12 Cosmic microwave background
The cosmic microwave background (CMB) is isotropic to a high degree. This tells
us that the early universe was very homogeneous at t = tdec, when the CMB was
formed. However, with precise measurements we can detect the small anisotropy of
the CMB, which reflects the small perturbations in the early universe.
This anisotropy was first detected by the COBE satellite in 1992, which mapped
the whole sky in three microwave frequencies. The angular resolution of COBE was
rather poor, 7◦, so only features larger than this were detected. Measurements with
better resolution, but covering only small parts of the sky, were then performed
using instruments carried by balloons to the upper atmosphere, and ground-based
detectors located at high altitudes.
The next full-sky map of the CMB was made by the Wilkinson Microwave
Anisotropy Probe (WMAP) satellite, in orbit around the L2 point of the Sun-Earth
system, 1.5 million kilometers from the Earth in the direction opposite to the Sun.
The satellite was launched by NASA in June 2001, and the results of the first year
of measurements were published in February 2003. The WMAP satellite made eight
years of measurements, and the data from the first seven have been made public.
The Planck satellite was launched by ESA in May 2009, and the first cosmological
results were released in March 2013. The polarisation data has not yet been released,
it is expected to be made public in December 2014.
Figure 1: The cosmic microwave background according to the DMR instrument aboard the
COBE satellite.
Figures 1 to 3 show the observed variation δT
T obs
in the temperature of the
CMB on the sky (in the first two plots, yellow and red mean hotter than average,
blue means colder than average).
The photons we see as the CMB have have travelled to us from where our past
light cone intersects the hypersurface where photons decouple at t = tdec. This
intersection forms a sphere that is called the last scattering surface1. We are at the
1
Or the last scattering sphere. The expression “last scattering surface” often refers to the entire
t = tdec time slice.
190
12 COSMIC MICROWAVE BACKGROUND 191
Figure 2: The cosmic microwave background according to WMAP 7-year results.
Figure 3: The cosmic microwave background according to Planck 1.5-year results.
12 COSMIC MICROWAVE BACKGROUND 192
Figure 4: The observed CMB temperature anisotropy gets a contribution from the last
scattering surface, (δT/T)intr = Θ(tdec, xls, ˆn) and from along the photon’s journey to us,
(δT/T)jour.
center of this sphere, which extends away from us both in space and in time.
The observed temperature anisotropy is due to two contributions, an intrinsic
temperature variation at the surface of last scattering and a variation in the redshift
the photons have suffered during their journey to us,
δT
T obs
=
δT
T intr
+
δT
T jour
. (12.1)
See figure 4. There are two ways to define what we mean by the CMB perturbation
δT. The first way is to just take the angular average of the temperature field and
call this the mean, ¯T ≡ T0 ≡ 1
4π dΩT, and defined the anisotropy as the difference
from the mean, δT = T − T0. This is the physically most correct way. However, in
the context of perturbed FRW models, it can be simpler to call the temperature in
the background model the mean temperature. The perturbations also contribute to
the mean temperature, so this is a bit misleading, but common. We will also use
the notation δT
T instead of δT
¯T
or δT
T0
, as is common, but it should be understood
that the temperature in the denominator is the mean temperature. (Of course, this
would only make a difference at second order.)
The first term in (12.1), δT
T intr
represents the temperature variation of the
photon gas at t = tdec. (It also includes the Doppler effect from the motion of this
photon gas.) At that time the largest scales we see on the CMB sky were still outside
the horizon. The separation of δT/T into two components is gauge-dependent. If
the time slice t = tdec dips further into the past in some location, it finds a higher
temperature, but the photons from there also have then a longer way to go and suffer
a larger redshift, so the two effects balance each other. We can calculate in any gauge
we want, getting different results for (δT/T)intr and (δT/T)jour depending on the
gauge, but their sum (δT/T)obs is gauge-independent, because it is an observed
quantity.
One might think that (δT/T)intr should be equal to zero, since in our earlier discussion
of recombination and decoupling we identified decoupling with a particular
temperature Tdec ∼ 3000 K. This kind of thinking corresponds to a particular gauge
choice where the t = tdec time slice coincides with the T = Tdec hypersurface. In
12 COSMIC MICROWAVE BACKGROUND 193
Figure 5: Depending on the gauge, the Tdec = const. surface may, or (usually) may not
coincide with the t = tdec time slice.
this gauge (δT/T)intr = 0, except for the Doppler effect (we are not going to use this
gauge). Anyway, it is not true that all photons have their last scattering exactly
when T = Tdec. Rather they occur during a rather large temperature interval and
time period. The zeroth-order (background) time evolution of the temperature of
the photon distribution is the same before and after last scattering, T ∝ a−1, so it
does not matter how we draw the artificial separation line, the time slice t = tdec
separating the fluid and free particle treatment of the photons. See figure 5.
12.1 Multipole analysis
The CMB temperature anisotropy is a function on a sphere. In analogy with Fourier
expansion in three-dimensional flat space, we separate out the contributions of different
angular scales by doing a multipole expansion,
δT
T0
(θ, φ) = aℓmYℓm(θ, φ) (12.2)
where the sum runs over l = 1, 2, . . . ∞ and m = −l, . . . , l, giving 2ℓ + 1 values of m
for each ℓ. The functions Yℓm(θ, φ) are the spherical harmonics (see figure 6), which
form an orthonormal set of functions over the sphere, so that we can calculate the
multipole coefficients aℓm from
aℓm = Y ∗
ℓm(θ, φ)
δT
T
(θ, φ)dΩ . (12.3)
This definition gives dimensionless aℓm. Often they are defined without the T0 =
2.725 K term in (12.2), and then they have the dimension of temperature and are
usually given in units of µK.
The coefficient al0 is real but the other alm are complex, and al,−m = a∗
lm. The
sum begins at ℓ = 1, since Y00 = const. and therefore we must have a00 = 0 for
a quantity which represents a deviation from average. The dipole part, ℓ = 1, is
dominated by the Doppler effect due to the motion of the solar system with respect
to the last scattering surface, and it is difficult to separate the cosmological dipole
caused by large scale perturbations. (This was done for the first time with Planck,
though not to great accuracy.) Therefore we are here interested only in the ℓ ≥ 2
part of the expansion.
Another notation for Yℓm(θ, φ) is Yℓm(ˆn), where ˆn is a unit vector whose direction
is specified by the angles θ and φ. (The hat denotes unit vector.)
12.1.1 Spherical harmonics
We list here some useful properties of the spherical harmonics. They are orthonormal
functions on the sphere, so
dΩ Yℓm(θ, φ)Y ∗
ℓ′m′ (θ, φ) = δℓℓ′ δmm′ . (12.4)
12 COSMIC MICROWAVE BACKGROUND 194
Summing over the m corresponding to the same multipole number ℓ we have the
closure relation
m
|Yℓm(θ, φ)|2
=
2ℓ + 1
4π
. (12.5)
We will also use the expansion of a plane wave in terms of spherical harmonics,
eik·x
= 4π
ℓm
iℓ
jℓ(kx)Yℓm(ˆx)Y ∗
ℓm(ˆk) . (12.6)
Here ˆx and ˆk are the unit vectors in the directions of x and k, and jℓ is the spherical
Bessel function.
12.1.2 The theoretical angular power spectrum
The CMB anisotropy is due to the primordial perturbations, and therefore it reflects
their Gaussian nature. Because we get the values of the aℓm from the other perturbation
quantities through linear equations (in first-order perturbation theory), the
aℓm are also (complex) Gaussian random variables. Since they represent deviation
from the average temperature, their expectation value is zero,
aℓm = 0 , (12.7)
and the quantity we want to calculate from theory is the variance |aℓm|2 to get a
prediction for the typical size of the aℓm. The isotropic nature of the random process
shows up in the aℓm so that these expectation values depend only on ℓ not m. (The
ℓ are related to the angular size of the anisotropy pattern, whereas the m are related
to “orientation” or “pattern”.) Since |aℓm|2 is independent of m, we can define
Cℓ ≡ |aℓm|2
=
1
2ℓ + 1 m
|aℓm|2
. (12.8)
The aℓm are independent random variables, so
aℓma∗
ℓ′m′ = δℓℓ′ δmm′ Cℓ . (12.9)
This function Cℓ (of integers l ≥ 1) is called the (theoretical) angular power spectrum.
It is analogous to the power spectrum P(k) of density perturbations. For
Gaussian perturbations, Cℓ contains all the statistical information about the CMB
temperature anisotropy. This is all we can predict from theory. Thus analysis of
the CMB anisotropy consists of calculating the angular power spectrum from the
observed CMB and comparing it to the Cℓ predicted by theory2.
2
In addition to the temperature anisotropy, the CMB also has another property, its polarisation.
There are two additional power spectra related to the polarisation, CEE
ℓ and CBB
ℓ , and one related
to the correlation between temperature and polarisation, CT E
ℓ . The spectra CEE
ℓ and CT E
ℓ have
been measured, while there is thus far no detection of a non-zero CBB
ℓ , only an upper bound. A
detection would indicate the presence of primordial gravitational waves. In the simplest inflationary
models, such as the m2
ϕ2
model, the amplitude of the gravitational waves produced during inflation
is large enough that it should be seen by Planck. In many other models, the amplitude is too small
to be detected by CMB experiments in the near future.
12 COSMIC MICROWAVE BACKGROUND 195
Figure 6: The three lowest multipoles ℓ = 1, 2, 3 of spherical harmonics. Left column: Y10,
Re Y11, Im Y11. Middle column: Y20, Re Y21, Im Y21, Re Y22, Im Y22. Right column: Y30,
Re Y31, Im Y31, Re Y32, Im Y32, Re Y33, Im Y33. Figure by Ville Heikkil¨a.
12 COSMIC MICROWAVE BACKGROUND 196
Just like the three-dimensional density power spectrum P(k) gives the contribution
of scale k to the density variance δ(x)2 , the angular power spectrum Cℓ is
related to the contribution of multipole ℓ to the temperature variance,
δT(θ, φ)
T
2
=
ℓm
aℓmYℓm(θ, φ)
ℓ′m′
a∗
ℓ′m′ Y ∗
ℓ′m′ (θ, φ)
=
ℓℓ′ mm′
Yℓm(θ, φ)Y ∗
ℓ′m′ (θ, φ) aℓma∗
ℓ′m′
=
ℓ
Cℓ
m
|Yℓm(θ, φ)|2
=
ℓ
2ℓ + 1
4π
Cℓ , (12.10)
where we used (12.9) and the closure relation (12.5).
Thus, if we plot (2ℓ + 1)Cℓ/4π on a linear ℓ scale, or ℓ(2ℓ + 1)Cℓ/4π on a logarithmic
ℓ scale, the area under the curve gives the temperature variance, i.e. the
expectation value for the squared deviation from the average temperature. It has
become customary to plot the angular power spectrum as ℓ(ℓ + 1)Cℓ/2π, which is
neither of these, but for large ℓ approximates the second case. The reason for this
custom is explained later.
Equation (12.10) represents the expectation value from theory and thus it is the
same for all directions θ,φ. The actual, “realised”, value of course varies from one
direction θ,φ to another. We can imagine an ensemble of universes, each representing
a different realisation of the same random process that produces the primordial
perturbations. Then represents the average over such an ensemble.
12.1.3 Observed angular power spectrum
Theory predicts expectation values |aℓm|2 from the random process responsible for
the CMB anisotropy, but we can observe only one realisation of this random process,
the set {aℓm} of our CMB sky. We define the observed angular power spectrum as
the average
Cℓ ≡
1
2ℓ + 1 m
|aℓm|2
(12.11)
of these observed values.
