Isocurvature perturbations in extra radiation

1.7 MB·

by  Masahiro Kawasaki

IPMU 11-0124
UT-11-22
Isocurvature perturbations in extra radiation
Masahiro Kawasakia,b, Koichi Miyamotoa, Kazunori Nakayamac
2 and Toyokazu Sekiguchid
1
0
2 aInstitute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
n bInstitute for the Physics and Mathematics of the Universe, University of Tokyo,
a Kashiwa 277-8568, Japan
J
cDepartment of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
3
2 dDepartment of Physics and Astrophysics, Nagoya University, Nagoya 464-8602, Japan

]
O
C
.
h
p Abstract
-
o
Recentcosmologicalobservations,includingmeasurementsoftheCMBanisotropy
r
t and the primordial helium abundance, indicate the existence of an extra radiation
s
a component in the Universe beyond the standard three neutrino species. In this
[
paper we explore the possibility that the extra radiation has isocurvatrue fluctu-

2 ations. A general formalism to evaluate isocurvature perturbations in the extra
v
radiation is provided in the mixed inflaton-curvaton system, where the extra radia-
2
6 tionisproducedbythedecayofbothscalarfields. Wealsoderiveconstraintsonthe
9 abundance of the extra radiation and the amount of its isocurvature perturbation.
4
Current observational data favors the existence of an extra radiation component,
.
7
but does not indicate its having isocurvature perturbation. These constraints are
0
1 applied to some particle physics motivated models. If future observations detect
1 isocurvature perturbations in the extra radiation, it will give us a hint to the origin
:
v of the extra radiation.
i
X
r
a
Contents
1 Introduction 2
2 Formalism 3
2.1 Nonlinear isocurvature perturbation . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Extra radiation and isocurvature perturbation . . . . . . . . . . . . . . . . 6
2.2.1 At the curvaton decay . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 At the neutrino freezeout . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 After the e± annihilation . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.4 Connecting to primordial perturbations . . . . . . . . . . . . . . . . 11
2.3 Non-Gaussianity in isocurvature perturbations of the extra radiation . . . 12
3 Observational signature of isocurvature perturbations in the extra radi-
ation 13
3.1 Perturbation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Initial conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Acoustic oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 CMB temperature power spectrum . . . . . . . . . . . . . . . . . . . . . . 21
4 Constraints from current observations 22
4.1 Fixed γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Fixed N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
eff
4.3 Varying N and γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
eff
5 Implications 28
5.1 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1.1 Uncorrelated case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.1.2 Totally anti-correlated case . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Particle physics model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.2.1 SUSY KSVZ axion model . . . . . . . . . . . . . . . . . . . . . . . 32
5.2.2 SUSY DFSZ axion model . . . . . . . . . . . . . . . . . . . . . . . 33
6 Conclusions and discussion 33
A Extra radiation production after BBN 34
1
1 Introduction
It is known that most of the energy content of the present Universe consist of “dark” com-
ponents: dark matter and dark energy. It is believed that the remaining components are
well-known stuff: baryons, photons and neutrinos. These are ingredients of the standard
cosmology supported by cosmological observations.
However, there is no reason to exclude the existence of additional component other
than listed above as long as its abundance is not large so that the cosmological evolution
scenarioisnotmuchaffected. Actually, theWMAPmeasurementofthecosmicmicrowave
background (CMB) anisotropy combined with observations of the baryon acoustic oscil-
lation (BAO) and the Hubble parameter (H0) constrains the effective number of neutrino
species N as N = 4.34+0.86 (68%C.L.) [1]. Including the Atacama Cosmology Tele-
eff eff −0.88
scope (ACT) data, the constraint is slightly improved as N = 4.56±0.75 (68%C.L.) [2].
eff
The recent results from South Pole Telescope (SPT), combined with WMAP, BAO and
H0indicateN = 3.86±0.42(68%C.L.)[3]. Thustheseobservationssuggestthepresence
eff
of an extra radiation component beyond the standard three neutrino species at the 2σ
level 1. (See also Refs. [6, 7, 9, 8] for former analyses.) Hereafter we call non-interacting
relativistic energy component other than neutrinos as “extra radiation”.
