JHEP08(2010)045 and Springer h June 3, 2010 July 20, 2010 : : August 10, 2010 : , e number of light , Received iverse a little bit darker. Accepted G. Tasinato ype-IIB string flux com- 10.1007/JHEP08(2010)045 . Published a bi-spectrum maximized nongaussianity at a level, f,g e generated in the standard doi: simple example of such post- verse. s broad category often have a milton ON, Canada tring cosmologies such as these he curvaton scenario. We argue lly relevant, whose contributions ry evolution of primordial fluctu- stitut der Bonn, Universit¨at Canada nary construction, using a K¨ahler Published for SISSA by F. Quevedo, rg, [email protected] d,e , [email protected] , [email protected] , M. G´omez-Reino, c M. Cicoli, a,b Strings and branes phenomenology Inflationary scenarios in string theory often involve a larg [email protected] i Dedicated to Lev Kofman, whose untimely passing makes the Un (10), potentially observable by the Planck satellite, with ≃ O [email protected] [email protected] Perimeter Institute for Theoretical Physics, Waterloo ON, Department of Physics & Astronomy, McMaster University, Ha Nußallee 12, D-53115 Bonn, Germany E-mail: Abdus Salam ICTP, Strada CostieraInstitut 11, Theoretische f¨ur Physik, Trieste Heidelbe Universit¨at 34014, Italy Philosophenweg 16 and 19, 69120Bethe Heidelberg, Center Germany for Theoretical Physics and Physikalisches In Avda. Calvo Sotelo 18,DAMTP/CMS, Oviedo, University Spain of Cambridge, Cambridge CB3 0WA, U.K Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany Theory Division, CERN, CH-1211 23, Gen`eve Department Switzerland of Physics, University of Oviedo, NL i b c e g d a h f nongaussianity in string inflation Non-standard primordial fluctuations and Abstract: C.P. Burgess, selection of scalars that areto light enough the to primordial be fluctuation cosmologica spectrumway can by compete the with inflaton.f thos These models consequently often predict scalar fields, whose presence canations enrich the generated post-inflationa during the inflationaryinflationary epoch. processing We provide within a anmodulus explicit as the string-inflatio inflaton within thepactifications. framework of We LARGE argue Volume T that inflationary models within thi that the observation of such athat signal predict would a robustly multi-field prefer dynamics s during theKeywords: very early uni by triangles with squeezed shape in a string realization of t I. Zavala Open Access JHEP08(2010)045 4 6 9 1 4 20 21 22 23 13 14 16 18 19 22 23 25 13 15 19 condi- smology, od agreement with the of the great wealth of large- e distribution of the Cosmic lar kind of initial conditions. tial promise was subsequently rved. s generated by quantum fluctua- NL f NL f – 1 – 1 4 χ φ ] that have recently revolutionized our understanding of co 2 ] was initially proposed as an elegant way of obtaining these , 5 1 – 3 6.1 First example:6.2 small volume, Second large example: larger volume, smaller 5.1 Amplitude of5.2 adiabatic fluctuations Nongaussianities 5.3 Constraints from Big-Bang nucleosynthesis 3.2 Dynamics of the curvaton field 4.1 First scenario 4.2 Second scenario 2.1 The field2.2 content The kinetic2.3 terms The potential 3.1 Dynamics of the inflaton field 1 Introduction Standard Hot Big Bang cosmologyscale provides a observations good [ description A String versus Einstein frame 6 Explicit set-ups 7 Conclusions 5 Dynamics after inflation: the curvaton mechanism 4 Moduli couplings to visible sector fields 3 Dynamics during inflation Contents 1 Introduction 2 The system under consideration tions as the outcomereinforced of by still-earlier the observation dynamics. thattions curvature But of perturbation the this inflaton ini fieldMicrowave can Background get (CMB) imprinted on inalmost-scale-invariant the the and temperatur Gaussian much spectrum later that universe, is in obse go but it only does soCosmic if inflation the [ universe is started off with a particu JHEP08(2010)045 g ]. ] for reviews). 43 42 , – ] — or a field from 42 34 28 , ].) We find we are able d so far is the effective V) scenario. (For other 27 onably good theoretical 55 progress, sparked by the uch models is the miscon- – ] (see [ e relative positions of BPS usually designed this way, lest single-field inflationary ard mechanism, the subse- gfully to cosmological data. of these models are usually stic picture of the at-present 53 33 rd to the present time with duli, it is possible to under- in this paper we take the first – ] between string inflation and tentially cosmologically active comes to thermal equilibrium ontrivial use of the presence of imordial fluctuations within a ima as the inflaton rolls. Such of string inflationary scenarios on they can acquire significant g from the inflaton field itself. ies required proved to be much ons for CMB observables. This t during and after the inflation- d cosmological history between 29 It is largely the removal of this 44 ry. This allows the construction ains plausibly independent of the ory, with the role of the inflaton tures are produced. onary models [ on can robustly convert into adi- nity. This prediction is sometimes ios see [ me perturbations, because they pre- ]. 52 ], or Wilson lines [ – 26 45 – – 2 – 20 of the relevant fields, and therefore to be sure that all agree in its predictions with single-field models, includin must ], or of a brane and antibrane [ 19 – 6 In order to investigate the robustness of such predictions, An unfortunate consequence of the focus for convenience on s Nowadays, various string inflationary models are under reas Obtaining the desired inflationary expansion within a reali One feature common to the string inflationary models explore in particular a prediction of vanishinglyheld small up nongaussia as a potential observational way to discriminate [ shortly after adiabatic perturbations with the desired fea isocurvature fluctuations which post-inflationaryabatic evoluti perturbations, swamping thoseAlthough contributions somewhat comin more history-dependentquent than evolution of is the the resulting adiabatic stand details fluctuations of rem cosmic evolution provided only that the universe steps towards exploring other mechanismsconcrete for string generating inflationary pr modeldiscussions based of nongaussianities on in a string LARGE inspiredto scenar construct Volume such (L string inflationary frameworksthe by large making number n of scalarary fields epoch. that are If generically these presen fields are sufficiently light during inflati ception that string inflation alternatives to inflation within string theory [ branes [ the closed-string sector — such as a geometrical moduluscontrol, [ and developed to aIn level particular, that can because be mechanismsstand compared now meanin the exist cosmological evolution to of stabilize mo ill-understood dynamics appropriate to theharder very than high expected, energ however.understanding The of modulus last stabilization decadeof within has calculable string seen theo inflationary some configurationsplayed within either string by the an open-string degree of freedom — such as th dict only adiabatic fluctuations,minimal which sensitivity can to be theinflation details evolved and forwa of now. the Itwell-captured, poorly-understoo ex is post because facto, of by this simple that single-field the inflati implications absence in them of isocurvaturedespite fluctuations the in fact that the most predicti scenariosscalar involve more field than during one po thewith inflationary all of epoch. the non-inflatonconstructions Indeed moduli greatly models simplify sitting the in are calculation their of local late-ti min the motion of fields otherpredictions than the for inflaton the do not evolutionpotential ruin of the theoretical curvature simp error perturbations. thatsufficiently now reliable makes for the comparisons predictions with observations. JHEP08(2010)045 y- regarded ]. 