The variance of the observed temperature anisotropy is the average of δT(θ,φ)
T
2
over the celestial sphere,
1
4π
δT(θ, φ)
T
2
dΩ =
1
4π
dΩ
ℓm
aℓmYℓm(θ, φ)
ℓ′m′
a∗
ℓ′m′ Y ∗
ℓ′m′ (θ, φ)
=
1
4π
ℓm ℓ′m′
aℓma∗
ℓ′m′ Yℓm(θ, φ)Y ∗
ℓ′m′ (θ, φ)dΩ
δℓℓ′ δmm′
=
1
4π
ℓ m
|aℓm|2
(2ℓ+1)Cℓ
=
ℓ
2ℓ + 1
4π
Cℓ . (12.12)
12 COSMIC MICROWAVE BACKGROUND 197
Figure 7: The observed angular power spectrum Cℓ according to the Planck satellite.
The observational results are the data points, with error bars representative of the cosmic
variance. The solid curve is the theoretical Cℓ from the best-fit ΛCDM model, and the gray
band around it represents the cosmic variance corresponding to this Cℓ.
Contrast this with (12.10), which gives the variance of δT/T at an arbitrary location
on the sky over different realisations of the random process which produced the
primordial perturbations; whereas equation (12.12) gives the variance of δT/T of
our given sky over the celestial sphere.
12.1.4 Cosmic variance
The expectation value of the observed spectrum Cℓ is equal to Cℓ, the theoretical
spectrum (12.8), i.e.
Cℓ = Cℓ ⇒ Cℓ − Cℓ = 0 , (12.13)
but its actual, realised, value is not, although we expect it to be close. The expected
squared difference between Cℓ and Cℓ is called the cosmic variance. We can calculate
it using the properties of (complex) Gaussian random variables (exercise). The
answer is
(Cℓ − Cℓ)2
=
2
2ℓ + 1
C2
ℓ . (12.14)
We see that the expected relative difference between Cℓ and Cℓ is smaller for
higher ℓ. This is because we have a larger (size 2ℓ + 1) statistical sample of aℓm
available for calculating the Cℓ.
The cosmic variance limits the accuracy of comparison of CMB observations with
theory, especially for large scales (low ℓ).
12.2 Multipoles and scales
12.2.1 Rough correspondence
The different multipole numbers ℓ correspond to different angular scales, low ℓ to
large scales and high ℓ to small scales. Examination of the functions Yℓm(θ, φ) reveals
that they have an oscillatory pattern on the sphere, so that there are typically ℓ
“wavelengths” of oscillation around a full great circle of the sphere. See figure 8.
12 COSMIC MICROWAVE BACKGROUND 198
Thus the angle corresponding to this wavelength is
θλ =
2π
ℓ
=
360◦
ℓ
. (12.15)
See figure 9. The angle corresponding to a “half-wavelength”, i.e. the separation
between a neighbouring minimum and maximum is then
θres =
π
ℓ
=
180◦
ℓ
. (12.16)
This is the angular resolution required of the microwave detector for it to be able to
resolve the angular power spectrum up to this ℓ.
For example, COBE had an angular resolution of 7◦ allowing a measurement up
to ℓ = 180/7 = 26, WMAP had resolution 0.23◦ reaching to ℓ = 180/0.23 = 783,
and the European Planck satellite has resolution 5′, which allows to measure Cℓ up
to ℓ = 21603.
The angles on the sky are related to actual physical or comoving distances via the
angular diameter distance dA, defined as the ratio of the physical length (transverse
to the line of sight) and the angle it covers, as discussed in chapter 3,
dA ≡
λphys
θ
. (12.17)
Likewise, we defined the comoving angular diameter distance dc
A by
dc
A ≡
λc
θ
(12.18)
where λc = (1/a)λphys = (1 + z)λphys is the corresponding comoving length. Thus
dc
A = (1/a)dA = (1 + z)dA. See figure 10.
Consider now the Fourier modes of our earlier perturbation theory discussion.
A mode with comoving wavenumber k has comoving wavelength λc = 2π/k. Thus
this mode should show up as a pattern on the CMB sky with angular size
θλ =
λc
dc
A
=
2π
kdc
A
=
2π
ℓ
. (12.19)
For the last equality we used the relation (12.15). From it we get that the modes
with wavenumber k contribute mostly to multipoles around
ℓ = kdc
A . (12.20)
12.2.2 Exact treatment
The above matching of wavenumbers with multipoles is rather naive, for two reasons:
1. The description of a spherical harmonic Yℓm having an “angular wavelength”
of 2π/ℓ is just a crude characterisation. See figure 8.
2. The modes k are not wrapped around the sphere of last scattering, but the
wave vector forms a different angle with the sphere at different places.
3
In reality, there is no sharp cut-off at a particular ℓ, the observational error bars just blow up.
12 COSMIC MICROWAVE BACKGROUND 199
Figure 8: Randomly generated skies containing only a single multipole ℓ. Staring from top
left: ℓ = 1 (dipole only), 2 (quadrupole only), 3 (octopole only), 4, 5, 6, 7, 8, 9, 10, 11, 12.
Figure by Ville Heikkil¨a.
Figure 9: The rough correspondence between multipoles ℓ and angles.
12 COSMIC MICROWAVE BACKGROUND 200
Figure 10: The comoving angular diameter distance relates the comoving size of an object
and the angle in which we see it.
The following precise discussion applies only for the case of a flat universe (K = 0
Friedmann model as the background), where one can Fourier expand functions on
a time slice. We start from the expansion of the plane wave in terms of spherical
harmonics, for which we have the result (12.6),
eik·x
= 4π
ℓm
iℓ
jℓ(kx)Yℓm(ˆx)Y ∗
ℓm(ˆk) , (12.21)
where jℓ is the spherical Bessel function.
Consider now some function
f(x) =
k
fkeik·x
(12.22)
on the t = tdec time slice. We want the multipole expansion of the values of this
function on the last scattering sphere. See figure 11. These are the values f(xˆx),
where x ≡ |x| has a constant value, the (comoving) radius of this sphere. Thus
aℓm = dΩxY ∗
ℓm(ˆx)f(xˆx)
=
k
dΩxY ∗
ℓm(ˆx)fkeik·x
= 4π
k ℓ′m′
dΩxfkY ∗
ℓm(ˆx)iℓ′
jℓ′ (kx)Yℓ′m′ (ˆx)Y ∗
ℓ′m′ (ˆk)
= 4πiℓ
k
fkjℓ(kx)Y ∗
ℓm(ˆk) , (12.23)
where we used the orthonormality of the spherical harmonics. The corresponding
result for a Fourier transform f(k) is
aℓm =
4πiℓ
(2π)3/2
d3
kf(k)jℓ(kx)Y ∗
ℓm(ˆk) . (12.24)
The jℓ are oscillating functions with decreasing amplitude. For large values of ℓ
the position of the first (and largest) maximum is near kx = ℓ (see figure 12). Thus
the aℓm pick a large contribution from the Fourier modes k for which
kx ∼ ℓ . (12.25)
12 COSMIC MICROWAVE BACKGROUND 201
Figure 11: A plane wave intersecting the last scattering sphere.
0 2 4 6 8 10 12 14 16 18 20
-0.1
0
0.1
0.2
0.3 j2
(x)
j3
(x)
j4
(x)
0 20 40 60 80 100
-0.1
0
0.1
0.2
0.3
180 190 200 210 220 230 240 250
-0.01
-0.005
0
0.005
0.01
0.015
0.02 j200
(x)
j201
(x)
j202
(x)
0 200 400 600 800 1000
-0.01
-0.005
0
0.005
0.01
Figure 12: Spherical Bessel functions jℓ(x) for ℓ = 2, 3, 4, 200, 201, and 202. Note how
the first and largest peak is near x = ℓ (but to be precise, at a slightly larger value). Figure
by R. Keskitalo.
12 COSMIC MICROWAVE BACKGROUND 202
In a flat universe the comoving distance x (from our location to the sphere of last
scattering) and the comoving angular diameter distance dc
A are equal, so we can
write this result as
kdc
A ∼ ℓ . (12.26)
The conclusion is that a given multipole ℓ acquires a contribution from modes
with a range of wavenumbers, but most of the contribution comes from near the value
given by (12.20). This concentration is tighter for larger ℓ. We will use equation
(12.20) for qualitative purposes.
12.3 Important distance scales on the last scattering surface
12.3.1 Angular diameter distance to the last scattering surface
In chapter 3 we derived the comoving angular diameter distance to redshift z in a
FRW model,
dc
A(z) =
1
√
−K
sinh
√
−K
z
0
dz′
H(z′)
= H−1
0
1
1
1+z
da
Ω0(a − a2) − ΩΛ0(a − a4) + a2
, (12.27)
where the second line holds for an FRW model that contains only matter and vacuum
energy (Ω0 = Ωm0 +ΩΛ0). In the real universe, the contribution of radiation is small,
since the radiation-dominated era ends early, when the universe is around 50 000
years old. Recall that Ω0 − 1 = −ΩK0 = K/H2
0 . We are interested in the distance
to the last scattering sphere, i.e. dc
A(zdec), where 1 + zdec ≈ 1090.
In the simplest case of the spatially flat matter-dominated universe, ΩΛ0 = 0,
Ωm0 = 1, the integral gives
dc
A(zdec) = H−1
0
1
1
1+z
da
√
a
= 2H−1
0 1 −
1
√
1 + zdec
= 1.94H−1
0 ≈ 2H−1
0 , (12.28)
where the last approximation corresponds to ignoring the contribution from the
lower limit.
We also consider two more general situations, of which the above is a special
case.
a) Open universe with no dark energy, ΩΛ0 = 0 and Ωm0 = Ω0 < 1. Now we have
dc
A(zdec) =
H−1
0√
1 − Ωm0
sinh 1 − Ωm0
1
1
1+z
da
(1 − Ωm0)a2 + Ωm0a
=
H−1
0√
1 − Ωm0
sinh


1
1
1+z
da
a2 + Ωm0
1−Ωm0
a


=
H−1
0√
1 − Ωm0
sinh 2 arsinh
1 − Ωm0
Ωm0
− 2 arsinh
1 − Ωm0
Ωm0
1
1 + zdec
≈
H−1
0√
1 − Ωm0
sinh 2 arsinh
1 − Ωm0
Ωm0
= 2
H−1
0
Ωm0
, (12.29)
12 COSMIC MICROWAVE BACKGROUND 203
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
1
2
3
4
5
6
7
8
9
10
ComovingdistanceinH0
-1
Horizon distance for = 0
distance
angular diameter distance
Figure 13: The comoving proper distance dc
P (z = ∞) (dashed) and the comoving angular
diameter distance dc
A(z = ∞) (solid) to the horizon in matter-only open universe. The
vertical axis is the distance in units of Hubble distance H−1
0 and the horizontal axis is the
density parameter Ω0 = Ωm0. The distances to last scattering, dc
P (zdec) and dc
A(zdec), are a
few per cent less.
where again the approximation ignores the contribution from the lower limit
(i.e., it actually gives the comoving angular diameter distance to the horizon,
dc
A(z = ∞)). In the last step we used sinh 2x = 2 sinh x cosh x =
2 sinh x 1 + sinh2
x. We show this result (together with the comoving proper
distance dc
P (z = ∞)) in figure 13.
b) Spatially flat universe with vacuum energy, ΩΛ + Ωm = 1. Here the integral
does not give an elementary function, but a reasonable approximation, which
we use in the following, is
dc
A(zdec) ≈
2
Ω0.4
m
H−1
0 . (12.30)
The distance dc
A(zdec) depends on the expansion history of the universe. For one,
the longer it takes for the universe to cool from Tdec to T0 (i.e., to expand by the
factor 1 + zdec), the longer distance the photons have time to travel. For spatially
curved universes the angular diameter distance gets an additional effect from the
geometry of the universe, which acts like a “lens” to make the distant CMB pattern
at the last scattering sphere to look smaller or larger (see figure 14).
