The CMBR fluctuations from HI perturbations prior to reionization

Abstract

Loeb and Zaldarriaga (2003) have recently proposed that observations of the CMBR brightness temperature fluctuations produced by the HI inhomogeneities prior to reionization hold the promise of probing the primordial power spectrum to a hitherto unprecedented level of accuracy. This requires a precise quantification of the relation between density perturbations and brightness temperature fluctuations. Brightness temperature fluctuations arise from two sources (1.) fluctuations in the spin temperature, and (2.) fluctuations in the HI optical depth, both of which are caused by density perturbations. For the spin temperature, we investigate in detail its evolution in the presence of HI fluctuations. For the optical depth, we find that it is affected by density perturbations both directly and through peculiar velocities which move the absorption features around in frequency. The latter effect, which has not been included in earlier studies, is similar to the redshift space distortion seen in galaxy surveys and this can cause changes of 50% or more in the birghtness temperature fluctuations.

Full text

Page 1

arXiv:astro-ph/0401206v2 31 Mar 2004
Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 February 2008 (MN LATEX style file v2.2)
The CMBR fluctuations from HI perturbations prior to
reionization
Somnath Bharadwaj⋆and Sk. Saiyad Ali†
Department of Physics and Meteorology
and
Centre for Theoretical Studies
IIT Kharagpur
Pin: 721 302 , India
2 February 2008
ABSTRACT
Loeb & Zaldarriaga (2003) have recently proposed that observations of the CMBR
brightness temperature fluctuations produced by the HI inhomogeneities prior to reion-
ization hold the promise of probing the primordial power spectrum to a hitherto un-
precedented level of accuracy. This requires a precise quantification of the relation
between density perturbations and brightness temperature fluctuations. Brightness
temperature fluctuations arise from two sources (1.) fluctuations in the spin tempera-
ture, and (2.) fluctuations in the HI optical depth, both of which are caused by density
perturbations. For the spin temperature, we investigate in detail its evolution in the
presence of HI fluctuations. For the optical depth, we find that it is affected by density
perturbations both directly and through peculiar velocities which move the absorption
features around in frequency. The latter effect, which has not been included in earlier
studies, is similar to the redshift space distortion seen in galaxy surveys and this can
cause changes of 50% or more in the birghtness temperature fluctuations.
Key words: cosmology: theory - cosmology: large scale structure of universe - diffuse
radiation
1 INTRODUCTION
The possibility of probing the universe at high redshifts
using the HI 21cm line has been the topic of exten-
sive theoretical investigation. This is perceived to be the
most promising window for studying the “dark ages”,
the era between the decoupling of the CMBR from the
primeval plasma at z∼
the first luminous objects at z
1979; Scott & Rees 1990; Madau, Meiksin & Rees 1997;
Tozz et al 2000; Barkana, & Loeb 2001; Iliev et al 2002;
Miralda-Escude 2003) After decoupling the gas tempera-
ture Tg is maintained at CMBR temperature Tγ through
collisions of the CMBR photons with the small fraction of
electrons that survive the process of recombination. The col-
lision process becomes ineffective in coupling Tg to Tγ at
z ∼ 200. In the absence of external heating at z < 200 the
gas cools adiabatically with Tg ∝ (1+z)2while Tγ ∝ (1+z).
The spin temperature Ts is strongly coupled to Tg through
the collisional spin flipping process until z ∼ 70. The colli-
sional process is weak at lower redshifts, and Ts again ap-
1000 and the formation of
∼ 20 (Hogan & Rees
⋆Email: somnath@cts.iitkgp.ernet.in
† Email: saiyad@cts.iitkgp.ernet.in
proaches Tγ. This gives a range of redshifts where Ts < Tγ.
We then have a window in redshift 30 ≤ z ≤ 200, or equiva-
lently in frequency ν = 1420MHz/(1 + z) where the HI will
produce absorption features in the CMBR spectrum.