The CMB measurement is sensitive to the expansion rate of the Universe, or the total
energy density of the Universe, at around the recombination epoch. On the other hand,
the big-bang nucleosynthesis (BBN) also gives a constraint on the expansion rate at the
cosmic temperature of T ∼ 1MeV, where the weak interaction is frozen and the neutron-
proton ratio is fixed. Thus the observation of the primordial helium abundance gives a
constraint on the extra radiation component at T ∼ 1MeV. Recently, it was reported
that the primordial helium abundance derived from the observations of the extragalactic
HII regions indicates an excess at the 2σ level compared with the theoretical expectation
using the baryon number obtained from the WMAP data [10]. (See also Ref. [11].)
Ref. [12] studied constraints on the effective number of neutrinos N and the mass of
eff
extra radiation component using CMB (including WMAP 7yr, ACBAR, BICEP and
QUaD), SDSS and H0 data, and similarly claimed the existence of an extra radiation
and that its mass should be smaller than ∼ 0.4 eV. If these observations are true, we
need a theory beyond the standard model in which a new light particle species exists.2
Some cosmological scenarios and particle physics models are proposed in order to explain
∆N (cid:39) 1 [17, 18, 19, 20, 21].
eff
The most literature treating the extra radiation so far (implicitly) assumes that the
extraradiationhastheadiabaticperturbation. However, thepropertyofthefluctuationof
theextraradiationdependsonitsproductionmechanism. Forexample, thereisascenario
that the decay of a scalar field nonthermally produces an extra radiation component [17,
20]. In this case the extra radiation has the fluctuation of the source scalar field, and
if the scalar field is light during inflation, it obtains large-scale quantum fluctuations.
Therefore, the extra radiation can have an independent perturbation from the adiabatic
1 There are several studies which discuss how significant the deviation (N (cid:54)= 3.04) is. For detailed
eff
discussion, we refer to Refs. [4, 5]
2Discrepancy between the prediction and observation of the primordial helium abundance may be
solved in the large lepton asymmetry scenario [13, 14, 15, 16]. We do not pursue this issue in this paper.
2
one: it is an isocurvature perturbation. Investigating the isocurvature perturbation in the
extra radiation may have a potential to distinguish the model of the extra radiation, if
future observations further confirm ∆N (cid:39) 1.
eff
Therefore, in this paper we study the effect of the isocurvature perturbation in the
extra radiation and derive constraints on them using the currently available datasets.
The effect on the CMB anisotropy is similar to that induced by the neutrino isocurvaure
perturbation. But we are mainly interested in the case where the energy density of the
extra radiation itself causes significant change in the Hubble expansion rate. These effects
of the extra radiation with isocurvature perturbation are phenomenologically taken into
account by scanning both the effective number of neutrino species N and the neutrino
eff
isocurvature perturbation S .
ν
First we develop formalism for calculating the primordial isocurvature perturbation
in the extra radiation in Sec. 2. For concreteness we restrict ourselves to the two-scalar
field case, which generalizes the curvaton model [22]. The extra radiation is assumed to
be produced from decay of either or both of the scalar fields. We also point out that a
large non-Gaussianity may exist in the isocurvature perturbation of the extra radiation.
A similar study for the case of CDM/baryon isocurvature perturbations, including their
non-Gaussianities, was done in Refs. [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33]. In Sec. 3,
we discuss the observational signatures of our model. In particular, we focus on the
CMB angular power spectrum, which would be the best probe of our model for the time
being. Then we derive observational constraints on such models with extra radiations
andisocurvatureperturbationsusingcurrentlyavailabledatasetsincludingWMAP,ACT,
BAO and H measurement in Sec. 4. They are applied to some particle physics-motivated
0
models in Sec. 5. We conclude in Sec. 6.
We note that author in Ref. [34] also considered a specific model for isocurvature
perturbations in extra radiation in the different context.
2 Formalism
We consider a cosmological scenario of two scalar fields, both of which obtain large-scale
quantum fluctuations and may generate isocurvature perturbations as well as adiabatic
ones. One is the inflaton field φ which initially dominates the energy density of the
Universe during and soon after inflation. We also introduce another scalar field σ. We
do not specify which of them is the dominant source of the adiabatic perturbation, since
it depends much on models. For convenience we call σ the curvaton even if it may not
dominantly generate the adiabatic perturbation.
All the components of the Universe, photon, neutrino, CDM, baryon and extra radi-
ation arise from both φ and σ. In general there exist isocurvature perturbations among
these components. But we particularly focus on the isocurvature perturbations between
the extra radiation and the standard radiation. We denote the extra radiation component
by X, photon by γ, and neutrino by ν. We also denote the Standard Model radiation
which exists prior to the neutrino decoupling by r, and the X together with ν by “DR”
(dark radiation). We assume X is produced by the decay of φ and/or σ and has no
interaction with the other components throughout the whole history of the Universe after
3
production.