12.3.2 Decoupling scale and the matter-radiation equality scale
Subhorizon (k ≫ aH) and superhorizon (k ≪ aH) scales behave differently. Thus
we want to know which of the structures we see on the last scattering surface are
12 COSMIC MICROWAVE BACKGROUND 204
Figure 14: The geometry effect in a closed (top) or an open (bottom) universe affects the
angle at which we see a structure of given size at the last scattering surface, and thus its
angular diameter distance.
subhorizon and which are superhorizon (at the time of last scattering). For that we
need to know the comoving Hubble scale aH at tdec.
We make the approximation that neutrinos are massless. The physical radiation
density today is then ωr ≡ Ωr0h2 ≈ 4.18 × 10−5, the photon contribution being
ωγ ≈ 2.47×10−5. We also make the approximation that the universe was completely
matter-dominated at tdec, i.e. we ignore the radiation contribution to the Friedmann
equation at tdec. This is not a terribly good approximation, since
ρm(tdec)
ρr(tdec)
=
ωm
(1 + zdec)ωr
≈ 22ωm ≈ 2.6 . . . 3.5 , (12.31)
for ωm = 0.12 . . . 0.16. The curvature and (for most dark energy models, including
vacuum energy) dark energy contributions are negligible at tdec. Thus we have
H2
dec ≈
8πG
3
ρm = Ωm0H2
0 (1 + zdec)3
, (12.32)
and we get for the comoving Hubble scale
k−1
dec ≡ (adecHdec)−1
= (1 + zdec)H−1
dec = (1 + zdec)−1/2 H−1
0√
Ωm0
=
1
√
Ωm0
91h−1
Mpc ,
(12.33)
using 1 + zdec = 1090. The scale which is entering at t = tdec is thus
kdec = adecHdec = (1 + zdec)1/2
Ωm0H0 , (12.34)
12 COSMIC MICROWAVE BACKGROUND 205
and the corresponding multipole number on the last scattering sphere is
ℓH ≡ kdecdc
A = (1 + zdec)1/2
Ωm0 ×
2/Ωm0 = 66.0 Ω−0.5
m0 (ΩΛ = 0)
2/Ω0.4
m0 = 66.0 Ω0.1
m0 (Ω0 = 1)
(12.35)
The angle subtended by a half-wavelength π/k of this mode on the last scattering
sphere is
θH ≡
π
ℓH
=
180◦
ℓH
=
2.7◦Ω0.5
m0
2.7◦Ω−0.1
m0 .
(12.36)
For the open model with Ωm0 = 0.3, we get 1.5◦, and for the spatially flat ΛCDM
model with Ωm0 = 0.3, we get ∼ 3◦.
Another important scale is keq, the scale which enters at the time of matterradiation
equality teq, since the transfer function T(k) is bent at that point. Perturbations
for scales k ≪ keq essentially maintain their primordial spectrum, whereas
scales k ≫ keq have lost relative power between their horizon entry and teq. With a
calculation similar to kdec (taking into account that ρtot(teq) = 2ρm(teq)), we get
k−1
eq = (aeqHeq)−1
≈ 14ω−1
m Mpc = 4.7 × 10−3
Ω−1
m0h−1
H−1
0 . (12.37)
For ωm = 0.14 we have keq = 100 Mpc. The corresponding multipole number is
ℓeq = keqdc
A = 214Ωm0h ×
2/Ωm0 = 430h (ΩΛ = 0)
2/Ω0.4
m0 ≈ 430h Ω0.6
m0 (Ω0 = 1) .
(12.38)
12.4 CMB anisotropy from perturbation theory
We began this chapter with the observation (12.1), that the CMB temperature
anisotropy is a sum of two parts,
δT
T obs
=
δT
T intr
+
δT
T jour
, (12.39)
and that this separation is gauge dependent. We shall consider this in the longitudinal
gauge, since the second part, δT
T jour
, the integrated redshift perturbation
along the line of sight, is easiest to calculate in this gauge. The calculation requires
more general relativity tools than we have available, so we just give the result.
δT
T jour
= −
o
dec
dΦ + vobs · ˆn +
o
dec
dt ˙Φ + ˙Ψ −
1
2
˙hij ˆni
ˆnj
= Φ(tdec, xls) − Φ(t0, 0) + vobs · ˆn +
o
dec
dt ˙Φ + ˙Ψ −
1
2
˙hij ˆni
ˆnj
Ψ≈Φ
= Φ(tdec, xls) − Φ(t0, 0) + vobs · ˆn + 2
o
dec
dt ˙Φ −
1
2
ˆni
ˆnj
o
dec
dt˙hij ,
(12.40)
where the integral is from (tdec, xls) to (t0, 0) along the path of the photon (a null
geodesic) and ˆn is a unit vector pointing in the direction the observer is looking
at. The observer’s location has been chosen as the origin 0. The term vobs · ˆn
is the Doppler effect from the observer’s motion (which is assumed nonrelativistic,
|vobs| ≪ 1), where vobs is the observer’s velocity. The subscript ls in xls indicates that
12 COSMIC MICROWAVE BACKGROUND 206
x lies somewhere on the last scattering sphere. In the matter-dominated universe
the Newtonian potential remains constant in time, ˙Φ = 04, so we get a contribution
from the integral only from epochs when the contributions of radiation, dark energy
of spatial curvature to the total energy density cannot be ignored.
We can understand the above result as follows. If the potential is constant in
time, the blueshift the photon acquires when falling into a potential well is canceled
by the redshift from climbing up the well. Thus the net redshift/blueshift caused
by gravitational potential perturbations is just the difference between the values of
Φ at the beginning and in the end. However, if the potential is changing while the
photon is traversing the well, this cancellation is not exact, and we get the integral
term to account for this effect.
The value of the potential perturbation at the observing site, Φ(t0, 0) is the same
for photons coming from all directions. Thus it does not contribute to the observed
anisotropy. It just produces an overall shift in the observed average temperature.
(Recall the discussion of the two ways of defining the mean temperature at the
beginning of the chapter.) This is included in the observed value T0 = 2.725 K, and
there is no way for us to separate it from the unperturbed value. Thus we will ignore
the monopole. The observer motion vobs causes a dipole (ℓ = 1) pattern in the CMB
anisotropy, from which it is difficult to disentangle the cosmological dipole on the
last scattering sphere. Therefore the dipole is usually removed from the CMB map
before analysing it for cosmological purposes. Accordingly, we ignore this term also.
We will also not consider the effect of gravitational waves. Our final result for the
journey part is therefore
δT
T jour
= Φ(tdec, xls) + 2
o
dec
˙Φdt . (12.41)
The other part, δT
T intr
, comes from the local temperature perturbation at t =
tdec and the Doppler effect, −v · ˆn, from the local (baryon+photon) fluid motion at
that time. Since
ργ =
π2
15
T4
, (12.42)
the local temperature perturbation is directly related to the relative perturbation in
the photon energy density,
δT
T intr
=
1
4
δγ − v · ˆn . (12.43)
We can now write the observed temperature anisotropy as
δT
T obs
=
1
4
δγ − v · ˆn + Φ(tdec, xls) + 2
o
dec
˙Φdt . (12.44)
Both the density perturbation δγ and the fluid velocity v are gauge dependent; we
use the longitudinal gauge only.
To make further progress we now
1. consider only adiabatic primordial perturbations and
4
In linear perturbation theory. In second and higher order perturbation theory we have ˙Φ = 0
even in a spatially flat matter-dominated universe.
12 COSMIC MICROWAVE BACKGROUND 207
2. make the (crude) approximation that the universe is already matter dominated
at t = tdec.
For adiabatic perturbations we have
δb = δc ≡ δm =
3
4
δγ . (12.45)
The perturbations stay adiabatic only on superhorizon scales. Once the perturbation
has entered horizon, different physics begin to act on different matter
components, so the adiabatic relation between their density perturbations is broken.
In particular, the baryon-photon perturbation is affected by photon pressure,
which damps its growth and causes it to oscillate, whereas the CDM perturbation
is unaffected and keeps growing. Since the baryon and photon components see the
same pressure, they evolve together and maintain their adiabatic relation until photon
decoupling. Thus, after horizon entry but before decoupling we have,
δc = δb =
3
4
δγ . (12.46)
At decoupling, the equality holds for scales larger than the photon mean free path
at tdec.
After decoupling, this connection between the photons and baryons is broken,
and the baryon density perturbation begins to approach the CDM density pertur-
bation,
δc ← δb =
3
4
δγ . (12.47)
We shall return to these issues when we discuss the shorter scales in sections 12.6
and 12.7. But let us first consider the scales which are still superhorizon at tdec, so
that (12.45) applies.
12.5 Large scales: Sachs–Wolfe part of the spectrum
Consider now the scales k ≪ kdec, or ℓ ≪ ℓH, which are still superhorizon at
decoupling. According to the adiabatic condition (12.45) we have
1
4
δγ =
1
3
δm ≈
1
3
δ , (12.48)
where the latter (approximate) equality comes from taking the universe to be matter
dominated at tdec, so that we can identify δ ≈ δm. For these scales the Doppler effect
from fluid motion is subdominant, and we can ignore it. This can be seen from (9.19):
Fourier transforming the equation we have ui ∼ kiΦ/(a2H). Thus (12.44) becomes
δT
T obs
=
1
3
δ + Φ(tdec, xls) + 2
o
dec
˙Φdt . (12.49)
On superhorizon scales we have δ = −2Φ and (12.49) becomes
δT
T obs
= −
2
3
Φ(tdec, xls) + Φ(tdec, xls) + 2
o
dec
˙Φdt
=
1
3
Φ(tdec, xls) + 2
o
dec
˙Φdt . (12.50)
12 COSMIC MICROWAVE BACKGROUND 208
This part of the CMB anisotropy is called the Sachs–Wolfe effect. The first part,
1
3 Φ(tdec, xls), is called the ordinary Sachs–Wolfe effect, and the second part, 2 ˙Φdt,
is called the integrated Sachs-Wolfe effect (ISW), since it involves integrating along
the line of sight. There are two contributions to the integrated Sachs–Wolfe effect,
the early Sachs–Wolfe effect and the late Sachs–Wolfe effect. The first is caused by
the effect of radiation at last scattering. In our approximation where we assume
that the universe is completely matter-dominated at t = tdec, this term is absent.
When dark energy becomes important at times close to today, Φ starts to evolve
again, which leads to the late ISW effect, which shows up as a rise in the smallest ℓ
of the angular power spectrum Cℓ. However, it is difficult to detect this effect due to
the large cosmic variance at small ℓ. The late ISW effect also leads to a correlation
between the CMB anisotropies and the galaxy distribution, which makes it easier to
detect its presence. The late ISW effect has been detected this way, from the crosscorrelation
of the CMB and large scale structure. We shall now for a while ignore
the ISW, which for ℓ ≪ ℓH is expected to be smaller than the ordinary Sachs–Wolfe
effect.
12.5.1 Angular power spectrum from the ordinary Sachs–Wolfe effect
We now calculate the contribution from the ordinary Sachs–Wolfe effect,
δT
T SW
=
1
3
Φ(tdec, xls) , (12.51)
to the angular power spectrum Cℓ. This is the dominant effect for ℓ ≪ ℓH.
