J. A. Rubiño-Martín1,2 - C. Hernández-Monteagudo1,
- R. A. Sunyaev1,3
1 - Max-Planck-Institut für Astrophysik,
Karl-Schwarzschild-Str.1, Postfach 1317, 85741 Garching, Germany
2 -
Instituto de Astrofísica de Canarias, 38200 La Laguna,
Tenerife, Spain
3 -
Space Research Institute (IKI), Russian Academy of Sciences,
Moscow, Russia
Received 25 February 2005 / Accepted 24 March 2005
Abstract
We explore the imprint of the cosmological hydrogen recombination
lines on the power spectrum of the cosmic microwave background (CMB).
In particular, we focus on the three strongest lines for
the Balmer, Paschen and Brackett series of hydrogen.
We expect changes in the angular
power spectrum due to these lines of about
for the H
line, being maximum at small angular scales
(
).
The morphology of the signal is very rich. It leads to relatively
narrow spectral features (
),
with several regions in the power spectrum showing a characteristic change
of sign of the effect as we probe different redshifts or different
multipoles by measuring the power spectrum at different frequencies.
It also has a very peculiar
dependence on the multipole scale, connected with the details of the
transfer function at the epoch of scattering.
In order to compute the optical depths for these transitions,
we have numerically evolved the populations of the levels of the
hydrogen atom during recombination, simultaneously treating the
evolution of helium. For the hydrogen atom, we follow
each angular momentum state separately, up to the level n=10.
Foregrounds and other frequency dependent contaminants such as Rayleigh
scattering may be a important limitation for these measurements,
although the peculiar frequency and angular dependences of the effect
that we are discussing might permit us to separate it.
Detection of this signal using future narrow-band spectral observations can
give information about the details of how the cosmic recombination
proceeds, and how Silk damping operates during recombination.
Key words: cosmic microwave background - cosmology: theory - early Universe - atomic processes
Existing and planned experiments devoted to the measurements of Cosmic Microwave Background (CMB) angular fluctuations will reach an unprecedent high sensitivity of measurements. Planck, SPT, ACT and QUEST will achieve sensitivities several orders of magnitude higher than the amplitude of the acoustic peaks which were the dream of theorists in the seventies (Peebles & Yu 1970; Doroshkevich et al. 1978; Sunyaev & Zeldovich 1970), and are observed with good precision now by Boomerang, Maxima, DASI, VSA, CBI, and more recently with very high accuracy by WMAP (Bennett et al. 2003).
The experimental progress demonstrates that we are entering the era of "precision cosmology'', and many of the effects that were obvious for theoreticians but of small amplitude become now accessible to experimentalists.
One of such problems is the direct detection of the hydrogen or helium lines from the
epoch of recombination of hydrogen in the Universe at redshift
.
In Russia, all the study of cosmological recombination grew up
from the question of Vladimir Kurt: "where are the Ly-
line photons
from recombination in the Universe?''. This question forced
Zeldovich et al. (1968, hereafter ZKS68) to study the detailed physics of recombination, and to show that two-photon decay of 2s level of hydrogen is more
important than the escape of redshifted Ly-
photons due
to expansion of the Universe in determining the rate at which recombination occurs.
It was shown that there are strong
distortions of the CMB spectrum due to two-photon decay, but
unfortunately these distortions are practically unobservable because they lie in
the distant Wien region of the CMB spectrum.
Peebles (1968, hereafter P68), and
Seager et al. (1999,2000)
came to the same conclusion. Dubrovich (1975) proposed to look
for CMB spectral distortions due to transitions between higher
levels (Balmer, Paschen and higher series), which should be broadened due to the delayed
recombination, and many papers were devoted to the computation of
these very small but extremely interesting effects
(Grachev & Dubrovich 1991; dell'Antonio & Rybicki 1993; Dubrovich & Shakhvorostova 2004).
Although several computations of the intensity of these
lines gave contradictory results (see e.g. dell'Antonio & Rybicki 1993 and
Dubrovich & Shakhvorostova 2004), it is obvious that the corresponding distortions of the CMB spectrum are extremely weak, and that
does not exceeds the level of 10-7,
which is two orders of magnitude below the sensitivity of COBE-FIRAS
experiment, but certainly will be the goal of future more precise experiments.
Another physical process with information about
the epoch of recombination is Rayleigh scattering on neutral hydrogen
and helium atoms (Yu et al. 2001).
