Gravity of Cosmological Perturbations in the CMB

Abstract

First, we establish which measures of large-scale perturbations are least afflicted by gauge artifacts and directly map the apparent evolution of inhomogeneities to local interactions of cosmological species. Considering nonlinear and linear perturbations of phase-space distribution, radiation intensity and arbitrary species' density, we require that: (i) the dynamics of perturbations defined by these measures is determined by observables within the local Hubble volume; (ii) the measures are practically applicable on microscopic scales and in an unperturbed geometry retain their microscopic meaning on all scales. We prove that all measures of linear overdensity that satisfy (i) and (ii) coincide in the superhorizon limit. Their dynamical equations are simpler than the traditional ones, have a nonsingular superhorizon limit and explicit Cauchy form. Then we show that, contrary to the popular view, the perturbations of the cosmic microwave background (CMB) in the radiation era are not resonantly boosted self-gravitationally during horizon entry. (Consequently, the CMB signatures of uncoupled species which may be abundant in the radiation era, e.g. neutrinos or early quintessence, are mild; albeit non-degenerate and robust to cosmic variance.) On the other hand, dark matter perturbations in the matter era gravitationally suppress large-angle CMB anisotropy by an order of magnitude stronger than presently believed. If cold dark matter were the only dominant component then, for adiabatic perturbations, the CMB temperature power spectrum C_l would be suppressed 25-fold.

Full text

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arXiv:astro-ph/0405157v5 4 Oct 2006
Gravity of Cosmological Perturbations in the CMB
Sergei Bashinsky
Theoretical Division, T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA and
International Centre for Theoretical Physics, Strada Costiera 11, Trieste, Italy
(Dated: October 4, 2006)
First, we establish which measures of large-scale perturbations are least afflicted by gauge arti-
facts and directly map the apparent evolution of inhomogeneities to local interactions of cosmological
species. Considering nonlinear and linear perturbations of phase-space distribution, radiation in-
tensity and arbitrary species’ density, we require that: (i) the dynamics of perturbations defined
by these measures is determined by observables within the local Hubble volume; (ii) the measures
are practically applicable on microscopic scales and in an unperturbed geometry retain their micro-
scopic meaning on all scales. We prove that all measures of linear overdensity that satisfy (i) and (ii)
coincide in the superhorizon limit. Their dynamical equations are simpler than the traditional ones,
have a nonsingular superhorizon limit and explicit Cauchy form. Then we show that, contrary to
the popular view, the perturbations of the cosmic microwave background (CMB) in the radiation
era are not resonantly boosted self-gravitationally during horizon entry. (Consequently, the CMB
signatures of uncoupled species which may be abundant in the radiation era, e.g. neutrinos or early
quintessence, are mild; albeit non-degenerate and robust to cosmic variance.) On the other hand,
dark matter perturbations in the matter era gravitationally suppress large-angle CMB anisotropy by
an order of magnitude stronger than presently believed. If cold dark matter were the only dominant
component then, for adiabatic perturbations, the CMB temperature power spectrum Cℓ would be
suppressed 25-fold.
Contents
I. Introduction
1
II. Measures of Perturbations
A. Phase-space distribution
B. Radiation intensity
C. Energy density
D. Uniqueness of superhorizon values
3
3
5
6
7
III. Dynamics in the Newtonian Gauge
A. Densities
B. Velocities
C. Metric
8
8
9
9
IV. Overview of Perturbation Evolution
A. Superhorizon evolution
B. Horizon entry
10
10
11
V. Features in the Angular Spectrum of the
CMB
A. Preliminaries
B. Radiation era
1. Radiation driving?
2. Gravitational impact of perturbations on
the CMB and CDM
C. Matter era
1. 25-fold Sachs-Wolfe suppression
2. Implications
11
11
12
12
13
14
14
16
VI. Summary
17
Acknowledgments
18
A. Perturbative Dynamics of Typical Species 18
1. Cold dark matter (CDM)
2. Neutrinos
a. General case
b. Ultrarelativistic limit
3. Photon-baryon plasma
a. Tight coupling limit
b. Photon intensity
c. Photon polarization
d. Baryons
4. Quintessence
5. All
18
18
18
19
19
19
19
20
20
21
21
B. CMB Transfer Functions and Cl’s
21
References
22
I.INTRODUCTION
Cosmological observations can be valuable probes of
matter species whose interactions are too feeble to be
studied by more traditional particle physics techniques.