Since Φ is evaluated at the last scattering sphere, we have from (12.23),
aℓm = 4πiℓ
k
1
3
Φkjℓ(kx)Y ∗
ℓm(ˆk) , (12.52)
In the matter-dominated epoch,
Φ = −
3
5
R , (12.53)
so that we have
aℓm = −
4π
5
iℓ
k
Rkjℓ(kx)Y ∗
ℓm(ˆk) . (12.54)
The coefficient aℓm is thus a linear combination of the independent random
variables Rk, i.e. it is of the form
k
bkRk , (12.55)
For any such linear combination, the expectation value of its absolute value squared
is
k
bkRk
2
=
k k′
bkb∗
k′ RkR∗
k′
=
2π
L
3
k
1
4πk3
PR(k) |bk|2
, (12.56)
12 COSMIC MICROWAVE BACKGROUND 209
where we used
RkR∗
k′ = δkk′
2π
L
3
1
4πk3
PR(k) (12.57)
(the independence of the random variables Rk and the definition of the power spectrum
P(k)).
Thus
Cℓ ≡
1
2ℓ + 1 m
|aℓm|2
=
16π2
25
1
2ℓ + 1 m
2π
L
3
k
1
4πk3
PR(k)jℓ(kx)2
Y ∗
ℓm(ˆk)
2
=
1
25
2π
L
3
k
1
k3
PR(k)jℓ(kx)2
. (12.58)
(Although all |aℓm|2
are equal for the same ℓ, we used the sum over m in order to
apply (12.5).) Replacing the sum with an integral, we get
Cℓ =
1
25
d3k
k3
PR(k)jℓ(kx)2
=
4π
25
∞
0
dk
k
PR(k)jℓ(kx)2
, (12.59)
the final result for an arbitrary primordial power spectrum PR(k).
The integral can be done for a power-law power spectrum, PR(k) = A2(k/kp)n−1.
In particular, for a scale-invariant (n = 1) primordial power spectrum,
PR(k) = const. = A2
, (12.60)
we have
Cℓ = A2 4π
25
∞
0
dk
k
jℓ(kx)2
=
A2
25
2π
ℓ(ℓ + 1)
, (12.61)
since ∞
0
dk
k
jℓ(kx)2
=
1
2ℓ(ℓ + 1)
. (12.62)
We can write this as
ℓ(ℓ + 1)
2π
Cℓ =
A2
25
= const. (independent of ℓ) (12.63)
The reason why the angular power spectrum is customarily plotted as ℓ(ℓ + 1)Cℓ/2π
is that it makes the Sachs–Wolfe part of the Cℓ flat for a scale-invariant primordial
power spectrum PR(k).
Present data shows that the spectrum has a small red tilt, n = 0.96 ± 0.007,
as expected from the simplest inflationary models. Since the spectrum is close to
scale-invariant, determining the spectral index requires observations over a range of
scales. However, determining the overall amplitude is possible just by observing the
few lowest multipoles, known as the Sachs–Wolfe plateau. The COBE satellite saw
12 COSMIC MICROWAVE BACKGROUND 210
only up to about ℓ = 25, so the COBE data in figure 1 is completely in this region.
The amplitude is
ℓ(ℓ + 1)
2π
Cℓ ≈ 10−10
. (12.64)
This gives the amplitude of the primordial power spectrum as
PR(k) = A2
≈ 25 × 10−10
= 5 × 10−5 2
. (12.65)
We already used this result (confirmed after COBE by other experiments) in chapter
10 as a constraint on the energy scale of inflation. Nowadays, the detailed structure
of the anisotropies has been measured: the latest data from Planck is shown in figure
7. Let us now discuss how the structure of peaks and troughs is generated.
12.6 Acoustic oscillations
Consider now the scales k ≫ kdec, or ℓ ≫ ℓH, which are subhorizon at decoupling.
The observed temperature anisotropy is, from (12.44)
δT
T obs
=
1
4
δγ(tdec, xls) + Φ(tdec, xls) − v · ˆn(tdec, xls) + 2
o
dec
˙Φdt . (12.66)
We concentrate on the three first terms, which correspond to the situation at the
point (tdec, xls) we are looking at on the last scattering sphere.
Before decoupling the photons are tightly coupled to the baryons. The perturbations
in the baryon-photon fluid are oscillating, whereas CDM perturbations
grow (logarithmically during the radiation-dominated epoch, and then ∝ a during
the matter-dominated epoch). Therefore CDM perturbations begin to dominate the
total density perturbation δρ and thus also Φ already before the universe becomes
matter-dominated and CDM begins to dominate the background energy density.
Thus we can make the approximation that Φ is given by the CDM perturbation.
The baryon-photon fluid oscillates in these potential wells caused by the CDM. The
potential Φ evolves at first but then becomes constant as the universe becomes
matter dominated.
We will not do a full calculation of the δbγ oscillations in the expanding universe,
that would require a bit more general relativity tools than we have at our disposal.
One reason is that ρbγ is a relativistic fluid, and we gave the equation for the density
perturbation for a nonrelativistic fluid only (the Jeans equation). The nonrelativistic
perturbation equation for a fluid component i is (this follows from (11.50) when we
replace the baryonic density contrast with the total density contrast in the driving
term)
¨δki + 2H ˙δki = −
k
a
2
c2
sδki + Φk . (12.67)
The generalisation of the (subhorizon) perturbation equations to the case of a
relativistic fluid is considerably easier if we ignore the expansion of the universe.
Then (12.67) becomes
¨δki + k2
c2
sδki + Φk = 0 . (12.68)
According to GR, the density of “passive gravitational mass” is ρ+p = (1+w)ρ, not
just ρ as in Newtonian gravity. Therefore the force on a fluid element of the fluid
12 COSMIC MICROWAVE BACKGROUND 211
component i is proportional to (ρi + pi)∇Φ = (1 + wi)ρi∇Φ instead of just ρi∇Φ,
and (12.68) generalises to the case of a relativistic fluid as5
¨δki + k2
c2
sδki + (1 + wi)Φk = 0 . (12.69)
In the present application the fluid component ρi is the baryon-photon fluid ρbγ
and the gravitational potential Φ is caused by the CDM. Before decoupling, the
adiabatic relation δb = 3
4δγ still holds between photons and baryons, and we have
the adiabatic relation between pressure and density perturbations,
δpbγ = c2
sδρbγ , (12.70)
so the sound speed of the fluid is given by
c2
s =
δpbγ
δρbγ
≈
δpγ
δρbγ
=
1
3
δργ
δργ + δρb
=
1
3
¯ργδγ
¯ργδγ + ¯ρbδb
=
1
3
1
1 + 3
4
¯ρb
¯ργ
≡
1
3
1
1 + R
. (12.71)
We defined
R ≡
3
4
¯ρb
¯ργ
. (12.72)
We can now write the perturbation equation (12.69) for the baryon-photon fluid as
¨δbγk + k2
c2
sδbγk + (1 + wbγ)Φk = 0 . (12.73)
The equation-of-state parameter for the baryon-photon fluid is
wbγ ≡
¯pbγ
¯ρbγ
=
1
3 ¯ργ
¯ργ + ¯ρb
=
1
3
1
1 + 4
3 R
, (12.74)
We can therefore write (12.73) as
¨δbγk + k2 1
3
1
1 + R
δbγk +
4
3(1 + R)
1 + 4
3R
Φk = 0 . (12.75)
We introduce the notation6
Θ0 ≡
1
4
δγ , (12.76)
which gives the perturbation in the photons, not in the baryon-photon fluid. The
two are related by
δbγ =
δρbγ
¯ρbγ
=
δργ + δρb
¯ργ + ¯ρb
=
¯ργδγ + ¯ρbδb
¯ργ + ¯ρb
=
1 + R
1 + 4
3R
δγ . (12.77)
Thus we can write (12.73) as
¨δγk + k2 1
3
1
1 + R
δγk +
4
3
Φk = 0 , (12.78)
5
Actually the derivation is more complicated, since also the density of “inertial mass” is ρi + pi
and the energy continuity equation is modified by a work-done-by-pressure term. Anyway, (12.69)
is the correct result.
6
The subscript 0 refers to the monopole (ℓ = 0) of the local photon distribution. Likewise,
the dipole (ℓ = 1) of the local photon distribution corresponds to the velocity of the photon fluid,
Θ1 ≡ vγ/3.
12 COSMIC MICROWAVE BACKGROUND 212
or
¨Θ0k + k2 1
3
1
1 + R
Θ0k +
1
3
Φk = 0 , (12.79)
or
¨Θ0k + c2
sk2
[Θ0k + (1 + R)Φk] = 0 , (12.80)
If we now take R and Φk to be constant, this is the harmonic oscillator equation
for the quantity Θ0k + (1 + R)Φk with the general solution
Θ0k + (1 + R)Φk = Ak cos(cskt) + Bk sin(cskt) , (12.81)
or
Θ0k + Φk = −RΦk + Ak cos(cskt) + Bk sin(cskt) , (12.82)
or
Θ0k = −(1 + R)Φk + Ak cos(cskt) + Bk sin(cskt) . (12.83)
We are interested in the quantity Θ0 + Φ = 1
4δγ + Φ, called the effective temperature
perturbation, since this combination appears in (12.66). It is the local temperature
perturbation minus the redshift photons suffer when climbing from the potential well
of the perturbation (negative Φ for a CDM overdensity). We see that this quantity
oscillates in time, and the effect of baryons (via R) is to shift the equilibrium point
of the oscillation by −RΦk.
In the preceding we ignored the effect of the expansion of the universe. The
expansion affects the result in several ways. For example, cs, wbγ and R change
with time. The potential Φ also evolves, especially at early times when radiation
dominates the expansion. However, the qualitative result of an oscillation of Θ0 +Φ,
and the shift of its equilibrium point by baryons, remains. The time t in the solution
(12.82) gets replaced by conformal time η, and since cs changes with time, csη is
replaced by
rc
s(t) ≡
η
0
csdη =
t
0
cs(t)
a(t)
dt . (12.84)
We call this quantity rc
s(t) the comoving sound horizon at time t, since it gives the
comoving distance sound waves have travelled to time t.
The relative weight of the cosine and sine solutions (i.e., the constants Ak and
Bk in (12.81) depends on the initial conditions. Since the perturbations are initially
at superhorizon scales, the initial conditions are determined there, and the present
discussion does not really apply. However, using the superhorizon initial conditions
gives the correct qualitative result for the phase of the oscillation.
We had that for adiabatic primordial perturbations, initially Φ = −3
5R and
1
4 δγ = −2
3Φ = 2
5R, giving us an initial condition Θ0 + Φ = 1
3Φ = −1
5R = const.
(At these early times R ≪ 1, so we can drop the factor 1 + R.) Thus adiabatic
primordial perturbations correspond essentially to the cosine solution. There are
effects at the horizon scale which affect the amplitude of the oscillations—the main
effect being the decay of Φ as it enters the horizon—so we can’t use the preceding
discussion to determine the amplitude, but we get the right result about the initial
phase of the Θ0 + Φ oscillations.
Thus we have that, qualitatively, the effective temperature behaves at subhorizon
scales as
Θ0k + (1 + R)Φk ∝ cos[krc
s(t)] , (12.85)
12 COSMIC MICROWAVE BACKGROUND 213
Consider a region where the primordial curvature perturbation R is positive. It
begins with an initial overdensity (as we assume perturbations are adiabatic, this
applies to all components: photons, baryons, CDM and neutrinos), and a negative
gravitational potential Φ. For the scales of interest for CMB anisotropy, the potential
stays negative, since the CDM begins to dominate the potential early enough and
the CDM perturbations do not oscillate, they just grow. The effective temperature
perturbation Θ0 + Φ, which is the oscillating quantity, begins with a negative value.