73 olume (LV) scenario ing post-inflationary ]; and the modulation eve it for the first time 58 us is that the epoch of consequence of the prop- on of a sizeable level of – e precision of this kind of 56 ly generated by the inflaton visible sector [ at form long, thin triangles: e to curvature perturbations estions about reheating and e present epoch (such as the ot tell us about microscopic me sort of multi-field system le degrees of freedom (d.o.f.). way familiar from single-field fferent ways by powers of the s underlying parameters, and he post-inflationary epoch. It string-inflationary framework. ry. But if such nongaussianity d curvature fluctuations, that erties are subject to a myriad ]. In particular, moduli K¨ahler els for which the string scale is vere constraints on the hidden ism [ ng bi-spectrum, consequently, is while those for large cycles tend 83 dentification of cosmic strings as istence of a suite of moduli, whose – ]. V ]. These models are convenient for / 80 78 p 79 M ]. Recent studies of reheating at the end 72 – 67 – 3 – ], as well as the utility of warping for channeling ], although we do not yet have an explicit example , in string units [ 66 6 s – 77 – /ℓ 63 74 = Vol 1 V (10); a level detectable by the Planck satellite. The underl ≃ O ] scenarios. In particular, the main models we present can be NL f 62 – 59 We perform our search for these mechanisms within the LARGE V In the models studied here the size of the nongaussianity is a The most interesting such difference is the generic predicti Because the observed primordial fluctuations are not direct Our search for models uses two generic mechanisms for achiev For a recent comprehensive review on nongaussianities see [ 1 this purpose for several reasons.masses First, they naturally predict come the ex withextra-dimensional a volume, hierarchical suppression infor di small cycles tend to arise with masses of order physics in this muchfluctuation generation detail. and its Whatsimilar aftermath they to are the most described ones likely by we so describe. would tell of modulus stabilization for Type IIB string vacua [ robustly predicted to bethe dominated by so-called triplets squeezed of limit. momenta th erties of the geometry ofshould the really extra dimensions be in observed string with theo these properties, they will n nongaussianity, ing imprinting of theis adiabatic characterized fluctuations by takes place agenerates in non-linear nongaussianities t relation of local between form. scalar an The correspondi are in particular notmodels. tied to This could the ultimately slow-rollnot allow parameters string in in inflationary conflict the mod with thesupersymmetry demands breaking of scale) particle [ physicswhich during does th so. energy into the observed low-energyof sector closed [ string inflationsector similarly dynamics reveal in the order need to to allow set an se efficient reheatingtheir of properties the in general depend differently on the variou in a fully calculableof string constraints set-up, imposed where by thesetup the required is underlying prop a UV necessarythe consistency. preliminary ultimate for transfer Th of asking energySimilar more from studies detailed the for qu inflaton brane-antibrane to inflationa observab allowed potential the late-epoch i signature [ isocurvature to adiabatic conversion: themechanism curvaton mechan [ as explicit realizations ofThe the idea curvaton that mechanism such within modulus a is dependent not effects could in contribut itself new. What we accomplish in this work is to achi JHEP08(2010)045 ]). 91 these These ], using 2 . 90 4 τ , that provides uce acceptable ] (see also [ 3 ] and [ ). These are: a τ 90 4 89 ] as closely as possible. – , τ 3 90 85 , τ 2 n [ o.f., which we take to be iew the field content and ion 5, where it is shown (10). Section 4 then gives , τ namics is of interest. We alabi-Yau manifold having bes the inflationary setup umes are measured by the 1 d so that the volumes are del may seem contrived, it O ded in order to exhibit the τ Moreover, after inflation its ( vature modes into adiabatic anisms of this type within a he modulus for a larger cycle f interest: aton, and so would plausibly e of the small blow-up cycles. ne of the models presented in ; and an inflaton V ices for underlying parameters ovided these fields are assumed V ons. This construction exploits el of nongaussianity). The spirit = ciated with moduli dynamics and ] or a large one [ 89 V – gned to follow ref. [ 85 . Because of the LV ‘magic’ this can be 3 > τ – 4 – 4 together with a blow-up mode, τ 2 ]. τ ≫ 90 or smaller. Second, the couplings of these moduli to 1 2 / 3 > τ V 2 / τ p 57. Our conclusions are briefly summarized in section 7. M ]. Finally, inflationary mechanisms are already known using ≃ 84 NL , f 81 , ; the volume modulus, 1 73 τ -dependence of the masses and couplings can be such as to prod V -dependence of the couplings of these fields to observable d. The paper is organized as follows. In section 2, we briefly rev To construct our models we splice the frameworks developed i V We apologize for the slightly opaque notation, which is desi 2 curvaton field, to get masses of the order adiabatic fluctuations. Sectionto 6 get then a explores feel several forthis cho the section range predicts that is possible for observables. O 2.1 The field content The model requires us toat choose a least compactification the based following on 4 a moduli, K¨ahler C whose dynamics are o have the desired hierarchystabilized of with masses the if hierarchy the fluxes are adjuste framework of the LVin compactifications. these models, Section which 3 minimally then involve descri 4 moduli: of our construction iswell-developed, to modern provide string set-up, an in existencestabilization which proof issues can asso for be mech analysed.actually uses Although the at minimaleffects first amount we sight of are the interested ingredients mo in. that are nee a modulus of a smallas (blow-up) cycle the as (curvaton) the field inflaton,the that keeping t fact acquires that isocurvature these fluctuati have moduli extra-Hubble like to fluctuations be imprinteddecay light on rate relative their to to the profiles. radiationfluctuations, infl has with the the correct right amplitude (and value a to sizeable convert lev isocur models, with the inflaton being either a small cycle [ done using hierarchies among thethe input fluxes that are at most The curvaton mechanism inthat the this framework is explored in sect localized on a brane wrapping either the curvaton cycle or on observable fields at late timesto can be reside plausibly on estimated a pr branevarious moduli (or [ branes) that wrap the cycles whose vol 2 The system underWe consideration start with afollow discussion throughout of the the conventions of system [ whose inflationary dy the standard LV stabilization mechanism for the volume JHEP08(2010)045 ]. d 92 to a and ]. Its at its 4 3 e and τ τ 89 V – e 85 on this field bilized at its ], and for this ould lead 93 It is heavy during uding a fifth cycle , During inflation its 4 , together with two 82 2 5 volume. d 4-cycle whose VEV (by string loop effects ing. However now the τ of 4 moduli K¨ahler and, tion. since these two cycles in- ount of nongaussianities. ongaussianities. 7-branes wrapping ector supported on a non- 2 ondensation on the hidden ery model-dependent way. τ and e to a tiny gauge coupling). D ios: 1 ]). The advantage is that now τ 7-branes [ which is heavy during inflation: ]. 96 (and D n-perturbative effects and chirality [ 1 5 73 τ τ ]). The potential depends on it through , it maximizes the strength of the coupling 1 89 τ reduces the strength of the coupling of the – 5 -terms as in [ 85 τ D – 5 – 7-branes needed to generate the string loop poten- and 7-branes wrapping this cycle. D 1 D τ , that plays the role of the inflaton field, as in [ 4 τ To have these four moduli we consider a Calabi-Yau three-fol , playing the role of curvaton field and wrapped by a stack of , that mainly controls the overall extra-dimensional volum 1 , that is an ‘assisting field’ required to stabilize the volum 2 τ 3 τ τ cannot be generated by an instanton since after inflation it w 4 τ The low-energy scalar potential first acquires a dependence This modulus is heavy during inflation, and remains well-sta 3 3 . 1 ]) or at the quiver locus (by τ 1, where we cannot trust the effective field theory [ 82 7-branes can support either a visible or a hidden sector in a v < i D 4 7-branes. τ h in this case wewhich need can to be make stabilized theas small system in either a [ bit in more the involved geometric incl regime of the curvaton toHowever, visible this d.o.f., is so a non-standard yieldingrigid realization the 4-cycle of which largest the tends am visible to be s stabilized large (giving ris minimum throughout inflation. tial for cannot support a visible sector due to the tension between no which is wrapped by a stack of tersect each other): this isdue the to case the with location the of the minimal visible number sector on minimum in the usual LV way (as in [ supporting a hidden sector that undergoes gaugino condensa we have a standardreproduces realization the of correct the ordergeometric of visible separation magnitude sector between of on the a gauge rigi coupl sector supported by aVEV stack is of few times the logarithm of the volume. non-perturbative contributions generated by a stack of inflation, and its VEV is proportional to the logarithm of the non-perturbative potential is again generated by gaugino c reason it likes to remain light during inflation. curvaton to visible d.o.f., so yielding a smaller amount of n D through string loop contributions sourced by the 3 τ These The potential for i) A fiber modulus, 2. Visible sector wrapped around a blow-up mode 1. Visible sector wrapped around the curvaton cycle 4 3 5 ii) A base modulus, iv) A second blow-up mode, iii) A blow-up mode The compactification. with a K3 fibration structure controlled by two moduli, We finally point out that we shall present two explicit scenar regime, JHEP08(2010)045 ent (2.4) (2.2) (2.1) (2.3) ], we heavy satisfy 90 3 i τ a being the N and ] and [ s we take the V by the constant 89 – . We choose the i 85 A sulting potential in a , with 4 and kinetic terms in the τ ants y flat regions of the poten- utions associated with the the fields are hierarchically stein frame): , 7 branes (with of refs. [ is the metric K¨ahler for the , and # to put the kinetic terms into D . 