry breaking effects which generate the inflatonpotential look like V e ± ϕ/f , yielding V = V (cid:0) − e − ϕ/f (cid:1) . The phenomenologicalimplications of this kind of potentials are: ǫ ≃ (cid:18) fM p (cid:19) η and η ≃ − (cid:18) M p f (cid:19) e − ϕ/f < ⇒ ǫ ≪ | η | ≪ r ≃ (cid:18) fM p (cid:19) ( n s − ⇒ r ≃ . (cid:18) fM p (cid:19) for n s ≃ . f , and so different predictions for r : • K¨ahler moduli inflation: f ∼ M p / √V ≪ M p ⇒ r ∼ − • Fibre inflation: f ∼ M p ⇒ r ∼ . • Poly-instanton inflation: f ∼ M p / ln V ⇒ r ∼ − Strings and power loss at large scales
The typical potential of models where the inflaton is a K¨ahler modulus involvesalso a steepening region for large values of ϕ : V = V (cid:0) − e − ϕ/f + δ e + ϕ/f (cid:1) . Inparticular the potential of Fibre inflation is very similar to the Starobinsky modelsince f = γf Staro with γ = 1 / √
2. The corresponding dual version is R − γ + R .Hence Fibre inflation provides a scalar-tensor theory which is the prototype of aworking UV completion of the Starobinsky model since δ ≃ g s ≪ R n> -terms are suppressed. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 12 Moreover the positive exponential term can provide an interesting explanation ofa possible power loss at large angular scales. In fact, after fitting Planck precisiondata at ℓ >
50, one can predict the CMB power at ℓ <
50, finding a suppressedpower at low- ℓ with around a 10% deficit at about 2 σ . This can be obtained instring inflationary models if the positive exponential becomes important just afterthe first 60 efoldings of inflation ( N e ≃ g − / s ≫ ). This gives adeparture from slow-roll and subsequently, for larger values of ϕ , the effective fieldtheory is not under control anymore. This steepening of the inflationary potentialgives a power loss at low- ℓ which can be intuitively understood by looking at theslow-roll expression of the amplitude A S (large scales) ≃ V / M p V ′ ≪ − .Interestingly, a power loss at large scales is a typical and generic feature of modelsof just enough inflation. In fact, a model-independent analysis of any non-slow-roll background evolution prior to slow-roll inflation has revealed a high degree ofuniversality since a power loss at large scales occurs for most common backgrounds:fast-roll ( w = 1), matter ( w = 0) and radiation dominance ( w = 1 /
4. Post-inflationary string cosmology4.1.
Reheating from moduli decay
After canonical normalisation the potential for the moduli around the minimum canbe written as V = m φ with m ∼ m / ∼ M soft ∼ O (1) TeV. During inflationthis potential receives an extra contribution of the form: V = 12 m φ + cH ( φ − φ ) ∼ cH ( φ − φ ) for m ≪ H inf Thus φ is displaced from φ = 0 during inflation. The equation of motion ¨ φ + 3 H ˙ φ + m φ = 0 shows that φ behaves as a harmonic oscillator with friction. At the end ofinflation the friction wins, and so φ is frozen at φ = φ . Reheating from the inflatondecay ( φ is the lightest modulus different from the inflaton) leads to a thermal bathwith temperature T and H ∼ T /M p . The Universe expands and cools down, andso H decreases. The field φ starts oscillating when H ∼ m and stores an energyof the order ρ φ ∼ m φ ∼ H M p ∼ T ∼ ρ rad . However φ redshifts as matter as ρ φ ∝ T while the thermal bath redshifts as radiation as ρ rad ∝ T . Thus φ quicklycomes to dominate the energy density of the Universe, and so dilutes everythingwhen it decays at H ∼ Γ ∼ m /M p giving rise to a reheating temperature of theorder T rh ∼ p Γ M p ∼ m p m/M p .This picture leads to a non-standard cosmology from strings. Focussing on m φ >
50 TeV to have T rh > T BBN ∼ φ causes several modifications: • Axionic dark matter is diluted if T rh < Λ QCD ≃
200 MeV. If T rh & T BBN onecan have f a ∼ GeV without tuning the initial axion misalignment angle. • Standard thermal LSP dark matter gets diluted if the reheating temperature isbelow the freeze-out temperature, i.e. T rh < T f ≃ m DM / ∼ O (10) GeV. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 13 • Baryon asymmetry produced before φ decay also gets diluted. This can be apromising effect for Affleck-Dine baryogenesis which tends to be too efficient. • Non-thermal dark matter gets produced from φ decay in two different ways: (1) Annihilation scenario for T rh close to T f : an abundant initial productionof dark matter is followed by an efficient annihilation. In this case the LSPhas to be Wino- or Higgsino-like.(2) Branching scenario for T rh close to T BBN : a smaller initial production ofdark matter is followed by an inefficient annihilation. In this case the LSPhas to be Bino-like.
