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3 January 2002

Physics Letters B 524 (2002) 5–14 www.elsevier.com/locate/npe

Generating the curvature perturbation without an inflaton

David H. Lyth a, David Wands b

a Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom b Relativity and Group, School of Computer Science and Mathematics,University of Portsmouth, Portsmouth PO1 2EG, United Kingdom Received 3 October 2001; accepted 26 October 2001 Editor: J. Frieman

Abstract

We present a mechanism for the origin of the large-scale curvature perturbation in our Universe by the late decay of a massive scalar field, the curvaton. The curvaton is light during a period of cosmological inflation, when it acquires a perturbation with an almost scale-invariant spectrum. This corresponds initially to an isocurvature density perturbation, which generates the curvature perturbation after inflation when the curvaton density becomes a significant fraction of the total. The isocurvature density perturbation disappears if the curvaton completely decays into thermalised radiation. Any residual isocurvature perturbation is 100% correlated with the curvature. The same mechanism can also generate the curvature perturbation in pre-/ekpyrotic models, provided that the curvaton has a suitable non-canonical kinetic term so as to generate a flat spectrum.  2002 Published by Elsevier Science B.V.

PACS: 98.80.Cq; 04.50.+h

1. Introduction with an almost flat spectrum, generating immediately the required curvature perturbation which is constant It is now widely accepted that the dominant cause of until the approach of horizon entry. This idea has the structure in the Universe is a spatial curvature pertur- advantage that the prediction for the spectrum is in- bation [1]. This perturbation is present on cosmologi- dependent of what goes on between the end of infla- cal scales a few Hubble times before these scales enter tion and horizon entry [1–3]. The spectrum depends the horizon, at which stage it is time-independent with only on the form of the potential and on the theory of an almost flat spectrum. One of the main objectives of gravity during inflation (usually taken to be Einstein theoretical cosmology is to understand its origin. gravity), providing, therefore, a direct probe of condi- The usual assumption is that the curvature perturba- tions during this era. On the other hand, the demand tion originates during inflation, from the quantum fluc- that inflation should produce the curvature perturba- tuation of the slowly-rolling inflaton field. As cosmo- tion in this particular way is very restrictive, ruling out logical scales leave the horizon, the quantum fluctua- or disfavouring several otherwise attractive models of tion is converted to a classical Gaussian perturbation inflation. In this Letter we point out that the primordial cur- vature perturbation may have a completely different E-mail address: [email protected] (D. Wands). origin, namely, the quantum fluctuation during infla-

