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The inflating curvaton

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Please note that terms and conditions apply. JCAP03(2012)022 σ ρ = 1. p can be and n ut a hysics P -folds of inflation e le h anomaly-mediated ic iverse t ligible interactions until d spectral index ar Tsukuba 305-0801, Japan , , Lancaster University, = 1 (2) that will probably be 10.1088/1475-7516/2012/03/022 David H. Lyth it is exceptionally easy to solve drives a few p b ζ doi: σ cillate while its energy density = 1 or 2 is allowed by the current ρ p ing strop may be generated by some curvaton A ζ 0013) for . 1 that may be observed, (iii) tensor fraction [email protected] ∼− , 0026 (0 . NL 0 f ≃ Kazunori Kohri, ′ a n for the first inflation; p φ ∝ V c,d [email protected] osmology and and osmology , C but the improvement in accuracy promised by Planck may rule o 1110.2951 n inflation, alternatives to inflation, physics of the early un

The primordial curvature perturbation [email protected] rnal of rnal , which is negligible during inflation and has more or less neg problem; one just has to make the curvaton a string axion, wit

σ ou η An IOP and SISSA journal An IOP and 2012 IOP Publishing Ltd and Sissa Medialab srl is probably too small to ever observed. Department of Physics, LancasterLancaster University, LA1 4YB, U.K. Laboratory of Physics, SaitamaFukaya, Institute Saitama of 369-0293, Technology, Japan E-mail: Consortium for Fundamental Physics, Department of Physics Lancaster LA1 4YB, U.K. Cosmophysics Group, Theory Center,and IPNS, The KEK, Graduate University for Advanced Study (Sokendai), [email protected] b c d a c obtained with a potential Received December 19, 2011 Accepted February 16, 2012 Published March 12, 2012 Abstract. Tomohiro Matsuda Konstantinos Dimopoulos, The inflating curvaton J uncertainty in the The predictions include (i) running susy breaking which may soon be tested at the LHC. The observe before the oscillation begins. In this scenario for generat field observed, (ii) non-gaussianity parameter it decays. In the current scenario, the curvaton starts to os is negligible. We explore the opposite scenario, in which ArXiv ePrint: r Keywords: JCAP03(2012)022 . ) ≡ ζ (1) only is the ]. As aH ζ 1 original k matter-dominated era. Repla ]) fields becomes a classical perturbation. According to the ζ ]. In this Letter we stay with the simpler curvaton scenario. 2 4 /M 1 − a . According to a standard calculation [ ], and fast-roll late inflation in [ 0 ) is the 8 = (8 ∝ a The primordial curvature perturbation H h contribute significantly to 0 P a δσ 15 ∆(ln ) can be determined, and one of the main tasks of theoretical c ) is the scale factor of the universe) when the vacuum fluctuat e is a negligible fraction of the total. Here we assume instead t is generated by the perturbation k ( ( ≡ = ζ σ a ζ and 2 ρ k N δφ Up till now, it has been assumed that the curvaton starts to os The generation of A second inflation has been discussed many times before, but a Instead of the curvaton scenario one can consider an inflaton The right hand side of eq. ( of a ‘curvaton’ field, that has practically no effect during in where of the second inflation. There is also thermal inflation [ ˙ during multi-field inflation, or by ato ‘modulating’ be field that inhomogeneous c [ Hubble parameter then and times before cosmologicalscales, scales its start Fourier to components enter the horizon initial condition the subsequent formation of large-scale dominate the totalFor while simplicity, it we demand is that still these slowly ‘cosmological scales’ varying, giving r density Introduction. denote the first (second) inflation thereafter. According to the curvaton scenario [ logical scales start outwhile within the the horizon. inflaton of Theflation the second for first infla this inflation ‘double is inflation’ still is calculated oscillating. for instanc Our scenario is different from allDuration of of these. the second inflation. or