2.1 Nonlinear isocurvature perturbation
Wedefineζ asthecurvatureperturbationontheuniform-densityslice. Afterthecurvaton
decay, itisthecurvatureperturbationontheslicewherethetotalradiationenergydensity
is spatially uniform.3 According to the δN formalism [35, 36], it is related to the local
e-folding number N(t ,t ;(cid:126)x) as
f i
a(t )
ˆ f
ζ((cid:126)x) = N(t ,t ;(cid:126)x)−ln , (1)
f i
a(t )
i
wherea(t)isthebackgroundscalefactor. Herethehypersurfaceatt = t , intheradiation
f
dominated era after the curvaton decays, is chosen to be the uniform density slice and
the initial one at t = t to be spatially flat slice. The curvature perturbation ζ on large
i
scales is conserved unless there are isocurvature perturbations, or non-adiabatic pressure.
It is expanded by the scalar field fluctuations δφ as
i
1
ˆ
ζ = N δφ + N δφ δφ +.... (2)
φi i 2 φiφj i j
In the two-field case, up to the second order, it is given by
1 1
ˆ
ζ = N δφ+N δσ + N δφδφ+N δφδσ + N δσδσ. (3)
φ σ φφ φσ σσ
2 2
Similarly we define the curvature perturbation of the fluid i, ζ , as the curvature
i
perturbationonthehypersurfacewheretheenergydensityofi-thfluidisspatiallyuniform.
ζ ’s are conserved on sufficiently large spatial scales as long as there no interactions or
i
energy exchange between fluids [36]. If the i-th fluid has the equation of state p = w ρ ,
i i i
ζ and ζ are related by
i
ρ ((cid:126)x) = ρ¯e3(1+wi)(ζi−ζ), (4)
i i
where ρ ((cid:126)x) is evaluated on the uniform density slice. Then we define the isocurvature
i
perturbation between i-th and j-th fluids as [37]
S ≡ 3(ζ −ζ ). (5)
ij i j
In this paper we are interested in the isocurvature perturbation in the extra radiation
X. For convenience, we define the isocurvature perturbation in the dark radiation, which
includes both X and neutrino, as
ˆ ˆ
S ≡ 3(ζ −ζ). (6)
DR DR
It is formally expanded by the scalar field fluctuations δφ as
i
1
ˆ
S = S δφ + S δφ δφ +.... (7)
DR φi i 2 φiφj i j
3For the reason discussed later, we distinguish the curvature perturbation at different epochs: ζ, ζ˜
and ζˆ. They correspond to the curvature perturbation after the curvaton decay, neutrino freezeout and
e± annihilation, respectively. It is ζˆthat is directly related to cosmological observations.
4
In the two-field case, up to the second order, it is given by
1 1
ˆ
S = S δφ+S δσ + S δφδφ+S δφδσ + S δσδσ. (8)
DR φ σ φφ φσ σσ
2 2
The power spectra of the curvature/isocurvature perturbations, and their correlation
are given by
(cid:104)ζˆ((cid:126)k )ζˆ((cid:126)k )(cid:105) ≡ (2π)3δ((cid:126)k +(cid:126)k )P (k ),
1 2 1 2 ζζ 1
(cid:104)ζˆ((cid:126)k )Sˆ ((cid:126)k )(cid:105) ≡ (2π)3δ((cid:126)k +(cid:126)k )P (k ), (9)
1 DR 2 1 2 ζSDR 1
(cid:104)Sˆ ((cid:126)k )Sˆ ((cid:126)k )(cid:105) ≡ (2π)3δ((cid:126)k +(cid:126)k )P (k ),
DR 1 DR 2 1 2 SDRSDR 1
where
P (k) = [N2 +N2]P (k),
ζζ φ σ δφ
P (k) = [N S +N S ]P (k), (10)
ζSDR φ φ σ σ δφ
P (k) = [S2 +S2]P (k),
SDRSDR φ σ δφ
where we have neglected higher order terms, and the power spectrum of δφ is defined as
i
(cid:104)δφ ((cid:126)k )δφ ((cid:126)k )(cid:105) ≡ (2π)3δ((cid:126)k +(cid:126)k )P (k )δ , (11)
i 1 j 2 1 2 δφ 1 ij
H2 (cid:18) k (cid:19)ns−1
P (k) = inf . (12)
δφ 2k3 k
0
Here H is the Hubble parameter during inflation, n is the scalar spectral index4 and
inf s
k is the pivot scale chosen as k = 0.002Mpc−1.