This process, due to the characteristic dependence of the scattering
cross-section on wavelength (
),
has a very strong frequency dependence of the power spectrum,
.
Unfortunately, this frequency dependence is rather similar to
the spectrum of dust emission from local dust and bright extragalactic
star-forming galaxies, and the amplitude of the effect becomes
important in frequencies where the dust emission is significant
(
GHz).
This will make a rather difficult task to separate it from nearby
and bright foreground dust contribution.
In this paper, we concentrate on the detection of
hydrogen lines, and we propose a different method to
detect the consequences of the overpopulated levels
of hydrogen atom at the epoch of recombination.
We propose to look for frequency dependent angular fluctuations
of the CMB in different angular scales and at frequencies in the
vicinity of the redshifted Balmer (and higher series) lines.
Here, we estimate how the small optical depths connected with these
transitions influences the angular distribution of the CMB,
arising mainly due to Thomson scattering on free electrons.
We show that one should
expect changes in the angular power spectrum due to these lines
of about 0.3
,
being maximum
at small angular scales (
).
These numbers are at the level of the planned sensitivity of the Planck
satellite but for large angular scales,
and certainly should be achieved by future
ground-based instruments looking for polarization of the CMB, and
future space missions like CMBPol. It is important to
mention that present day technology permits one to achieve sensitivities
at least 50 times better than that of Planck (Church 2002).
The maximum signal that we are discussing corresponds to the
redshifted H
recombination line, which unfortunately is
located in a frequency region where the contribution
from dust emission is very important, and where the Rayleigh scattering
is 100 times larger in
(10 times in temperature).
![]() |
Figure 2:
Distortion of the power spectrum due to scattering in the
H![]() |
Coherent scattering in the H
line produces a
signal in the angular distribution of
the CMB, whereas spectral distortions in the same line are
close to
10-8-10-7. Currently,
experimentalists put their main effort in observing angular
fluctuations. Therefore, the signal under discussion in this paper might be
observed sooner or later.
This effect is small, but it leads to relatively narrow spectral
features in the power spectrum (
).
In addition, there are several regions in the angular power spectrum
at high multipoles where the sign
of
(=
)
connected with this process depends on the
wavelength, i.e. it changes from positive to negative when the
frequency is shifted from 350 GHz to 450 GHz (Fig. 1).
The effect also has a very peculiar dependence on
at all
frequencies (Fig. 2),
and these peculiarities in frequency and angular scale
dependence might permit future observers to separate coherent scattering on
during the epoch of recombination from foregrounds and even
from the cosmic variance of the effect of Rayleigh scattering.
The discovery of these spectral features in the CMB angular distribution will permit us to measure directly the tail of the recombination process and its position with good precision, and may give additional information about cosmological parameters of the Universe and their evolution with time.
This may also give direct spectroscopic evidence of recombination.
If Rayleigh scattering probes the existence of neutral hydrogen
immediately after the peak of the
visibility function, the profile of the Balmer lines features in the power spectrum gives us the possibility to check the dynamics of recombination when the electron density changes from
to 10-2, and correspondingly,
the optical depth in H
and the population of the second level are traced.
The characteristic physical length producing fluctuations in H
in that
epoch corresponds to smaller scales than those for Rayleigh scattering.
It is also important that for the hydrogen lines,
each observed frequency corresponds to a given
redshift z, but the "picture'' from Rayleigh scattering is the same
for all frequencies (there is only a change in amplitude).
Thus, when the line is placed inside recombination, the shape of the angular power spectrum for these two effects differs strongly.
Computations of the
connected with recombinational
lines are direct, because detailed estimates of the
recombination process show (SSS99)
that the assumption of ZKS68 about the Saha equilibrium
of all levels above level two is valid with good precision.
This gives us the possibility to find the optical depth in
each line of interest, and include it as an additional
opacity coefficient in the CMBFast code (Seljak & Zaldarriaga 1996), as it was done
for the lines in the case of resonant scattering
by Zaldarriaga & Loeb (2002) (lithium doublet), and
Basu et al. (2004, hereafter BHMS04) for fine structure lines of neutral atoms and ions of carbon, nitrogen and oxygen.
Let us mention here that the computations of the line
intensities (i.e. spectral distortions of CMB) require the
detailed calculation of tiny deviations from Saha equilibrium,
which are connected to the process of recombination
controlled by 2s-level decay.