Non-thermal dark matter
In order to understand if the lightest modulus decay leads to thermal or non-thermaldark matter, one has to ask what is the generic value of T rh from strings. This ques-tion can be answered by considering generic features of string compactifications: (1) SUSY breaking generates m φ ;(2) Moduli mediate SUSY breaking to the MSSM via gravitational interactions,and so M soft = k m φ with k a model-dependent constant of proportionality;(3) Since m φ >
50 TeV, one can get TeV-scale SUSY only for k ≪ k ∼ O (10 − ) from loop suppressionfactors or k ∼ O (10 − − − ) from sequestering effects; (5) For M soft ∼ O (1) TeV, the reheating temperature can be written as: T rh ≃ m q m/M p ∼ k − / M soft q M soft /M p ∼ k − / O (10 − ) MeVFor 10 − ≤ k ≤ − this gives 10 MeV ≤ T rh ≤
10 GeV which is below thefreeze-out temperature for LSP masses between O (100) GeV and O (1) TeVsince: 10 GeV ≤ T f ≃ m DM ≤
100 GeVHence we conclude that string compactifications tend to give rise to non-thermaldark matter. Let us have a look at its production mechanism in the annihilationscenario. The decay of φ dilutes thermal dark matter enlarging the underlyingparameter space, and reproduces dark matter non-thermally as follows: n DM s = (cid:16) n DM s (cid:17) obs h σ ann v i thf h σ ann v i f (cid:18) T f T rh (cid:19) where (cid:0) n DM s (cid:1) obs ≃ · − (cid:16) m DM (cid:17) and h σ ann v i thf ≃ · − cm s − . Clearly,in order to reproduce the observed value one needs h σ ann v i f = h σ ann v i thf ( T f /T rh ).Since T rh < T f , we have to consider the case with h σ ann v i f > h σ ann v i thf leadingto Wino/Higgsino-like LSP dark matter. For Bino-like LSP we have h σ ann v i f < h σ ann v i thf which yields dark matter overproduction. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 14 Non-thermal CMSSM
Let us now study the phenomenological consequences of a non-standard cosmolog-ical history in the CMSSM case with non-thermal dark matter. After imposing:(1) Radiative EW symmetry breaking and a Higgs mass around 125 GeV;(2) No dark matter overproduction;(3) Present bounds from colliders (LHC), CMB (Planck), direct (LUX) and indirect(Fermi) dark matter searches;the observed dark matter content turns out to be saturated for T rh = 2 GeV and a300 GeV Higgsino-like LSP. Moreover the masses of the supersymmetric particlesresemble a typical natural SUSY spectrum: m ˜ g ∼ − m ˜ t ∼ − χ , ˜ χ and the chargino ˜ χ +1 are almost degenerate. This model hasa clear LHC signature: neutralino production via vector boson fusion. All thiscan be realised in string models with sequestered SUSY breaking. Axionic dark radiation
A generic feature of string compactifications is the presence of light axionic degreesof freedom which is unavoidable in string models where not all the moduli are fixedby non-perturbative effect. This leads to the production of axionic dark radiationfrom the decay of the lightest modulus. In fact, the moduli are gauge singlets,and so they do not prefer to decay into visible sector fields and might have non-negligible branching ratios into light axions. This results in a non-zero contributionto the effective number of neutrino-like species N eff which parameterises the energydensity of radiation as: ρ rad = ρ γ (cid:18) (cid:19) / N eff ! However there are tight bounds from observations, N eff = 3 . +0 . − . at 95% CL, which would give a central value of order ∆ N eff ≃ .