0 0
The correlation parameter between the curvature and isocurvature perturbations is
defined by
P (k ) N S +N S
γ ≡ ζSDR 0 = φ φ σ σ . (13)
(cid:112) (cid:113)
Pζζ(k0)PSDRSDR(k0) (Nφ2 +Nσ2)(Sφ2 +Sσ2)
The uncorrelated isocurvature perturbation corresponds to γ = 0, and totally (anti-
) correlated one to γ = (−)1. The effect of isocurvature perturbations on the CMB
anisotropy depends on its magnitude as well as the correlation parameter. Thus what we
need is to express N ,S , and so on, in terms of model parameters.
φ φ
Forlateruse,wealsodefinethedimensionlesspowerspectrumasP (k) ≡ (k3/2π2)P (k)
AB AB
where the subscripts A and B are either ζ or S . Using the quantities given above, they
DR
are expressed as
(cid:18)H (cid:19)2(cid:18) k (cid:19)ns−1
P (k) = [N2 +N2] inf ,
ζζ φ σ 2π k
0
(cid:18)H (cid:19)2(cid:18) k (cid:19)ns−1
P (k) = [N S +N S ] inf , (14)
ζSDR φ φ σ σ 2π k
0
(cid:18)H (cid:19)2(cid:18) k (cid:19)ns−1
P (k) = [S2 +S2] inf .
SDRSDR φ σ 2π k
0
In the following we formulate a method for calculating the extra radiation isocurvature
perturbation in a typical cosmological setup.
4 The scalar spectral indices for φ and σ do not coincide in general. In the following we assume they
are the same just for simplicity.
5
epoch component energy transfer
Γ < H φ, σ φ → X(φ) +r(φ)
φ
Γ < H < Γ X(φ), r(φ), σ σ → X(σ) +r(σ)
σ φ
Γ < H < Γ X, r r → ν +r
ν σ e
Γ < H < Γ X, ν, r (DR = X +ν) e± → γ
e± ν e
H < Γ X, ν, γ (DR = X +ν)
e±
Table 1: Energy components of the Universe at each epoch except for CDM and baryon.
Here Γ (Γ ) is the decay rate of the inflaton (curvaton), and Γ denotes the neutrino
φ σ ν
interaction rate at the neutrino freezeout. Γ denotes the Hubble parameter at the e±
e±
annihilation. Here r denotes the plasma consisting of γ and e±.
e
2.2 Extra radiation and isocurvature perturbation
Now let us move to our concrete setup. Here we consider φ as the inflaton field. We want
to know the final dark radiation (“DR”) isocurvature perturbation, and hence we must
connect it to the primordial perturbations of φ and σ through some transition points
where constituents of the cosmological fluids change. Thus we evaluate the curvature
perturbations step-by-step in the following. The evolution of the fluids in our model is
summarized in Table. 1. We assume that the σ decays well before BBN begins. Con-
sidering that the decay of σ after BBN might produce significant energy in the form of
radiations and hadrons and upset the standard BBN, it is a reasonable assumption. In
the most part of this paper we take this assumption.5
In general, the CDM and baryon can have isocurvature perturbations depending on
theirorigins. However, includingthemmakestheanalysistoocomplicated. Thuswefocus
on the case where the extra radiation has an isocurvature perturbation and CDM/baryon
do not. For example, if CDM/baryon are created from thermal bath after the curvaton
decays, they only have adiabatic perturbations and our assumption is justified.
2.2.1 At the curvaton decay
Let us take the uniform density slice at the curvaton decay : H((cid:126)x) = Γ , where Γ is
σ σ
the decay rate of σ. We make use of the sudden decay approximation in the following
analysis [38]. On this slice, we have following relations
ρ(φ)((cid:126)x)+(1−r )ρ ((cid:126)x) = ρ ((cid:126)x), (15)
X σ σ X
ρ(φ)((cid:126)x)+r ρ ((cid:126)x) = ρ ((cid:126)x), (16)
r σ σ r
ρ(φ)((cid:126)x)+ρ(φ)((cid:126)x)+ρ ((cid:126)x) = ρ ((cid:126)x)(= ρ¯ ), (17)
r X σ tot tot
where r is the branching fraction of σ into particles other than the extra radiation X.
σ
Superscript (φ) means that the corresponding component comes from the inflaton decay.
5 The BBN constraint can be avoided if the σ decays dominantly to X particles even if its decay
occurs after BBN. For completeness, we will also discuss such a case in Appendix.