4 ˆ ξ 2 1 T τ 4 ! tten and are likely to be absent a + ) 3 2 ) − / V 3 e i M π expansion compete with the leading 4 ( " τ i (2 ′ χ A γ 2 α 3 s + g 2 ln (3) 4 =3 3 2 i ζ − X corrections) for the effective low-energy 4D T ]: 3 ′ − a − = α ]. However, for the inflationary analysis our 97 − 2 ′ = e – 6 – τ 80 α 3 1 [ τ 2 A / ξ δK √ 3 s W + 1). g

. We assume the Calabi-Yau volume when expressed + 0 , 4 α 0 τ 1 ≡ W . K ˆ ξ = ≃ and ≃ V 3 W , since in this region the potential likes to become independ τ K ), so that terms in the 3 3 τ ( > τ , relative to those of the small cycles, 4 2 τ τ ≫ V ≃ O are non-rigid cycles. The superpotential is characterized 1 and 2 1 corrections are controlled by the quantity — as usual in LV models — to be order one, and the parameters τ τ > τ ′ , at least before string-loop contributions are included. ) is the Euler number of the compact manifold. In application since they arise due to gaugino condensation on 0 to lie in the interval (0 2 4 α τ τ W ξ M and ( χ 1 π/N τ and In the superpotential we neglect non-perturbative contrib Following the LV program, our interest is in the form of the re We now use these expressions to compute the scalar potential and the non-perturbative corrections are weighted by const 1 = 2 τ 0 i large cycles, a since these are negligiblesince relative to those explicitly wri W additional blow-up modes, where the The potential K¨ahler (including the leading expect such a regime todifferent: arise in the region of field space where interest is not in thetial local along LV which minimum. the Instead potential we is seek nearb shallow as a function of where quantity quantity non-perturbative contributions from as a function of these moduli has the form [ supergravity in this case is (we work throughout in the 4D Ein rank of the associated gauge group). regime where ln enough to sit at their local minima. Following the reasoning of desired regime. 2.2 The kineticIn terms this section wecanonical investigate form. the field The redefinitions starting needed point in the regime of interest JHEP08(2010)045 , ) 3 τ 2.6 (2.6) (2.8) (2.5) (2.9) (2.7) (2.10) (2.11) (2.12) 2 4 ∂τ 1 ∂τ 3 e to the ones ∂τ 2 V ∂τ 2 / ∂ / 3 4 1 1 3 , i τ τ 2 , τ 3 2 2 i i ∂τ , , τ         V τ γ 4 2 2 ∂τ 4 √ V V 4 τ τ 4 1 1 ∂ τ 1 V 1 3 τ τ 2 ∂τ 2 4 4 are much larger than 2 , and so we rewrite ( τ τ 4 τ 1 / =3 τ ∂ 3 √ τ γ √ √ i 2 3 X 1 2 V √ 3 V i √ τ 4 , τ ∂τ / 6 ∂τ τ γ 4 4 γ 4 4 + 4 − 1 , , 2 γ 2 γ 3 4 V 1 2 2 √ 2 ) 2 3 2 3 αγ α τ 1 τ 2 ∂τ ) i ) γ 1 3 2 − V 3 i 9 τ i 3 9 − − 2 τ V 2 V αγ ∂τ ( ∂τ 2 + ∂τ √ 3 ( ( 3 τ 3 1 − i kin O , 4 τ τ τ i 2 i 4 2 3 2 ) τ 2 / L i τ τ i τ √ τ 4 2 γ √ √ 2 ∂τ ) 3 =3 3 1 i 3 V √ X τ 3 τ V √ √ V + γ αγ  αγ 4 3 γ bχ 3 ) ∂ V 2 √ V 3 − 3 2 2 γ 3 2 1 ( 2 , 8 8 3 αγ α + 2 4 τ − 3 − ) 2 9 ) ∂τ 1 γ − 2 V i 4 4 1 ( 3 V =3 =3 i i X X + + 2 2 γ kin O 4 ∂τ aχ cχ / 2 i ( 3 1 4 L 1 + + – 7 – α + τ i τ 4 ∂τ i 2 9 τ 2 2 2 ∂τ + 2 γ ) ) ) / V 2 √ 2 + 3 1 1  V αγ ∂ 2 = τ (1) , in the limit in which V 3 / ∂ i i = exp ( ∂τ = exp ( 3 ∂τ 8 3 ( O kin V τ ( τ ) τ ( 2 1 3 L 2 V 2 2 √ 2 √ g τ 2 − γ 4 1 =3 1 4. V τ 1 V i i for i 3 τ V V X − = , ( 2 2 2 2 2 8 2 1 2 2 ′′ ′′ ′′ ′′ ′′ ′′ τ τ kin O √ αγ αγ τ = + + 3 3 kin L = = 3 )         2 2 L 1 i ) ) g g 4 4 − − 2 =3 =1 2 1 1 (1) ∀ i i X X V 1 τ = 1 unless otherwise stated. − − ( O kin 4 2 ∂τ ∂τ p − − kin O √ √ ( ( L /τ L = 2 2 i 1 1 πM − τ 3 1 τ τ ¯  0 i − 8 4 p (1) the transformation K ]) we systematically drop terms that are suppressed relativ = = O 90 g kin − L √ ). At ) and 2 V − − / V ( (1 The kinetic Lagrangian to leading order therefore becomes We use units with 8 O O 6 . It is convenient to canonically normalize order by order in 4 τ where the last equality trades at while the subleading terms are where (as in [ moduli, which is given by the following symmetric matrix: at as: shown by factors where the leading term is JHEP08(2010)045 ) 1 1 4. = kj χ , V 2.9 ≪ 1 ], the M − ) ik (2.17) (2.15) (2.18) (2.14) (2.13) (2.19) = 3 2 73 / K j 1 k ∀ − V j ): P ( τ = 2. We can now 2.12 2 , ij ∼ O with j M φ = 1 V i . ) and ( also arises that is not ), where: 2 c ) 1 i , , . The first term in ( − χ 2.11 ) ), for   3 2 1 ows ∂φ b, 2 i V 4 (2.16) ( − , φ on , − r V s signs. As is shown in [ i ( 4 are neglected. Once string-loop =3 (ln 2 a, 4 kin O V =3 j = X ) j X = 3 I. 2 L − , 1 2 ions to ( j 2 3 = ∼ O ∂χ   ∀ + ), ( i 2 i + 2 χ φ M 2 ) + ( i , · χ 3 2 , c 4 =3 / ) a, b, c ! ) is subleading because 2 3 j X 2 3 ∂χ 4 j 1 ( 1 2 1 − φ 9 4 r r ∂χ 3 − 2 2.18 =1 ( / + 4 to distinguish these from the large fields, i + ), ( X – 8 – = h 2 , c 2 1 3 4 2 1 2 1  1 2 χ ), we have − χ j − 3 2 = 3 = 3 b, = V 2

) and ( ) αγ 3 j − √ 2.12 1 r · 4 g − (1) are obtained from the condition that the matrix a, T     −  , b V O kin 2.17 ) is similarly diagonalized by mixing ( c 3 √ L M kin O 2 with = ), ( √ 2.9 L j − ) and ( j τ φ and + = = exp = exp b 1 a, b, c a V 2.11 (1) τ , O kin a L ) into canonical form 2.8 turn out to also diagonalize the mass-squared matrix, 2 χ 4, while from ( and , satisfies 1 . The second term in ( 2 χ ! = 3 χ Next we diagonalize the next-order kinetic term, c j 0 a b Explicitly, introducing the following subleading correct and for concreteness we shall choose the first one will all plu in the limit wherecorrections string-loop corrections are to included the potential a subdominant dependence of puts expression ( gives to this order important for our purposes. becomes diagonal once we rescale the two small moduli as foll fields where we use the notation

This has four solutions: ( where the coefficients Notice that the last term in eqs. ( and for JHEP08(2010)045 ) 4, , 2.2 (2.21) (2.22) (2.25) (2.24) (2.20) (2.23) = 3 . i # 2 0 , for 3 i . , . ˆ ) slightly, into: ξW φ V 4 4 4 4 , appearing in the β , , 3 β 2.20 = 3 = 3 = 3 + i i i i τ ∀ i ∀ ∀ a subleading contributions ) i-de Sitter minimum to a − 2 in terms of , e i t is given by the expression 4 3 i i 4 τ 4 3 of the scalar potential, again he form of the kinetic terms.  i τ φ 2 h 2 φ i ) 4 2 τ i nd superpotential of eqs. ( , and due to additional uplifting   ess all quantities in the 4D Ein- V , a 3 2 4 The constant 3 2 j 4 3 2  τ i τ φ   h i 7 φ . 4   A =3 a A i i − 2 j ,   4 j 4 4 a 2 j φ X 6= 2 τ 0 0 i φ h − 3 4 4 =3 2 3 W j X a =3 V After minimizing the axion directions, the (1 ˆ ξ W , 4 j 9 4 3 4 π 4 X i 6= =3 β is removed by modifying ( i i X 1 + φ s 32 + ) 4 3 (1 + g 2 4   – 9 – 2   3 2 − 2 j and directly express χ / − # − V 3 φ ( 2 i ≡ 1 i 3 2 . V τ i kin O χ 4 i 0 =3   τ φ a L r 2 3 4 j V h ˆ 2 ξ X 4 # 6= −   i 2 r ]. e " 4 9 χ αγ ) in front of the potential is a consequence of an overall 90  π exp i 2 3 16 − = 1. τ   (8 exp V 1 − / r cs √ 3 2 3 2 K "   cs =   e  K i ) to eliminate 2 3 i 2 i e β i s V exp 3 A 3 αγ g αγ 2 i 3 3 2 2 4 αγ 2.16 4 a 3    8  i i 3 3 ≃ = 4 αγ αγ =3 ) in ( i X ], as is explained in detail in appendix 4 4 i " τ   84 2.18 cs ≃ ≃ K π i e ) (and neglecting subleading powers of large moduli) is 8 τ s g -independent, term, 2.4 τ Finally, the off-diagonal term in = From now on we shall set 7 V last, terms needed to upliftnearly the Minkowski minima vacuum. of Its the value can potential be from easily an found, ant and i substitute ( following the discussion of ref. [ The field redefinitions we have determined render canonical2.3 t The potential We next chase these field redefinitions through the definition The potential without loopscalar corrections. potential constructed usingand ( the potential K¨ahler a includes contributions due to the stabilization of the field stein frame [ normalization of the superpotential, that is needed to expr Notice that passagecontrolled from by the higher order first powers to of the second line neglects obtaining The overall factor of JHEP08(2010)045 ), ]: and 90 1 2.26 (2.31) (2.28) (2.27) (2.26) (2.29) (2.30) τ ], spoil 7-branes 90 D s as free to (the amount ], whose size . , 82 0 4 4 , τ 4 a = 3 − B > e i if ∀  2 4 , at the following values ) does not depend at all τ 3 V V rocedure identical to [ / ,  2 i i 0. It is important to notice tic approximation is valid in 4 τ  2.23 h ing loop level [ t follows. , i A V a 2 2 4 0 e B A a V π 4 i W B > 0 i 2 π s τ ,  8 g W h 2 s , with respect to its value in eq. (