0370-2693/02/$ – see front  2002 Published by Elsevier Science B.V. PII:S0370-2693(01)01366-1 6 D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14 tion of a light scalar field which is not the slowly- side, we know that the curvature perturbation pro- rolling inflaton, and need have nothing to do with the vides the principle origin of structure. On the theoret- fields driving of inflation. We call this field the cur- ical side, late-decaying scalars are routinely invoked vaton. The curvaton creates the curvature perturbation in cosmology, and are ubiquitous in extensions of the in two separate stages. First, its quantum fluctuation of physics. Also, one is now during inflation is converted at horizon exit to a clas- aware of inflation models whose only defect is their sical perturbation with a flat spectrum. Then, after in- failure to generate the curvature perturbation from the flation, the perturbation in the curvaton field is con- inflaton, some of which will be mentioned later. verted into a curvature perturbation. In contrast with Before ending this introduction, we need to empha- the usual mechanism, the generation of curvature by sise that the curvaton can produce a curvature pertur- the curvaton requires no assumption about the nature bation without any accompanying isocurvature pertur- of inflation, beyond the requirement that (if the cur- bation at late times. This is the reason why we include vaton has a canonical kinetic term) the Hubble para- in our setup the requirement that the curvaton decays meter is practically constant. Instead, it requires cer- before decoupling. If it decays later, the cur- tain properties of the curvaton and of the cosmology vature perturbation may be accompanied by a signif- after inflation so that the required curvature perturba- icant isocurvature neutrino perturbation as discussed tion will be generated. We shall explore the simplest recently by Hu [5]. setup, consisting of the following sequence of events. First, the curvaton field starts to oscillate during some radiation-dominated era, so that it constitutes matter 2. The curvature perturbation with an isocurvature density perturbation. Second, the oscillation persists for many Hubble times so that a The spatial curvature perturbation is of interest only significant curvature perturbation is generated. Finally, on comoving scales much bigger than the Hubble scale before neutrino decoupling, the curvaton decays and (super-horizon scales). To define it one has to specify the curvature perturbation remains constant until the a foliation of spacetime into spacelike hypersurfaces approach of horizon entry. (slicing), and the most convenient choice is the slicing We shall show that under these conditions the of uniform energy density (or the slicing orthogonal quantum fluctuation of the curvaton during inflation to comoving worldlines, which is practically the same is converted into a curvature perturbation after decay on super-horizon scales). The curvature perturbation according to the formula on uniform-density slices [6–8] is given by the metric ζ ∼ perturbation , defined with a suitable coordinate ζ rδ, (1) choice by the line element where δ is the isocurvature fractional density pertur- 2 = 2 − i j bation in the curvaton before it decays and r is the d a (t)(1 2ζ)dx dx . (2) fraction of the final radiation that the decay produces. On cosmological scales, the spectrum Pζ of ζ at The mechanism that we are describing may suc- the approach of horizon entry is almost flat, with cinctly be described as the conversion of an isocurva- magnitude of order 10−10. ture perturbation into a curvature perturbation. It was The time-dependence of ζ on large scales is given actually discovered more than a decade ago by Moller- by [8] ach [4], who corrected the prevailing misconception H that no conversion would occur. At the time the con- ˙ =− ζ + δPnad, (3) version was regarded as a negative feature, because ρ P the focus was on finding a good mechanism for gen- where H ≡˙a/a is the Hubble parameter, ρ is the erating an isocurvature perturbation. For this reason, energy