Cmb part2

Cosmic Microwave Background Radiation
Lecture 2 : Statistics of CMB
Jayanti Prasad
Inter-University Centre for Astronomy & Astrophysics (IUCAA)
Pune, India (411007)
October 17, 2013
1 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
2 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
CMB Anisotropies
2 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
CMB Anisotropies
Acoustic Oscillations
An overview
Angular Power Spectrum
2 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
CMB Anisotropies
An overview
Cosmological Parameters
Spatial Curvature
Cosmological Constant
Baryons
Dark Matter
Primordial Power Spectrum
2 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
CMB Anisotropies
An overview
Spatial Curvature
Baryons
Dark Matter
Boltzmann Codes - CMBFAST and CAMB
2 / 32

Plan of the Talk
Horizons
Horizon Problem
Inﬂation
Perturbations
CMB Anisotropies
An overview
Spatial Curvature
Baryons
Dark Matter
Boltzmann Codes - CMBFAST and CAMB
CMB Missions
2 / 32

The maximum distance between two points which can have
causal relationship at any epoch is given by the size of the
comoving horizon η (conformal time):
η =
da
a
1
a H(a )
(1)
3 / 32

η =
da
a
1
a H(a )
(1)
Einstein-Boltzmann equation describe the evolution of Fourier
modes.
3 / 32

η =
da
a
1
a H(a )
(1)
Einstein-Boltzmann equation describe the evolution of Fourier
modes.
A mode k is said to be outside the horizon if 1/kη > 1 or
kη < 1 and it is said to be inside the horizon if kη > 1.
3 / 32

Note that two points which are outside the comoving horizon
can never causal relationship, however, two points outside the
Hubble region cannot now but they can have in future.
4 / 32

We can also use the comoving Hubble scale 1/aH(a) to know
the maximum distance at any epoch within which signals can
establish causal relationship.
4 / 32

We can also use the comoving Hubble scale 1/aH(a) to know
the maximum distance at any epoch within which signals can
establish causal relationship.
Any scale larger than comoving Hubble length is also said to
be outside horizon.
Problem 1
Show that the comoving Hubble length during radiation dominated
era and exponential expansion evolve as a and 1/a respectively and
also ﬁnd out how it evolve during matter dominated era.
4 / 32

Horizon Problem
At present we ﬁnd that CMB is very isotropic at all scales
observed which is diﬃcult to explain since the largest scales
observed have entered the horizon just recently, long after
decoupling.
5 / 32

Horizon Problem
decoupling.
Before decoupling, the wavelengths of these modes are so
large that no causal physics could force deviations from
smoothness to go away.
5 / 32

Horizon Problem
decoupling.
After decoupling, the photons do not interact at all; they
simply free stream.
5 / 32

Horizon Problem
decoupling.
simply free stream.
So even though it is technically possible that photons reaching
us today from opposite directions had a chance to
communicate with each other and equilibrating to the same
temperature, practically this could not have happened.
5 / 32

Horizon Problem
decoupling.
simply free stream.
So even though it is technically possible that photons reaching
us today from opposite directions had a chance to
communicate with each other and equilibrating to the same
temperature, practically this could not have happened.
Why then is the CMB temperature so uniform ?
5 / 32

Inﬂation
In order to solve horizon problem we need the comoving
Hubble radius decrease with time:
d
dt
1
aH(a)
< 0 (2)
or
d
dt
[aH(a)] =
d2a(t)
dt
> 0 (3)
This is called inﬂation.
6 / 32

Inﬂation
d
dt
1
aH(a)
< 0 (2)
or
d
dt
[aH(a)] =
d2a(t)
dt
> 0 (3)
During inﬂation, comoving Hubble distance become so small
that all the cosmologically relevant scales comes within it and
can establish causal relations.
6 / 32

Inﬂation
d
dt
1
aH(a)
< 0 (2)
or
d
dt
[aH(a)] =
d2a(t)
dt
> 0 (3)
Models which leave the horizon re-enter the horizon after
inﬂation ends.
6 / 32

Inﬂation
d
dt
1
aH(a)
< 0 (2)
or
d
dt
[aH(a)] =
d2a(t)
dt
> 0 (3)
Models which leave the horizon re-enter the horizon after
inﬂation ends.
Evolution of a mode depend on the epoch in which it enters
the horizon i.e., radiation dominated, matter dominated etc..
6 / 32