For our purposes, it is enough to use the equilibrium distribution
because the generation of new angular fluctuations
is connected with just re-distribution of radiation over angles, so
it has no relation to the difference between excitation
temperature and color temperature of radiation.
Optical depths in all lines, including Balmer lines, is very small
(e.g.
10-5 for H
line at z=1000).
But as it was shown by Hernández-Monteagudo & Sunyaev (2005, hereafter HMS05) and BHMS04,
correlation effects with the existing radiation fluctuation field amplifies the effect
in such a way that
s are effectively proportional to
,
instead
of
.
It should be noted that the signal that we are discussing is smaller
than the cosmic variance level, but as is explained by HMS05 and BHMS04,
multi-frequency observations permit one to avoid the limit imposed by the
cosmic variance associated with the intrinsic CMB fluctuations, and
reveal signals below this threshold.
Finally, it is important to mention that these detections strongly rely on a accurate cross-calibration between different channels of a given experiment. However, there is a possibility to use the thermal SZ effect (Sunyaev & Zeldovich 1972) on clusters of galaxies and sum of blackbodies approach for such a calibration. Both methods are discussed by Chluba & Sunyaev (2004).
The outline of the paper is as follows. In Sect. 2, we review the
basic equations which describe the cosmic recombination process, and
we present a code that we have developed to
compute the optical depths in the lines for the required transitions.
In Sect. 3, we show how to derive the
quantities
for a given line, and present our results for the considered transitions.
Section 4 presents a discussion about the foreground contamination and
the amplitude of the effect compared to Rayleigh scattering.
Conclusions are given in Sect. 5.
In this section, we review the basic equations describing the process of cosmic recombination of hydrogen and helium, and we show how to use them to infer the optical depths for the recombination lines. Since the first computations by ZKS68 and P68, many refinements have been introduced (for a review, see SSS00). SSS00 has the most detailed calculations to date, including 300 levels for the hydrogen atom, 200 for HeI and 100 for HeII. When compared with the standard "effective three-level computation'', their calculation results in a roughly 10% change in the electron fraction at low redshift for most of the cosmological models, plus a delay in HeI recombination. All these results were incorporated in the RECFAST code (SSS99), which consists in a modification of the effective three-level model to reproduce the new values.
For this paper, we are interested in computing the optical depths associated with the hydrogen recombination lines of the Balmer, Paschen and Brackett series. We will not discuss here the helium lines (see Dubrovich & Stolyarov 1997, for a discussion on the spectrum of primordial HeI and HeII recombination lines). In order to obtain the optical depths, we need to know the populations for all levels in the hydrogen atom which contribute to the transitions we want to study. We will do this following two different approaches.
On the one hand (method I), we will follow SSS00 and we will evolve the level
populations for a hydrogen atom with 10 levels, separating levels
both by principal quantum number n and angular momentum l.
In method II, we will adopt the ionisation fraction from
RECFAST and compute, using the Saha equation relative to the continuum,
the population of the excited levels (e.g. Liubarskii & Sunyaev 1983).
This method is a valid approximation for those
levels above the second one and for redshifts ,
but
as we will see, this is indeed our range of interest because at lower
redshifts the visibility function is very small.
The detailed formalism and the equations describing the evolution of the population of the levels of hydrogen and helium atoms with cosmic time is presented in SSS00. Here, we enumerate the assumptions of our work and the differences to that paper.
Values for the physical constants for hydrogen and helium
were taken from SSS99 and SSS00.
The values for the oscillator strengths and the corresponding
Einstein coefficients for instantaneous emission were computed following
Green et al. (1957).
The value for the two photon decay transition
is essential, given that this is the dominant mechanism driving the
cosmological recombination. We adopt here the latest
value,
,
from Goldman (1989).
The photoionization/recombination rates were computed in two different ways. The first one was by using the photoionization cross-sections from Storey & Hummer (1991); Hummer (1994). The second one uses the analytical expressions from Burgess (1958), which are valid for small values of the energy of the ejected electron. In both cases, radiative recombination rates included spontaneous and induced transitions. Our final results were obtained using the second method, which is computationally faster than the first one. Nevertheless, we checked that both methods give similar results for the populations of the levels.
