New Astronomy Reviews 47 (2003) 793–796

www.elsevier.com/locate/newastrev

Curvaton model constraints from WMAP

Christopher Gordon a,*, Antony Lewis b

a DAMTP, CMS, Wilberforce Road, Cambridge CB3 0WA, UKb CITA, 60 St. George St, Toronto, Ont., Canada M5S 3H8

Abstract

Simple curvaton models can generate a mixture of correlated primordial adiabatic and isocurvature perturbations.

Working numerically we use the latest WMAP observations and a variety of other data to constrain the curvaton

model. We find that models with an isocurvature contribution are not favored relative to simple purely adiabatic

models. However, a significant primordial totally correlated baryon isocurvature perturbation is not ruled out. Certain

classes of curvaton model are thereby ruled out, other classes predict enough non-Gaussianity to be detectable by the

Planck satellite.

� 2003 Elsevier B.V. All rights reserved.

1. Introduction

Recent detailed measurements of the acoustic

peaks in CMB anisotropy power spectrum by theWMAP satellite (Hinshaw et al., 2003; Peiris et al.,

2003) are consistent with the standard model of a

predominantly adiabatic, approximately scale in-

variant primordial power spectrum in a spatially

flat Universe. Frequently it is assumed the initial

power spectrum is entirely adiabatic, though there

is still no compelling justification for this as-

sumption. In particular, the recently proposedcurvaton model uses a second scalar field (the

�curvaton�) to form the perturbations (Mollerach,

1990; Lyth and Wands, 2002; Lyth et al., 2003).

The motivation for this is it makes it easier for

otherwise satisfactory particle physics models of

* Corresponding author.

E-mail addresses: c.gordon@damtp.cam.ac.uk (C. Gordon),

antony@antonylewis.com (A. Lewis).

1387-6473/$ - see front matter � 2003 Elsevier B.V. All rights reserv

doi:10.1016/j.newar.2003.07.021

inflation to produce the correct primordial spec-

trum of perturbations (Dimopoulos and Lyth,

2002).

The curvaton scenario also has the feature ofbeing able to generate isocurvature perturbations

of a similar magnitude to the adiabatic perturba-

tion without fine tuning, and therefore is open to

observational test.

We use the CMB temperature and temperature-

polarization cross-correlation anisotropy power

spectra from the WMAP 1 (Kogut et al., 2003;

Hinshaw et al., 2003; Verde et al., 2003) observa-tions, as well as seven almost independent tem-

perature band powers from ACBAR 2 (Kuo et al.,

2002) on smaller scales. In addition we use data

from the 2dF galaxy redshift survey (Percival

et al., 2002), HST Key Project (Freedman et al.,

2001), and nucleosynthesis (Burles et al., 2001)

1 http://lambda.gsfc.nasa.gov/.2 http://cosmologist.info/ACBAR.

ed.

794 C. Gordon, A. Lewis / New Astronomy Reviews 47 (2003) 793–796

using a slightly modified version of the Cos-

moMC 3 Markov-Chain Monte Carlo program, as

described in (Lewis and Bridle, 2002).

For simplicity we assume a flat universe with a

cosmological constant, uninteracting cold dark

matter, and massless neutrinos evolving accordingto general relativity.

2. Constraining the curvaton model

Primordial correlated isocurvature modes can

be generated if the baryons or CDM are generated

by, or before, the curvaton decays (Gordon andLewis, 2002). If one or both were created before the

curvaton decays, the current model assumes that

the curvaton had a negligible density when they

decayed (Lyth et al., 2003). We assume that the

curvaton is the only cosmologically relevant scalar

field after inflation decay, and that the perturba-

tions in the inflation field are negligible. Generi-

cally such models predict a very small tensor modecontribution, which we assume can be neglected.

The baryon and CDM isocurvature modes

predict proportional results (Gordon and Lewis,

2002), so we can account for Sc (the CDM iso-

curvature perturbation) by using just an effective

baryon isocurvature perturbation

Seffb ¼ Sb þ

Rc

Rb

Sc; ð1Þ

where Sb is the baryon isocurvature perturbation.

The baryon and CDM isocurvature perturba-

tions are completely correlated (or anti-correlated)

with each other and the adiabatic perturbation, so

Seffb ¼ Bf, where B measures the isocurvature mode

contribution and is taken to be scale independent 4

and f is the adiabatic perturbation. Note that our

number of degrees of freedom is actually less thangeneric inflation, because although we have in-

troduced B we now no longer have the amplitude

and slope of the tensor component to consider.

The slope of the isocurvature perturbation is pre-

3 http://cosmologist.info/cosmomc.4 Our sign convention for B differs from that in Amendola

et al. (2002). In our convention B > 0 corresponds to a positive

correlation and the modes contribute with the same sign to the

large scale CMB anisotropy.

dicted to be the same as the adiabatic perturbation

and the tensors are predicted to be negligible in the

curvaton scenario (Lyth et al., 2003).