6
The curvature perturbation of each component is related to its background value as
ρ(φ)((cid:126)x) = ρ¯(φ)e4(ζφ−ζ), (18)
X X
ρ(φ)((cid:126)x) = ρ¯(φ)e4(ζφ−ζ), (19)
r r
ρ ((cid:126)x) = ρ¯ e4(ζX−ζ), (20)
X X
ρ ((cid:126)x) = ρ¯ e4(ζr−ζ), (21)
r r
ρ ((cid:126)x) = ρ¯ e3(ζσ−ζ). (22)
σ σ
From these equations we obtain
R(φ)e4(ζφ−ζ) +R(σ)e3(ζσ−ζ) = R e4(ζX−ζ), (23)
X X X
R(φ)e4(ζφ−ζ) +R(σ)e3(ζσ−ζ) = R e4(ζr−ζ), (24)
r r r
(1−R )e4(ζφ−ζ) +R e3(ζσ−ζ) = 1, (25)
σ σ
where we have defined
R ≡ ρ¯ /ρ¯ at σ decay, (26)
σ σ tot
and
R(φ) ≡ ρ¯(φ)/ρ¯ = (1−r )(1−R ),
X X tot φ σ
R(σ) ≡ ρ¯(σ)/ρ¯ = (1−r )R ,
X X tot σ σ (27)
R(φ) ≡ ρ¯(φ)/ρ¯ = r (1−R ),
r r tot φ σ
R(σ) ≡ ρ¯(σ)/ρ¯ = r R ,
r r tot σ σ
where r is the branching fraction of φ into particles other than the extra radiation X,6
φ
and R = R(φ) +R(σ) = ρ¯ /ρ¯ , R = R(φ) +R(σ) = ρ¯ /ρ¯ . Notice that R +R = 1.
X X X X tot r r r r tot X r
All of these quantities are evaluated at the curvaton decay, H = Γ . As is already stated,
σ
these quantities are not constants in time because of the changes in the relativistic degrees
of freedom g . Solving these equations up to the second order in ζ and ζ , we obtain
∗ φ σ
1
ζ = ζ +R(ζ −ζ )+ R(1−R)(3+R)(ζ −ζ )2,
φ σ φ φ σ
2
1 (cid:104) (cid:105)
ζ = (4R(φ) +(1−R)R(σ))ζ +(3+R)R(σ)ζ
X 4R X X φ X σ
X
(3+R)R(σ) (cid:104) (cid:105)
+ 8R2 X (1+R)(3−R)RX(φ) +(1−R)RRX(σ) (ζφ −ζσ)2, (28)
X
1
(cid:2) (cid:3)
ζ = (4R(φ) +(1−R)R(σ))ζ +(3+R)R(σ)ζ
r 4R r r φ r σ
r
(3+R)R(σ)
+ r (cid:2)(1+R)(3−R)R(φ) +(1−R)RR(σ)(cid:3)(ζ −ζ )2,
8R2 r r φ σ
r
6Precisely speaking, 1−r is not the branching ratio of φ into X (B ). It is given by 1−r =
φ φ→X φ
B (cid:15)/(1−B (1−(cid:15))) where (cid:15)≡[g (H =Γ )/g (H =Γ )]1/3 with Γ denoting the inflaton decay
φ→X φ→X ∗ σ ∗ φ φ
width, taking into account the change of relativistic degrees of freedom between the inflaton decay and
curvaton decay.
7
where we have defined
3R 4R
σ
R ≡ ↔ R = . (29)
σ
4−R 3+R
σ
Note that ζ and ζ are conserved quantities at superhorizon scales [37]. In general, the
X r
curvature perturbation ζ is not conserved in the presence of non-adiabatic pressure. In
the present case, both X and r behave as radiation, and hence one may consider that
ζ is conserved on sufficiently large scales. More precisely, however, the radiation energy
density ρ does not scale as a(t)−4 because of the changes in the relativistic degrees of
r
freedom g . Therefore, ζ is not conserved even after the curvaton decay. The changes in
∗
ζ at the changes in g depends on the fraction in which the X dominates the Universe.