Once the populations of all levels are obtained at all redshifts,
the optical depth associated with any permitted
transition
connecting the levels i=(n,l) and
j=(m,l') (with m>n) can be computed using the Sobolev formula as
In Fig. 3 we show our result for the
electron density evolution (
)
as a function of
redshift
,
compared with the standard calculation using RECFAST. These
results are consistent with those obtained by SSS00
when they use n=10 levels, but it should be noted that in our calculation
we do a detailed follow-up of the population of the different angular momentum states.
When more levels are included in the computation, SSS00 show that the
residual electron density at low redshifts becomes smaller,
and for
it converges as the atom becomes complete in terms of energy levels.
However, for the purposes of this paper, it is enough to consider
a 10-level hydrogen atom to achieve good precision in the region of interest.
![]() |
Figure 3:
Multilevel hydrogen recombination as computed in this paper
(using 10 levels for quantum number n, and following independently all the states
of angular momentum), compared with the standard "effective three level''
computation from RECFAST. It is shown ![]() ![]() ![]() ![]() |
As suggested by ZKS68, and confirmed with the computations of SSS00,
the approximation of equilibrium of all levels above the second one
yields very good results, and deviations greater than 10% are only
found at redshifts .
Thus, we can derive (to a good approximation) the population of the
level in two steps. We first use the standard RECFAST computation, and
derive the evolution of the electron density. From here, and using the
Saha equation relative to the continuum, we can derive
the population of all levels above the second one, and
can compute all the optical depths for Balmer lines and
higher series. It should be noted that this approximation
fails for redshifts below
because most of the atoms
have recombined and the transition rates are not enough to keep
equilibrium between high levels.
Transition | m |
![]() ![]() |
fnm |
![]() |
![]() |
Balmer lines (n=2) | |||||
H![]() |
3 | 6562.8 | 0.6407 | 456.3 | 4.53 ![]() |
H![]() |
4 | 4861.3 | 0.1193 | 616.1 | 6.14 ![]() |
H![]() |
5 | 4340.5 | 0.0447 | 690.0 | 2.09 ![]() |
Paschen lines (n=3) | |||||
P![]() |
4 | 18751.0 | 0.8421 | 159.7 | 1.17 ![]() |
P![]() |
5 | 12818.1 | 0.1506 | 233.6 | 1.47 ![]() |
P![]() |
6 | 10938.1 | 0.0558 | 273.8 | 4.75 ![]() |
Brackett lines (n=4) | |||||
B![]() |
5 | 40512.0 | 1.0377 | 73.9 | 2.57 ![]() |
B![]() |
6 | 26252.0 | 0.1793 | 114.1 | 3.42 ![]() |
B![]() |
7 | 21655.0 | 0.0655 | 138.3 | 1.09 ![]() |
Thus, the population of the level n is derived in this second method as
Although we have computed many more transitions, in this paper
we shall concentrate on those lines whose expected contribution
is largest. Thus, we investigate here the
three first transitions (labelled as
)
for the Balmer
(n=2), Paschen (n=3) and Brackett (n=4) series.
Table 1 presents the wavelengths and oscillator strengths
for these lines. For illustration, we show in the last two columns the
redshifted frequency and the
optical depth in the line evaluated at redshift z=1000, and
for a cosmological model with parameter values taken
from Bennett et al. (2003),
i.e.
,
and reduced Hubble constant h=0.71.
Throughout this paper we have used these values, unless otherwise stated.
Figure 4 shows the optical depth in the lines ()
derived with our method I, and for this cosmological model.
Using the method II, we obtain these same values for the optical
depths, but only for redshifts above
.
To illustrate this point, we also display in Fig. 4
the optical depth for the H
line using the method II (dotted line).
Method II reproduces the non-equilibrium computation above
,
but fails below this
point, showing a divergent behaviour. The reason for this
was pointed out above. At these redshifts, the radiation field
has not enough photons to maintain the equilibrium of the second
level with the continuum. The photoionization rates become smaller than
the photorecombination rates, and the Saha equilibrium between the second
level and the continuum is no longer valid (e.g. ZKS68).
Moreover, the populations of
and
levels start to show strong departures
from their equilibrium ratio.
Given that the cosmological recombination is proceeding much more slowly
than expected from a Saha recombination,
method II (which uses the Saha equation relative to a continuum
level which has been computed using the effective
three level calculation) predicts very high values for the
population of the second level in this redshift range.
This is the reason for the divergent behaviour at
of the H
optical depth computed with method II.