density, P is the pressure and δPnad is the and also because cosmology involving late-decaying pressure perturbation on uniform-density slices (the scalars was not considered to be very likely, the con- non-adiabatic pressure perturbation). version mechanism has received little attention. The In the usual scenario where ζ is generated during in- situation now is very different. On the observational flation through the perturbation of a single-component D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14 7 inflaton field, it becomes practically time-independent (Throughout we adopt the convenient notation that the soon after horizon exit and remains so until the ap- absence of an argument denotes the unperturbed quan- proach of horizon entry. The mechanism that we are tity.) Like any cosmological quantity the spatial depen- proposing starts instead with a negligible curvature dence of δσ can be Fourier-expanded in a comoving perturbation, which is generated later through a non- box much larger than the observable Universe, but it is adiabatic pressure perturbation associated with the unnecessary and in fact undesirable for the box to be curvaton perturbation. indefinitely large. Failure to limit the box size leads among other things to an indefinitely large fluctua- tion for any quantity with flat spectrum. It is a source 3. The curvaton field of confusion in the usual case of inflaton-generated curvature and the same would be true for curvaton- The curvaton field σ lives in an unperturbed Ro- generated curvature. bertson–Walker spacetime characterised by the line The unperturbed curvaton field satisfies element σ¨ + 3H σ˙ + Vσ = 0, (8) 2 = 2 − 2 i j ds dt a (t)δij dx dx (4) where H ≡˙a/a is the Hubble parameter, and a and its Lagrangian is subscript σ denotes ∂/∂σ. We are interested in the perturbation δσk,wherek 1 2 1 2 Lσ = σ˙ − (∇σ) − V(σ). (5) denotes the comoving momentum. It is conveniently 2 2 defined on the spatially-flat slicing. In general, the The potential V depends of course on all scalar fields scalar field perturbations on this slicing satisfy to first but we exhibit only the dependence on σ which is order the set of coupled equations [9] assumed to have no significant coupling to the fields k2 driving inflation. ¨ + ˙ + δφi 3H δφi 2 δφi The initial era for our discussion is the one which  a     3 · begins several Hubble times before the observable 1 a ˙ ˙ + Vφ φ − φiφj δφj i j M2a3 H Universe leaves the horizon during an inflationary i P phase, and ends several Hubble times after the smallest = 0. (9) cosmological scale leaves the horizon. During this era, we assume that the Hubble parameter H ≡˙a/a is For simplicity we assume that σ is sufficiently decou- almost constant, that is pled from the other perturbations that the latter can be ignored, leading to ≡− ˙ 2    H H/H 1. (6) ¨ ˙ 2 δσk + 3H δσk + (k/a) + Vσσ δσk = 0. (10) In the usual slow-roll paradigm with Einstein gravity, We assume that the curvaton potential is sufficiently 2 2H (MPV /V) where V(φ) is the inflationary flat during inflation, potential, φ is the slowly rolling inflaton and MP = 2 2 × 1018 GeV is the reduced Planck mass. However, |Vσσ|H , (11) for our mechanism we need not assume any specific and that on cosmological scales each Fourier compo- paradigm for inflation. nent is in the vacuum state well before horizon exit. We assume that the curvature perturbations is negli- The vacuum fluctuation then causes a classical per- gible during inflation. For slow-roll inflation with Ein- turbation δσk well after horizon exit, which satisfies  −5 1/2 1/4  stein gravity this requires H 10 H MP or V Eq. (10) with negligible gradient term, −2 1/4 10 H MP. In any case, an inflation model with Ein- δσ¨ + 3H δσ˙ + V δσ = 0. (12) −5 k k σσ k stein gravity requires H  10 MP from the cosmic microwave background limit on gravitational waves. The perturbation is Gaussian, and in the limit where We write at any given time Eq. (11) is very well satisfied its spectrum given by H∗ = + P1/2 = . (13) σ(x) σ δσ(x). (7) δσ 2π 8 D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14