After half an oscillation period it is at its positive extreme value. This increase of
Θ0 + Φ corresponds to an increase in δγ; from its initial positive value it has grown
to a larger positive value. Thus the oscillation begins by the initially overdense
baryon-photon fluid element falling deeper into the potential well, and reaching
maximum compression after half a period. After this maximum compression the
photon pressure pushes the baryon-photon fluid out from the potential well, and
after a full period, the fluid reaches its maximum decompression in the potential
well. Since the potential Φ has meanwhile decayed (horizon entry and the resulting
potential decay always happens during the first oscillation period, since the sound
horizon and the Hubble length are close to each other, as the sound speed is close to
the speed of light), the decompression does not bring the δbγ back to its initial value
(which was overdense), but the photon-baryon fluid actually becomes underdense in
the potential well (and overdense in the neighbouring potential “hill”). And so the
oscillation goes on until photon decoupling.
These are standing waves and they are called acoustic oscillations. See figure 15.
Because of the potential decay at horizon entry, the amplitude of the oscillation is
larger than Φ, and thus also Θ0 changes sign in the oscillation.
These oscillations end at photon decoupling, when the photons are liberated.
The CMB shows these standing waves as a snapshot7 at their final moment t = tdec.
At photon decoupling we have
Θ0k + (1 + R)Φk ∝ cos[krc
s(tdec)] . (12.86)
At this moment oscillations for scales k which have
krc
s(tdec) = nπ (12.87)
(n = 1, 2, 3, . . .) are at their extreme values (maximum compression or maximum
decompression). Therefore we see strong structure in the CMB anisotropy at the
multipoles
ℓ = kdc
A(tdec) = nπ
dc
A(tdec)
rc
s(tdec)
≡ nℓA (12.88)
corresponding to these scales. Here
ℓA ≡ π
dc
A(tdec)
rc
s(tdec)
≡
π
θs
(12.89)
is the acoustic scale in multipole space and
θs ≡
rc
s(tdec)
dc
A(tdec)
(12.90)
7
Actually, photon decoupling takes quite a long time. Therefore this “snapshot” has a rather
long “exposure time” causing it to be “blurred”. This prevents us from seeing very small scales in
the CMB anisotropy.
12 COSMIC MICROWAVE BACKGROUND 214
Figure 15: Acoustic oscillations. The top panel shows the time evolution of the Fourier
amplitudes Θ0k, Φk, and the effective temperature Θ0k + Φk. The Fourier mode shown
corresponds to the fourth acoustic peak of the Cℓ spectrum. The bottom panel shows δbγ(x)
for one Fourier mode as a function of position at various times (maximum compression,
equilibrium level, and maximum decompression).
is the sound horizon angle, i.e., the angle at which we see the sound horizon on the
last scattering surface. This is the CMB anisotropy quantity which is determined
with most accuracy from the data. Analysis of the 5-year data from the WMAP
satellite and data from the ACBAR ground-based CMB experiment gives the modelindependent
value θs = 0.593◦ ± 0.001◦, a precision of 0.3% [1].
Because of these acoustic oscillations, the CMB angular power spectrum Cℓ has
a structure of acoustic peaks on subhorizon scales. The centers of these peaks are
located approximately at ℓn ≈ nℓA. An exact calculation shows that they actually
lie at somewhat smaller ℓ due to a number of effects. The separation of Neighbouring
peaks is closer to ℓA than the positions of the peaks are to nℓA.
These acoustic oscillations involve motion of the baryon-photon fluid. When the
oscillation of one Fourier mode is at its extreme, e.g. at the maximal compression in
the potential well, the fluid is momentarily at rest, but then it begins flowing out of
the well until the other extreme, the maximal decompression, is reached. Therefore
those Fourier modes k which have the maximum effect on the CMB anisotropy via
the 1
4 δγ(tdec, xls)+Φ(tdec, xls) term (the effective temperature effect) in (12.66) have
the minimum effect via the −v · ˆn(tdec, xls) term (the Doppler effect) and vice versa.
Therefore the Doppler effect also contributes a peak structure to the Cℓ spectrum,
but its peaks are in the locations where the effective temperature contribution has
troughs.
The Doppler effect is subdominant to the effective temperature effect, and therefore
the peak positions in the Cℓ spectrum is determined by the effective tempera-
12 COSMIC MICROWAVE BACKGROUND 215
ture effect, according to (12.88). The Doppler effect just partially fills the troughs
between the peaks, weakening the peak structure of Cℓ. See figure 18. The calculation
involves some approximations which allow the description of Cℓ as just a sum
of these contributions, and is not as accurate as a full calculation using e.g. the
CAMB code8.)
Figure 16 shows the values of the effective temperature perturbation Θ0 + Φ (as
well as Θ0 and Φ separately) and the magnitude of the velocity perturbation (Θ1 ∼
v/3) at tdec as a function of the scale k. This is a result of a numerical calculation
which includes the effect of the expansion of the universe, but not diffusion damping.
12.7 Diffusion damping
For small enough scales the effect of photon diffusion and the finite thickness of
the last scattering surface (∼ the photon mean free path just before last scattering)
smooth out the photon distribution and the CMB anisotropy. This effect is characterised
by the damping scale k−1
D , which is the distance that photons have travelled
up to last scattering. The photon density and velocity perturbations at scale k are
damped at tdec by
e−k2/k2
D , (12.91)
and the Cℓ spectrum is damped as
e−ℓ2/ℓ2
D . (12.92)
We can estimate kD and ℓD as follows (see [2], page 129, for a bit more details).
Before decoupling photons are scattering from the electrons in the plasma. The
typical time between collisions (i.e. the photon mean free path) at time t is λγ =
tc(t) = Γ−1 = (ne(t)σT )−1, where σT = 8π
3
α2
m2
e
is the Thomson cross-section. The
free electron density depends on the ionisation fraction x (see section 5.6). For
simplicity, we take x = 1. (In fact, the ionisation fraction drops, and tc grows,
rapidly during decoupling.) The photon direction changes randomly at each collision
and independently of the previous collision, so photons undergo a random walk with
uncorrelated steps. The number of steps the photons has taken up to time t is
N = t/tc (taking tc to be constant for simplicity), and the total distance it has
travelled at decoupling is
dD =
√
Ntc =
√
tdectc ≈ 14 kpc , (12.93)
where we have put in tdec = 380 000 yr, tc = tc(tdec) and used Tdec = 3000 K,
η = 6 × 10−10. The comoving diffusion wavenumber is given by
k−1
D = (1 + zdec)dD ≈ 15 Mpc , (12.94)
using zdec = 1090. This corresponds to multipole moment
ℓD ∼ kDdc
A(zdec) ≈ 900 , (12.95)
where we have put in dc
A(zdec) = 13.8 Gpc (see section 12.9.2).
8
CAMB is a publicly available code for precise calculation of the Cℓ spectrum. See
http://camb.info/ .
12 COSMIC MICROWAVE BACKGROUND 216
-0.5
0
0.5
1
Φ,Θ0
-0.5
0
0.5
(Θ0
+Φ)
-0.4
-0.2
0
0.2
0.4
Θ1
0
0.1
0.2
0.3
0.4
0.5
0.6
(Θ0
+Φ)
2
0 200 400 600 800 1000
k/H0
0
0.05
0.1
0.15
Θ1
2
ωm
= 0.10
ωm
= 0.20
ωm
= 0.30
Figure 16: Values of oscillating quantities (normalised to an initial value Rk = 1) at the
time of decoupling as a function of the scale k, for three different values of ωm, and for
ωb = 0.01. Θ1 represents the velocity perturbation. The effect of diffusion damping is
neglected. Figure and calculation by R. Keskitalo.
12 COSMIC MICROWAVE BACKGROUND 217
0 500 1000 1500 2000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ωm
= 0.20
ωm
= 0.25
ωm
= 0.30
ωm
= 0.35
Undamped spectra
Including damping
Figure 17: The angular power spectrum Cℓ, calculated both with and without the effect of
diffusion damping. The spectrum is given for four different values of ωm, with ωb = 0.01.
(This is a rather low value of ωb, about half the real value, so ℓD < 1500 and the damping
is quite strong.) Figure and calculation by R. Keskitalo.
This calculation is rather approximate, because of the rapid growth of the photon
mean free path (and the typical time between collisions) during recombination,
and a more accurate calculation involves an integral over time to take into account
this effect. However, the order of magnitude ℓD ∼ 1000 is correct, as we see from figure
17, which shows the result of a numerical calculation with and without diffusion
damping.
Of the cosmological parameters, the damping depends most strongly on ωb, since
increasing baryon density shortens the photon mean free path before decoupling.
Thus for larger ωb the damping moves to shorter scales, i.e. ℓD becomes larger.
The time evolution of λγ before decoupling, and the diffusion scale, is different
for different ωb. For small ωb, tc has already become quite large through the slow
dilution of the baryon density by the expansion of the universe, and the growth of
λγ relies less on the fast reduction of free electron density during recombination.
12.8 The complete Cℓ spectrum
As we have discussed the CMB anisotropy has three contributions (see 12.66), the
effective temperature effect,
1
4
δγ(tdec, xls) + Φ(tdec, xls) , (12.96)
the Doppler effect,
−v · ˆn(tdec, xls) , (12.97)
and the integrated Sachs–Wolfe effect,
2
o
dec
˙Φdt . (12.98)
12 COSMIC MICROWAVE BACKGROUND 218
Because Cℓ is quadratic in the perturbations, it includes cross-terms between these
three effects.
The calculation of the full Cℓ proceeds much as the calculation of just the ordinary
Sachs–Wolfe part (which the effective temperature effect becomes at superhorizon
scales) in section 12.5.1, but now with the full δT/T. Since all perturbations
are proportional to the primordial perturbations, the Cℓ spectrum is proportional
to the primordial perturbation spectrum PR(k) (with integrals over the spherical
Bessel functions jℓ(kx), like in (12.59), to get from k to ℓ).
The difference is that instead of the constant proportionality factor (δT/T)SW =
−(1/5)R, we have a k-dependent proportionality resulting from the evolution (including
e.g. the acoustic oscillations) of the perturbations.
In figure 18 we show the full Cℓ spectrum and the different contributions to it.
Because the Doppler effect and the effective temperature effect are almost completely
off-phase, their cross term gives a negligible contribution.
Since the ISW effect is relatively weak, it contributes more via its cross terms
with the Doppler effect and effective temperature than directly. The cosmological
model used for figure 18 has ΩΛ = 0, so there is no late ISW effect (which would
contribute at the very lowest ℓ), and the ISW effect shown is the early ISW effect
due to radiation contribution to the expansion law. This effect contributes mainly
to the first peak and to the left of it, explaining why the first peak is so much higher
than the other peaks. It also shifts the first peak position slightly to the left and
changes its shape.
12.9 Cosmological parameters and CMB anisotropy
Let us finally consider the effect of the various cosmological parameters on the Cℓ
spectrum. The Cℓ provides perhaps the most important single observational data set
for determining (or constraining) cosmological parameters, since it has a rich structure
which we can measure with an accuracy that other cosmological observations
cannot match, and because it depends on several different cosmological parameters.
The latter is both a strength and weakness: the CMB has only a couple of features
(overall amplitude and the positions and heights of the peaks and troughs), so typically
you cannot hope to determine more than a handful if independent parameters
from the data. This is because different parameters affect the same features in similar
ways, so that we only get a constraint on their combination. Such parameters
are called degenerate. Other cosmological observations are needed to break these
degeneracies.