Although we have used the exact (non-equilibrium) computation for all the
results presented in this paper, we wanted to show that using a very simple
approximation (method II) it is possible to correctly estimate the
values for the optical depths in the range of interest (as we show below,
the peak of the effect we are investigating occurs at
).
In Fig. 4, we finally note that for Balmer transitions, the shape of the
optical depth as a function of frequency is similar for all lines.
This can be easily understood because the population of the
second level is much larger than the population of higher levels, and
thus the optical depth is directly proportional to this population (
)
for all Balmer lines.
Figure 5 shows the optical depth in the H line, together
with the Thomson visibility function (Sunyaev & Zeldovich 1970),
,
and the
function
.
Here,
is the Thomson optical depth
and
is the conformal time. As we can see from this figure,
the maximum optical depth for this transition is reached beyond the
peak of the visibility function, around
,
but there
the Thomson optical depth is very large.
This is why the maximum contribution to the
effect we are discussing comes from lower redshifts (
).
As we show in the next section, the generation of new anisotropies is conected
to the term
,
also shown in
Fig. 5.
The imprint of line transitions on the CMB spectrum has been examined by Zaldarriaga & Loeb (2002) for the case of lithium recombination, and for other ions and atoms in BHMS04. We extend these works here and we consider the case in which the coherent scattering occurs inside recombination.
The drag force induced on CMB photons by the scattering in the hydrogen recombination lines was already discussed by Loeb (2001). They showed that due to the low population of the excited levels, the characteristic time over which the peculiar velocity of the gas is damped due to the drag force on the hydrogen atoms is much longer than the Hubble time at that epoch. We repeated this computation and found that it is at least five orders of magnitude longer. Thus the drag force can be safely neglected when computing the effect of coherent scattering in these lines on the power spectrum of the CMB, as it was done in Zaldarriaga & Loeb (2002) and BHMS04.
Before hydrogen is significantly recombined in the Universe,
the optical depth for
electron scattering is very high, so every scattering leads to a
broadening of the line. For z=1000, the thermal (Doppler) width of
the lines is close to
10-5.
Subsequent electron scattering of these photons might increase this
width up to
10-3.
The characteristic width of recombination in redshift space,
,
is
much bigger than this electron broadening (from Fig. 5,
the visibility function has a width of
,
so
).
On the other hand, the optical depths in the lines
under discussion are also changing slowly. The redshift
interval
in which
changes significantly (i.e.
)
is about 0.07 for z=1000. Thus, we expect that both Doppler and
electron broadening of the lines will not change our effect
significantly. We have checked this point by repeating the computations of this
paper for different widths for the lines.
The opacity in each line (
)
is computed as
One important consequence that can be determined from here is that every
frequency observed today corresponds to a given redshift z.
With electron scattering, we are averaging all the effects well before
the peak of the visibility function, and we are losing information about the
redshift in which a given part of the signal was produced.
However, the study of these lines will permit us to check
the velocities and optical depths at any stage of recombination.
In order to do this, it will be necessary to optimise the observing frequencies
of the detectors, and to have observing bandwidths narrower than
the widths of the features (we will see below that
typical widths are
).
Unfortunately, present day and planned experiments (like Planck)
have widths of the channels broader than this,
and even broader than the width of recombination,
so these effects are averaged inside the observing bandwidths.
But in principle, future ground based experiments or experiments like
CMBPol might be adapted to the width and frequency dependence of the features that we are discussing, and it could be possible to trace using this
tomography the overall behavior of recombination.
In this subsection, we present the equations and the method of computation of the effect of coherent scattering in hydrogen lines. To perform the computations for this paper, we have used the code from BHMS04, which was a modification of CMBFAST including the presence of a resonant line at a given redshift. We will follow here their notation.
In the conformal Newtonian gauge, the Boltzmann equation for the
evolution of the k-mode of the photon
temperature fluctuation can be formally integrated to give
(Seljak & Zaldarriaga 1996)
Using this modification inside the CMBFAST code, and neglecting the drag
force induced by this process on the CMB photons, it is possible to compute the effect of the coherent scattering on the power spectrum. The important point for us is that, as shown
by HMS05, the effect of lines on the CMB spectrum is
amplified due to correlation with the existing radiation field.