∗
2.2.2 At the neutrino freezeout
Next let us take the uniform density slice at the freezeout of neutrinos H = Γ (T ∼
ν
1MeV). On this slice we have the following relations
(1−c )ρ ((cid:126)x) = ρ ((cid:126)x), (30)
ν r γ
c ρ ((cid:126)x) = ρ ((cid:126)x), (31)
ν r ν
ρ ((cid:126)x)+ρ ((cid:126)x) = ρ ((cid:126)x)(= ρ¯ ), (32)
r X tot tot
where c is the energy fraction of neutrinos. For the standard three neutrino species, it
ν
is given by
ρ¯ 21
ν
c = = at neutrino freezeout. (33)
ν
ρ¯ 43
r
Here the symbol γ should be interpreted as a thermal plasma consisting of photons elec-
trons and positrons. The curvature perturbation of each component is related to its
background value as
ρ ((cid:126)x) = ρ¯ e4(ζν−ζ˜), (34)
ν ν
ρ ((cid:126)x) = ρ¯ e4(ζγ−ζ˜), (35)
γ γ
˜
where ζ denotes the curvature perturbation at this epoch and it is different from ζ in
Eq. (28). From Eqs. (30) and (31) we find
ζ = ζ , ζ = ζ , (36)
γ r ν r
which holds nonlinearly, as is expected. We also define dark radiation (DR) as the sum
of extra radiation X and neutrino. It satisfies
ρ ((cid:126)x)+c ρ ((cid:126)x) = ρ ((cid:126)x)+ρ ((cid:126)x) ≡ ρ ((cid:126)x), (37)
X ν r X ν DR
on the uniform density slice at the neutrino freezeout. The curvature perturbation of DR,
ζ , is given by
DR
ρ ((cid:126)x) = ρ¯ e4(ζDR−ζ˜). (38)
DR DR
8
From Eq. (37), we find at first order
˜ ˜
ρ¯ ρ¯ R c R
X ν X ν r
ζ = ζ + ζ = ζ + ζ , (39)
DR ρ¯ X ρ¯ ν R˜ X R˜ r
DR DR DR DR
where we have defined R˜ ≡ R˜(φ) +R˜(σ) = ρ¯ /ρ¯ | , with
DR DR DR DR tot H=Γν
R˜(φ) = R˜(φ) +c R˜(φ),
DR X ν r (40)
R˜(σ) = R˜(σ) +c R˜(σ).
DR X ν r
Here quantities with a tilde means that they are evaluated at the neutrino freezeout.
Relations between quantities with and without a tilde are given by
(cid:16) (cid:17)1/3
R g∗(H=Γσ)
R˜ = RX , R˜ = r g∗(H=Γν) . (41)
X (cid:16) (cid:17)1/3 r (cid:16) (cid:17)1/3
R +R g∗(H=Γσ) R +R g∗(H=Γσ)
X r g∗(H=Γν) X r g∗(H=Γν)
This is approximated as
(cid:18)g (H = Γ )(cid:19)1/3
˜ ∗ ν ˜
R (cid:39) R , R (cid:39) R , (42)
X X r r
g (H = Γ )
∗ σ
if ρ (H = Γ ) (cid:28) ρ (H = Γ ). Here g (H = Γ ) = 10.75 is the relativistic degrees of
X ν r ν ∗ ν
˜
freedom at the neutrino freezeout. Using these quantities, we find that ζ is given by
(cid:32) (cid:33)
3+R R˜ R(σ) R˜ R(σ)
ζ˜= ζ + r r + X X
φ
4 R R
r X
(43)
(cid:34) (cid:40) (cid:32) (cid:33)(cid:41) (cid:35)
(1+R)(3−R) 3+R R˜ R(σ) R˜ R(σ)
× (ζ −ζ )+ − r r + X X (ζ −ζ )2 .
σ φ σ φ
2 2 R R
r X
˜ ˜
It can be checked that it coincides with ζ in Eq. (28) for R = R and R = R .
r r X X
Therefore, as is expected, the relative difference between the evolutions of ρ and ρ
X r
results in the non-conservation of the curvature perturbation.
Up to the second order in ζ and ζ , the DR isocurvature perturbation is given by
φ σ
˜ ˜
S ≡ 3(ζ −ζ)
DR DR
(cid:32) (cid:33) (cid:34)
3+RR˜ R˜ R(σ) R(σ)
= 3 r X r − X (1−c ) −(ζ −ζ )
4 R˜ R R ν σ φ
DR r X
(cid:40) (cid:32) (cid:33)(cid:41) (cid:35)
(R−3)(R+1) 3+R R˜ R(σ)(c +R˜ ) R˜ R(σ)(1+R˜ )
+ + r r ν DR + X X DR (ζ −ζ )2 .
2 2R˜ R R σ φ
DR r X
(44)
9