The isocurvature modes have little effect on

small scales, but they can either raise or lower the

Sachs Wolfe plateau relative to the acoustic peaksdepending on the sign of B (Gordon and Lewis,

2002). This is in contrast to tensor perturbations

which can only raise the Sachs Wolfe plateau rel-

ative to the acoustic peaks.

We find the ratio of themean likelihood allowing

for isocurvature modes to that for purely adiabatic

models is about 0.7 (Gordon and Lewis, 2002) (for

discussion of mean likelihoods see (Lewis and Bri-dle, 2002)). Thus, the isocurvature modes do not

improve the already good fit to the data of the

standard purely adiabatic case. By the same token,

the current data is still consistent with a significant

isocurvature contribution, with the 95% marginal-

ized confidence interval�0:53 < B < 0:43 (Gordon

and Lewis, 2002). If new data favored B > 0 this

would be largely degenerate with a tensor contri-bution predicted by standard single field inflation-

ary scenarios, and would be hard to distinguish

without good CMB polarization data. Evidence for

B < 0 would be a smoking gun for an isocurvature

mode, though the large scale polarization data has

large enough cosmic variance that to distinguish it

from an adiabatic model with an unexpected initial

power spectrum shape would be difficult.If the CDM is created before the curvaton de-

cays, and while the curvaton still has negligible

energy density, its density is essentially unper-

turbed. After the curvature perturbation is gener-

ated there is therefore a relative isocurvature

perturbation, given by Lyth et al. (2003)

Sc � �3f: ð2ÞIf the curvaton decays before its energy density

completely dominates, a CDM isocurvature per-

turbation is produced (Lyth et al., 2003)

Sc � 31� rr

!f; ð3Þ

where r measures the transfer function from

fcurvaton (perturbation of the curvaton) before

curvaton decay to f after decay, f ¼ rfcurvaton.Lyth et al. (2003) find the approximate result

C. Gordon, A. Lewis / New Astronomy Reviews 47 (2003) 793–796 795

r � qcurvaton=qtotal, where q is the energy density at

curvaton decay, to an accuracy of about 10%

(Malik et al., 2003). The same formulas, Eqs. (2)

and (3), apply for the baryons with Sc replaced by

Sb. If either the CDM or the baryon number wascreated after the curvaton decayed then there

would be no isocurvature perturbation in that

quantity (Lyth et al., 2003). If both were created

after the curvaton decayed there would be no

isocurvature modes.

Various permutations of when the baryons and

CDMwere created can be considered. It was found

that models in which either the baryon number orCDM was created before the curvaton dominated

the energy density are ruled out unless counter-

balanced by the other species being created by the

curvaton decay (Gordon and Lewis, 2002).

The amount of non-Gaussianity in the CMB is

dependent on r with the conventional governing

parameter (Lyth et al., 2003)

fnl �5

4r; ð4Þ

Fig. 1. Plots of the un-normalized posterior probability distri-

bution for the amount of non-Gaussianity, fnl. The dotted line

is for CDM created before curvaton decay and baryon number

by curvaton decay. The other lines are for the other possible

permutations of the times of curvaton and baryon creation

(Gordon and Lewis, 2002).

assuming fnl � 1. Using this equation we can

convert the likelihood for r into those for fnl as isshown in Fig. 1. The current one year WMAP data

has fnl < 134 (95%) and is predicted to reach

fnl < 80 (95%) with the four year WMAP data(Komatsu et al., 2003). So if WMAP eventually

detects non-Gaussianity it will rule out all the

models considered here. The Planck satellite is

predicted to ultimately be able to detect fnl J 5

(Komatsu and Spergel, 2001). If this is realized,

Planck will be able to distinguish between the case

where the CDM is created before curvaton decay

and the baryon number by curvaton decay and theother possibilities.

3. Conclusions

The curvaton model provides a simple scenario

that can give rise to correlated adiabatic and iso-

curvature modes of similar size. The current datado not favor a large isocurvature contribution, but

a significant amplitude is still allowed.

Numerically, we found that the data was con-

sistent at the two sigma level with the presence of

an effective correlated baryon isocurvature per-

turbation of about 50% the magnitude of the adi-

abatic perturbation. The individual baryon and

CDM isocurvature modes can be even larger if theycompensate each other. Models in which either the

baryon number or CDM was created before the

curvaton dominated the energy density are ruled

out unless counter-balanced by the other species

being created by the curvaton decay. The levels of

non-Gaussianity expected for the various scenarios

were evaluated and in the case of the CDM being

created before the curvaton decayed and the bar-yon number by the curvaton decay, could be high

enough to detectable by the Planck satellite.

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