Whenever such “dark” species, including dark energy,
dark matter, or neutrinos, constitute a non-negligible en-
ergy fraction of the universe, these species leave gravita-
tional imprints on the distribution of more readily observ-
able matter, notably, the cosmic microwave background
(CMB) and baryonic astrophysical objects. In the frame-
work of a perturbed cosmological expansion, the dark
species influence the visible matter by affecting both the
expansion rate, controlled by their average energy, and
the metric inhomogeneities, sourced by species’ pertur-
bations.

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The gravitational signatures of the perturbations are
particularly informative as they are sensitive to the in-
ternal (kinetic) properties of the dark species. These sig-
natures are generally dissimilar for species with identical
background energy and pressure but different dynamics
of perturbations, e.g., for self-interacting particles, un-
coupled particles, or classical fields [1–5]. Perturbations
of dark species can affect the observed CMB and matter
power spectra by an order of magnitude. For example,
the gravitational impact of dark matter perturbations
suppresses the CMB power at low multipoles ℓ ? 200 by
up to a factor of 25 (Sec. VC).
Despite the value and prominence of the gravitational
impact of perturbations, most of the existing descriptions
of cosmological evolution are misleading in matching dy-
namics of perturbations in dark sectors to features in
observable distributions. One long-investigated source of
ambiguities and erroneous conclusions is the dependence
of the apparent evolution of perturbations and the in-
duced by them gravitational fields on the choice of space-
time coordinates (metric gauge), e.g. [6, 7]. Of course,
the CMB and large scale structure observables should
be identical in any gauge. However, in various gauges
the features of the observable distributions may appear
to be generated in different cosmological epochs and by
different mechanisms.
For example, given the standard adiabatic initial con-
ditions, the perturbation of the CMB temperature δT/T
grows monotonically on superhorizon scales in the his-
torically important and still popular synchronous gauge,
e.g. [8]; δT/T generally remains constant beyond the
horizon in a single cosmological epoch but changes dur-
ing the transition to another epoch in the calculationally
convenient and intuitive Newtonian gauge, e.g. [9, 10];
or, δT/T is strictly frozen beyond the horizon but evolves
differently since horizon entry in the spatially flat gauge.
These descriptions naively suggest different separation
of the presently measured CMB temperature anisotropy
into the inhomogeneities of primordial (inflationary) ori-
gin and those generated by the gravitational impact of
perturbations in various species (CMB, neutrinos, dark
matter, dark energy, etc.)
The use of gauge-invariant perturbation variables1[6,
7, 11, 12] does not remove this ambiguity [13].
deed, the perturbations of species’ density and veloc-
ity or the metric in any fixed gauge can be written as
gauge-invariant expressions, which may be called “gauge-
invariant” (though, clearly, non-unique) definitions of
In-
1A “gauge-invariant perturbation variable” generally cannot be
measured by a local observer. (For example, the local values of
the gauge-invariant Bardeen potentials Φ and Ψ [6] are deter-
mined by spacetime curvature outside the observer’s past light
cone.)Thus the gauge-invariant perturbations should be dis-
tinguished from gauge-invariant local observables or from local
physical tensor quantities, such as the energy-momentum ten-
sor Tµνor the curvature tensor Rµ
ναβ.
these perturbations [13]. Hence, the variety of gauge-
invariant perturbation variables is at least as large as the
variety of gauge-fixing methods.
Nevertheless, without contradicting general covari-
ance, much of this descriptional ambiguity can be
avoided. We show that in cosmological applications the
perturbed evolution can be described by variables whose
change is necessarily induced by a local physical cause.
Moreover, when the equations of perturbation dynamics
are presented in terms of such variables, the structure of
the equations simplifies considerably. The simpler equa-
tions allow tractable analytical analysis of perturbation
evolution for realistic cosmological models.