less directly by observation. They range from explore models of the early universe that can generate it. inflation model. In that case, when the second inflation begins. was considered in [ starting during radiation domination might be calledζ late i a result, to be outside the horizon, and in [ the horizon when the second inflation starts corresponds to scenario, where where the first inflation generates at least a significant part a δσ inflation and then of the radiation dominationterm by is matter positive domination reduces because th when its energy density becomes a significant fractionboth of the around with spectrum more scalar (or vector [ JCAP03(2012)022 2 − (2) (3) ion ) is a and ,t P x ( , where ) where M r σ ,t p 30 k − ( is described + ) is a unique ); that is the σ 10 σ ζ ζ ,t ,t ed by observa- ) p tion-dominated x x t > ( For the inflating ( ( a unique function σ ρ 2 = f . the locally defined p 4 or any other mech- σ H p r − will become constant ρ a gy continuity equation ) = to vary significantly. if and only if σ + ζ ,t ∝ ζ and σ σ k r ρ ρ ( r , ρ ζ ρ ) quation is valid locally. As a = t ) ( + ) that we are denoting simply 2 p ,t . We begin with the following σ is constant if ,t ) as H 2 k ρ 2 t k 2 ( P ugh it is generally very difficult to σ ( )+ ζ . h of these make ˙ t ζ = δρ M = ( ]. In this paper, we just work to first ρ ρ t ). There is supposed to be negligible 1 3 11 p does not cause ) = 0 16. r ) is a unique function of ) does not vary between curvaton decay at = + σ becomes constant only after the oscillation GeV corresponding to ρ ]: The curvature perturbation ( becomes radiation before it is a significant ,t ,t ) ′ 1 ρ , with 3 3 to σ ( r x k x ζ ) ,t = 0). When cosmological scales start to enter V ( ( / ρ t 10 ) k p ζ ( 45 p ( ˙ + σ ρ – 2 – > p allowing it to correspond to the oscillation of a ˙ It might even decrease more slowly, say like < is a unique function of σ 2 are the curvaton contributions. For the original δρ 4 / 2 ) / 3 ]. + σ H t 2 1 2 − p ( N ρ σ σ 13 a ρ H + 3 ( ) for each formalism as in [ − 3 ∝ ¨ )+ ˙ σ is generated while ) redshift away that quantity has negligible spatial gradi- ≡ σ r is defined on the slicing with uniform locally-defined scale ( ζ δN ,t ρ ) is somewhat less than 1MeV and we know that the Universe )= t 3) or matter ( V x ( ( δρ T ,t − f σ ρ/ = 0 (attractor solution). We expect that soon after the inflat k ( t = ζ = σ p p ) and σ p ]. Then 0. We now argue that for the inflating curvaton, 3 ). For the original curvaton, + 2 and , since that would give an isocurvature perturbation exclud ≃ ,t MeV, which is guaranteed if the universe is completely radia / σ ζ the proper time. Also, ). 2 σ x ρ p ( t might be matter, p ( σ ∼ after the curvaton inflation, so that is constant during any era when σ / cannot be matter and is taken to be pure radiation, + r ) (flat slicing). The second equality corresponds to the ener ) ρ T ζ ρ , as in [ r ρ σ ρ ,t We therefore require roughly )+ ˙ ρ ( ], is the pressure, and δρ x σ δρ ( responsible for the first inflation. ( H ( 12 p and ]. That requires something like ). According to the curvaton scenario, a 3 16. ]. As a result it satisfies eq. ( V 3 Keeping only super-horizon scales, the energy continuity e In these cases though, we demand that CDM and baryon number cannot be created before the curvaton ( In any curvaton scenario, To facilitate an analytic calculation, one writes eq. ( k φ giving indeed the required attractor solution. d 1 ≡ − ( t < = ζ σ σ 2 = = σ ζ ˙ begins, when function of result [ is practically pure radiation giving a time-independent ρ the horizon, the temperature non-perturbatively through the exchange of energy between the two components, so that factor soon after the second inflation begins at the epoch 3 case for pure radiation ( by where equation, valid in the absence of perturbations [ anism) creates order in quantity and begins it will be slow-roll or fast-roll (see below), and bot unique up to the choice of throughout that era. tion [ N of t ent [ fraction of Since sub-horizon modes of Calculating the curvature perturbation. ρ curvaton curvaton, corresponding to a frustratedachieve cosmic frustration string for network, altho cosmic strings [ field JCAP03(2012)022 ]) In (6) (7) (5) (4) the 17 = 1 [ p ), we where p 0 1. But . [Note H 1 -folds of becomes 2 is nearly 0 e − ≪ for a ). To first These give ( H 2 10 4. p t to be linear δσ 3) ( ]. N / 2 to V/∂σ 14 g r to during the first 2 19 σ r ≡ ρ [ 3˙ 3 ∂ ρ φ | ≪ ≃ 1 ′ / (4 term is negligible. . ) otherwise until the / n ∼ gument for the at- N ≡ 2 | ) ) , where the equality N 4 p 2 1 2 n of ,t η − σ of its value at horizon 2 N x (˙ a flation ( /H - g and H σ − P ∝ ≃ N 1 2 ) gives different from the inflaton , δρ ) M / 1 − t 1 ) is the number of 5 ) σ ( η are the fields responsible for 1 k − 10 f and 3 ( ]. Defining while cosmological scales leave ) = /ρ 1 N /H x + 2  ] which will be very close to 1, σ ≪ ( r P r 1 on to aH ζ . ln(10 ρ ρ (corresponding to monodromy ζ φ, H 014. M 4 3 1 1 2 . ǫ π 5 3) P = and 0 2 2 H / / − inflation or + − . h k 1 − = 1 to have the canonical kinetic term with ] ± 1 2 )) ) ˙ ), where H 2 p σ 60 φ = 2 18 t + (4 k [ ) to be a function t 037. While one is free to postulate any ( ≪  ln(10 ( this is compatible with . ( 2 k 037 ≡ P σ . . σ ) 0 2 ρ ( − N 0 ,t d [ n 1 ˙ ′ ln r σ ) = 1 gives t 4 = 3 / x ≃ − ( d N V ( σ 60 t – 3 – 2 1 f /d ′ p/ ( ρ σ 3 . g H f P ≃ . This formula is obtained taking leaves the horizon [ ǫ ≃ ] a large change in the inflaton field ) 1= ≃ 5 /H 1 | 2 ≃ ln − 1 ) k ˙ 16 − N d 2 H d to be time-independent while cosmological scales leave H is absent, we would probably need | t / 10 /H ǫ n ( 1 2 ζ 1 1 2 ≡ f . At horizon exit during the first inflation × P H 2 N . H ≡ 1 5 σ . Allowing − ] gives 3˙ H ln( ǫ 1 / π ) ≃ ] gives we need [ ) 2 2 1 2 . Then k 1 x p / ) and ( 2 15 may be ruled out when Planck reduces the uncertainty. For the ( 1 / n we have H ,t 1 were constant during both inflations and ζ ǫ H Aφ d x 037, leading to three important consequences. δσ with good justification are P ′ ] . 15 + ρ t ( 1 0 respectively ] σ = 1 p V ) during the first inflation where to get the required 2 )= ≃ 15 p 28 p φ 1 ( corresponding to ‘extranatural’ ) = H 14 V ǫ x and ( ≃ ≃ σ,φ, ζ decays at ( 1 , observation [ ) =2( though , 14 ) is a universal formula, applying whenever a field k σ . It can easily happen (as in our case, see below) that the last V N ], but that might not apply because although the contributio were completely negligible we could invoke a more general ar p 6 ζ δσ ≃ ln 2, r φ,σ 14 = ρ ( 1 ). 1 or oscillating curvaton, where Taking into account the uncertainty in need inflation. To achieve that one usually takes V and N the first inflation after the scale observed matter-dominated era. Combining this with eq. ( would be exact if only choices of V If Taking into account the time-dependence of Eq. ( Keeping only super-horizon modes, dn/ t ( 2. A standard calculation [ 1. To get the required σ ≡ ′ n small soon after the second inflation begins, its contributi that that inflation.] Assuming a tensor fraction with the right hand side evaluated at horizon exit tractor [ just before in Observation gives [ generates Then we need 2 may be dominant at least initially. While that is happening, contrast, the oscillating curvaton can have the horizon one finds [ exit, but taking both the horizon, we get the scale-independent spectrum order in gaussian with spectrum JCAP03(2012)022 . 1 ]. 3 1. δσ (8) (9) − 24 (10) (11) (13) | ∼ ) the ≪ 10 4 ) gives ζ NL . f 10 P | s / φ is given by ζ ) contributes ] if σ 6 ζ 22 1; for P ≡ P . This prediction 1 This choice is im- s roll approximation | ≫ . Using eq. ( /N 2 2 2. ) to second order in scenario, one expects NL big enough to observe / ), and for the inflaton H n f ible, 2 σ 1 ′ | 3 σ / − n H δρ ore the non-gaussianity of 2 σ ′′ ǫ 2 is ever to be detected [ ) ) (12) ]. V m σ ] for 3 P 3˙ 1 = (1 . Then we get NL ≃ ≡ | ≪ ′ ρ f 1 , /M erm which means that we need ) n 2 ′ η . n δσ/ σ η | . The tensor fraction will not be g ′ ≪ ) ( P 2 r )( . (The equalities hold for a single 1 if V ′ V p P cannot dominate. ρ ( ], P π 0026 we expand M 1 will be probably be tested in the future . . 