PART 2 OF 3 OF A COURSE I HAVE GIVEN AT BIT PILANI ON CMB

1. 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

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

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

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

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

6. Plan of the Talk
Horizons
Horizon Problem
Inflation
Perturbations
CMB Anisotropies
Acoustic Oscillations
An overview
Angular Power Spectrum
Cosmological Parameters
Spatial Curvature
Cosmological Constant
Baryons
Dark Matter
Primordial Power Spectrum
Boltzmann Codes - CMBFAST and CAMB
2 / 32

7. Plan of the Talk
Horizons
Horizon Problem
Inflation
Perturbations
CMB Anisotropies
Acoustic Oscillations
An overview
Angular Power Spectrum
Cosmological Parameters
Spatial Curvature
Cosmological Constant
Baryons
Dark Matter
Primordial Power Spectrum
Boltzmann Codes - CMBFAST and CAMB
CMB Missions
2 / 32

8. 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

9. 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)
Einstein-Boltzmann equation describe the evolution of Fourier
modes.
3 / 32

10. 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)
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

11. 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

12. 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.
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

13. 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.
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

14. 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

15. 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.
Before decoupling, the wavelengths of these modes are so
large that no causal physics could force deviations from
smoothness to go away.
5 / 32

16. 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.
Before decoupling, the wavelengths of these modes are so
large that no causal physics could force deviations from
smoothness to go away.
After decoupling, the photons do not interact at all; they
simply free stream.
5 / 32

17. 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.
Before decoupling, the wavelengths of these modes are so
large that no causal physics could force deviations from
smoothness to go away.
After decoupling, the photons do not interact at all; they
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

18. 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.
Before decoupling, the wavelengths of these modes are so
large that no causal physics could force deviations from
smoothness to go away.
After decoupling, the photons do not interact at all; they
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

19. 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

20. 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.
During inflation, comoving Hubble distance become so small
that all the cosmologically relevant scales comes within it and
can establish causal relations.
6 / 32

21. 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.
During inflation, comoving Hubble distance become so small
that all the cosmologically relevant scales comes within it and
can establish causal relations.
Models which leave the horizon re-enter the horizon after
inflation ends.
6 / 32

22. 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.
During inflation, comoving Hubble distance become so small
that all the cosmologically relevant scales comes within it and
can establish causal relations.
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

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

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

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

26. 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

27. 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.
The transfer function Tf (k) describes the evolution of
perturbations through the epochs of horizon crossing and
radiation/matter transition.
8 / 32

28. 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.
The transfer function Tf (k) describes the evolution of
perturbations through the epochs of horizon crossing and
radiation/matter transition.
The growth factor D(a) describes the wavelength-independent
growth at late times.
8 / 32

29. 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.
The transfer function Tf (k) describes the evolution of
perturbations through the epochs of horizon crossing and
radiation/matter transition.
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

30. 9 / 32

31. 10 / 32

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

33. 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.
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

34. 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.
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.
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

35. 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.
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.
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)
The initial conditions for matter, both δ and δb, depend upon
the nature of the primordial perturbations
11 / 32

36. 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.
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.
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)
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

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

38. Adiabatic Perturbations
The initial conditions for δ and δb depend on the nature of
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

39. Adiabatic Perturbations
The initial conditions for δ and δb depend on the nature of
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.
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

40. Adiabatic Perturbations
The initial conditions for δ and δb depend on the nature of
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.
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

41. 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

42. 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

43. 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

44. 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)
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

45. Angular Power Spectrum : the Cl
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

46. Angular Power Spectrum : the Cl
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

47. 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

48. 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)
The first term show the contribution due to fluctuations in
CMB photon density ργ at the time of last scattering.
[Rubakov & Vlasov (2012)]
16 / 32

49. 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)
The first term show the contribution due to fluctuations in
CMB photon density ργ at the time of last scattering.
The second term reflects the fact that photons get redshifted
(blueshifted) when they escape from potential wells (humps).
[Rubakov & Vlasov (2012)]
16 / 32

50. 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)
The first term show the contribution due to fluctuations in
CMB photon density ργ at the time of last scattering.
The second term reflects the fact that photons get redshifted
(blueshifted) when they escape from potential wells (humps).
The first and second contributions together are called the
Sachs Wolfe effect.
[Rubakov & Vlasov (2012)]
16 / 32

51. 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)
The first term show the contribution due to fluctuations in
CMB photon density ργ at the time of last scattering.
The second term reflects the fact that photons get redshifted
(blueshifted) when they escape from potential wells (humps).
The first and second contributions together are called the
Sachs Wolfe effect.
The third term is due to Doppler effect.
[Rubakov & Vlasov (2012)]
16 / 32

52. 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)
The first term show the contribution due to fluctuations in
CMB photon density ργ at the time of last scattering.
The second term reflects the fact that photons get redshifted
(blueshifted) when they escape from potential wells (humps).
The first and second contributions together are called the
Sachs Wolfe effect.
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.
[Rubakov & Vlasov (2012)]
16 / 32