We impose two requirements on a measure of dynam-
ical cosmological perturbations:
I. The dynamics of the measure is determined com-
pletely by locally identifiable physical phenomena
within the local Hubble volume;
II. The measure is universally applicable to all scales:
from superhorizon (governed by general relativity)
to subhorizon (governed by microscopic kinetics).
The goals of this paper are, first (Secs. II and III), to
provide a full dynamical description of cosmological in-
homogeneities in terms of measures which satisfy these
two requirements. Second (Secs. IV and V), to establish
the origin of features in the CMB angular spectrum, as
revealed by this more direct description, and to explore
the utility of these features for probing the dark sectors.
Most of the best-known formalisms for the dynamics
of CMB and matter inhomogeneities are formulated in
terms of perturbations of physical observables, such as
radiation temperature T(xµ, ˆ n) or proper energy den-
sity ρ(xµ) [10, 14–17].When these perturbations are
evaluated in the Newtonian gauge, nonsingular on small
scales, then the small-scale dynamics in these formalisms
does reduce to special-relativistic kinetics and Newtonian
gravity. Then, however, the perturbation evolution be-
yond the Hubble scale is nonlocal2and subject to gauge
artifacts. The artifacts are caused by the nonlocality of
defining the motion of coordinate observers, hence, infer-
ring the components of tensors and the splitting of dy-
namical variables into their background values and per-
turbations from the gauge conditions. While some gauge
2For illustration, a perturbation of energy density ρ in a dust (zero
pressure) universe evolves in the Newtonian gauge (26) as
δ ˙ ρ + 3˙ a
aδρ + ρ∇ivi= 3ρ˙Φ,
with
Φ = 4πGa21
∇2
?
δρ − 3˙ a
aρ
1
∇2∇ivi
?
.
Although the first equation above is obtained by linearization
of a causal relation T0µ
equation signal that after elimination of Φ the evolution of δρ
becomes nonlocal. This does not violate causality because δρ
cannot be measured locally.
;µ= 0, the operators 1/∇2in the second

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conditions, e.g. synchronous, can be imposed locally, the
resulting descriptions tend to be singular, contain gauge
modes, and fail to reduce to Newtonian gravity on small
scales.
Studies of the connection of the observed cosmological
inhomogeneities to quantum fluctuations of an inflaton
field during inflation have led to an extensive list of vari-
ables which under certain conditions freeze (become con-
stant in time) beyond the horizon. The best known exam-
ples are the Bardeen curvature ζ [6, 18] (conventionally
interpreted either as a perturbation of intrinsic curva-
ture on uniform-density hypersurfaces or as density per-
turbation on spatially flat hypersurfaces) and the curva-
ture perturbation R on comoving hypersurfaces, e.g. [19].
Both ζ and R are frozen for adiabatic superhorizon per-
turbations. For the general, nonadiabatic, superhorizon
perturbations, the curvature perturbations ζa[20] on the
hypersurfaces of uniform energy density of an individ-
ual minimally coupled perfect fluid a were shown to be
also frozen (“conserved”).3Finally, perturbed cosmolog-
ical evolution has been described in terms of conserved
spatial gradients of various quantities [22–24]. The evo-
lution of any of these variables is more robust to gauge
artifacts on superhorizon scales than, for example, that of
δT/T, δρ/ρ, or (the Bardeen) potentials Φ and Ψ in the
Newtonian gauge. However, neither the uniform-density,
nor comoving, nor spatially flat hypersurfaces reduce to
Newtonian hypersurfaces on subhorizon scales. Nor does
the evolution of gradients tend to the familiar picture of
particles and fields propagating in Minkowski spacetime.
In Sec. II we consider natural measures for nonlinear
perturbations of phase-space distribution and radiation
intensity and for linear perturbations of species’ density
that conform to both requirements I and II. Moreover, we
prove in Sec. IID that, although there is no “physically
preferred” description of perturbed evolution, the over-
all change of density perturbations of any species during
horizon entry is the same in any description which satis-
fies certain conditions formalizing requirements I and II.
This change is different in most of the traditional for-
malisms, violating these requirements.