Perturbations
Inﬂation not only solves horizon problem, it also gives initial
condition for the subsequent evolution of the Universe.
7 / 32

Perturbations
In particular, inflation provides potential fluctuations which
give rise fluctuations in matter filed which work as seed for
structure formation and also give CMB anisotropies.
7 / 32

Perturbations
In particular, inflation provides potential fluctuations which
give rise fluctuations in matter filed which work as seed for
structure formation and also give CMB anisotropies.
We represent the primordial perturbations a scale invariant
power spectrum:
PΦ(k) = As
k
k∗
ns −1
(4)
where As and ns are parameters.
7 / 32

Evolution of Primordial perturbations
We can relate the perturbations at present to the primordial
perturbations in the following way:
Φ(k, a) = ΦP(K) × Tf (k, a) × D(a) (5)
where Tf (k) and D(a) are called the Transfer function and
Growth factor respectively.
8 / 32

Φ(k, a) = ΦP(K) × Tf (k, a) × D(a) (5)
The transfer function Tf (k) describes the evolution of
perturbations through the epochs of horizon crossing and
radiation/matter transition.
8 / 32

Φ(k, a) = ΦP(K) × Tf (k, a) × D(a) (5)
The growth factor D(a) describes the wavelength-independent
growth at late times.
8 / 32

Φ(k, a) = ΦP(K) × Tf (k, a) × D(a) (5)
The growth factor D(a) describes the wavelength-independent
growth at late times.
In the context of inﬂation Φp is drawn from a Gaussian
distribution with mean zero and variance given by its power
spectrum.
8 / 32

Einstein-Boltzmann equations for super horizon mode
For the modes outside horizon (at early times) in
Einstein-Boltzmann equation we can terms which are
multiplied by k.
11 / 32

multiplied by k.
Before recombination, baryon-photon plasma behaved like
perfect ﬂuid ( can be described by density and velocities) and
higher multipoles can be ignored since they are quite small as
compared to the monopole.
11 / 32

multiplied by k.
For kη << 1 limit we get the following evolution equations for
Θ, N, δ and δb.
˙Θ0 + ˙Φ = 0 (6)
˙N0 + ˙Φ = 0 (7)
˙δ = −3 ˙Φ (8)
˙δb = −3 ˙Φ (9)
11 / 32

multiplied by k.
Θ, N, δ and δb.
˙Θ0 + ˙Φ = 0 (6)
˙N0 + ˙Φ = 0 (7)
˙δ = −3 ˙Φ (8)
˙δb = −3 ˙Φ (9)
The initial conditions for matter, both δ and δb, depend upon
the nature of the primordial perturbations
11 / 32

multiplied by k.
Θ, N, δ and δb.
˙Θ0 + ˙Φ = 0 (6)
˙N0 + ˙Φ = 0 (7)
˙δ = −3 ˙Φ (8)
˙δb = −3 ˙Φ (9)
The initial conditions for matter, both δ and δb, depend upon
the nature of the primordial perturbations
Note that in the limit kη < 1, the velocities are comparable to 11 / 32

Adiabatic Perturbations
The initial conditions for δ and δb depend on the nature of
perturbations.
12 / 32

perturbations.
From the Boltzmann equations for Θ0 and δ in super-horizon
limit we:
δ = 3Θ0 + constant (10)
The perturbations are said to be adiabatic when the constant
is zero and otherwise they care called isocurvature.
12 / 32

perturbations.
limit we:
Adiabatic perturbations have the same matter-to-radiation
ratio everywhere i.e., constant entropy.
ndm
nγ
=
n
(0)
dm
n
(0)
γ
1 + δ
1 + 3Θ0
(11)
12 / 32

perturbations.
limit we:
Adiabatic perturbations have the same matter-to-radiation
ratio everywhere i.e., constant entropy.
ndm
nγ
=
n
(0)
dm
n
(0)
γ
1 + δ
1 + 3Θ0
(11)
There are models based on isocurvature perturbations, but
these have not been very successful to date so adiabatic initial
conditions are generally considered.
12 / 32

CMB Anisotropies
The temperature anisotropies of the sky can be expanded in
spherical harmonics:
∆T
T
(ˆn) =
∞
l=0
m=l
m=−l
almYlm(ˆn) (12)
and
alm = dˆn
∆T
T
(ˆn)Y ∗
lm(ˆn) (13)
13 / 32