The star denotes the epoch of horizon exit, k = a∗H∗, When the oscillation starts, the mean-square pertur- and, by virtue of Eq. (6), Pσ is almost flat. To be more bation of σ is given by precise, the spectral tilt of the perturbation is given by kmax ˙  dk dlnP H∗ 2 (V )∗ 2 = P ≡ σ = + σσ (δσ) σ (k) . (18) nσ 2 2 2 . (14) k dlnk H∗ 3 H∗ kmin As discussed in [10], the short distance cutoff at

4. Oscillating phase the epoch when the oscillation starts is kmax ∼ (a˜H) where a tilde denotes this epoch, subhorizon modes We now move on to the epoch when the curvaton having red-shifted away. Also, since we are working field starts to oscillate around the minimum of its in a box not too much bigger than the observable potential. We suppose that the oscillation starts during Universe the long distance cutoff is kmin ∼ a0H0. some radiation-dominated era. It may not be the one Assuming that Pσ is flat this gives the estimate   in which nucleosynthesis occurs, but if it is we require 2 (δσ)2 H∗ that the oscillation starts well before cosmological = 2 ln(kmax/kmin) scales enter the horizon. We assume that Einstein σ 2πσ∗ 2 gravity is valid from the oscillation time onwards, so ∼ (H∗/σ∗) . (19) that the total energy density is ρ = 3H 2M2. P (If P increases dramatically on small scales, the We assume that the curvaton continues to satisfy σ estimate has to be increased appropriately.) Eq. (8), and that its perturbation continues to satisfy If H∗  σ∗, the field perturbation is small and Eq. (12), after inflation. Assuming that the potential V(σ)is quadratic, V = m2σ 2/2, then oscillations start δσ δ = 2 . (20) at the epoch H ∼ m. Also, Eq. (12) for σ and Eq. (8) σ for δσ are then the same and the ratio δσ/σ remains This is a time-independent Gaussian perturbation with fixed on super-horizon scales. This assumption can a flat spectrum given by easily be relaxed, and in particular, one can handle 1/2 the situation where the curvaton may initially be near P ∗ P1/2 = 2 σ = H  δ 1. (21) a maximum of the potential [10]. Any evolution of σ πσ∗ δσ on super-horizon scales leads to an overall scale- In the opposite regime, H∗  σ∗, the perturbation is independent factor which will not spoil the flatness bigger than the unperturbed value and of the spectrum. Even if the potential is not quadratic at the onset of oscillation, it will become practically (δσ)2 δ = . (22) quadratic after a few Hubble times as the oscillation (δσ)2 amplitude decreases. This is again time-independent, but is now the square We are interested in the curvaton energy density of a Gaussian quantity (a χ2 quantity). Its spectrum is ρσ (x) = ρσ + δρσ (x) (15) flat up to logarithms [10] 2 and in the density contrast Pσ (kmin) Pδ(k) = 4ln(k/kmin) ∼ 1. (23) δρ (δσ)2 δ ≡ σ . (16) ρσ  We shall show that the curvature perturbation is Since the spatial gradients are negligible on super- a multiple of δ, which means that it is Gaussian in horizon scales, the oscillation is harmonic at each the regime H∗  σ∗ but a χ2 non-Gaussian quantity point in space and in the opposite regime. A χ2 curvature perturbation is strongly forbidden by observation [11], which, 1 2 2 ρσ (x) = m σ (x), (17) therefore, requires 2 where σ(x) is the amplitude of the oscillation. H∗  σ∗. (24) D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14 9

5. Generating the curvature perturbation In the opposite case, the curvaton density just before decay is some fraction r<1 of the radiation density. Once the curvaton field starts to oscillate the energy Making the approximation r  1,     density becomes a mixture of matter (the curvaton) 2 1/2 1 σ∗ m and radiation. According to Eq. (3) the generation r , (31) of the curvature perturbation begins at that point, 6 MP Γ because the pressure perturbation corresponding to and the final curvature perturbation is this mixture is non-adiabatic. It ends when the pressure 1 perturbation again becomes adiabatic, which is at the ζ = rδ. (32) 4 epoch of curvaton matter domination, or the epoch of Dropping the prefactors 1/3and1/4, the spectrum curvaton decay, whichever is earlier. of the curvature perturbation is given in the Gaussian The curvature perturbation finally generated could regime, H∗  σ∗,by be precisely calculated from Eq. (3) knowing the de- H∗ cay rate Γ of the curvaton, but for an estimate it P1/2 r . (33) is enough to assume that the decay occurs instanta- ζ πσ∗ neously at the epoch H = Γ . In that case one can The scale dependence of the spectrum is the same avoid the use of Eq. (3) altogether by considering sepa- as that of δσ. From Eqs. (6) and (14) its spectral index rately the curvature perturbations ζr and ζσ on, respec- n is given by tively, slices of uniform radiation- and matter density. P ˙ 2 2 These are separately conserved [8] as the radiation and − ≡ dln ζ = H∗ + 2 m n 1 2 2 2 . (34) matter are perfect non-interacting fluids. The curvature dlnk H∗ 3 H∗ perturbations are given by [8] If we relax the demand that the potential is quadratic, 2 2 = δρ m is replaced by the effective mass-squared m∗ ζ =−H , (25) (V )∗ which can be either positive or negative. The ρ˙ σσ observational constraint at 95% confidence level is δρr 1 δρr = ± ζ =−H = , n 0.93 0.13 [12], which requires r ˙ (26) ρr 4 ρr 2 |m∗| δρσ 1 δρσ 1  =− = ≡ 2 0.1. (35) ζσ H δ, (27) H∗ ρ˙σ 3 ρσ 3 The completely non-Gaussian regime H∗  σ∗ where the density perturbations are defined on the flat is strongly forbidden by observation [11], but the slicing of spacetime. (Note that the constancy of ζ σ intermediate regime is allowed provided that the non- is equivalent to the constancy of δ for the oscillating Gaussian component is small. From Eq. (17), the field which we noted earlier.) Using these results the curvature perturbation in that case is of the form [10] curvature perturbation is     r δσ δσ 2 4ρrζr + 3ρσ ζσ ζ = 2 + , (36) ζ = . (28) 4 σ σ 4ρr + 3ρσ with Before the oscillation begins, ζ = ζr which we are H∗ supposing is negligible. It follows that P1/2 = δσ/σ (37) 2πσ∗ ρ ζ = σ δ. (29) and, since the first term dominates 4ρr + 3ρσ 10−5 This calculation applies until the curvaton decays, r . (38) P1/2 after which ζ is constant. If the curvaton dominates δσ/σ the energy density before decay, the final value of ζ is Observational constraints on non-Gaussianity can place 1 strong upper limits on the small ratio H∗/σ∗,whichit ζ = δ. (30) 3 would be interesting to evaluate. 10 D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14