In a recent paper Loeb & Zaldarriaga (2003) propose
that observations of the angular fluctuations in the CMBR
brightness temperature Tb arising from the HI absorption
can be used to study the power spectrum of density fluctua-
tions at small scales to a level of accuracy far exceeding those
achievable by any other means. The enormous wealth of in-
formation arises from the fact that observations at differ-
ent frequencies which are sufficiently separated will provide
independent estimates of the power spectrum at the same
wave number k. These observations will probe the power
spectrum before the epoch of structure formation, and they
hold the possibility of revealing the entire primordial power
spectrum down to very small scales. In another recent paper
Gnedin & Shever (2003) have studied the linear fluctuations
in the 21 cm emission from the pre-reionization era. They
show that that it should be possible to detect this signal
against the foreground contaminations in the frequency do-
main. This signal is expected to constrain the equation of
state of the universe at high z.

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2 S. Bharadwaj and S. S. Ali
The CMBR brightness temperature is related to Ts
and the HI number density nH as Tb ∝ (1 − Tγ/Ts)nH.
Fluctuations in Tb arise from fluctuations in nH directly
and also through fluctuations in Ts which in turn arise
from fluctuations in nH. In calculating the fluctuations in
Ts, Loeb & Zaldarriaga (2003) consider only one process,
namely the change in the collision rate arising from fluctua-
tions in nH. Perturbations in nH will also produce perturba-
tion in Tg, which in turn will affect Ts. This effect has not
been taken into account in their work.
Density perturbations produce peculiar velocities. This
causes the frequency of the HI absorption features to be
shifted by the line of sight component of the peculiar ve-
locity. This effect will rearrange the HI absorption features
in frequency space where converging velocity patterns ap-
pear as enhancements in the HI number density and di-
verging velocity patterns appear as decrements in the HI
number density. It may be noted that this is the familiar
linear redshift space distortion (Kaiser effect, Kaiser 1987)
seen in galaxy redshift surveys. This effect has been stud-
ied by Bharadwaj, Nath and Sethi (2001) in the context of
cosmological HI emission from z ∼ 3.5.
It is important to identify and take into account all
possible contribution to the brightness temperature fluctu-
ations, if these are to be used to extract precise information
about the power spectrum and the equation of state of the
universe. In this paper we study two effects which will con-
tribute to brightness temperature fluctuations, namely (1.)
perturbations in the gas temperature produced by density
fluctuations, and (2.) the effect of redshift space distortions.
To the best of our knowledge, these effects have not been
included in earlier works.
We next present an outline of the paper. In Section
2. we discuss the processes involved in determining the
brightness temperature fluctuations and present the rele-
vant equations. In Section 3, we present our results and dis-
cuss their consequences. It may be noted that we have used
(Ωm0,ΩΛ0,Ωb0h2,h) = (0.3,0.7,0.02,0.7) whenever specific
values have been needed for the cosmological parameters.
2 CALCULATING THE BRIGHTNESS
TEMPERATURE FLUCTUATIONS.
We first consider the evolution of the gas temperature after
the recombination era (z ∼ 1000) when it becomes largely
neutral. This is governed by the equation
∂Tg
∂z
−2Tg
3nH
∂nH
∂z
=−9.88 × 10−8
Ωbh2
(1 + z)3/2(Tγ − Tg)(1)
The third term in the above equation represents the energy
transfer from the CMBR to the gas through the collisions
with the residual electrons (Peebles 1993). This terms tries
to maintain the gas temperature at the CMBR temperature
as the universe expands. The second term is the change in
Tg in adiabatic expansion. If the HI is uniformly distributed,
then nH ∝ (1+z)3and in the absence of the CMBR heating
we have Tg ∝ (1 + z)2. Collisions are able to maintain Tg =
Tγ = 2.73K(1+z) up to a redshift z ∼ 200 (Figure 1) after
which Tg ∝ (1 + z)2.