In Sec. III we describe a closed formulation of lin-
ear dynamics of scalar perturbations in terms of the
above measures evaluated in the Newtonian gauge. We
find that this formulation has several technical advan-
tages, ultimately related to its tighter connection be-
tween the perturbation measures and the causality of
3The perturbations ζa[20] are conserved for uncoupled perfect flu-
ids [20] or species whose perturbations are internally adiabatic [4]
only if gravitational decays [21] of species into another type of
species are negligible. This should be a reasonable assumption
for realistic applications to the evolution of the CMB and large
scale structure.
the perturbed cosmological dynamics. The proposed ap-
proach has broad scope of applicability, providing an eco-
nomical and physically adequate description of phenom-
ena which involve inhomogeneous evolution of multiple
species and non-negligible general-relativistic effects. Ex-
amples include the physics of inflation, reheating, the
CMB, and cosmic structure. In this paper we focus on
the last two topics.
After a concise review of the evolution of perturba-
tions on superhorizon scales and during horizon entry in
Sec. IV, we apply in Sec. V the developed formalism to
study the gravitational signatures of various species in
CMB temperature anisotropy and large scale structure.
Contrary to a popular view, e.g. [17, 25–29], based on
the study of traditional proper perturbations in the New-
tonian gauge, we find that CMB perturbations in the
radiation era are not resonantly boosted by their self-
gravity (Sec. VB1). As a consequence, the gravitational
signatures of dark species in the radiation era, such as
neutrinos or a dynamical scalar field (quintessence [30,
31]), are moderate; as these species, even when they are
abundant, do not untune any physical resonant amplifi-
cation. Fortunately, due to low cosmic variance on the
corresponding scales and the existence of characteristic
nondegenerate signatures (Sec. VB2), we can still ex-
pect meaningful robust constraints on the nature of the
dark radiation sector.
On the other hand, the gravitational impact of dark
sectors’ perturbations on the CMB in the matter era is
found to be an order of magnitude stronger than the
traditional interpretations suggest (Sec. VC).
We summarize our results in Sec. VI.
presents linear dynamical equations for scalar perturba-
tions of typical cosmological species in terms of the sug-
gested measures. Appendix B summarizes the formulas
for the scalar transfer functions and angular power spec-
tra of CMB temperature and polarization. The main
notations to be used in this paper are listed in Table I.
Appendix A
II.MEASURES OF PERTURBATIONS
A.Phase-space distribution
Our first example of a quantity whose dynamics can be
fully determined by physics in the local Hubble volume
is the one-particle phase-space distribution of classical,
possibly relativistic, point particles [photons, neutrinos,
or cold dark matter (CDM)] f(τ,xi,Pi) = dN/d3xid3Pi.
(By default, Latin indices range from 1 to 3 and Greek
from 0 to 3.) We consider f as a function of the co-
ordinate time τ ≡ x0, spatial coordinates xi, and, cru-
cially for the following results, canonically conjugate mo-
menta Pi. The distribution f evolves according to the
Boltzmann equation
˙f +dx
dτ
i∂f
∂xi+dPi
dτ
∂f
∂Pi
= C (1)

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Symbol
τ
x
Pi
P
ni
f(xµ,Pi)
df(xi,Pi)
I(xµ,ni)
ι(xµ,ni)
fαβ, Iαβ, ιαβ Describe polarized photons
ρ(xµ), p(xµ) Energy density and pressure of species
Perturbation of species’ coordinate number density
d(xµ)