 p g ] which, as pointed out in [ 3 can evolve independently in field space. i ≃ 1 0 τ 2 1 at the minimum. 95 2 KK − 1 2 τ V . However it is possible to fine-tune the
– W τ h 4 Cτ and the volume 4 ]). Hence we shall focus only on the 4 C τ τ We next identify that part of the potential τ  s 4 93 4 KK + 1 W 73 i a τ Each cycle wrapped by a stack of i 12 2 , and C 1 αg A , − i s τ 3 – 10 – αγ 4 82 e αC g a τ B τ 3 √ 4  to their minima, and follow the dependence of the V  4 = = 4 = 2 3 τ τ V − = 0 or √ A C B 2 1 A  τ , and so and hVi 2 4  4 4 B < τ are constants that depend on the details of the string loop A V + 2 4 if ]. It does not do so because the dominant contribution to the a π αγ , s 90 3 is at: g WW [ 2 2 3 . What is noteworthy is that eq. ( 3 1): 2 1 / C 1 / ! 2 τ + τ 3 i ≫ 0  ˆ i ξ V /a αJ V -dependent loop correction in order to render it negligible τ ] for more details). In what follows we regard these constant i C 2 i 4 γ 2 B = τ , and a 90

are given by − W =3 12 V 4 i we denote the value of the field =  C C i P i 4 , , i τ τ h B i ≃ does not depend on = h 1 KK i 1 , i τ a 1 h J C A τ h This potential completely stabilizes The minimum for -dependent loop corrections which can be estimated using a p 2 coefficient of the (here we assume of fine-tuning needed has been estimated in [ In the following, for definiteness, we consider the case where by but this small correction does not modifyThe the canonically discussion normalized tha potential. relevant to inflation. We set be fixed using phenomenological requirements. The potential with loop corrections. the flatness of the inflationary potential for on the fibre modulus, potential of large moduliwe such now as estimate. these first arises at the str String loop corrections also shift the minimum for that where rest of the potential on the remaining two fields. This adiaba τ where where corrections (see [ receives 1-loop open string corrections [ JHEP08(2010)045 . 3 φ (2.39) (2.34) (2.41) (2.33) (2.40) (2.36) (2.32) (2.42) (2.43) (2.38) (2.35) whose 4 φ , ), that they , we expect #) C 3 / , 2.35 4 4 is parametrically i φ and 1 1 arger than those of 3 ˆ χ χ / B 3 3 2 4 / , √ .  2 A − 4 tionary potential breaks  e 2 4 V 2 αγ = 0 3 φ A/B . 4 C 4 4 2  ) (2.37) V + C ≡ 1 αγ 4 3 2 1 4 a q χ ˆ χ (ˆ " − 3  1 √ 1 − cur 2 ≃ − , V C ( , thereby ensuring that it is 1 e . 1 , 4 ˆ  χ 3 1 3 + 2 3 χ 2 φ / / 3 3 ) + 2 1 C 4 φ 0 with exp √ / 4 B − ensures, within the limit ( 2 − + ˆ C e 4 9 φ 3 − i ( / i 1 ≪ 1  4 4 1 − ˆ C q A q B q i χ ⇒ = 0 at the minimum of the potential, it follows inf χ φ χ 1 3 , 1 h h 2 – 11 – V 1 ) √ 3 χ = = =  AC / h χ V e = 2 0 2 1 q 3 0 3 appearing in the constants 32 1 = 0 / are given in terms of their canonically normalized  ) = C C C 2 2 C 1 χ s √ 4 h 4 ˆ g neglects the subleading dependence on the modulus χ  V τ  =0 , 2 4 αγ 4 3 3 3 ln ( 1 4 φ / τ 4 ˆ 2 χ 0 the inflaton because the potential for φ √

4 10 (  exp / 4 W and  V relative to that for V V 4 3 φ s αγ 3 / 1 1 g π 4 A V 2 τ cur = 1 2 ˆ 8 4 χ / V  a V ∂ i ∂V 0 1 π = = ) = χ  2 W h 1 4 1 s τ τ χ g (ˆ − cur 0 V V has been defined such that ˆ ]) ≃ limit because the masses of these fields are parametrically l 1 the curvaton and ) 90 χ 4 1 V φ χ ( inf Recall that the fields With this information, the leading contribution to the infla Keeping in mind the factors of Since ˆ V with (see [ and the approximate equality for the large- where satisfy the fields whose motion we consider. where we define into a sum of terms for the would-be inflaton and curvaton counterparts by We call suppressed by powers of 1 and energy dominates the cosmic expansion. in weak coupling, and in this case one finds that the dependence of the constants on JHEP08(2010)045 ), ant 2 3 0 / 2.47 W (2.47) (2.48) (2.45) (2.44) 10 t V C ansion of π s 8 sses of the g . 2 p ≡ ] M 2 at the potential’s 2 3 0 C / W 10 have masses that are ) can be set to their t . Considering, as an 3 V 3 4 C emporarily reintroduce + 16 φ π φ φ e above formulae shows s 1 4 nsidering that the infla- g C ∼ and − 2 p 2 0 χ C . M , 1 ] ) as its field moves away from its [4 as it is at its minimum, eq. ( τ 2 2 p 0 one finds 2 0 3 C √ V / M 3 ), our benchmark during inflation is 2 / 2.46 W 0 , are moved away from their minima 3 10 n 1 V s n > 2 − 0 4 (2.46) W V g χ + 16 π , (which is mostly 3+ s V π 4 W 1 ( g , the fields V 3 very small, for which the exponentials in s C τ V 24 π g ∼ = 3 and 1 4 i − 1 , with χ 4 ∼ O = 2 χ 0 ∀ V ≃ φ – 12 – 4 1 C τ 4 ) and 2 χ 4 that is the largest, this is the field whose evolution 2 φ [4 2 a ) ln have masses that are smaller, justifying the picture 4 m χ − m n 3 1 φ e 4 2 φ , − ≡ 2 p ]). (2 + t with M remain light enough to have cosmic fluctuations imprinted C and 89 ∝ V > 2 – 1 4 , m 1 2  4 φ χ 2 p ˆ 85 inf τ 0 χ 4 , is absorbable into a subdominant contribution to the const V V 1 M 0 W a 2 χ 2 3 2 0 and  V m cur, 1 W s V π π s , while χ g mass changes and becomes smaller for larger ). We check in our later applications that this quadratic exp 4 4 g + H For inflationary applications our interest is whether the ma 0 mass remains of the same order in 1 4 ∼ ∼ φ 1 2.38 i 2 . Relative to this consider the following masses, evaluated cur, χ 2 φ 2 χ 2 V / m 3 m (which is mostly given by ≃ − V ) V ) can be expanded up to quadratic order, 1 2 p χ then their masses arethat potentially the modified. Inspection of th If the inflation and curvaton fields, so the inflaton mass is reduced relative to eq. ( example, a regime minimum (as in ref. [ while the If all the fields sitthe at dependence their minima, on the the mass Planck spectrum mass): is (we t (ˆ M 2.39 We see from these estimates that for large • • in formula ( cur ∼ V 0 minimum: H The constant piece, In the following we work in regimes with ˆ Field masses. the potential suffices in the regime of interest. which defines the new constant V various fields areton larger potential or is smaller of than order the Hubble scale. Co much larger than dictates the end of inflation and so earns the name inflaton. eq. ( minima while both wherein on them. Since it is the potential for JHEP08(2010)045 1 χ ole , (3.5) (3.1) (3.2) (3.3) (3.4) enario ]. The #) 3 / 89 4 4 – φ ]. , become 85 3 4 ). If the field / 60 (3.6) . Within this τ 2 90 0 acquires a scale  V ≥ 4 2.39 1 4 V χ are already at their αγ 3 , dτ ] and [ 4 4 3 4 ple because these two ) τ parameter becomes of τ φ  4 4 89 during inflation can be 4 ntial is a τ ǫ – 4 ) and ( a 4 4 − 2 a a τ e 85 " − 4 s of the field and τ e r
− 2.38 4 the isocurvature fluctuations − 2 2 bove regime, together with a 4 a 2 d by the value of the volume, ulable amplitude, during this τ ) χ e ( .( (1 2 4 4 arameters remain small. In the τ a 4 4 τ exp a √ in 4 + 4 3 τ − / 4 4 , in 4 4 4 τ such that the a τ end 4 (1 4 φ V τ a 4 3 Z ≤ 9 / τ end 4 2 4 4 τ τ 2 ln − √ much smaller than one, we choose at horizon 4 γ  4 2 4 1 0 4 η A a ≫ A 4 4 V – 13 – 2 4 αγ ≤ 3 a 4 φ ˆ ξW in 4 4 a τ 2 and τ 2 4  V √ 0 2 ǫ that drives inflation, as in the model of [ αβ end a 4 0 drives inflation, and how the field 4 V 9 τ W 4 64 4 4 A 4 2 − φ 2 4 a 3 φ A β β W a V 4 2 V 0 = 4 ˆ a ξ π 2 ). W ˜ α φ ˆ 16 ξ γ 512 s d 4 g α γ need not be. We then consider the evolution of the moduli 9 2.24 ′ has the properties required for it to realize the curvaton sc inf inf − 1 are lighter than the Hubble parameter. The former plays the r 27 − V V 1 χ 2 0 1 χ 3 = = χ -foldings is given by the integral in 4 e end 4 φ V ˆ ǫ ξW η φ in eq. ( and π Z β 0 and 4 s 32 V φ g 4 = 3 φ e N ) = 4 . As we pointed out before, the analysis is comparatively sim φ 4 into adiabatic perturbations after inflation ends. ( φ The number of The corresponding slow-roll parameters, expressed in term We start with the hypothesis that the massive moduli In this section we recap how 1 inf acquires a large value, the dominant term in the inflaton pote χ V 4 fields evolve almost independently: see the potential in eqs in this system. We find these two fields combine the results of [ order one. These considerations imply that the field range fo showing again that thebeing scale given of by inflation is mainly controlle parametrized as We next discuss thediscussion properties of of whether slow-roll inflation in the a 3 Dynamics during inflation exit limit of large volume, in order to have Slow-roll inflation lasts as long as the previous slow-roll p regime, both inflationary potential is φ and On the other hand, inflation ends at a value of inflaton field, while the latter is the candidate curvaton. 3.1 Dynamics of the inflaton field independent spectrum ofinflationary isocurvature epoch. fluctuations, The of nextof calc sections discuss how to convert minima, while In the scenario just described it is JHEP08(2010)045 1 Q ), ∼ ∼ χ χ 1 1 e at (3.9) (3.7) (3.8) χ ≤ 2.44 (3.11) (3.10) δχ 1 χ than the in the right ≤ H when ). This im- , 9 7 3.8 − ], but our approach ation the field 10 98 × . and that the inflaton 7 much lower . . On the other hand, Q ) and ( 2 π ]. In one Hubble time χ 2 h must be supplemented 3.6 / 98 inate the classical motion. , ∼ ≪ ⋆ s quadratic, as in eq. ( cenarios that satisfy all the lowing constraint: 2 1 0 ons is H nge the field value by ∆ in 3 4 χ from the successful realization τ 6 V r 4 cause it is so light it also under- ∼ ˆ , ξ W a V π 2 to lie in the interval 0 . 0 1 β e 6 4 1 32 s / W δχ 2 − 7 χ g lar the value of the power of i for discussions on this point. 2 .  V / 10 3 3 ⋆ t 0 4 ≃ ) ˆ C H ξ A × W 2 ⋆ 8 β (  2 H ≃ . 5 parameterizing the displacement from the minimum, 2 2 ) / 1