We next consider the evolution of the spin temperature
Ts which is defined through the relation
T
T
T
z
T [K ]
γ
s
g
10
100
1000
10100
1000
Figure 1. This shows the evolution of the CMBR temperature,
the gas temperature and the spin temperature as the universe
expands.
n1
n0
=g1
g0e−T⋆/Ts
(2)
where n1, n0 are the population densities and g1 = 3, g0 = 1
the spin degeneracy factors of the excited and the ground
states of the HI 21 cm transition. In this equation T⋆ =
hpνe/kB = 0.068K, where hP is the Planck’s constant, νe =
1420MHz is the frequency corresponding to the 21 cm line
and kBis Boltzmann’s constant. The evolution of the ground
state population density is governed by two processes, one
collisional and the other radiative
?∂
a
+n1A10+ (n1B10− n0B01)Iνe
where a(t) is the scale factor, C01 and C10 are the collisional
excitations and de-excitation rates of the hyperfine levels,
A10 is the Einstein spontaneous emission coefficient, B01
and B10are the Einstein B coefficients and Iνeis the specific
intensity of the background radiation at νe.
In the regime of our interest Ts, Tg, Tγ ≫ T⋆ and we
can use the approximation e−T⋆/T= 1 − (T⋆/T) through-
out. Also, the fact that in equilibrium the collisional pro-
cesses and the radiative process are separately balanced
gives us the relations C01 = 3(1 − T⋆/Tg)C10 and B01 =
3B10 = (3λ3
(Rybicki & Lightman 1979). The collisional de-excitation
rate can be written as C10 = (4/3)κ(1 − 0)nH where
the values of κ(1 − 0) is tabulated as a function of Tg
(Allison & Dalgarno 1969). Using these and equation (3) we
obtain an equation for the redshift evolution of Ts
∂t+ 3˙ a
?
n0
=n1C10− n0C01
(3)
e/2hpc)A10 where A10 = 2.85 × 10−15s−1
∂
∂z
?1
Ts
?
??
= −
4
H(z)(1 + z)×
?
1
Tg−
1
Ts
C10+
?
1
Tγ
−
1
Ts
?
Tγ
T⋆A10
?
(4)
where H(z) is the Hubble parameter. The collisional term
tries to set the spin temperature at the same value as the gas
temperature while the radiative term tries to set it at the
CMBR temperature, which process dominates being decided
by the rate coefficients. At high redshifts the collisional pro-
cess dominates and the spin temperature closely follows the
gas temperature. At lower redshifts nH falls substantially,
the collisional process looses out to the radiative process

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The CMBR fluctuations from HI perturbations prior to reionization3
g(z)
s(z)
Z
g(z),s(z)
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
10 100
1000
Figure 2. This shows the evolution of the functions g(z) and s(z)
defined in the text.
and the spin temperature approaches the CMBR tempera-
ture. Figure 1 shows the evolution of the spin temperature
as the universe expands.
We next shift our attention to the effect of HI density
perturbations ∆H(x,z) = (nH(x,z) − ¯
will produce fluctuations in the gas temperature. If the gas
were undergoing adiabatic expansion, the fluctuations in the
gas temperature ∆g(x,z) = (Tg(x,z)−¯
be related to ∆H through ∆g = (2/3)∆H. This will get mod-
ified because of the energy that is pumped into the gas from
the CMBR and ∆g = 0 during the era when Tg = Tγ. We
define a function g(z) =
∂∆H. such that ∆g(z) = g ∆H(z).
Using this in equation (1) we obtain
nH(z))/ ¯nH(z). These
Tg(z))/¯
Tg(z) would
∂∆g
dg
dz=9.88 × 10−8Tγ
Ωbh2Tg
(1 + z)3/2g + (2
3− g)
1
∆H
∂∆H
∂z
(5)
The first term in RHS arises from the coupling of the gas
to the CMBR and it tries to set g(z) → 0 while the second
term corresponds to adiabatic expansion and it tries to make
g(z) → 2/3. The quantity g(z) is expected to evolve from
g(z) = 0 to g(z) = 2/3 as the universe expands and the
contribution of the heat pumped into the gas goes down.