5. Planck 2015 data reducedthe inferred amount of dark radiation to N eff = 3 . ± .
32 at 68%CL. Howeverone should take into account that Planck 2015 data are in slight tension with theHST value of H together with the fact that N eff is positively correlated with thevalue of the Hubble constant.Thus when φ decays, it produces both SM particles and axionic dark radiation.These axions are relativistic, and so behave as radiation even if they are not inthermal equilibrium with SM particles since they are very weakly (gravitationally)coupled. Hence they free-stream to present day. Given that the temperature ofthe thermal bath is T γ ∼ T rh ∼ m φ p m φ /M p while the energy of the axions is E a = m φ /
2, the ratio between the two energies is: E a T γ ∼ s M p m φ ∼ (cid:18) GeV m φ (cid:19) / pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 15 This ratio is retained through all cosmic history. Therefore, if the lightest modulusmass is around 10 GeV (often associated with low-energy SUSY in many stringmodels), for T γ ∼ − eV these axions today have an energy of order 100 eV.Hence we have the prediction of a Cosmic Axion Background (CAB) with energiesin the soft X-ray wavebands. Let us stress that this prediction comes from verygeneral properties of string moduli since it relies just on the existence of massiveparticles with only gravitational couplings to ordinary matter.This CAB can be revealed via axion-photon conversion in coherent magneticfields induced by a Lagrangian of the form: L = − F µν F µν + 12 ∂ µ a∂ µ a − m a a − a M F µν ˜ F µν Notice that M ≥ GeV from supernovae cooling bounds. The axion-photonconversion probability in a plasma with frequency ω pl is given by ( L is the coherencelength of the magnetic field): (1) P a → γ ∼ (cid:0) B LM (cid:1) for m a < ω pl (2) P ′ a → γ ∼ P a → γ (cid:16) ω pl m a (cid:17) ≪ P a → γ for m a ≫ ω pl In order to have a large conversion probability we need therefore large values of B and L . Promising astrophysical objects where this condition is satisfied are galaxyclusters which have a typical size of order R cluster ∼ B ∼ − µG with L ∼ −
10 kpc. The ICM plasma frequency is of order ω pl ∼ − eV, implyingthat axions with m a ≫ − eV (like the QCD axion) give rise to a negligibleconversion probability. CAB evidence in the sky
A substantial soft X-ray excess in galaxy clusters above the thermal emission fromthe ICM has been observed since 1996 by several missions (EUVE, ROSAT, XMM-Newton, Suzaku and Chandra). Its statistical significance is very large and atpresent there is no astrophysical explanation which is completely satisfactory. Thetypical excess luminosity is about L excess ∼ erg s − . Given that the CABenergy density is ρ CAB = 1 . · erg Mpc − (cid:0) ∆ N eff . (cid:1) , the soft X-ray luminosityfrom axion-photon conversion becomes: L a → γ = ρ CAB P cluster a → γ = 3 . · erg s − (cid:18) ∆ N eff . (cid:19) (cid:18) B √ µG GeV M (cid:19) (cid:18) L (cid:19) This can match the data for ∆ N eff ≃ . m a < − eV and M ∼ GeV. The 3.5 keV line
Recently several missions have claimed the detection of a 3 . pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 16 (2) Perseus and Andromeda (XMM-Newton); (3) Perseus (Suzaku); On the contrary, this 3 . (2) Stacked galaxies (XMM-Newton and Chandra); The origin of this line could be astrophysical if it is due to a new atomic transitionin the ICM plasma. On the other hand, the simplest particle physics interpretationinvolves a dark matter particle with mass m DM ∼ F i DM → γ ∝ Γ DM → γ ρ i DM ⇒ F i DM → γ F j DM → γ ∝ ρ i DM ρ j DM fixedNonetheless the signal strength from Perseus is larger than for other galaxyclusters, and Coma, Virgo and