1 ≪ 1 Q χ χ in – 14 – 4 ≪  χ 2 τ 0 ( s π 4 2 3 0 g 10 8 ′ a ′ W inf cur 2 t  W V / V V − 3 3 inf C 3 π p ) ≃ 1 V s 8 ˆ ξ , given by ]. In this work, we follow the prescription of [ g M Q Q β ). Fluctuations dominate classical evolution in 99 4 χ ( 2 χ ⋆ ≃ τ from the field ˆ 4 4 1 H = a 2 χ (3 1 α γ hat q m / χ 4 2 can fluctuate by an amount 3 , is lighter than the Hubble parameter during inflation, sinc / ′ cur 1 3 4 1 V a χ χ 6 − 8 4 =  ⋆ s π ). See for example [ t g 8 ’ indicates a quantity evaluated at horizon exit. During infl  ) ∆ 3.10 ⋆ ⋆ ). H (3 2.34 / , the light field , which occurs when We now estimate when fluctuations dominate, following [ Because we seek the dominant contribution elsewhere, we dem A successful model must satisfy both of the constraints ( In the present case, approximating the curvaton potential a ′ This estimate has been debated in the literature, in particu From now on we drop the 1 1 cur 9 8 − χ ⋆ V hand side of ( − ∆ slowly rolls classically towards itsgoes minimum at quantum zero, fluctuations but that be in some circumstances can dom where the ‘ Substituting the potential, we find at horizon exit can be adapted to different possibilities. We thank Sami Nurm one finds H 3.2 Dynamics of the curvaton field poses conditions on someby the of constraints the derived parameters inof of the the following the curvaton sections mechanism. model, coming conditions whic We to discuss have in a section successful 6 curvaton model. explicit s contribution to the poweramplitude measured spectrum by of the curvature COBE satellite. perturbati This gives the fol large volume during the same time interval a classical slow roll would cha During inflation quantum fluctuations cause the field with uniform probability. Then, its typical value is of orde in eq. ( The curvaton field, JHEP08(2010)045 1 ]. χ 84 , (3.15) (3.14) (3.13) (3.12) 81 , le or even 73 cannot decay 1 that justify the χ V / ], and to determine ]. ate to visible gauge 73 trum for the curvaton 73 ure fluctuations into adi- which is suppressed with rvaton) and all the other e it has at horizon exit: ψg bution to its mass. Thus . these fluctuations are never- . ating forces to have for each (see the previous discussion). ads e to directly compute the cou- r inflation and reheating have ψ 1 powers of 1 6 . inflation [ 3 / χ Q / 7 nd curvaton decay rates into vis- → χ 1 t V . 1 V = 0 . / C ≃ ˆ χ ξ ⋆ 2 1 ⋆ = 0  χ δχ 1 π β 1 which are massless before the EW phase , one finds the following equation for the ′ χ 1 cur ′′ δχ ≃ cur ψ V V  δχ ⋆ ] that since the fermions are massless, there + ⋆ satisfy the same equation, their ratio does not + ′′ 73 1 = cur H π χ 1 1 ˙ – 15 – χ V 2 χ ˙  δχ H ≃ 1 1 7-branes wrapped on internal 4-cycles [ = H by a phase space factor. In addition χ D δχ is suppressed by 1 and ′ 1 + 3 cur 1  1 /χ gg Q + 3 and fermions 2 1 ¨ δV χ χ / 1 δχ g 1 δχ → ¨ P δχ 1 χ , but only a 3-body decay ]. The classical evolution equation for the curvaton field is ψ ψ = 1 SYM theory that develops a mass gap [ 100 → N 1 is so light that it cannot decay to any supersymmetric partic χ . But because 1 ⋆ χ H ≫ Q χ In the next section we discuss how to convert these isocurvat We now estimate in more detail the amplitude of the power spec In the case of the curvaton, we have to focus only on its decay r An important feature of the LVplings framework is between that the it moduli is possibl (amongvisible which or the hidden inflaton d.o.f. and localized the cu on abatic curvature fluctuations when thealready curvaton taken decays place. afte 4 Moduli couplings to visible sector fields expansion of the curvaton potential up to second order in and so where in the last approximate equality we suppose fluctuations, following [ inhomogeneous curvaton fluctuation at superhorizon scales is no direct decay respect to the 2-body decay ible d.o.f. allowing us towhether understand reheating a at curvaton the mechanism end can of be successfully developed bosons. In fact This is a necessary ingredient for calculating the inflaton a Then the power spectrum of fractional field perturbations re to the Higgs since this receives a large SUSY breaking contri Since, for a quadratic potential, theless very small at large volume. A posteriori, it is these evolve in time. This means that this ratio keeps the same valu Making the first order expansion transition. However it has been shown in [ can only decay to gauge bosons to light hidden d.o.f.hidden since sector the a pure requirement of a viable rehe JHEP08(2010)045 i 7- D (4.4) (4.5) (4.2) (4.1) (4.3) , this 5 , using g ow to fix the ) mass. tter in the adjoint s see in section e fixed by the use of M e coupling as ( O of the field theory living l gauge kinetic function . ions. Indeed, the previous 25, we find constraints on n scenario with the smallest µν ≃ e following advantages: G maximizes the strength of the . 2 1 µν , and go to the canonically nor- 3 τ / τ G , 2 7s of interest wrap a 4-cycle whose i π/g  + ˆ τ µν . 4 D h i V p ˆ F τ τ µν V B 10 A 125 M µν F 4 i 4 F → h coming from constraints on the size that is τ  p = h − 1 τ τ τ M ≃ p A µν B 1 4 − – 16 – τ G cycle, and analyze the decay of the inflaton and = 2 = = µν 2 τ µν G 2 π decouple from the EFT getting an 4 1 g 4 G 2 gauge τ − and L 1 = τ we imagine the observable sector to be localized on a stack defined by gauge L 2.1 µν around its minimum G (which can be any of our moduli): the couplings with the modul τ τ ]). In full generality, the kinetic terms read: 81 7-branes, we proceed as follows. The ), we have D (see [ 2.31 τ = expected for the observed gauge coupling. Denoting the gaug eq. ( brane deformation moduli thatrepresentation. would Here give we rise shallbackground to fluxes. assume unwanted that ma these moduli can b the parameters that characterize therelation string implies loop contribut coupling of the curvaton to visible gauge bosons. As we shall Focusing for definiteness on a GUT theory, will yield the largest amount of nongaussianities. number of moduli K¨ahler which is 4; 2 7-branes wrapped on the In order to analyse the coupling of moduli to the gauge bosons Assuming that the gauge bosons on 2. There is a constraint on the volume of 1. The K3 fiber is not a rigid cycle and so one has to worry about h 2. The geometric localization of the visible sector on 1. It represents the simplest example of multi-field curvato D 10 π/g can be worked out4 from the moduli dependence of the tree-leve volume is given by However there are also some shortcomings: of on a stack of Doing so, we obtain: curvaton fields into visible gauge bosons. This set-up has th and we must expand malized field strength As explained in section 4.1 First scenario JHEP08(2010)045 , l, : V g and (4.7) (4.9) (4.6) (4.8) esent 1 and χ 1 τ 11 cycle. ]: 1 whose decay τ 73 4 ˆ φ o an extremely and fewer ˆ , 2 4 on the ), we obtain ˆ χ φ ) into gauge bosons 2 / ) we find [ s, this condition is rela- 2.29 1 ϕ 4 arge hierarchies between i rms with both , p , the masses of the fields V ∀ ble gauge bosons living on 1 ( lowing. 3 4 ), ( M i e amplitude of modulation- ) χ 2 τ O = 3 / V dominated by 1 i + ˆ ological consequences. It is for + V 2.28 ∀ i i 3 i 3 (ln a W ˆ 12 , φ τ V 2 i ) h C , ˆ 2 φ / = 3 ϕ 1 ): α i fluctuations. Expanding the canonical 2 in eqs ( s − m p τ g (1) µν π 1 4 V -particles plus some ˆ 2 is the total number of gauge bosons: for M ( B τ 125 4 F 2 λ 64 ˆ g ˆ 2 3 φ χ O g , = N p 5 N + q and 2 V 2 M