The presence of coherent scattering in lines produces an angular
redistribution of the CMB photons. For the lines considered in
BHMS04, the net effect is generation of anisotropies at large
scales due to Doppler motion of scattering atoms, and suppression of
power at small scales. Summarizing, the amplification effect consists in that
the change in the power spectrum (
)
due to a given transition
is proportional to
(and not to
)
for small values
of the optical depths. Thus, in the optically thin limit we have
Equation (5) can be used in our problem, in which the scattering line
is embedded inside recombination, although there are some differences
to the case in which the line is placed between recombination and us.
As was shown in BHMS04, the generation and blurring
terms tend to cancel each other at a multipole value
,
with
,
and
the conformal times at the
coherent scattering, at recombination and the present day, respectively.
From this, one can expect that if coherent scattering takes place not
far from recombination, both generation and blurring will cancel
over a wider range of multipoles.
We illustrate this in Fig. 6, where we show the
dependence of the linear term (
)
for four different cases of a hypothetical line placed at redshifts
z=850, 950, 1050 and 1150.
Thus, in other words, the presence of this strong decrease in the power
at angular scales
is
direct evidence that the line is located at the epoch of recombination in the Universe.
Unfortunately, because of this decrease, the detection at large
angular scales of this signal will be more difficult.
Only at redshifts lower than the redshift of recombination is the
blurring term dominant at high multipoles: when the line
is well within the peak of the Thomson visibility function, both blurring and generation terms
cancel each other, at least to a level of a few percent.
This is seen in Fig. 6, so for redshifts in the
tail of the visibility function (z=850) the linear term is equal
to
at high multipoles. But if we move to higher redshifts,
this behavior disappears.
For illustrational purposes, we discuss how the visibility function is modified when we
consider coherent scattering in the hydrogen lines.
In the optically thin limit, we can write to first order in the
optical depth ()
that
![]() |
= | ![]() |
|
![]() |
![]() |
(6) |
![]() |
Figure 7:
Correction to the visibility function in the presence of
coherent scattering in the H![]() ![]() ![]() |
Line |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
[GHz] | [GHz] | [
![]() |
|||||
H![]() |
873 | 140 | 1010 | 452 | 56 | 0.28 |
![]() |
H![]() |
873 | 140 | 1010 | 610 | 76 | 0.10 |
![]() |
H![]() |
873 | 140 | 1010 | 683 | 85 | 0.06 |
![]() |
P![]() |
888 | 121 | 1050 | 152 | 16 |
![]() |
![]() |
P![]() |
888 | 121 | 1050 | 223 | 24 |
![]() |
![]() |
P![]() |
888 | 121 | 1050 | 261 | 28 |
![]() |
![]() |
B![]() |
891 | 116 | 1060 | 70 | 7 |
![]() |
![]() |
B![]() |
891 | 116 | 1060 | 108 | 11 |
![]() |
![]() |
B![]() |
891 | 116 | 1060 | 130 | 13 |
![]() |
![]() |
Our main result for the nine transitions under discussion
can be summarized in Table 2, where we present the angular scale and the
redshift at which we obtain the maximum signal for each of the
lines. From these values, we see that the best angular scale to look for this
effect is placed in the region of the third acoustic peak, and that
the maximum signal comes from redshifts
.
These
values lie close to the peak of the visibility function, but
in the low-redshift tail. We also present the full widths of the regions around the maxima where the
signal is within 70% of the peak value, both in multipole (
)
and frequency (
). We note that
is of the order of 10%, which is
hundreds of times larger than the Doppler broadening of the lines.
This will help in the detection of these features. We also show that the width of the feature in multipole space is of the order of 140, so a full sky coverage is not necessary to detect
the signal.
The last two columns in Table 2 show the amplitude of
the signals at these maxima, both in temperature and the fractional
change in temperature with respect to the intrinsic CMB power
spectrum. Although the values for
are small, the present day sensitivity for
the detection of angular fluctuations is better than that for spectral
measurements, so in principle these features should be easier to
detect than the distortions of the spectrum.
We now investigate in more detail the angular and frequency
dependences of this effect. We will concentrate here on the H
line, which gives the strongest signal. Predictions for the other transitions can be derived in a similar way.
Figure 8 shows the angular dependence of the effect, for the
case of H line, and for redshift values z=900, 1010 and 1150, which correspond to (redshifted) present day frequencies of 507, 452 and 397 GHz, respectively.