An interesting feature is that g(z) depends on the
growth rate of density fluctuations. For example, it follows
from equation (5) that a static density perturbation which
does not evolve in time will not produce fluctuations in
the gas temperature. Here we assume that ∆H follows the
dark matter perturbation and grows as ∆H ∝ a(z). The
result of integrating equation (5) is shown in figure 2. We
see that g(z) ∼ 0.3 in the redshift range of our interest
(30 ≤ z ≤ 100).
We finally come to the fluctuations in the spin tem-
perature ∆s(x,z) = (Ts(x,z) −¯
by density perturbations. From equation (4) we see that
fluctuations in Ts can arise from changes in Tg and from
changes in the collision rate. The changes in collision rate
C10 = (3/4)κ(1 − 0)nH will come about directly through
changes in nH and also through changes in Tg which will
affect the value of κ(1 − 0). Taking into account both these
effect we have ∆C10 = (1+(2/3)dlnκ/dlnTg)C10∆H. Defin-
ing a function s(z) =
∂∆H. such that ∆s(z) = s∆H(z) and
Ts(z))/¯
Ts(z) produced
∂∆s
using equation (4) we obtain
ds
dz= −s1
∆H
∂∆H
∂z
+
4
H(z)(1 + z)
??
Ts
Tg(s − g)
+
?Ts
Tg− 1
??
1 +dlnκ
dlnTg
??
C10+ sTs
T⋆A10
?
(6)
Here again, the evolution of s(z), like that of g(z), de-
pends on the time evolution the density fluctuations. Fig-
ure 2 shows the evolution of s(z) under the assumption
∆H ∝ a(z). We find that s(z) > 0 for z > 90 ie. a posi-
tive density perturbation causes the spin temperature to go
up, and the effect is opposite at z < 90.
During the era when Ts < Tγ the HI along a line of
sight n reduces the CMBR brightness temperature at the
frequency ν by an amount
Tb(n,ν) = (Ts− Tγ)τ/(1 + z).
Here τ is the optical depth of the 21 cm HI absorption given
by
(7)
τ =3nHhP c2A10a2(z)
32π kBTsνe
|∂r
∂ν| (8)
where r is the comoving distance to the HI whose 21 cm
absorption is redshifted to ν.
We are interested in the angular fluctuations of the
brightness temperature Tb(n,ν). HI density fluctuations will
produce fluctuations in Tb(n,ν) through the fluctuations in
the spin temperature discussed earlier. Density fluctuations
will also directly affect Tb through variations in the opti-
cal depth which we now calculate. The relation between the
comoving distance r and the frequency ν is given by
r =
?1
νe(1−v/c)
ν
cda
a2H(a)
(9)
where v is the line of sight component of the peculiar velocity
of the HI which produces the absorption. Density perturba-
tions will, in general, be accompanied by velocity perturba-
tions and these will move around the HI absorption features
in frequency. Here we assume that the HI traces the dark
matter and that we can use linear perturbation theory to
relate the peculiar velocity to the density perturbations. In-
corporating the effect of both the density fluctuations and
the peculiar velocity we have
τ =
3 ¯nHhP c3A10
32π kBTsν2
eH(z)
?
1 + ∆H−
1
H(z)a(z)
∂v
∂r
?
.(10)
Here we have dropped terms of order v/c in the final expres-
sion. Also, we have retained terms only to linear order in v.
There is also the effect of our own motion which we have
not included. These effects are not expected to be impor-
tant. The term involving the derivative of the peculiar ve-
locity is the dominant effect, particularly at the small scales
of interest here.
Combining the effects of the fluctuations in the optical
depth and in the spin temperature we can write the fluctu-
ations in the CMBR brightness temperature as
δTb(n,ν) =¯T
??
1 −Tγ
Ts
??
∆H−
1
H a
∂v
∂r
?
+Tγ
Tss∆H
?