A – 17 – = χ p ≃ KK 1 1 M gg (1)ˆ 2 1 2 ˆ 3 p C χ 3 χ → O √ ϕ M m . + Γ π 1 W 12 1 is introduced once string loop corrections are included. 4 χ µν C 1 -dependence of couplings and masses to use the modula- )ˆ 1 χ 3 = / (1) χ 1 F and − gg = 12 as in the MSSM. We obtain, for our set of fields, V → g (1) µν ( 1 ], at the end of inflation, due to the steepness of the potentia KK ] to generate the primordial fluctuations. Although in the pr 1 ˆ F N χ C Γ 61 106 ∼ O , ) around the global minimum ( 4 ˆ τ 59 stops oscillating just after two or three oscillations due t 2.22 4 τ . Couplings between moduli and gauge bosons for a field theory is the coupling listed in table 1 and λ tively easy to satisfy. We dothe not parameters have to choose unnaturally l As we see in the following, when discussing explicit example from which, using the definitions of As studied in [ We can now derive the total decay rate of a generic modulus The following table summarizes the moduli couplings to visi Because the light curvaton field mixes through its kinetic te Table 1 The subleading dependence on ˆ (denoting the corresponding field strength as -particles. Therefore the energy density of the Universe is 11 3 1 ˆ φ realising that the Universe is mostly filled with efficient non-perturbative particle production of do. However, it turnsgenerated out fluctuations that is too in smallthis all to reason cases have that we interesting we investigated cosm focus th on the curvaton mechanism in the fol to visible d.o.f. is responsible for reheating. τ instance the couplings do not depend on the fluctuations of normalization ( the inflaton where one might hope to use the definiteness we choose tion mechanism [ JHEP08(2010)045 4 τ , the . (4.15) (4.10) 5 (4.12) (4.13) (4.14) (4.11) 2 n total τ 1 / 1 χ Γ can shrink and ∝ 5 τ 1 τ . ]). In this case . . NL 0 ( 2 f p p p 2 / 82 , with the following 5 which is stabilized V V 5 M M M → 5 V 5 τ in [ τ ale as: 5 ≃ ≃ . τ ≃ 4 , RH RH of sudden thermalization too small (given that the RH ole in the viability of the T T a curvaton scenario with a T volume dependence. Notice ocus = 3 e inflaton decay rate, in the h the visible sector localized tion between an in the first scenario. This j . ∀ p ⇒ ⇒ 2 ee in section ⇒ is wrapped by the visible sector. V M zzo 4-cycle without any constraint 4 , τ , , , ≃ p p p 4 4 2 p 5 2 V V V / M M M V M 1 so that the non-perturbative corrections  ≃ ≃ ≃ , p 5 ≃ ). However, the blow-up mode supporting 2 τ 2 p p j V / M gg gg gg 3 φ 9 M M due to the tension between chirality and non- → → 7-branes wrapped around a combination of gg V → m 4 4 7-branes wrapped around V 4 ˆ ˆ 4 – 18 – φ φ D ˆ → φ τ D 3 2 4 ≃ Γ Γ Γ ˆ ) φ ]). The inflaton and curvaton total decay rates to 2 2 Γ p or V 3 χ π 96  , , 3 M , m τ 64 3 3 3 ≃ / / p p / p π 1 17 17 27 (ln 8 M 20 M ]: M RH V V V T = = 73 ≃ ≃ ≃ gg gg gg gg gg → → j 2 → → -terms as in [ → ˆ ˆ 1 1 ]. Hence we need to introduce a fifth modulus φ χ 1 ˆ ˆ χ χ ˆ Γ Γ D χ is not wrapped by the visible sector. The inflaton and curvato 92 Γ Γ Γ 4 ]: τ 73 with chiral intersections only on 5 are not destroyed. In this case the inflaton τ 4 τ to zero size by in the geometric regime (for example by string loop effects as decay rates to gauge bosons scale as: gauge bosons scale as: the inflaton in The inflaton and curvaton total decay rates to gauge bosons sc and 1. Visible sector built with a stack of 3. Visible sector built via fractional branes at the quiver l 2. Visible sector built with a stack of It is interesting to notice that, due to the geometric separa limit of largemechanism. volume. The This reheating observationturns temperature plays out to an in be important the [ r approximation where we have emphasized, in thethat extreme the right, curvaton the decay dominant rate is suppressed with respect to th the visible sector cannot be either three possible brane set-ups [ coupling of the curvaton toyields visible a gauge lower level bosons of is nongaussianities weaker since, th as we shall s VEV of blow-up moduli does not depend on 4.2 Second scenario In this section weon briefly a present a small different blow-upstandard brane cycle, realization set-up showing of the wit that visible it sectoron on is the a possible rigid overall del-Pe to volume build to keep the gauge coupling from getting perturbative effects [ JHEP08(2010)045 , i de φ me time . 2 on the relative − llate coherently a reaches a region of t fall this fast. For 4 φ is essentially set by the rastically towards the end at, recall, in our set-up is h a conversion depends on ensity into radiation, after rvaton one. In this regime, ell below the Hubble scale. h happens once the Hubble ike non-relativistic matter: . Since this is the dominant verting isocurvature fluctu- hile the curvaton oscillates. er falls like 4 tion at the point where the , the two fields are very light the adiabatic fluctuations of hat the energy density of the − vaton masses are much smaller nflationary model. Moreover, in this process focusing on the : the inflaton is only few times a ons that had been stored in the aton field ∝ sses of these two fields turn out to , then in a sudden decay approxi- γ γ ρ /ρ . However it is easy to re-formulate our cur 1 ρ τ ) — and so these moduli are also the first of – 19 – ), that receives contributions from the uplifting, 4.11 2.23 )). At this point the curvaton energy density also converts 4.9 during inflation, while the inflaton and curvaton fields provi 0 V dependence of the decay rates found above, the small moduli, dominates the energy density of the Universe, while at the sa V in the potential ( 4 φ falls below the curvaton field’s mass this field starts to osci 0 V H . Since this is much slower than the energy density of radiati 3 − a ∼ Energy tied up in the curvaton field, on the other hand, need no Because of the The total size of the adiabatic fluctuations inherited by suc This continues until the curvaton field starts to decay, whic 1 χ than the Hubble parameter.constant The piece value of the Hubble parameter analysis for the second scenario. 5.1 Amplitude of adiabatic fluctuations During the inflationary process, both the inflaton and the cur have the largest decay rate — see eq. ( the potential, where itsthe mass inflaton becomes field much larger than the cu instance, once the curvaton energy density remains subdominant with mass w In this section, we summarizeations the into curvaton adiabatic, mechanism curvature forwe fluctuations con estimate in the resulting the levelfirst above of scenario i nongaussianity with produced the visible sector localized on 5 Dynamics after inflation: the curvaton mechanism be not too different duringmore massive the than slow-roll the inflationary curvaton. period and In evolve any independently case, one duringof inflation from inflation: the slow-roll other. conditions Things are change violated, d and the infl negligible contributions. Typically, in our set-up, the ma curvaton field. This convertsthe the radiation energy curvaton density. fluctuations into the size of thecurvaton decays. curvaton energy Denoting this density fraction relative by to Ω the = radia into radiation, bringing with it any isocurvature fluctuati the moduli to decay.which This its decay energy converts density falls the with inflaton the energy scale d factor like and from the stabilizationuniverse of is additional dominated fields. by This means t component of the energy, after this point the Hubble paramet around its minimum, duringρ which its energy density scales l proportion of curvaton energy to radiation energy can growparameter w becomes comparable tothe the most suppressed: curvaton see decay eq. rate ( (th JHEP08(2010)045 ent (5.8) (5.3) (5.4) (5.5) (5.6) (5.7) (5.2) (5.1) . The ), and 1 : χ 5.6 NL f ngaussianity ), allows to ex- of eq. ( ζ 5.3 , , a . , 1 1 # arly well-suited to the 1 3 0 ), indeed, we implicitly