Ophiuchus. (2) Inconsistent morphology of the signal: one would expect a non-zero signal fromeverywhere in the dark matter halo but the signal is stronger from the centralcool core of Perseus, and Ophiucus and Centaurus. (3) Non-observation in dwarf spheroidal galaxies: dwarf galaxies are dominated bydark matter, and so they should give the cleanest dark matter decay line butno line has been observed from these astrophysical objects. Alternative explanation: DM → ALP → γ An alternative explanation of the 3 . . m DM ∼ The dark matterdecay into axions could be induced by couplings of the form:a) ΦΛ ∂ µ a∂ µ a ⇒ Γ Φ = 132 π m Λ b) ∂ µ a Λ ¯ ψγ µ γ χ ⇒ Γ ψ → χa = 116 π ( m ψ − m χ ) m ψ Λ The predicted photon flux is: F i DM → γ ∝ Γ DM → a P ia → γ ρ i DM ⇒ F i DM → γ F j DM → γ ∝ ρ i DM P ia → γ ρ j DM P ja → γ ∝ (cid:18) B i B j (cid:19) It is interesting to notice that observational data can be matched for the samevalues which reproduce the soft X-ray excess: m a < − eV and M ∼ GeV. pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 17 Moreover this line with a B -dependent strength can explain all the anomalies listedabove:(1) Since the photon flux depends on both dark matter density and B -field, astronger signal from Perseus is explained by its large magnetic field(2) The morphology of the signal is explained by the fact that the B -field peakesat the central cool core in galaxy clusters(3) The non-observation from dwarf galaxies is due to the fact that L and B aresmaller than in galaxy clusters. This has been predicted in Ref.15 and after-wards confirmed in Ref. 39(4) The non-observation in galaxies is again due to the fact that L and B aresmaller than in galaxy clusters. This effects has also been predicted in Ref.15and afterwards confirmed in Ref. 40(5) The observation of the line in Andromeda could be due to the fact that thisgalaxy is almost edge on to us, and so axions have significant passage throughits disk enhancing their conversion probability before reaching us.
5. Conclusions
Let us list the main topics discussed in this talk: • Globally consistent chiral models with full closed string moduli stabilisation; • dS vacua compatible with the presence of chiral matter; • Phenomenological applications: SUSY breaking, TeV-scale soft terms, inflation,dark matter and dark radiation • Difficulty to build robust inflationary models with detectable tensor modes sincethis requires ∆ φ > M p • K¨ahler moduli as promising inflaton candidates due to the presence of an effec-tive shift symmetry from the extended no-scale structure • Largest value of tensor-to-scalar ratio of order r ≤ .
01 in models where theinflaton is a K¨ahler modulus • Generic power loss at large scales for models of just enough inflation • Reheating driven by the decay of the lightest modulus • Non-standard cosmology characterised by the dilution of thermal dark matter • Production of non-thermal dark matter • Non-thermal CMSSM with a 300 GeV Higgsino-like LSP saturating the darkmatter content for T rh = 2 GeV • Generic production of axionic dark radiation • Prediction of a cosmic axion background with E a ∼
200 eV that is detectablevia axion-photon conversion in astrophysical magnetic fields • The soft X-ray excess and the 3 . pril 5, 2016 0:27 WSPC Proceedings - 9.75in x 6.5in Cicoli page 18 Acknowledgments
I would like to thank R. Allahverdi, L. Aparicio, C. Burgess, J. Conlon, B. Dutta,K. Dutta, S. Downes, D. Klevers, S. Krippendorf, A. Maharana, C. Mayrhofer, D.Marsh, F. Muia, F. Pedro, F. Quevedo, M. Rummel, K. Sinha, R. Valandro, A.Westphal and M. Williams for fruitful collaboration on the topics covered in thistalk.
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