CMB Anisotropies
The temperature anisotropies of the sky can be expanded in
spherical harmonics:
∆T
T
(ˆn) =
∞
l=0
m=l
m=−l
almYlm(ˆn) (12)
and
alm = dˆn
∆T
T
(ˆn)Y ∗
lm(ˆn) (13)
Under the assumption that our theory has no preferred
direction on the sky, and the ﬂuctuations on temperature
follow Gaussian statistics the cross-correlation of temperature
anisotropies contain all the physical information:
C(θ) =
∆T
T
(ˆn)
∆T
T
(ˆn ) with θ = ˆn.ˆn (14)
13 / 32

Angular Power Spectrum : the Cl
Fourier transformation of the angular correlation function
C(θ) is called the angular power spectrum, or multipoles Cl ,
which is the most important object in the analysis of CMB
anisotropies:
C(θ) =
1
4π
∞
l=2
(2l + 1)Cl Pl (cos θ) (15)
where Pl (cos θ) are Legendre Polynomials an:
Cl = a∗
lmalm (16)
[White & Cohn (2002)] 14 / 32

Fourier transformation of the angular correlation function
C(θ) is called the angular power spectrum, or multipoles Cl ,
which is the most important object in the analysis of CMB
anisotropies:
C(θ) =
1
4π
∞
l=2
(2l + 1)Cl Pl (cos θ) (15)
where Pl (cos θ) are Legendre Polynomials an:
Cl = a∗
lmalm (16)
Since we are trying to “estimate” Cl s from a ﬁnite number of
samples, so there is error (cosmic variance):
∆Cl
Cl
=
2
2l + 1
(17)
factor of 2 is because Cl is the square of Gaussian random
variable.
[White & Cohn (2002)] 14 / 32

For historical reasons the quantity which is generally called
(temperature-temperature) spectrum is deﬁned as:
∆T2
=
l(l + 1)
2π
Cl T2
CMB (18)
and measured in micro Kelvin square.
15 / 32

For historical reasons the quantity which is generally called
(temperature-temperature) spectrum is deﬁned as:
∆T2
=
l(l + 1)
2π
Cl T2
CMB (18)
and measured in micro Kelvin square.
∆T2 represents the power per logarithmic interval in l and is
expected to be uniform (nearly) in inﬂationary models (scale
invariant) over much of the spectrum.
15 / 32

For scalar perturbations we can write CMB anisotropy in the
following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
[Rubakov & Vlasov (2012)]
16 / 32

following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
The ﬁrst term show the contribution due to ﬂuctuations in
CMB photon density ργ at the time of last scattering.
16 / 32

following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
The second term reﬂects the fact that photons get redshifted
(blueshifted) when they escape from potential wells (humps).
16 / 32

following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
The ﬁrst and second contributions together are called the
Sachs Wolfe eﬀect.
16 / 32

following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
The third term is due to Doppler eﬀect.
16 / 32

following form:
δT(n)
T
=
1
4
δργ
ργ
(x) + Φ(x) + n.vγ(x) +
t0
tr
dt( ˙Φ − ˙Ψ) (19)
The third term is due to Doppler effect.
The fourth term is called Integrated Sachs Wolfe effect (ISW)
and it shows that that as photons propagate through the
Universe, they get red- or blueshifted due to time-dependent
gravitational field.
16 / 32

For scalar perturbations, both Doppler and integrated
Sachs-Wolfe eﬀects are numerically rather small.
17 / 32

Up to a good approximation, before decoupling baryons and
photons together make a single ﬂuid (perfect): they are
tightly coupled due to intense scattering of photons oﬀ free
electrons and intense Coulomb interaction between electrons
and proton.
17 / 32

Up to a good approximation, before decoupling baryons and
photons together make a single fluid (perfect): they are
tightly coupled due to intense scattering of photons off free
electrons and intense Coulomb interaction between electrons
and proton.
Motion of photon-baryon fluid cause acoustics oscillations.
17 / 32

Acoustics Oscillations
Before decoupling there were density ﬂuctuations in the
matter (dark) present and baryons could fall in the
gravitational potential of those.
[Hu et al. (1996)]
18 / 32