(11)
where

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4 S. Bharadwaj and S. S. Ali
¯T = 2.67 × 10−3K
Ωbh2
0.02
(1 + z)1/2
Ω1/2
m0h
(12)
It is convenient to express this in Fourier space where
∆H(x,z) =
?
d3k
(2π)3e−ik·x∆(k,z) (13)
and the Fourier transform of the peculiar velocity is given
by v(k,z) = −iH(z)a(z)k∆(k,z)/k2. Using this we express
the fluctuations in brightness temperature as
δTb(n,ν)=
¯T
?
1 −Tγ
d3k
(2π)3e−ikrµ∆(k,z) ×
??1 + µ2?
??
Ts
+Tγ
Tss
?
(14)
where µ is the cosine of the angle between the comoving
wave vector k and the line of sight n.
We now calculate the angular power spectrum of
the brightness temperature fluctuations resulting from the
density fluctuations ∆(k,z). The statistical properties of
∆(k,z) are specified through the 3D power spectrum de-
fined as
?∆(k,z)∆(k
where ?...? denotes ensemble average and δD() is the Dirac
delta function.
The angular power spectrum is calculated by decompos-
ing the angular dependence of δTb into spherical harmonics
with expansion coefficients alm(ν) and using these to cal-
culate the angular power spectrum Cl(ν) = ?| alm |2?. The
angular power spectrum can be expressed in terms of the
3D power spectrum as
′,z)? = (2π)3δ3
D(k − k
′)P(k,z) (15)
Cl(ν) = 4π¯T2
?
d3k
(2π)3
P(k,z)
??
1 −Tγ
?2
Ts
?
Jl(kr)
+Tγ
Tssjl(kr) (16)
where jl(kr) are the spherical Bessel functions and
Jl(kr)=
?
−l(l − 1)
4l2− 1jl−2(kr) +2(3l2+ 3l − 2)
(l + 2)(l + 1)
(2l + 1)(2l + 3)jl+2(kr)
4l2+ 4l − 3
?
jl(kr)
−
(17)
3 RESULTS AND DISCUSSION.
In this paper we have investigated in detail the CMBR fluc-
tuations produced by HI priot to the epoch of reionization.
As proposed by Loeb and Zaldarriaga (2003), this holds the
promise of allowing the power-spectrum of density fluctua-
tions to be probed to a high level of precision.
HI density perturbations produce fluctuations in the
decrement of the CMBR brightness temperature by two
means (1.) through fluctuations in the optical depth, and
(2.) through fluctuations in the spin temperature. The ef-
fect of changes in the optical depth in response to a pos-
itive density perturbation (curve A of Fig. 3) is such that
it reduces Tb and enhances the decrement in the brightness
temperature. This effect is maximum at z ∼ 80. This effect
is enhanced by peculiar velocities.
The change in brightness temperature produced by HI
B
C
A
z
δΤbΤ
|
∆Η
D
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
10 100
1000
Figure 3. This shows how the CMBR brightness temperature
fluctuations (in units of¯T) respond to HI density perturbations.
A. shows (1 − Tγ/Ts) which is how it responds through changes
in the optical depth, ignoring the effect of peculiar velocities. Pe-
culiar velocities will enhance this effect. B. shows sTγ/Ts which
is how it responds through changes in the spin temperature. C.
shows the total response for µ2= 2/3. D is the same as B without
the gas temperature fluctuations.
density perturbations through changes in the spin tempera-
ture varies with z (curve B of Fig. 3). Density perturbations
increase the spin temperature and the brightness temper-
ature in the redshift range z > 100. Here the collisional
process is very efficient and Ts closely follows Tg. A positive
density perturbation increases Tg which causes Tsto also in-
crease. The effect of density perturbations on the brightness
temperature acting through changes in the optical depth and
through the spin temperature have opposite signs. The effect
of changes in the optical depth is larger and the brightness
temperature decrement is enhanced by a positive density
perturbation (curve C of Fig. 3).