≪ ≫ . 3 W V 2 0 / 0 5.3 3 6 Ω 2 Ω ) / s that e measured by COBE then V ˆ 1 s W ξ n the fluctuation of g 2 2 β V βξW / ) ( f curvature fluctuations, in the local form definition of 5 t 2 ˆ ⇔ ξ re fluctuations. Indeed, it reads ⇔ g eq ( 2 / 5 / C 3 t β en scalar fluctuations, and curva- 1 s . ( 2 on given in eq. ( g C nce case. 2 ) 0 , / 1 2 1 2 1 s 2 = 10 G W χ g ζ √ δχ 768 . 2 ( 3 ) π " 1 / ˆ ξ 2 1 2 NL χ V + χ f β √ V ≃ 3 ρ ( δρ 1 576 3 5 ξ 4 1 s 2 a g / 3 χ 3 δχ Ω 1 s # + 2 ≃ β g 1 2 / – 20 – 0 5 G t 1 =  ζ 16 C W 1 = 2 /χ χ ζ 1 χ = ≃ m 1 Γ 2 1 1 δχ χ 679 2 3 ζ χ t  ), one can read the following expression for P ρ δρ C 2 Ω ≃ 5.5  ]: 2 3 p ⋆ 5 χ 4 Ω M 3 for curvaton dominance = 100 /  2 1 ) substitutes the value of the various quantities in the pres ζ = 1 6 P = 4 = 1 for radiation dominance 5.1 , which imposes the constraint " 1, is [ 5 a NL a f − ≃ ≪ ( 10 Ω × ], it is not difficult to provide an estimate for the amount of no 8 . 100 is a Gaussian curvature fluctuation. This Ansatz is particul = 4 1 2 G in which Ω ζ ζ P 12 It is easy to re-express each quantity in the curvaton domina 12 5.2 Nongaussianities Following [ Demanding this converted amplitude agreegives with the amplitud mation, we find: complete expression, generalizing the linear order relati comparing with the Ansatz in ( express the curvature fluctuation as a first orderhibit expansion the i non-linear connection between scalar and curvatu where predicted in this scenario. We focus on nongaussianities of scenario. The resulting expression forlimit the power spectrum o In our case, since we work with a quadratic potential, one find present context, since there isture a perturbations non-linear produced relation betwe after inflation ends. In writin Consequently, including this second order expansion in the The last equality in ( with: JHEP08(2010)045 , NL g (5.9) (5.11) (5.12) )). But in in order to 1 5.7 , for example turn out to be /χ potential, small )–( NL 1 f NL δχ 5.6 g owing inequality or n the volume of order ), becomes important in ameter ration NL ) and ( NL ]. It would be interesting 5.7 f BBN takes place, at around τ iency of the conversion pro- . gaussianity. 5.3 3 ( ided in the literature: see for 5 urbations. The value of 10 103 e for curvature perturbations, / p 5 ts on the curvaton model. This is inversely proportional to the 1 , V  M 3 0 NL ≃ − GeV (5.10) 102 ! f W 3 24 0 2 NL / 2 − 1 / is the typical one for models where only W 5 − 2 10 ) / A π 3 NL t ∼ 2 τ , in formula eq. ( C 1 B 2 ). This expression quantifies the amount of / 2 16(2 ], or in which one obtains a sizeable running of 3 / s /χ BBN 3 s – 21 – 1 g that characterize the trispectrum. Expressions 5.4 g

104 ), we obtain  δχ from WMAP7 [ 7 > H NL , g ≃ g 4.9 10 gg NL 2 ]. NL gg f f × → within the approximation we are considering, we get an → 1 1 and 7 ˆ χ 105 ˆ 1 3 χ 36 25 Γ Γ − < NL = τ V B q , turns out to be too low for being detectable by Planck, given ). The expression for NL ≃ NL τ f 5.8 t C ] for a recent review. For our curvaton model, with quadratic 101 given in eq. ( 1 MeV. In order to satisfy this constraint, we impose the foll . 8 NL ∼ f 10 It is also possible to analyse nongaussianity beyond the par BBN large, e.g. as in the model discussed in [ with Now, recalling that where in the last step we use relation ( upper bound on the volume, T 5.3 Constraints from Big-BangBesides nucleosynthesis the requirements ofBig-Bang providing nucleosynthesis the (BBN) correct imposes further amplitud since constrain we must ensure that the curvaton field decays by the time to extend the model above such asnongaussianity, as to analysed find in set-ups [ in which comparison with the linear term, implying an increase of non decay rate Ω and in the sudden decay approximation, one finds nongaussianity in this set-up. Notice that the size of this case, the quadratic contribution in For standard valuesV of ≤ the parameters, this imposes a bound o discussing the parameters account for the observed amplitude of fluctuations (see eqs. conversion factor Ω. Thiscess, is by expected: decreasing Ω, if we we have decrease at the the effic same time to increase the for these parameters, inexample curvaton [ scenarios, have been prov one species contributes to the generation of curvature pert Using the expression for Γ given in ( being proportional to the already stringent bounds on JHEP08(2010)045 , - ) 4 5.8 (6.1) s. Be- ). Since COBE ζ . The COBE 142. P . Also, eq. ( W 12 2 8 ∼ C − t 10 C 10 ≃ parameter is of order t ≃ ǫ C us sections suggest that d in example 1 below. ongaussianity it predicts: le becomes too low) then inf ζ 4 we never allow ourselves to ≤ V ≤ ngaussianities of small size. γ 20 P e present two representative ctuations using the curvaton ameters, but the requirement 3 3 ( / Plugging these parameters in 1 m must satisfy in order to fur- ypical values of the parameters preliminary sense of how much m in this regard. Consequently 4 ng theory, discussed in section ) then fixes V 1 10 A / ly high in this example. Next, the 1 5.4 56). Moreover, there is a small infla- ∝ 6 α 2 s ≃ . This example is characterized by not- e . 2 /ℓ 0 ′ NL 1 10). 10 N α f W / 57 ≃ = 1 s 1 – 22 – , which in turn implies g . 100 4 NL 2 3 f a
10 1 ξ KK 1 10 ≃ C 4 min 1 is fairly robust in this scenario with not too large volume 10 a V in Planck units, and by a relatively small string coupling, NL 3 = 10 3 ) cannot be satisfied for volumes that are too large without re f 10 corresponds to a gauge group with large rank in the non- V 10 / 5.3 W 12 is therefore easiest when choosing relatively small volume = 10 C = 1 V NL 4 f ), and imposing that inflation starts when the a 3.8 . Set of parameters with relatively small volume, considere . Also ) and high rank gauge group ( 2 3 ) and ( − 10 3.6 , we find a sufficient number of e-foldings ( There is a simple first observation. The results of the previo The most important feature of this model is the high level of n Table 2 4 = 10 ≃ − s V Consider the choice of parameters given in table perturbative contribution toeqs. the ( inflaton superpotential. 6.1 First example: small volume, large g consider volumes smaller than once volumes are tooit large becomes (and difficult so tomechanism. the obtain Indeed, inflationary adequately eq. large Hubble ( primordial sca flu 6 Explicit set-ups The previous sections present thenish conditions a that realization our of syste parameter a choices that curvaton satisfy scenario. allobservable quantities the In vary. constraints, this to section, get a w imposes the condition too-large a volume, a curvaton scenario has ashows chance that for very volumes in largeObtaining the volumes range a are 10 usually large associatedcause with the underlying no expansion is in powers of quiring other parameters to acquire unnatural values. For t 10 ( using the previous results we find normalization condition for the curvaton fluctuations ( This value can be slightly changedof by satisfying tuning all the the choice of constraintsthe par does order not of leave magnitude much for freedo ton contribution to the amplitude of adiabatic fluctuations conditions of having an acceptable size for the gauge coupli the volume is relatively small, the scale of inflation is fair JHEP08(2010)045 . f- NL (6.2) f ). Af- COBE ζ P 3 , and imposing − 4 2 below. 10 ≃ f the inflaton and the construct a controlled e must be other fields, are the same. Plugging inf ζ The hierarchy of volume- eld slow roll, it becomes 4 P a en slow-roll conditions are aton one. These appear to 4 hange as the volume grows. ate primordial fluctuations. nflation ends. ome moduli to have masses γ 10 that are fixed by the require- ude for the power spectrum. ns ( nd providing the same amplitude ssed in section er is only few times more mas- ns into adiabatic fluctuations. istances. The second ingredient 4 ve very different values for 1 10 A 66) and a small contribution of the These tend to have the local form 306. The amount of nongaussianity NL ≃ f ∼ e α 10 t N C , these features are compatible with the , 0 2 . In this example, the volume is larger with 10 5.1 W 3 ≃ during the inflationary epoch in order to have – 23 – s NL 1 g f 100 H ), with the same criteria of the previous example, we ≪ ξ 1 ), we find that 3.8 m 4 1 8 5.4 a strongly depends on the choice of underlying parameters. Di 6 ) and ( -foldings in this model ( V NL 10 e 3.6 f . Set of parameters with large volume, considered in example Table 3 The key ingredients for any such a scenario are twofold. Ther In both the previous examples, the ratio between the masses o We find that both ingredients are possible in the LV scenario. besides the inflaton, with masses In this paper westring-inflation use model LARGE that Volume doesBecause string not the compactifications use dynamics to the cannotpossible inflaton to be to generate captured gener observably by largein non-gaussianities. a the model simple examined single-fi because they are generated well after i requisites for having a succesful curvaton mechanism. 