6. An isocurvature density perturbation? the momentum distribution of each species specified by the densities of the number and the three At the epoch when perturbations first become ob- numbers. In this situation, if the decay radia- servable as cosmological scales approach the horizon, tion thermalises with the existing radiation, and gen- the curvature perturbation seems to be the dominant erates no baryon or lepton number, the curvaton can cause of structure but it may not be the only one. only generate a dark matter isocurvature perturbation In particular, there may be an isocurvature perturba- if some small fraction decays into decoupled dark tion (one present on slices of uniform total energy matter with an exceedingly small branching ratio. If density) in the density of one or more of the con- on the other hand the decay radiation thermalises but stituents of the Universe relative (conventionally) to does possess some baryon or lepton number asymme- the density. In the standard picture the con- try, this can generate a baryon or neutrino isocurva- stituents are the photon plus (i) the cold dark mat- ture perturbation, which might be big enough to ob- ter (ii) the and (iii) the three of the serve if the asymmetry is not too small. An exam- Standard Model which have travelled freely since they ple of the latter possibility is given in [16], which, fell out of equilibrium shortly before nucleosynthe- however, neglects the dominant curvature perturba- sis. There could be an isocurvature density perturba- tion. tion in any or all of these three components. As de- If the curvaton decays after neutrino decoupling, scribed for instance in [13–15], these isocurvature per- the pre-existing radiation consists of and neu- turbations could be a significant fraction of the to- trinos which are now decoupled. In that case the tal as far as present microwave background observa- curvaton decay will cause a perturbation in the rel- tions are concerned, though the PLANCK satellite will ative abundance of neutrinos and photons, no mat- rule out (or detect) them at something like the 10% ter whether it decays to photons or to neutrinos. level if their spectrum is flat. Going beyond the stan- In other words it will (conventionally) generate a dard picture, the dark matter might have non-trivial neutrino isocurvature perturbation. At least if the properties and there may be neutrinos or other free- curvaton decays to neutrinos, this isocurvature per- streaming matter with a non-thermal momentum dis- turbation should be big enough to observe in the tribution. forseeable future [5]. (Curvaton decay after nucle- An isocurvature density perturbation may origi- osynthesis can also significanatly alter the epoch of nate as the quantum fluctuation of a scalar field dur- matter-radiation equality, which again may be observ- ing inflation. However, such a field cannot be the able.) inflaton, and in the usual scenario where the lat- A distinctive prediction of curvaton decay that gen- ter generates the curvature perturbation it is hard erates both the curvature perturbation and an isocur- to see why the effect of an isocurvature pertur- vature perturbation at late times is that the two bation should be big enough to be observable. A perturbations, arising from a single initial curvaton priori one expects that either it will be dominant, perturbation, must be completely correlated. Current which is forbidden by observation, or else negli- microwave background data alone cannot rule out a gible. In contrast, if the curvature perturbation is significant contribution from an isocurvature pertur- generated by a curvaton field, that same field may bation correlated with the curvature perturbation [14, also generate an isocurvature perturbation. In other 15], but future data will give much tighter con- words, the isocurvature density perturbation of the straints [13]. curvaton field might be converted into a mixture of a curvature and a correlated isocurvature perturba- tion. 7. The curvaton as a flat direction A study of this possibility is outside the scope of the present Letter, but we offer some brief comments. Consider first the case that the curvaton decays be- We have still to consider the nature of the curvaton fore neutrino decoupling. In that case the pre-existing in the context of . In particular, we radiation is most likely in thermal equilibrium, with did not examine the fundamental assumption that the D.H. Lyth, D. Wands / Physics Letters B 524 (2002) 5–14 11 curvaton potential satisfies the flatness requirement of occurring rather naturally in the context of par- Eq. (11). ticle physics. A recent example, which actually ap- Let us first suppose that the curvaton is a generic peared when the present Letter was almost written, field, running over an indefinitely large range σ>0 is Ref. [16] in which σ is the scalar super-partner of with the lower end the fixed point of a symmetry. The a right-handed neutrino. (Note though that this Letter potential will be sufficiently flat only over some range does not take account our isocurvature-curvature con- σ<σmax, beyond which it rises too steeply. If inflation version mechanism, supposing instead that the decay lasts long enough we may expect σ∗ in our location of the sneutrino will set up only an isocurvature baryon to have equal probability of being anywhere in the density perturbation.) range 0 <σ∗ <σmax, leading to the rough estimate σ∗ ∼ σmax. For simplicity assume that the symmetry forbids 8. The curvaton as a pseudo-Goldstone odd powers of σ in the potential. Adopting the usual paradigm of , the renormalizable To achieve better control of the curvaton mecha- (quadratic and quartic) terms of the potential can be nism, one can suppose that σ is a pseudo-Goldstone eliminated at the level of global supersymmetry (the boson, so that σ → σ +const under the action of some curvaton can be chosen as a flat direction in field spontaneously broken global symmetry. In the limit space). However, at the supergravity level the typical where this symmetry is exact the potential would be effective mass-squared m2(t), of a generic field in the exactly flat which makes the approximate flatness we early Universe is of order ±H 2, the true mass m being require technically natural. 1 Also, in this case σ runs relevant only after H falls below m. This is marginally over some finite range 0 <σ