(for a fluid of particles, δncoo/ncoo with ncoo = dN/d3x)
vi(xµ) Normal bulk velocity of species
v′i(xµ)Coordinate bulk velocity
u(xµ)Velocity potential of scalar perturbations, vi = −∇iu
σ(xµ)Scalar potential of anisotropic stress
dl(xµ)Scalar multipole potentials of ι (in particular, d0 = d, d1 = u, d2 = σ)
pl(xµ) Scalar multipole potentials of photon polarization
φ(xµ)A classical scalar field (quintessence)
gµν
Metric tensor, with the signature (−,+,+,+)
aScale factor in the FRW background
hµν
General perturbation of the metric, δgµν/a2
D and ǫGeneral scalar perturbations of the spatial metric gij
Φ and ΨScalar perturbations of the metric in the Newtonian gauge
ζa
Reduced curvature, D +1
overdot,˙∂/∂τ
H
γ4πGa2(ρ + p) in the FRW background
ιeff and Θeff Effective intensity and temperature perturbations
cs
Speed of sound in the photon-baryon plasma
Rb
Baryon to photon enthalpy ratio, 3ρb/(4ργ)
S
ϕPhase of acoustic oscillations, kS(τ)
τc
Mean τ of a photon collisionless flight
δT/T|in
∆T/TPresently observed perturbation of CMB temperature
MeaningDefinition
Sec. IIA [eq. (26)]Coordinate time, x0[Conformal time in the FRW background]
Spatial coordinates, xi[Comoving coordinates in the FRW background]
Canonical momenta
(?3
Direction of propagation, Pi/P
Phase-space distribution
Canonical perturbation of f, f(xi,Pi) −¯f(P)
Conformal intensity of radiation
Perturbation of radiation intensity, I/¯I − 1
[eq. (26)]
Sec. IIA
eq. (3)
eq. (3)
Sec. IIA
Sec. IIA, eq. (A3)
eq. (7)
eq. (9)
Sec. A3c
p. 6
eq. (20) or (27)
(p. 7)
eq. (17)
eq. (21)
eq. (30)
eq. (32)
eq. (A8)
eq. (A21)
Sec. A4
i=1P2
i)1/2in any metric
eq. (14)
eq. (14)
eq. (29)
eq. (26)
Sec. IIIA
3∇2ǫ = −1
4a2(∇−2)(3)R, of 3-slices ρa = const
Coordinate expansion rate [conformal, in the FRW background]Sec. IIA [˙ a/a]
eq. (34)
eqs. (47) and (48)
eq. (A10)
Sec. VA
Sec. VB2
eq. (50)
eq. (A14)
eq. (57)
eq. (49)
Acoustic horizon,?τ
0csdτ (when Rb≪ 1, τ/√3)
Primordial superhorizon perturbation of CMB temperature,
1
3dγ
??
k≪H
TABLE I: Summary of the main notations.
(for a systematic formulation of kinetic theory in gen-
eral relativity see, e.g., [32].)
only gravitationally, their canonical momenta Pi coin-
cide with the spatial covariant components of the parti-
cle 4-momenta Pµ: Pi= giµPµ, where gµν is the metric
tensor. Then dxi/dτ = Pi/P0, dPi/dτ is given by the
geodesic equation, and C, accounting for two-particle col-
lisions, vanishes. Hence,
If the particles interact
˙f +Pi
P0
∂f
∂xi+gµν,iPµPν
2P0
∂f
∂Pi
= 0.(2)
We stress that this equation and the following observa-
tion apply to the fully nonlinear general-relativistic dy-
namics.
Let us consider the minimally coupled particles in
the perturbed spacetime, possibly populated by addi-
tional species. Let us also suppose that in certain co-
ordinates the spatial scale, λ, of inhomogeneities in the
particle distribution and in the metric is much larger
than the temporal scale of local cosmological expansion,
H−1, where for nonlinear theory H ≡
Then from eq. (2), where both the second and third
terms contain spatial gradients (∂f/∂xiand gµν,i), we
see that˙f/f = O(λ−1) ≪ H. In the limit of super-
horizon inhomogeneities (λH → ∞) the phase-space dis-
tribution f(τ,xi,Pi) of minimally coupled particles be-
comes time-independent (frozen). Then its background
value,¯f, and perturbation, df ≡ f(xµ,Pi) −¯f(P), are
also frozen.
1
6
d
dτln(detgij).
Conversely, an apparent temporal change of df (at
fixed xiand Pi in any regular gauge) can always be
attributed either to the gravity of physical inhomo-
geneities within the local Hubble volume or to local non-
gravitational interaction.