The CMB anisotropy pattern is set by the physics at decoupling, and it is then
modified as the CMB passes through the universe to be observed today. The CMB
pattern at decoupling is determined by the primordial spectrum, and the densities of
photons, neutrinos, baryons and cold dark matter. The photon density is precisely
known from the CMB mean temperature, and (assuming zero neutrino chemical
potential) this also fixes the density of neutrinos. In the case of many inflationary
models, the primordial spectrum is a power-law, characterised by a an amplitude
and a spectral index. In summary, the physics at decoupling is determined by
• ωb ≡ Ωb0h2 the physical baryon density
• ωm ≡ Ωm0h2 the physical matter density
12 COSMIC MICROWAVE BACKGROUND 219
0
0.05
0.1
0.15
0.2
Θ0
+Φ
-0.001
0
0.001
(Θ0
+Φ)×Θ1
0
0.02
0.04
0.06
0.08
Θ1
-0.02
0
0.02
0.04
0.06
0.08ISW
×(Θ0
+Φ)
ISW
×Θ1
0 500 1000 1500 2000
l
1e-05
0.0001
0.001
0.01
ISW
0 500 1000 1500 2000
l
0
0.1
0.2
0.3
2l(l+1)Cl
/2π
Full spectrum
Θ0
+ Φ
Θ1
ISW
ISW cross terms
(Θ0
+Φ)×Θ1
-0.02
0
0.02
0.04
0.06
0.08ISW
×(Θ0
+Φ) + ISW
×Θ1
0 500 1000 1500
0
0.01
0.02
Figure 18: The full Cℓ spectrum calculated for the cosmological model Ω0 = 1, ΩΛ = 0,
ωm = 0.2, ωb = 0.03, A = 1, n = 1, and the different contributions to it. Here Θ1 denotes
the Doppler effect. Figure and calculation by R. Keskitalo.
12 COSMIC MICROWAVE BACKGROUND 220
• A amplitude of primordial scalar perturbations (at pivot scale kp)
• n spectral index of primordial scalar perturbations
The angular scale at which the pattern is seen changes as the universe evolves,
and this is the main effect of the physics after decoupling on the CMB anisotropy. In
addition, the CMB photons scatter off free charges after reionisation. In principle,
this effect is determined in terms of the physical parameters at decoupling, but the
physics involved in stellar formation and other relevant processes is too complicated
to calculate from first principles. Therefore the effect of reionisation is encoded in an
effective parameter τ called the optical depth (discussed in section 12.9.6). Roughly
speaking, τ gives the probability that a given photon scatters at least once between
decoupling and today. We could therefore take the model-independent CMB postdecoupling
CMB parameters as
• dc
A(zdec) comoving angular diameter distance to the last scattering surface
• τ optical depth
The angular diameter distance is a general model-independent quantity. In a
given FRW model, it is determined by the spatial curvature and the expansion
history, as we have discussed. In the ΛCDM model, where there is vacuum energy
and spatial curvature, the angular diameter distance can be replaced by these two
parameters, so we have
• Ω0 total density parameter
• ΩΛ0 vacuum energy density parameter
• τ optical depth
In addition to changing the angular diameter distance, vacuum energy and spatial
curvature also contribute to the CMB anisotropy via the ISW effect, as discussed
earlier. The decoupling and post-decoupling parameters add up to a total of seven
parameters. Since spatial curvature is not needed to explain the observations and
there is no indication for it, it is usually put to zero, i.e. Ω0 = 1. Usually references
to the ΛCDM model, or the “standard cosmological model”, or the concordance
model refer to the model parametrised by the six parameters above, without spatial
curvature. We will keep spatial curvature in the discussion in order to show what
effect it would have.
There are other possible cosmological parameters (“additional parameters”) which
might affect the Cℓ spectrum, e.g.
• mνi neutrino masses
• w dark energy equation of state parameter
• dn
d ln k scale dependence of the spectral index
• r, nT relative amplitude and spectral index of tensor perturbations
• B, niso amplitudes and spectral indices of primordial isocurvature perturbations,
• Acor, ncor and their correlation with primordial curvature perturbations
12 COSMIC MICROWAVE BACKGROUND 221
We assume here that these additional parameters have no impact, i.e., they have
the “standard” values
mνi = r =
dn
d ln k
= B = Acor = 0 , w = −1 (12.99)
to the accuracy which matters for Cℓ observations. Apart from the neutrino masses,
there is no sign in the present-day CMB data for non-zero values of these parameters.
On the other hand, significant deviations from zero can be consistent with the data,
and may be discovered by future CMB (and other) observations, in particular the
Planck satellite. The primordial isocurvature perturbations refer to the possibility
that the primordial scalar perturbations are not adiabatic, and therefore are not
completely determined by the comoving curvature perturbation R.
The assumption that these additional parameters have no impact leads to a
determination of the standard parameters with an accuracy that may be too optimistic,
since the standard parameters may have degeneracies with the additional
parameters.
12.9.1 Independent vs. dependent parameters
The above is our choice of independent cosmological parameters. Ωm0, Ωb0 and H0
(or h) are then dependent (or “derived”) parameters, since they are determined by
Ω0 = Ωm0 + ΩΛ0 ⇒ Ωm0 = Ω0 − ΩΛ0 (12.100)
h =
ωm
Ωm0
=
ωm
Ω0 − ΩΛ0
(12.101)
Ωb0 =
ωb
h2
=
ωb
ωm
(Ω0 − ΩΛ0) (12.102)
In particular, the Hubble constant H0 is a dependent parameter. The CMB has
no sensitivity to H0 except via the angular diameter distance to the last scattering
surface.
Different choices of independent parameters are possible within our 7-dimensional
parameter space (e.g. we could have chosen H0 to be an independent parameter and
let ΩΛ0 to be a dependent parameter instead). They can be though of as different coordinate
systems in this seven-dimensional space. It is not meaningful to discuss the
effect of one parameter without specifying what is the set of independent parameters!
Some choices of independent parameters are better than others. The above choice
represents standard practice in cosmology today.9 The independent parameters have
been chosen so that they correspond as directly as possible to physics affecting
the Cℓ spectrum and thus to observable features in it. We want the effects of
our independent parameters on the observables to be as different (“orthogonal”) as
possible in order to avoid parameter degeneracy.
In particular,
• ωm (not Ωm0) determines zeq and keq, and thus e.g. the magnitude of the
early ISW effect and which scales enter during matter- or radiation-dominated
epoch.
9
There are other choices in use, that are even more geared to minimising parameter degeneracy.
For example, the sound horizon angle θs may be used instead of ΩΛ0 as an independent parameter,
since it is directly determined by the acoustic peak separation, and thus less subject to degeneracies.
However, the determination of the dependent parameters from it is in turn more complicated.
12 COSMIC MICROWAVE BACKGROUND 222
• ωb (not Ωb0) determines the baryon/photon ratio and thus e.g. the relative
heights of the odd and even peaks.
• ΩΛ0 (not ΩΛ0h2) determines the late ISW effect.
There are many effects on the Cℓ spectrum, and parameters act on them in different
combinations. Thus there is no perfectly “clean” way of choosing independent
parameters.
In the following plots made with CAMB we see the effect of these parameters on
Cℓ by varying one parameter at a time around a reference model, whose parameters
have the following values.
Independent parameters:
Ω0 = 1 ΩΛ0 = 0.7
A = 1 ωm = 0.147
n = 1 ωb = 0.022
τ = 0.1
which give for the dependent parameters
Ωm0 = 0.3 h = 0.7
Ωc0 = 0.2551 ωc = 0.125
Ωb0 = 0.0449
The meaning of setting A = 1 is just that the resulting Cℓ still need to be multiplied
by the true value of A2. (In this model the true value should be about A = 5×10−5
to agree with observations.) If we really had A = 1, perturbation theory of course
would not be valid! This is a relatively common practice, since the effect of changing
A is so trivial, it doesn’t make sense to plot Cℓ separately for different values of A.
12.9.2 Sound horizon angle
The positions of the acoustic peaks of the Cℓ spectrum provides us with a measurement
of the sound horizon angle
θs ≡
rc
s(tdec)
dc
A(tdec)
We can use this in the determination of the values of the cosmological parameters,
once we have calculated how this angle depends on those parameters. It is the ratio
of two quantities, the sound horizon at photon decoupling, rc
s(tdec), and the angular
diameter distance to the last scattering, dc
A(tdec).
Angular diameter distance to last scattering
The angular diameter distance dc
A(tdec) to the last scattering surface we have
already calculated and it is given by (12.27) as
dc
A(tdec) = H−1
0
1
√
1 − Ω0
sinh 1 − Ω0
1
1
1+zdec
da
Ω0(a − a2) − ΩΛ0(a − a4) + a2
.
(12.103)
12 COSMIC MICROWAVE BACKGROUND 223
from which we see that it depends on the three cosmological parameters H0, Ω0 and
ΩΛ0. Here Ω0 = Ωm0 + ΩΛ0, so we could also say that it depends on H0, Ωm0, and
ΩΛ0, but it is easier to discuss the effects of these different parameters if we keep
Ω0 as an independent parameter, instead of Ωm0, since the “geometry effect” of the
curvature of space, which determines the relation between the comoving angular
diameter distance dc
A and the comoving distance dc, is determined by Ω0.
1. The comoving angular diameter distance is inversely proportional to H0 (directly
proportional to the Hubble distance H−1
0 ).
2. Increasing Ω0 decreases dc
A(tdec) in relation to dc(tdec) because of the geometry
effect.
3. With a fixed ΩΛ0, increasing Ω0 decreases dc
A(tdec), since it means increasing
Ωm0, which has a decelerating effect on the expansion. With a fixed present
expansion rate H0, deceleration means that expansion was faster earlier ⇒
universe is younger ⇒ there is less time for photons to travel as the universe
cools from Tdec to T0 ⇒ last scattering surface is closer to us.
4. Increasing ΩΛ0 (with a fixed Ω0) increases dc
A(tdec), since it means a larger
part of the energy density is in dark energy, which has an accelerating effect
on the expansion. With fixed H0, this means that expansion was slower in
the past ⇒ universe is older ⇒ more time for photons ⇒ last
scattering surface is further out ⇒ ΩΛ0 increases dc
A(tdec).
Here 2 and 3 work in the same direction: increasing Ω0 decreases dc
A(tdec), but the
geometry effect (2) is stronger. See figure 13 for the case ΩΛ0 = 0, where the dashed
line (the comoving distance) shows effect (3) and the solid line (the comoving angular
diameter distance) the combined effect (2) and (3).
However, now we have to take into account that, in our chosen parametrisation,
H0 is not an independent parameter, but
H−1
0 ∝
Ω0 − ΩΛ0
ωm
,
so that via H−1
0 , Ω0 increases and ΩΛ0 decreases dc
A(tdec), which are the opposite
effects to those discussed above. For ΩΛ0 this opposite effect wins. See Figs. 21 and
22.
Sound horizon
To calculate the comoving sound horizon,
rc
s(tdec) = a0
tdec
0
cs(t)
a(t)
dt =
tdec
0
dt
a
cs(t) =
adec
0
da
a · (da/dt)
cs(a) , (12.104)
we need the speed of sound from (12.71),
c2
s(a) =
1
3
1
1 + 3
4
¯ρb
¯ργ
=
1
3
1
1 + 3
4
ωb
ωγ
a
, (12.105)
where the upper limit of the integral is adec = 1/(1 + zdec).