[30] A. Buonanno, K.A. Meissner, C. Ungarelli, G. Veneziano, F. Vernizzi, A. Melchiorri, R. Durrer, Phys. Rev. D 63 (2001) Phys. Rev. D 57 (1998) 2543, hep-th/9706221; 063501, astro-ph/0008232. A. Buonanno, K.A. Meissner, C. Ungarelli, G. Veneziano, [34] K. Enqvist, M.S. Sloth, hep-ph/0109214. JHEP 9801 (1998) 004, hep-th/9710188. [35] D. La, P.J. Steinhardt, Phys. Rev. Lett. 62 (1989) 376. [31] R. Brustein, M. Hadad, Phys. Lett. B 442 (1998) 74, hep- [36] A.R. Liddle, D.H. Lyth, Phys. Lett. B 291 (1992) 391, astro- ph/9806202. ph/9208007. [32] R. Durrer, M. Gasperini, M. Sakellariadou, G. Veneziano, [37] T. Banks, M. Dine, L. Motl, JHEP 0101 (2001) 031, hep- Phys. Lett. B 436 (1998) 66, astro-ph/9806015; th/0007206. R. Durrer, M. Gasperini, M. Sakellariadou, G. Veneziano, [38] L. Covi, D.H. Lyth, astro-ph/0008165, to appear in MNRAS. Phys. Rev. D 59 (1999) 043511, gr-qc/9804076. [39] D.H. Lyth, E.D. Stewart, Phys. Rev. D 53 (1996) 1784, hep- [33] A. Melchiorri, F. Vernizzi, R. Durrer, G. Veneziano, Phys. Rev. ph/9510204. Lett. 83 (1999) 4464, astro-ph/9905327;