The majority of contemporary formalisms for cosmo-
logical evolution in phase-space, e.g. [10, 15, 17, 33, 34],
work not with the canonical momentum of particles Pi

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5
but rather with proper momentum p, measured by coor-
dinate or by normal observers. These formalisms typi-
cally consider the phase-space distribution as a function
of q ≡ ap, the proper momentum rescaled by the back-
ground scale factor. The evolution of the correspond-
ing perturbation δf(xµ,q) ≡ f(xµ,q) −¯f(q), as well as
of perturbations of integrated proper densities and in-
tensities, depends on the nonlocal procedure for defining
the observer’s frame from the gauge condition. In typ-
ical fixed gauges δf(xµ,q), unlike df(xµ,Pi), generally
changes even in the absence of local physical inhomo-
geneity or local non-gravitational coupling.4
Thus the stated in the introduction criteria I and II
for a measure of cosmological inhomogeneities are ful-
filled for a perturbation of one-particle phase-space dis-
tribution f(xµ,Pi), provided Piare the particle canonical
momenta. A description in terms of phase-space distribu-
tions is, however, unnecessarily detailed for most cosmo-
logical applications. Instead, it is more convenient to use
integrated characteristics of the species, such as inten-
sities (for radiation of photons or other ultrarelativistic
species) or energy densities and momentum-averaged ve-
locities (for arbitrary species, including non-relativistic
particles, fluids, or classical fields). In the next two sub-
sections we discuss the general-relativistic measures of
perturbations in intensity and density that also satisfy
criteria I and II.
B. Radiation intensity
We start by considering inhomogeneities in the inten-
sity of any type of cosmological radiation, such as CMB
photons or relic neutrinos at the redshifts at which the
kinetic energy of the particles dominates their mass. We
describe the direction of particle propagation by
ni≡Pi
P
whereP2≡
3
?
i=1
P2
i,(3)
so that
3
?
i=1
n2
i= 1.(4)
We also introduce n0 ≡ P0/P (for the ultrarelativistic
particles gµνnµnν = 0) and nµ≡ gµνnν = Pµ/P. The
4For
gauge (26), qi= Pi/(1 − Ψ) [10, 17, 34]. Hence,
example,for linearperturbationsintheNewtonian
δf(xµ,q) = df(xµ,Pi) − ΨPi
∂¯f
∂Pi
.
While the first term on the right-hand side evolves causally on
superhorizon scales, the Newtonian potential Ψ does not, and so,
neither does δf(xµ,q).
motivation for the noncovariant definitions (3) is to de-
scribe radiation transport in terms of variables which are
fully specified by the “dynamical” quantity f(τ,xi,Pi)
but are unmixed with the gauge-dependent and, in gen-
eral, noncausally evolving metric gµν.
The energy-momentum tensor of radiation equals
?
Tµ
ν=
d3Pi
√−g
PµPν
P0
f. (5)
Substituting Pµ= nµP and Pµ= nµP, we can rewrite
it as
?
Tµ
ν=
d2ni
√−g
nµnν
n0
I(6)
–a directional average of an expression which depends
only on the local value of the metric and the quantity
?∞
0
I(xµ,ni) ≡P3dP f(xµ,niP).(7)
The variable (7), to be called the conformal intensity
of radiation, inherits such useful properties of f(xµ,Pi)
as time-independence for superhorizon inhomogeneities
of non-interacting particles and reduction to the proper
intensity, dE/(dV d2ˆ n), in the Minkowski limit.
The dynamics of I(xµ,ni) for minimally coupled ra-
diation in a known metric is given by a closed equa-
tion [35] which is straightforwardlyderived by integrating
the Boltzmann equation (2) over P3dP and remembering
that for the ultrarelativistic particles gµνnµnν= 0:5
?
nµ∂I
∂xµ= nµnνgµν,i
2niI −1
2
∂I
∂ni
?
.(8)
This equation confirms that for decoupled particles which
are perturbed only on superhorizon scales (the gradients
∂I/∂xiand gµν,iare negligible) the conformal intensity
freezes:˙I = 0.
Now we can describe perturbation of the intensity by
a variable
ι(xµ,ni) ≡I
¯I− 1,(9)
where¯I is a time-independent background value of I. As
intended, the perturbation ι is time-independent in full
(nonlinear) general relativity for superhorizon perturba-
tions of uncoupled radiation. This variable also reduces
5In the last terms of eqs. (8) and (10), the partial derivatives
with respect to the components of ni= Pi/P, constrained by
condition (4), can be naturally defined as
?P∂µα
∂
∂ni
≡
?
α=1,2
∂Pi
?
∂
∂µα,
where µα= (µ1,µ2) are any two independent variables parame-
terizing ˆ n.