12 COSMIC MICROWAVE BACKGROUND 224
The other element in the integrand of (12.104) is the expansion law a(t) before
decoupling. We have
a
da
dt
= H0 ΩΛ0a4 + (1 − Ω0)a2 + Ωm0a + Ωr0 . (12.106)
In the integral (12.103) we dropped the Ωr0, since it is important only at early times,
and the integral from adec to 1 is dominated by late times. Integral (12.104), on the
other hand, includes only early times, and now we can instead drop the ΩΛ0 and
1 − Ω0 terms (i.e., we can ignore the effect of curvature and dark energy in the early
universe, before photon decoupling), so that
a
da
dt
≈ H0 Ωm0a + Ωr0 = H100
√
ωma + ωr =
√
ωma + ωr
2998 Mpc
, (12.107)
where we have written
H0 ≡ h · 100
km/s
Mpc
≡ h · H100 =
h
2997.92 Mpc
. (12.108)
Thus the sound horizon is given by
rc
s(a) = 2998 Mpc
a
0
cs(a)da
√
ωma + ωr
= 2998 Mpc ·
1
√
3ωr
a
0
da
1 + ωm
ωr
a 1 + 3
4
ωb
ωγ
a
.
(12.109)
Here
ωγ = 2.4702 × 10−5
and (12.110)
ωr = 1 +
7
8
Nν
4
11
4/3
ωγ = 1.6904 ωγ = 4.1756 × 10−5
(12.111)
are accurately known from the CMB temperature T0 = 2.725 K (and therefore we
do not consider them as cosmological parameters in the sense of something to be
determined from the Cℓ spectrum).
Thus the sound horizon depends on the two cosmological parameters ωm and ωb,
rc
s(tdec) = rc
s(ωm, ωb)
From (12.109) we see that increasing either ωm or ωb makes the sound horizon at
decoupling, rc
s(adec), shorter:
• ωb slows the sound down
• ωm speeds up the expansion at a given temperature, so the universe cools to
Tdec in less time.
The integral (12.109) can be done and it gives
rc
s(tdec) =
2998 Mpc
√
1 + zdec
2
√
3ωmR∗
ln
√
1 + R∗ +
√
R∗ + r∗R∗
1 +
√
r∗R∗
, (12.112)
12 COSMIC MICROWAVE BACKGROUND 225
where
r∗ ≡
¯ρr(tdec)
¯ρm(tdec)
=
ωr
ωm
1
adec
= 0.0459
1
ωm
1 + zdec
1100
(12.113)
R∗ ≡
3¯ρb(tdec)
4¯ργ(tdec)
=
3ωb
4ωγ
adec = 27.6 ωb
1100
1 + zdec
. (12.114)
For our reference values ωm = 0.147, ωb = 0.022, and 1 + zdec = 110010 we get r∗ =
0.312 and R∗ = 0.607 and rc
s(tdec) = 143 Mpc for the sound horizon at decoupling.
Summary
The angular diameter distance dc
A(tdec) is most naturally discussed in terms of
H0, Ω0, and ΩΛ0, but since these are not the most convenient choice of independent
parameters for other purposes, we shall trade H0 for ωm according to (12.101). Thus
we see that the sound horizon angle depends on 4 parameters,
θs ≡
rc
s(ωm, ωb)
dc
A(Ω0, ΩΛ0, ωm)
= θs(Ω0, ΩΛ0, ωm, ωb) . (12.115)
If we keep ωm and ωb fixed, we have rc
s(tdec) = 143 Mpc. From the observed
model-independent value θs = 0.593◦ ± 0.001◦ [1] we then have dc
A = 13.8 Gpc
≈ 4.6hH−1
0 ≈ 3H−1
0 , where in the last equality we have taken h = 0.7. For the
Einstein-de Sitter model we have dc
A(1090) ≈ 1.97H−1
0 ≈ 8.4 Gpc, so the observed
distance to the last scattering surface is about 50% larger than predicted by the
FRW model without dark energy or spatial curvature.
We get a rough estimate of the angular diameter distance from the observed
angular size of the extrema on the CMB sky as follows.
dc
A(zdec) =
rc
s(tdec)
θs
≈
1√
3
dhor(tdec)
θs
(1 + zdec)
≈
180◦
πθs(◦)
√
3tdec(1 + zdec) ≈ 21 Gpc , (12.116)
where we have approximated rs = dhor/
√
3 and dhor = 3t, and θs(◦) is θs in degrees.
This value is within a factor of 2 of the real result. However, the difference between
the observation and the Einstein-de Sitter result for dc
A is only 50%, so this rough
approximation cannot be used to rule out the Einstein-de Sitter model, we have to
use a more precise value for the sound horizon.
12.9.3 Acoustic peak heights
There are a number of effects which affect the heights of the acoustic peaks:
1. The early ISW effect. The early ISW effect raises the first peak. It is
caused by the evolution of Φ because of the effect of the radiation contribution
on the expansion law after tdec. This depends on the radiation-matter ratio at
that time; decreasing ωm makes the early ISW effect stronger.
10
Photon decoupling temperature, and thus 1 + zdec, depends somewhat on ωb, but since this
dependence is not easy to calculate (recombination and photon decoupling were discussed in chapter
5), we have mostly ignored this dependence and used the fixed value 1 + zdec = 1100.
12 COSMIC MICROWAVE BACKGROUND 226
2. Shift of oscillation equilibrium by baryons. (Baryon drag.) This makes
the odd peaks (which correspond to compression of the baryon-photon fluid
in the potential wells, decompression on potential hills) higher, and the even
peaks (decompression at potential wells, compression on top of potential hills)
lower.
3. Baryon damping. The time evolution of R ≡ 3¯ρb/4¯ργ causes the amplitude
of the acoustic oscillations to be damped in time roughly as (1 + R)−1/4. This
reduces the amplitudes of all peaks.
4. Radiation driving.11 This is an effect related to horizon scale physics that
we have not tried to properly calculate. For scales k which enter during the
radiation-dominated epoch, or near matter-radiation equality, the potential Φ
decays around the time when the scale enters. The potential keeps changing as
long as the radiation contribution is important, but the largest change in Φ is
around horizon entry. Because the sound horizon and Hubble length are comparable,
horizon entry and the corresponding potential decay always happen
during the first oscillation period. This means that the baryon-photon fluid is
falling into a deep potential well, and therefore is compressed by gravity by a
large factor, before the resulting overpressure is able to push it out. Meanwhile
the potential has decayed, so it is less able to resist the decompression phase,
and the overpressure is able to kick the fluid further out of the well. This
increases the amplitude of the acoustic oscillations. The effect is stronger for
the smaller scales which enter when the universe is more radiation-dominated,
and therefore raises the peaks with a larger peak number n more. Reducing
ωm makes the universe more radiation dominated, making this effect stronger
and extending it towards the peaks with lower peak number n.
5. Diffusion damping. Diffusion damping lowers the heights of the peaks. It
acts in the opposite direction than the radiation driving effect, lowering the
peaks with a larger peak number m more. Because the diffusion damping
effect is exponential in ℓ, it wins for large ℓ.
Effects 1 and 4 depend on ωm, effects 2, 3, and 5 on ωb. See Figs. 19 and 20 for the
effects of ωm and ωb on peak heights.
12.9.4 Effect of Ω0 and ΩΛ0
These two parameters have only two effects:
1. they affect the sound horizon angle and thus the positions of the acoustic peaks
2. they affect the late ISW effect
See Figs. 21 and 22. Since the late ISW effect is in the region of the Cℓ spectrum
where the cosmic variance is large, it is difficult to detect. Thus we can in practice
only use θs to determine Ω0 and ΩΛ0. Since ωb and ωm can be determined quite
accurately from Cℓ acoustic peak heights, peak separation, i.e., θs, can then indeed
be used for the determination of Ω0 and ΩΛ0. Since one number cannot be used
11
This is also called gravitational driving, which is perhaps more appropriate, since the effect is
due to the change in the gravitational potential.
12 COSMIC MICROWAVE BACKGROUND 227
0 500 1000 1500 2000
l
0
0.2
0.4
0.6
l(l+1)Cl
/2π
ωm
= 0.10
ωm
= 0.20
ωm
= 0.30
ωm
= 0.40
ωb
= 0.01
0 500 1000 1500 2000
l
0
0.2
0.4
0.6
ωb
= 0.03
Figure 19: The effect of ωm. The angular power spectrum Cℓ is here calculated without
the effect of diffusion damping, so that the other effects on peak heights could be seen more
clearly. Notice how reducing ωm raises all peaks, but the effect on the first few peaks is
stronger in relative terms, as the radiation driving effect is extended towards larger scales
(smaller ℓ). The first peak is raised mainly because the ISW effect becomes stronger. Figure
and calculation by R. Keskitalo.
0 500 1000 1500 2000
l
0
0.1
0.2
0.3
0.4
0.5
0.6
2l(l+1)Cl
/2π
ωb
= 0.01
ωb
= 0.02
ωb
= 0.03
ωb
= 0.04
Figure 20: The effect of ωb. The angular power spectrum Cℓ is here calculated without
the effect of diffusion damping, so that the other effects on peak heights could be seen more
clearly. Notice how increasing ωb raises odd peaks relative to the even peaks. Because of
baryon damping there is a general trend downwards with increasing ωb. This figure is for
ωm = 0.20. Figure and calculation by R. Keskitalo.
12 COSMIC MICROWAVE BACKGROUND 228
to determine two, the parameters Ω0 and ΩΛ0 are degenerate. CMB observations
alone cannot be used to determine them both. Other cosmological observations (like
the power spectrum Pδ(k) from large scale structure, or the SNIa redshift-distance
relationship) are needed to break this degeneracy.
A fixed θs together with fixed ωb and ωm determine a line on the (Ω0, ΩΛ0)
-plane. See figure 23. Derived parameters, e.g., h, vary along that line. As you can
see from Figs. 21 and 22, changing Ω0 (around the reference model) affects θs much
more strongly than changing ΩΛ0. This means that the orientation of the line is such
that ΩΛ0 varies more rapidly along that line than Ω0. Therefore using additional
constraints from other cosmological observations, e.g. the Hubble Space Telescope
determination of h based on the distance ladder, which select a short section from
that line, gives us a fairly good determination of Ω0, leaving the allowed range for
ΩΛ0 still quite large.
Therefore it is often said that CMB measurements have determined that Ω0 ∼ 1,
i.e. that the universe is spatially flat. However, this is misleading. First, the
CMB only determines the angular diameter distance to the last scattering surface.
Determining the spatial curvature from this requires knowing the expansion history
H(z), in other words the constraints on the spatial curvature are model-dependent.
Even restricting to the ΛCDM model, we need to use some other cosmological data
to fix H0. So the correct statement is that assuming that the universe is described
by the ΛCDM model, and given constraints on the Hubble parameter today, the
CMB data shows the universe to be close to spatially flat.
12.9.5 Effect of the primordial spectrum
The effect of the primordial spectrum is simple: raising the amplitude A raises the
Cℓ also, and changing the primordial spectral index tilts Cℓ. See Figs. 24 and 25.
12.9.6 Optical depth due to reionisation
When radiation from the first stars reunites the intergalactic gas, CMB photons may
scatter from the resulting free electrons. The optical depth τ due to reionisation
is the expectation number of such scatterings per CMB photon. It has a value of
about τ = 0.09±0.02, i.e., most CMB photons do not scatter at all. The rescattering
causes additional polarisation of the CMB, and CMB polarisation measurements are
actually the best way to determine τ.
Because of this scattering, not all CMB photons come from the location on
the last scattering surface they seem to come from. The effect of the rescattered
photons is to mix up signals from different directions and therefore reduce the CMB
anisotropy. The reduction factor on δT/T is e−τ and on the Cℓ spectrum e−2τ .
However, this does not affect the largest scales, scales larger than the area from
which the rescattered photons reaching us from a certain direction originally came
from. Such a large-scale anisotropy has affected all such photons the same way, and
thus is not lost in the mixing. See figure 26.