1 2 /g & 2 σ 0 ) M ζ NL / ′′ ′ / H | 2 πM 1 2 2 1 f g ′ NL n P / s 2  f 1 . 5 H 5) NL 1 where 1 ) /g 1 and ( n f / − is given by ′′ 2 | 0007. For the inflating curvaton, we have for H ) the curvaton contribution NL . / rs g 037 ′ ǫ − formalism as in [ 1 f is reliable only [ 1 10 . ( ≪ 0 )(1 + 2 n 13 4 0 1 too large but let us anyway see what it implies. 2 − | ≪ ′ 2 σ + (3 – 4 – ) − √ × 2 if the first inflation has inflaton(s) with canonical ≃ . Even if the second term of eq. ( / 2  2 2 1 δN η ′ 2 (1 + ǫ δρ 6 | 10 /V N ) σ − 2 ≥ n = ′′ 3˙ η 74 which means that the second term of eq. ( 2 ′ . × 10 / V /V φ n ) and ( ′ 1 2 ζ = (2 = < 1 1 9 δσ/ . . . We need σ ′ V P , and it will probably never be observed [ 2 1 H ζ 1 and ( 1 / r V NL NL 2 1 P P ζ − f ≤ P ≪ M = (5˙ 10 1 which means that 5) and gravitino 0 2 . . flation, with 0 σ V ) gives during the second m σ [3(1 3 irement on the parameters is achieved if from some field 1 m/H ≪ P ≫ 1 e. Choosing instead evere because it exists during ζ σ ∼ πσ H M ] that , ] in the early universe 6 is adequate for ].) curvaton scenario the 3 /m . 2 2 2 . Then, in the regime 1 ]. Then 1 ∼ 2 26 1  P ∼ m H H 0 27 0 [ 1 M σ 1 and σ πσ + 1 ∼− -folds of inflation. Setting 3 m/H ∼ πσ 9 4  H e ≫ 2 is exceptionally small we generally need would typically be too big and this choice 2 0 ) 2 ≃ 2 ]. (If the latter difficulty is ignored we can | s 2 while cosmological scales leave the horizon excludes any radiation contribution. It is a σ  N 2 N 27 + /m F 2 nr , in that the generic value − 2 ρ 2 3 m 1+cos H 10 to be constant, eq. ( – 5 – FH  ≡ ∂V/∂χ FH 0 | ( 2 .  V 1, but 2 ≡− nr | 1 3 This corresponds to anomaly mediation, which gives − H 2 1 χ H . η 2 problem exists only during inflation when P | = = between the two evolutions corresponding to a highly P | ≪ 2 1 and just a few η )= 2 M 2 The required value M η Any scheme for generating 1 will be negligible if / σ | NL 1 ζ ( but we are interested in the ‘fast-roll’ [ 14 f − |∼ ). P − 2 V ). σ , F 2 2 n 5) P 2 η and 3 V/∂σ is a linear function and we get / H | 10 2 to g 2 (3 ∂ nr /M ) is self-consistent if ∼ ≪ FH 1 to arrive at ‘Natural Inflation’ [ 2 0 will be quickly driven to a minimum of , and taking H η σ 15 σ 3 φ ≃ χ m 2 σ,φ,... / )( 1. (The approximation ( ˙ ) 2 σ m GeV 2 2 giving V and (like any version of susy) may soon be tested at the π problem for the curvaton one can take it be a pNGB with the pote & 4 − / 3 ) by 2 / η 10 F σ = 14 ) is linear, (2 2 small enough, (ii) unless V/∂χ 3 GeV ′ & 2 | this is a mild requirement which we will take to be satisfied, s m 5 ≃ V ∂ 1 m ( − ]. The curvaton and gravitino have to decay before they can up 2 ] 10 m ] 0 − ≡ 27 V in eq. ( 29 [ 28 to n /m | 2 σ nr χ g 2 ≃ 4 1 at all times or η problem; that a generic supergravity theory gives [ H m What we need for the inflating curvaton is Writing Since eq. ( The contribution of For the inflaton scenario, the To avoid the V 10 η | ≪ ∼ χ η the gives practically independent of other field values. It is known [ axion with gravity- or anomaly mediated susy breaking. The slow-roll regime is The curvaton a string axion. corresponding to have requires [ inflation [ replace holds. lating curvaton, but it is unclear how to motivate such a valu where mass m anyhow seems impossible within string theory [ problem for two reasons; (i) we need would give a second inflation with the fast-roll regime, eq. ( to keep such a low | it is oftenof ignored the and supergravity could inflatonmay be potential. be regarded more For as severe the a because oscillating fine-tuning it requ may exist for a long time after in decreasing. For theboth inflating inflations with curvaton very it different is values for definitely more s a very strong evolution of non-trivial potential JCAP03(2012)022 ] , p 24. φ and ) in- , but GeV, . ∝ 7 5 σ 4 ) 2 ) and a 10 and the del with t φ δρ 10 ( ( to have a , 1 & FN 1, we have V 0 ∼ H imple early- ]. 