Since baryons were coupled to photons, radiation pressure
opposed gravitational infall of baryons which created acoustic
oscillations, compression and rarefaction.
[Hu et al. (1996)]
18 / 32

At decoupling different modes were caught at different phases
of their oscillations which resulted in different temperature of
the decoupled photons i.e., anisotropies - pekes and troughs in
Cl .
[Hu et al. (1996)]
18 / 32

At decoupling different modes were caught at different phases
of their oscillations which resulted in different temperature of
the decoupled photons i.e., anisotropies - pekes and troughs in
Cl .
Oscillation in the CMB temperature can be described by the
following equation:
meff
¨Θ +
1
3
k2
cΘ ≈ meffg (20)
[Hu et al. (1996)]
18 / 32

Acoustic Oscillation
Where the eﬀective mass:
meﬀ = 1 + R = 1 +
3ρb
4ργ
(21)
19 / 32

meﬀ = 1 + R = 1 +
3ρb
4ργ
(21)
Eﬀective acceleration due to gravity:
g = −
1
3
k2
c2
Ψ − ¨Φ (22)
19 / 32

meﬀ = 1 + R = 1 +
3ρb
4ργ
(21)
g = −
1
3
k2
c2
Ψ − ¨Φ (22)
Frequency of oscillation:
ω =
kc
√
3meﬀ
= kcs (23)
19 / 32

meﬀ = 1 + R = 1 +
3ρb
4ργ
(21)
g = −
1
3
k2
c2
Ψ − ¨Φ (22)
Frequency of oscillation:
ω =
kc
√
3meﬀ
= kcs (23)
The phase of oscillation is given by integrating over conformal
time:
φ = ωdη = krs (24)
where:
rs = csdη (25)
19 / 32

Problem 1
Show that the angular diameter distance of the sound horizon
at decoupling is given by:
dA(adec) =
c
H0
1
adec
[(1 − Ω)x
2
+ ΩΛx
1−3w
+ Ωmx + Ωr ]
−1/2
dx (26)
You can download programs from here :
http://gyudon.as.utexas.edu/ komatsu/CRL/
20 / 32

Problem 1
dA(adec) =
c
H0
1
adec
[(1 − Ω)x
2
+ ΩΛx
1−3w
+ Ωmx + Ωr ]
−1/2
dx (26)
Show that the expression for the size of sound horizon at
decoupling is given by:
rs (adec) =
c
H0
√
3
adec
0
1 +
3ρb
4ργ
[(1 − Ω)x
2
+ ΩΛx
1−3w
+ Ωmx + Ωr x
2
]
−1/2
dx (27)
20 / 32

Problem 1
dA(adec) =
c
H0
1
adec
[(1 − Ω)x
2
+ ΩΛx
1−3w
+ Ωmx + Ωr ]
−1/2
dx (26)
Show that the expression for the size of sound horizon at
decoupling is given by:
rs (adec) =
c
H0
√
3
adec
0
1 +
3ρb
4ργ
[(1 − Ω)x
2
+ ΩΛx
1−3w
+ Ωmx + Ωr x
2
]
−1/2
dx (27)
Using the angular diameter distance as distance for the
distance to the sound horizon at decoupling show that size of
sound horizon at decoupling is around 1 degree for standard
cosmological parameters (WMAP 9).
20 / 32

Evolution of the baryon-photon ﬂuid oscillations (of size less than
the horizon size) is given by:
δργ
ργ
= A cos(krs) (28)
21 / 32

δργ
ργ
= A cos(krs) (28)
The value of the density perturbation at recombination oscillates as
a function of wavelength.
21 / 32

δργ
ργ
= A cos(krs) (28)
The main eﬀect of the baryonphoton perturbations on CMB comes
from the oscillatory ﬁrst term in equation (19) i.e., δργ/ργ.
21 / 32

δργ
ργ
= A cos(krs) (28)
The absolute values of perturbations in the baryon photon plasma
are maximal at conformal momenta:
knrs = nπ (29)
21 / 32

δργ
ργ
= A cos(krs) (28)
The absolute values of perturbations in the baryon photon plasma
are maximal at conformal momenta:
knrs = nπ (29)
Peaks in CMB angular power spectrum are expected to be at:
ln ≈ πn
η0
rs
(30)
where η0 is the the conformal time interval between the present and
last scattering epochs, i. e., the comoving distance to the sphere of
last scattering. 21 / 32