12.9.7 Effect of ωb and ωm
These parameters affect both the positions of the acoustic peaks (through θs) and
the heights of the different peaks. The latter effect is the more important, since
12 COSMIC MICROWAVE BACKGROUND 229
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
0 = 0.9
0 = 1.0
0 = 1.1
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
0 = 0.9
0 = 1.0
0 = 1.1
Figure 21: The effect of changing Ω0 from its reference value Ω0 = 1. The top panel
shows the Cℓ spectrum with a linear ℓ scale so that details at larger ℓ where cosmic variance
effects are smaller can be better seen. The bottom plot has a logarithmic ℓ scale so that
the integrated Sachs-Wolfe effect at small ℓ can be better seen. The logarithmic scale also
makes clear that the effect of the change in sound horizon angle is to stretch the spectrum
by a constant factor in ℓ space.
12 COSMIC MICROWAVE BACKGROUND 230
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
= 0.60
= 0.70
= 0.80
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
= 0.60
= 0.70
= 0.80
Figure 22: The effect of changing ΩΛ0 from its reference value ΩΛ0 = 0.7.
12 COSMIC MICROWAVE BACKGROUND 231
0.2 0.4 0.6 0.8 1 1.2 1.4
0
0.5
1
1.5
Ω
m
Ω
Λ
50
50
100
100
200
200
250
250
300
300
350
400
500
600
open −−− closed
accel −−− decel
no big
bang
Distance between successive acoustic peaks (∆ l)
ωb
= 0.022, ωm
= 0.147, h is derived parameter
Figure 23: The lines of constant sound horizon angle θs on the (Ωm0,ΩΛ0) plane for fixed
ωb and ωm. The numbers on the lines refer to the corresponding acoustic scale ℓA ≡ π/θs
(∼ peak separation) in multipole space. Figure by J. V¨aliviita. See his PhD thesis[5], p.70,
for an improved version including the HST constraint on h.
both parameters have their own signature on the peak heights, allowing an accurate
determination of these parameters, whereas the effect on θs is degenerate with Ω0
and ΩΛ0.
Especially ωb has a characteristic effect on peak heights: Increasing ωb raises the
odd peaks and reduces the even peaks, because it shifts the balance of the acoustic
oscillations (the −RΦ effect). This shows the most clearly at the first and second
peaks.12 Raising ωb also shortens the damping scale k−1
D due to photon diffusion,
moving the corresponding damping scale ℓD of the Cℓ spectrum towards larger ℓ.
This has the effect of raising Cℓ at large ℓ. See figure 27.
Increasing ωm makes the universe more matter dominated at tdec and therefore
it reduces the early ISW effect, making the first peak lower. This also affects the
shape of the first peak.
The “radiation driving” effect is most clear at the second to fourth peaks. Reducing
ωm makes these peaks higher by making the universe more radiation-dominated
at the time the scales corresponding to these peaks enter, and thus strengthening
this radiation driving. The fifth and further peaks correspond to scales that have
anyway essentially the full effect, whereas for the first peak this effect is anyway
weak. (We see instead the ISW effect in the first peak.) See figure 28.
12
There is also an overall “baryon damping effect” on the acoustic oscillations which we have not
calculated. It is due to the time dependence of R ≡ 3¯ρb/4¯ρm, which reduces the amplitude of the
oscillation by about (1 + R)−1/4
. This explains why the third peak in figure 27 is no higher for
ωb = 0.030 than it is for ωb = 0.022.
12 COSMIC MICROWAVE BACKGROUND 232
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
A = 0.9
A = 1
A = 1.1
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
A = 0.9
A = 1
A = 1.1
Figure 24: The effect of changing the primordial amplitude from its reference value A = 1.
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
n = 0.9
n = 1
n = 1.1
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
n = 0.9
n = 1
n = 1.1
Figure 25: The effect of changing the spectral index from its reference value n = 1.
12 COSMIC MICROWAVE BACKGROUND 233
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4(+1)C/2
= 0
= 0.10
= 0.20
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
= 0
= 0.10
= 0.20
Figure 26: The effect of changing the optical depth from its reference value τ = 0.1.
12 COSMIC MICROWAVE BACKGROUND 234
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4(+1)C/2
b = 0.015
b = 0.022
b = 0.030
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
b = 0.015
b = 0.022
b = 0.030
Figure 27: The effect of changing the physical baryon density parameter from its reference
value ωb = 0.022.
12 COSMIC MICROWAVE BACKGROUND 235
0 200 400 600 800 1000 1200 1400
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4(+1)C/2
m = 0.100
m = 0.147
m = 0.200
2 5 10
1
2 5 10
2
2 5 10
3
2
0.0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
(+1)C/2
m = 0.100
m = 0.147
m = 0.200
Figure 28: The effect of changing the physical matter density parameter from its reference
value ωm = 0.147.
12 COSMIC MICROWAVE BACKGROUND 236
12.10 Best values of the cosmological parameters
The most important cosmological data set for determining the values for the cosmological
parameters is the Planck satellite data on the CMB anisotropy. For high ℓ,
it can be supplemented with CMB measurements from ground-based and balloonborne
instruments with higher resolution but poorer sensitivity and sky coverage.
The most accurate measurements for the higher multipoles to date are from the
Arcminute Cosmology Bolometer Array Receiver (ACBAR) [3] and the South Pole
Telescope (SPT) [4].
Because of degeneracies of cosmological parameters in the CMB data, most importantly
the fact that the CMB is sensitive to the vacuum energy and spatial
curvature mostly via the angular diameter distance, CMB observations have to be
supplemented by other cosmological data for a good determination of the main cosmological
parameters.
Large scale structure surveys, i.e. the measurement of the 3-dimensional matter
power spectrum Pδ(k) from the distribution of galaxies, mainly measure the combination
Ωm0h, since this determines where Pδ(k) turns down. The turn is at keq
which is proportional to ωm ≡ Ωm0h2, but since in these surveys the distances to
the galaxies are deduced from their redshifts (therefore these surveys are also called
galaxy redshift surveys), which give the distances only up to the Hubble constant
H0, these surveys determine h−1keq instead of keq. This cancels one power of h.
Having Ωm0h2 from CMB and Ωm0h from the galaxy surveys, gives us both h and
Ωm0 = Ω0 − ΩΛ0, which breaks the Ω0-ΩΛ0 degeneracy.
These measurements of Pδ(k) are now so accurate that the small residual effect
from the baryon acoustic oscillations (BAO) before photon decoupling can be seen
as a weak wavy pattern [6]. This is the same structure which we see in the Cℓ
but now much fainter, since now the baryons have fallen into the CDM potential
wells, and the CDM was only mildly affected by these oscillations in the baryonphoton
fluid. The half-wavelength of this pattern, however, corresponds to the
same sound horizon distance rc
s(tdec) in both cases.13 The redshift at which the
pattern is seen is however much smaller, so this gives a measurement of dc
A(z) at a
different redshift14. Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital
Sky Survey (SDSS). Another way to break the Ω0-ΩΛ0 degeneracy is to use the
redshift-distance relationship from Supernova Type Ia surveys.
However, the more datasets one puts together, the more assumptions are involved
in the analysis, so constraints from large combinations of data should be treated
cautiously. In Table I we give values for the standard parameters from the analysis
of the 1.5 year Planck data [7]. It has been assumed that Ω0 = 1. The first column
gives the mean value and the error bars15 for the Planck 1.5-year data only, and in
the second column a measurement of polarisation from the WMAP satellite (WP),
large multipole data from ground-based CMB experiments (highL) and data from
baryon acoustic oscillations (see below) has also been used. In Table II we list some
13
To be accurate, the best tdec value to represent the effect in Pδ(k) is not exactly the same as
for Cℓ, since photon decoupling was not instantaneous, and in the galaxies are looking at the effect
on matter and in the CMB the effect on photons.
14
In fact, the BAO signal gives a combination of dA(z) and H(z).
15
The upper and lower limits are “16- and 84-percentiles” which means that there is some relation
to having a formal 68% probability that the correct value is in this range. The probability
interpretation has some subtleties however; we will not go into the matter here.
12 COSMIC MICROWAVE BACKGROUND 237
Table I: Standard parameters
Planck only
Planck + WP +
HighL + BAO
100ωb 2.217 ± 0.033 2.214 ± 0.024
ωc 0.1186 ± 0.0031 0.1187 ± 0.0017
n 0.9635 ± 0.0094 0.9608 ± 0.0054
ΩΛ0 0.693 ± 0.019 0.692 ± 0.010
τ 0.089 ± 0.032 0.092 ± 0.013
Table II: Derived parameters
Planck only
Planck + WP +
HighL + BAO
Ωm0 0.307 ± 0.019 0.308 ± 0.010
100h 67.9 ± 1.5 67.8 ± 0.77
related derived parameters, and in Table III we give limits on some non-standard
parameters. Note that in table III the error bars are the 95% confidence limits
(instead of the usual 68% confidence limits), and the first column is Planck data
plus WMAP polarisation data.
The BBN limit 0.019 ≤ ωb ≤ 0.024 has not been used here, but we see that the
constraint on the baryon density coming from the CMB is consistent with the BBN
value. The agreement between these two independent datasets (the abundances
of light elements and anisotropies on the microwave sky) one of which probes the
physics around a couple of minutes and the other at around 400 000 years is remarkable.
This increases our confidence that the basic physical picture of the evolution
of the universe is correct. Indeed, BBN and CMB are two of the most important
pieces of observational support for the standard cosmological model.
The parameters in Table III are derived under the assumption that the nonstandard
parameters other than the one being considered remain zero. The CMB
alone does not give good constraints on the spatial curvature or the dark energy
equation of state (since they mostly only affect dc
A(zdec), and are thus degenerate
with ΩΛ0). In fact, the CMB data is consistent with a closed universe without dark
energy, with Ω0 = Ωm0 ≈ 1.3, and h ≈ 0.3. The upper limits given for the sum of
neutrino masses mν and the ratio r ≡ A2
T /A2 of tensor perturbations to scalar
perturbations are 95% confidence limits. We see that there is no indication in the
data for a deviation of these additional parameters from their standard values.
In conclusion, almost all cosmological data are consistent with a “vanilla” universe,
i.e. a spatially flat ΛCDM model with adiabatic and Gaussian primordial
density perturbations, described by the six cosmological parameters ΩΛ0, ωm, ωb,
A, n, τ.
Simplest inflationary models predict an amplitude for gravity waves that Planck
would be able to observe using the polarisation of the CMB. This data will be
released in 2014.
12 COSMIC MICROWAVE BACKGROUND 238
Table III: Additional parameters
Planck + WP
Planck + WP +
highL + BAO
mν < 0.933 eV < 0.230 eV
w −1.49+0.65
−0.57 −1.13+0.23
−0.25
ΩK0 −0.037+0.043
−0.049 −0.0005+0.0065
−0.0066
dn
d ln k −0.013 ± 0.018 −0.014+0.016
−0,017
r < 0.12 < 0.111
Figure 29: Constraints on the scalar perturbation spectral index n, and the tensor/scalar
ratio r from Planck satellite data. Figure from [8].
REFERENCES 239
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[astro-ph.CO].
[2] David H. Lyth and Andrew R. Liddle: The Primordial Density Perturbation
(Cambridge University Press 2009).
[3] C. Reichardt et al., High Resolution CMB Power Spectrum from the Complet
ACBAR Data Set, arXiv:0801.1491.
[4] K.T. Story et al, A Measurement of the Cosmic Microwave Background Damping
Tail from the 2500-square-degree SPT-SZ survey, arXiv:1210.7231.
[5] J. V¨aliviita, PhD thesis, University of Helsinki 2005.
[6] W.J. Percival et al., Measuring the Baryon Acoustic Oscillation scale using the
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