1 V ≫ H & /H 4. 2 . P arXiv:0809.1055 ential ]; there we might 4 [ 1 N ≃ SPIRE 1 M ]. Taking that dis- that is probably too H IN p r is realistic one would GeV gives 30 & n current observational 4 0 2 e supported (in part) by V 10 he early stage of this work. 4. cience (CPIS) of Sokendai ), the late-time probability /H & from UNILHC23792, Euro- ). Since 1 may also be testable, but t & 1 (2009) 013 2 r the Ministry of Education, ( . raction 4 1 Statistical anisotropy of the an Nos. 21111006, 22244030, 1 2 H 2 /σ 05 H ntal Physics under STFC grant H z, ∼ − P tion with the correct FN > H M h anomaly-mediated susy breaking NL 1 f ]; ln( JCAP H ) holds we expect , ) is near the top of the potential, given -folds of inflation with 8 ∼ 1 e t implies ( 2 SPIRE σ P IN may eventually be detectable ) with FN ] [ M 16 6 NL − – 6 – f 10 to be constant, we would need that eventually be observed and will decide between ∼ ′ 1 . The former bound would probably make eq. ( we have 1 n P H 0 H σ M is different for the oscillating curvaton [ 7 . Using eq. ( ′ − ∼ 2 hep-ph/0110002 The primordial density perturbation: cosmology, inflation is near the top. This requires [ n it seems that 10 Generating the curvature perturbation without an inflaton σ ) with σ FN & 16 . negligible, and even if eq. ( 1 σ ≪ ′ 2 H n ) are roughly the same as those of the oscillating curvaton mo (2002) 5 , Cambridge University Press, Cambridge U.K. (2009) [ 17 may be strongly modified by the correction of second order in = 1. But to know whether the estimate m/H at the end of the first inflation can be calculated [ with p GeV) NL 1 The hypothesis that the curvaton is a string axion leads to a s σ is created. The main inflation takes place earlier, with a pot B 524 4 f η 2 ζ ) and ( 10 ]. ≃ 16 > , hence 1 2 2 1. But the result for − SPIRE = 1 or 2 needed to reproduce the observed spectral index withi H ≃ FN IN n One may worry about the assumption that Eqs. ( Since inflation ends at p [ Phys. Lett. curvature perturbation from vector field perturbations origin of structure ) d ≪ t ( [2] K. Dimopoulos, M. Karciauskas, D.H. Lyth and Y. Rodrigue [3] D.H. Lyth and D. Wands, [1] D.H. Lyth and A.R. Liddle, universe scenario. The curvaton generates a few Conclusion. distribution of during which The result for have tribution to apply and also taking compatible with f range of initial values of Going the other way, eq. ( uncertainity. The hypothesis requires low-energywhich may susy soon wit be tested at the LHC. It predicts that a tensor f with that the first inflation may be of long duration. For a given small ever to observe, but a running barring a strong cancellation significant probability that have to calculate the evolution of the probability distribu the linear and quadratic potentials. A third prediction (since F the accuracy of the calculation needs to be improved. Acknowledgments TM and KK thank Anupam MazumdarDHL for thanks valuable discussions Michael in t Dinepean for Research correspondence, and Training and Networkthe has (RTN) grant. Lancaster-Manchester-Sheffield support Consortium KD for andST/J000418/1. Fundame DHL ar KK is partlyCulture, supported Sports, by Science the23540327, and Grant-in-Aid and fo Technology, Government by of(1HB5806020). the Jap Center for the Promotion of IntegratedReferences S JCAP03(2012)022 ] , ] n ]. , ation , , in ] y gh a SPIRE ] , IN ] [ (2008) 003 ]. (2011) 18 hep-ph/0110096 08 astro-ph/0003278 [ in the cosmic 192 Multiple inflation, ]; , SPIRE (2008) 023513 hep-th/0501125 [ ]. IN JCAP hep-ph/9606387 [ [ , SPIRE hep-ph/0702260 (2002) 303] [ D 78 ]. [ , IN SPIRE Extra natural inflation ]. ]; ultamaki, ] [ IN ]. ]; (2000) 043527 ] [ ]. (2005) 067 all, Non-Gaussianity, spectral index and N-flation ]. SPIRE ]. (1992) 623 B 539 IN SPIRE SPIRE (1997) 1861 05 SPIRE ] [ IN SPIRE IN D 62 Phys. Rev. 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