CMB anisotropy caused both by inhomogeneities in the early
Universe and by the eﬀects of propagation of CMB photons
through the Universe at later time.
22 / 32

CMB anisotropies can be used to probe the physics of the
Early universe like inﬂation.
22 / 32

As CMB photons passe through interstellar plasma they get
scattered by free electrons which distorts CMB angular
spectrum at small angular scales. This eﬀect is called Sunyaev
Zeldovich eﬀect.
22 / 32

Zeldovich effect.
The photons travel freely for rather long time, so they diffuse
away from over-dense regions to regions of lower density, and
this also smears out small inhomogeneities. This effect is
called Silk damping.
22 / 32

Zeldovich effect.
The photons travel freely for rather long time, so they diffuse
away from over-dense regions to regions of lower density, and
this also smears out small inhomogeneities. This effect is
called Silk damping.
Because of these effects, the CMB anisotropy spectrum is not
particularly informative from the cosmological point of view at
angular scales of a few arc minutes and smaller.
22 / 32

The inhomogeneities in the following components of cosmic
plasma contribute to the CMB anisotropy:
Baryon - electron - photon plasma.
23 / 32

Dark matter.
23 / 32

Dark matter.
Neutrinos.
23 / 32

Dark matter.
Neutrinos.
Gravitational waves.
23 / 32

Cosmological Parameters: Spatial curvature
The dependence on spatial curvature comes from the fact that the
angular size of a standard ruler viewed from the same distance
depend on the geometry of space.
24 / 32

25 / 32

Location of the ﬁrst peak which represents the angular size of the
sound horizon at decoupling depend on the spatial geometry of the
Universe.
26 / 32

Cosmological Parameters:Cosmological Constant
Given that we know precisely the epoch of the last scattering and
the present (a(t0)/a(tr )), how fast the Universe has expanded since
then can make the last scattering surface closer or farther from us
and so can change the θ.
27 / 32

Given that we know precisely the epoch of the last scattering and
the present (a(t0)/a(tr )), how fast the Universe has expanded since
then can make the last scattering surface closer or farther from us
and so can change the θ.
The expansion rate of the Universe strongly depends on various
species of energy, in particular on the value of cosmological constant
so it has strong imprints on CMB angular power spectrum.
27 / 32

Eﬀect of cosmological constant on CMB angular power spectrum
28 / 32

Cosmological Parameters : Dark Matter
For photon-baryon plasma the hydrostatic equilibrium condition can
be written as:
p = (p + ρ) Φ (31)
or
δp = −(p + ρ)Φ (32)
now using pγ = 1
3 ργ and ργ ∝ T4
γ we get:
δT
T
=
−3
4
ρb
ργ
ΦDM (33)
29 / 32

be written as:
p = (p + ρ) Φ (31)
or
δp = −(p + ρ)Φ (32)
now using pγ = 1
γ we get:
δT
T
=
−3
4
ρb
ργ
ΦDM (33)
From equation (19) we can write the anisotropy due to Sachs -
Wolfe eﬀects:
∆T
T
=
1
4
ρb
ργ
ΦDM (34)
29 / 32

be written as:
p = (p + ρ) Φ (31)
or
δp = −(p + ρ)Φ (32)
now using pγ = 1
γ we get:
δT
T
=
−3
4
ρb
ργ
ΦDM (33)
From equation (19) we can write the anisotropy due to Sachs -
Wolfe eﬀects:
∆T
T
=
1
4
ρb
ργ
ΦDM (34)
The sum of the contributions of the baryonPhoton and dark matter
perturbations to the CMB temperature ﬂuctuation has the following
form:
∆T
T
= A cos(krs) −
3
4
ρb
ργ
ΦDM (35)
29 / 32

Cosmological Parameters: Dark Matter
The eﬀect of baryons of CMB angular power spectrum
30 / 32

Cosmological Parameters : Primordial Power spectrum
31 / 32

References
Hu, W., Sugiyama, N., & Silk, J. 1996, ArXiv Astrophysics e-prints
Hu, W., & White, M. 1996, Astrophys. J. , 471, 30
Rubakov, V. A., & Vlasov, A. D. 2012, Physics of Atomic Nuclei, 75,
1123
White, M., & Cohn, J. D. 2002, ArXiv Astrophysics e-prints
32 / 32

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