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

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

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

56. 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

57. Acoustics Oscillations
Before decoupling there were density fluctuations in the
matter (dark) present and baryons could fall in the
gravitational potential of those.
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

58. Acoustics Oscillations
Before decoupling there were density fluctuations in the
matter (dark) present and baryons could fall in the
gravitational potential of those.
Since baryons were coupled to photons, radiation pressure
opposed gravitational infall of baryons which created acoustic
oscillations, compression and rarefaction.
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

59. Acoustics Oscillations
Before decoupling there were density fluctuations in the
matter (dark) present and baryons could fall in the
gravitational potential of those.
Since baryons were coupled to photons, radiation pressure
opposed gravitational infall of baryons which created acoustic
oscillations, compression and rarefaction.
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

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

61. Acoustic Oscillation
Where the effective mass:
meff = 1 + R = 1 +
3ρb
4ργ
(21)
Effective acceleration due to gravity:
g = −
1
3
k2
c2
Ψ − ¨Φ (22)
19 / 32

62. Acoustic Oscillation
Where the effective mass:
meff = 1 + R = 1 +
3ρb
4ργ
(21)
Effective acceleration due to gravity:
g = −
1
3
k2
c2
Ψ − ¨Φ (22)
Frequency of oscillation:
ω =
kc
√
3meff
= kcs (23)
19 / 32

63. Acoustic Oscillation
Where the effective mass:
meff = 1 + R = 1 +
3ρb
4ργ
(21)
Effective acceleration due to gravity:
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

64. 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

65. 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)
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)
You can download programs from here :
http://gyudon.as.utexas.edu/ komatsu/CRL/
20 / 32

66. 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)
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).
You can download programs from here :
http://gyudon.as.utexas.edu/ komatsu/CRL/
20 / 32

67. Acoustics Oscillations
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

68. Acoustics Oscillations
Evolution of the baryon-photon fluid oscillations (of size less than
the horizon size) is given by:
δργ
ργ
= A cos(krs) (28)
The value of the density perturbation at recombination oscillates as
a function of wavelength.
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69. Acoustics Oscillations
Evolution of the baryon-photon fluid oscillations (of size less than
the horizon size) is given by:
δργ
ργ
= A cos(krs) (28)
The value of the density perturbation at recombination oscillates as
a function of wavelength.
The main effect of the baryonphoton perturbations on CMB comes
from the oscillatory first term in equation (19) i.e., δργ/ργ.
21 / 32

70. Acoustics Oscillations
Evolution of the baryon-photon fluid oscillations (of size less than
the horizon size) is given by:
δργ
ργ
= A cos(krs) (28)
The value of the density perturbation at recombination oscillates as
a function of wavelength.
The main effect of the baryonphoton perturbations on CMB comes
from the oscillatory first term in equation (19) i.e., δργ/ργ.
The absolute values of perturbations in the baryon photon plasma
are maximal at conformal momenta:
knrs = nπ (29)
21 / 32

71. Acoustics Oscillations
Evolution of the baryon-photon fluid oscillations (of size less than
the horizon size) is given by:
δργ
ργ
= A cos(krs) (28)
The value of the density perturbation at recombination oscillates as
a function of wavelength.
The main effect of the baryonphoton perturbations on CMB comes
from the oscillatory first term in equation (19) i.e., δργ/ργ.
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

72. Cosmological Parameters
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

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

74. Cosmological Parameters
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.
CMB anisotropies can be used to probe the physics of the
Early universe like inflation.
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

75. Cosmological Parameters
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.
CMB anisotropies can be used to probe the physics of the
Early universe like inflation.
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.
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

76. Cosmological Parameters
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.
CMB anisotropies can be used to probe the physics of the
Early universe like inflation.
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.
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

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

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

79. Cosmological Parameters
The inhomogeneities in the following components of cosmic
plasma contribute to the CMB anisotropy:
Baryon - electron - photon plasma.
Dark matter.
Neutrinos.
23 / 32

80. Cosmological Parameters
The inhomogeneities in the following components of cosmic
plasma contribute to the CMB anisotropy:
Baryon - electron - photon plasma.
Dark matter.
Neutrinos.
Gravitational waves.
23 / 32

81. 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

82. Cosmological Parameters: Spatial curvature
25 / 32

83. Cosmological Parameters: Spatial curvature
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

84. 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

85. 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.
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

86. Cosmological Parameters:Cosmological Constant
Effect of cosmological constant on CMB angular power spectrum
28 / 32

87. 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

88. 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)
From equation (19) we can write the anisotropy due to Sachs -
Wolfe effects:
∆T
T
=
1
4
ρb
ργ
ΦDM (34)
29 / 32

89. 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)
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

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

91. Cosmological Parameters : Primordial Power spectrum
31 / 32

92. 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