In the redshift range z < 100 positive density per-
turbations lower the spin temperature which enhance the
brightness temperature decrement. In this regime the colli-
sional process slowly looses out to the radiative process and
Ts → Tγ. A positive density perturbation enhances the colli-
sion rate which pulls the spin temperature down toward the
gas temperature. The two processes which contribute toward
brightness temperature fluctuations both act to enhance the
decrement. Curve C of Fig. 3 shows the combined effect of
both these processes for the value µ2= 2/3. We find that
the response of the brightness temperature to density per-
turbations peaks at z ∼ 55. This is somewhat smaller than
the value obtained by Loeb & Zaldarriaga (2003). Curve D
of Figure 3 shows the contribution to brightness tempera-
ture fluctuations arising from the spin temperature if the
effect of density perturbations on the gas temperature is
not taken into account (eg. Loeb & Zaldarriaga 2003). We
find that including the gas temperature makes a significant
change, particularly at z > 100 where there is a qualitative
difference between curves B and D.
We have calculated the angular power spectrum Cl(ν) of
the brightness temperature fluctuations for the COBE nor-
malized ΛCDM model (Bunn & White 1996). The power
spectrum has been suppressed beyond the arbitrarily cho-
sen value k = 14hMpc−1using a Gaussian cut-off. Our re-
sults are in qualitative agreement with Loeb & Zaldarriaga
(2003). We find that the signal peaks at z ∼ 50 (Fig. 4)

Page 5

The CMBR fluctuations from HI perturbations prior to reionization5
[l (l+1) Cl/ 2 π]1/2[ mK]
z=50
z=100
z=150
with redshift distortions
without redshift distortions
l
.
1e−05
0.0001
0.001
0.01
0.1
1
10
1 10 1001000 10000 1000001e+061e+07
Figure 4. This shows the angular power spectrum of the CMBR
brightness fluctuations at various redshifts with and without the
effects of peculiar velocities.
where the product of the growing mode of density perturba-
tions and the response of brightness temperature to density
perturbations is maximum.
To gauge the effect of peculiar velocities, we have calcu-
lated Cl(ν) without incorporating this effect. This is easily
done by replacing Jl(kr) with jl(kr) in equation (16). The
results are shown in Fig. 4. We find that the peculiar veloc-
ities increase√Cl by more than 50%.
We have also quantified the effect of gas temperature
fluctuations. We find that for z < 100 the values of
are ∼ 10% lower if the gas temperature is not taken into
account, and the effect is reversed at z > 100 where√Cl is
∼ 30% higher.
The ability to probe the dark matter power spectrum
using the Cl(ν)s will be restricted to scales k < kJ where
kJ is the Jeans wave number. This has a nearly constant
value ∼ 500hMpc−1in the redshift range of interest. The
HI power spectrum on scales smaller than the Jeans length
scale is interesting in its own right. The HI perturbations will
undergo acoustic oscillations on these scales. The spin tem-
perature fluctuations and the peculiar velocities produced by
density perturbations will be quite different from the situa-
tion considered here. On scales slightly larger than the Jeans
lengthscale (2π/kJ) the density fluctuations are mildly non-
linear with rms. values in the range .7 > σ > .1, and the
Z’eldovich approximation may give a better description (eg.
Hui & Gnedin 1997).
The low frequency cut-off imposed by the earth’s iono-
sphere restricts observations to frequencies more than ∼
10 − 20MHz. Extracting the HI signal from the contam-
inations arising from the galactic and extragalactic fore-
grounds is going to be a big challenge. The foregrounds are
mostly continuum sources whose contribution varies slowly
with frequency. It will be necessary to combine both the
angular fluctuations and the frequency domain properties
of the CMBR brightness temperature fluctuations in order
to detect it (eg. Shaver et al. 1999, Di Matteo et al. 2002,
Oh & Mack 2003, Di Matteo et al. 2004).
√Cl
ACKNOWLEDGMENTS
SSA would like to acknowledge financial support through
a junior research fellowship of the Council of Scientific and
Industrial Research (CSIR), India.
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