7 Conclusions find a sufficient number of COBE normalization condition ( respect to the previous example, while the string coupling a these parameters in eqs. ( ferent models characterized byfor different the volumes, spectrum although of adiabatic fluctuations, nevertheless gi As we discussed at the beginning of section curvaton is comparable during slow-rollsive inflation: than the the form latter.violated, the Instead, inflaton towards mass thebe becomes general end much features larger of of than our inflationments the set-ups, wh of and curv depend having on sufficient parameters e-foldings, and the correct amplit in this case is small: isocurvature fluctuations be generated overis extra-Hubble a d mechanism for converting these isocurvature fluctuatio suppressed modulus masses enjoyed by this scenario allows s showing that the value of ter requiring to have an acceptable gauge coupling, as discu inflaton sector to the COBE amplitude of adiabatic fluctuatio 6.2 Second example:Choosing larger a volume, different smaller set ofConsider parameters the shows set how of the parameters listed results in c table JHEP08(2010)045 ples of this type. A po- be expected also to man- ssions on related topics a -standard ways to generate ommon such models might sually good starting points ith current CMB measure- nflation. Indeed, should lo- e many moduli to work, this nto ordinary matter, assum- e of the cycles whose moduli cause of the potentially large ncouraging that non-inflaton more, contrary to most mod- nd Andrew Tolley for helpful o lower the string scale while nsity carried by the curvaton uge coupling, efficient reheat- ions, such as the modulation cale during inflation, thereby eory asks to be more than just ld be telling us: the dynamics s timely and worth pursuing. ed by particle phenomenology re mode to first accumulate as s all observable issues and not tive taken in this article is that erting its fluctuations into adia- constraints, such as the location consistent with having the right ization of the curvaton mechanism ianity for future experiments. The since this makes it easier to have a l cosmologically active fields rather ing after the inflaton has decayed to – 24 – (10) — if the amplitude is the one observed). O , naturally leading this fraction to be a small (and nongaus- V / It is worth noticing that even though these scenarios requir Because additional light fields are present these models can Ultimately, the reason such a construction is possible is be These states also plausibly have a hierarchy of decay rates i Acknowledgments We wish to thank Tony Riottoprimordial for encouraging fluctuations, us and to to explore Neildiscussions. non Constable, Sami GT Nurmi a would also like to thank Eran Palti for discu only a subset of them.generation Even with of these primordial constraints, we perturbations findamount it appears e of possible, inflation requiredments, by with potentially later observable features cosmology, like agreementimminent nongauss start w of Planck observations makes these question but may not capture the dynamicsels of of the string generic cosmology, case. we also Further of consider the phenomenological standard-model brane, theing, value and of so on. the present-day Wea believe ga this model to of be crucial inflation: because string string th scenarios must therefore addres in the later universe. Asbe ever, in it the would string be landscape. useful to know how c is the generic case insimplicity string arguments compactifications. using the The minimum perspec number of fields are u ifest other nonstandard mechanismsmechanism, for although generating we fluctuat dotential not benefit yet of these have kindsstill explicit of obtaining working models acceptably exam large might primordial belower fluctuations, the supersymmetry-breaking ability scale, t as seems to be preferr number of fields thatcal can nongaussianity be be cosmologically observed, active thisgenerating during is primordial LV probably fluctuations i what likely it involvesthan severa wou just one. sianities to be comparatively large — radiation. It can then itselfbatic decay perturbations. at much The later resulting times, picturefor conv provides a string real inflationary models.is The suppressed fraction by powers of of the 1 energy de providing a source of isocurvature fluctuations. ing that ordinary matter isappear localized in on the a low-energy brane theory. thatan overall This wraps fraction on allows of the the total isocurvatu energy density, by oscillat that are parametrically suppressed relative to the Hubble s JHEP08(2010)045 (A.3) (A.1) (A.4) (A.5) (A.2) ,      {  3 S ¯ G · 3 . G 12 Re  ) with the Einstein- ) 3 , E , ¯ ( 6 G 3!  A.4 g 09-237920). ·  tic terms for the moduli · . ¯ d by the Province of On- Ω Sciences and Engineering 3 3 of the metric of the form S q 2 E er Institute is supported in ¯ ∧ al Physics. We also thank x G G t) and the European Union V p pected but undivided time. orted by the SFB-Tansregio 6 · ). MGR acknowledges CERN π Ω d oretical Cosmology, Perimeter − flux 3 M for the the support of, and the }| 4 owing only the relevant terms): V ) frame has been explicitly shown Z G s 12 Re Z √ ): tor of the superpotential starting ( 10 anta Barbara, McMaster Univer- i 0  R − − = ) iC φ ) ) from 10D to 4D then yields:  E s 2 1 ( 4 l E + ( 10 − g ln e A.3 φ R ≡ −  − − s  , we find: ) e q ) ) s ) ( ¯ 10 x M S E E = 4 g ( ( 10 d g + − R S ) − – 25 – S z Z q E ( x q and g − ln( x 10 − E d − 10 d p . Comparing the first term in ( E E Z x 6 s V ol 2 s V V l 4 Z ) l 4 ′ π d E E 8 π 4 s ( 4 α l R 2 7 2 ln 1 R ) = ) − ≡ V 2) π ⊃ E 2 / p ) ( 4 (2 2 = ) p E D g ( M 6 E 2 p ( 10 − g E M ⊃ S K M q q D ) = ( s x x ( 10 4 6 . The dimensional reduction of ( ′ S d d : EH α R ) S Z √ E ( MN = π      g 2 8 π E s l 2 = 2 φ/ e s V ol l ⊃ ]. Given that the potential K¨ahler that reproduces the kine = ) D E 84 ( 4 ) s S ( MN where from the 10D type IIB supergravity action in string frame (sh here we shall briefly review just the derivation of the prefac A String versus EinsteinThe frame correct prefactor of the scalarin potential [ in 4D Einstein Research Council (NSERC) of Canada.part Research by at the the Perimet Government oftario Canada through through the Industry Ministry Canada, of Research an Theory and Division Information for (MRI financialTR33 support. ”The Dark IZ Universe” was7th (Deutsche partially network Froschungsgemeinschaf program supp ”Unification in the LHC era” (PITN-GA-20 pleasant environs provided by, the Cambridge CenterInstitute, for the The Kavli Institute forsity Theoretical and Physics the in S AbdusEyjafjallajokull Salam for International helping CenterCB’s to for research provide Theoretic was some supported of in part us by with funds unex from the Natural couple of years ago. Various combinations of us are grateful Hilbert action in 4D Einstein frame is known to be (with where g The action in Einstein frame is obtained via a Weyl rescaling JHEP08(2010)045 and and (A.6) (A.9) (A.7) (A.8) s E (A.16) (A.11) (A.10) (A.13) (A.12) (A.15) (A.14) K K ,  , ¯ Ω .  ∧ i E Ω Ω W ∧ E Z corrections to 3 i , ′ W G . − cs α h 2 p . . !  3 , 4 K p 4 Z s p M is reproduced by: cs can be worked out from: ln 2 s + /g M 1 l M ) ame can be worked out from: − K s ,  − (  i flux  s E = T ) + 2 V ! cs i g cs . ¯ ) 0 S a  K W π  E K π − ( s j e ¯ i ! , 8 W e 2 + 8 e s ) g T 4 s i D Ω i g E  g S ( a i E A + ln  ∧ T −  i e 3 ln( W a i ! i i − + ln and G X 2 − A e D / ), and then working out the form of j ¯ 3 i s  s ξ # i + E Z ⇒ g i 2 2 ξ ⇒ A 0 , 2 / 2 s /g = X K -term scalar potential in 4D Einstein frame can p l = 3 ) – 26 –  s i + W  F ( + +  i i 2 p X s =

and the complex structure moduli via background ¯ 0 T ¯ Ω S E V 2 3 /M p E 2 | 1 V W 2  2 | ∧ = p 2 + E p π − s E s , the Ω + W

M )

4 K g ) reduce to: Ω S 2 E M M E e h W W in 4D string frame can be derived by transforming the ∧ / π ( | | 3 2 ln √ i p 3 ) s  = W 3 4 2 2 s p p 2 ln g T Z A.9 − E ξ 2 M ( i 4 G i √ − W ) g /M /M = = − s + S E Z − = = s K  2 K s p E 2 s e 1 e q l 2 and E p V W E ). Therefore, including the leading order ln Re( K ) and ( M x h can be found from requiring that "

π s 4 W K M − 4 d p π K 3 A.8 p √ 4 = 2 ln ( Z M √ / − cs 3 p = K = = M V 2 p E E = W M K p that reproduce such a potential. We obtain: s fluxes at tree-level, ( the correct prefactor Writing the superpotential in 4D Einstein frame as: where: Stabilising the dilaton non-perturbative corrections to The expressions for obtaining Hence the prefactor of the scalar potential in 4D Einstein fr Thus the prefactor of the scalar potential in 4D string frame scalar potential (recalling that W be derived from: JHEP08(2010)045 , , ]. , , metry , ommons SPIRES (2009) 330 ][ brane inflation (2004) 129907] 7 180 the horizon, D ]. / Brane inflation, ]. n and flatness 3 ]. D 70 (1999) 72 D ]. , rmann, SPIRES ]. SPIRES mmercial use, distribution, inflationary model and SPIRES ][ B 450 r(s) and source are credited. ][ hep-th/0501185 [ rmann, [ 7 oblems ]. D SPIRES / ][ 3 SPIRES Erratum ibid. D [ Astrophys. J. Suppl. ][ SPIRES , Phys. Lett. ]. ]. ]. ]. (2005) 009 ][ (1982) 1220 , ]. 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