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 Cosmology 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-big bang/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 matter 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 Standard Model of particle 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 neutrino 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.