Given the small values of the optical depth,
the linear term gives the dominant contribution to the effect,
so the shape of the plot is similar to Fig. 6,
but the amplitude is modulated by the optical depth for the transition at every redshift.
As shown in Table 2, the maximum signal is obtained
at redshift z=1010. Prior to this redshift, the shape of the
mimics that of the primordial
at high multipoles,
because the suppression term dominates here
(and hence the
are negative).
When the line is placed at higher redshifts, well inside the
recombination, the cancellation of the generation and
blurring terms at high multipoles
produces a number of regions where the
quantities
have different signs. The exact position of the zeros strongly depends on the
particular cosmological model, and as pointed above, it could be used
to investigate in detail the shape of the transfer function at all
redshifts.
In Fig. 9 we present the same redshift slices as in
Fig. 6, but now showing
for the H
transition.
The relative value of the distortion of the power spectrum is of the
order of
10-5-10-4 in band power (
10-2 in temperature).
It should be also noted that
is growing at large multipoles, and
approaches the asymptotic value of
(shown here as a
dotted line), although the multipole region where we reach this
behavior depends on the considered redshift slice.
When the line is placed at higher redshifts inside recombination, the
asymptotic behavior is reached at higher multipoles
in the damping tail region of the spectrum.
Moreover, when the redshift is high enough, this asymptotic region
disappears, and we find a complicated structure
of positive/negative features due to the aforementioned cancellation.
Thus, this method could give us information about the "Silk damping''
(Silk 1968), or processes of dissipation of short wavelength adiabatic
perturbations due to radiative viscosity
and thermal conductivity during the recombination.
![]() |
Figure 11:
Same as Fig. 10, but only for the H![]() ![]() ![]() ![]() |
We now consider the frequency dependence of the effect.
Figure 10 shows the relative
change in the power spectrum due to the presence of the hydrogen
recombination lines of Balmer series, as a function of the
present day redshifted frequency.
For each Balmer line, we show three different panels corresponding to
three different angular scales (
and 800).
As pointed above, the signal has a very peculiar frequency dependence,
and for example at angular scales of
and
,
we can
see a characteristic change of sign of the effect, related to the
positions of the cancellations of the generation and blurring terms.
For the most intense line, H
,
we show five more angular scales
in Fig. 11, where we explicitly use a linear scale
in the vertical axis to show the change of sign. The case of
,
which is giving the maximum signal (in absolute value) is
shown as a solid line in that figure. The frequency dependence of the effect
is unique and completely different from that of foregrounds (e.g. dust emission), and thus it might
be used to separate these signals from other contaminants, as we will see below.
![]() |
Figure 12:
Angular power spectrum arising from the P![]() ![]() |
The figures for the Paschen and Brackett lines are similar to those for Balmer lines.
For illustration, in Fig. 12
we present the angular and frequency dependence for the B
and the P
lines.
The amplitude of the signal for all the transitions considered in this
paper is presented in Table 2.
Summarizing this subsection, we conclude that the best strategy to observe these
features and perform this tomography would be to use bandwidths (
)
of the order of 10%, although bandwidths of 1% would trace this
signal much better. These values are narrower than the width of the
visibility function in the line, but still wide enough to keep enough photons.
It is clear that the major contribution to the effect during the epoch
of recombination is coming from high
multipoles, and that
the widths of the features is of the order of
,
so this opens the possibility to use not only satellites but
balloons or ground based experiments to look for this signal.
In this section we discuss how the presence of foregrounds and the effect associated with Rayleigh scattering might affect the study of the lines. We shall adopt the middle-of-the-road model of Tegmark et al. (2000, hereafter T00), and we consider the contribution of five foreground sources, namely synchrotron radiation, free-free emission, dust emission, thermal SZ effect associated with filaments and superclusters of galaxies, and Rayleigh scattering. Point sources are not considered here, so we assume that resolved sources can be extracted from the maps, and that the contribution of unresolved sources can be lowered down to roughly the noise level of the observing instrument.
For the first three components, synchrotron, free-free and dust,
the angular dependence of the power spectra was approximated by a
power law. The frequency dependence for free-free and synchrotron was
taken to be a power law, whereas a modified black-body spectrum was
used for the dust component.
For the SZ effect, the frequency dependence is well-known.
Details of the modeling of the
angular dependence can be found in T00. This
neglects the contribution from the SZ effect generated by
resolved clusters of galaxies, which we assume can be removed
from the maps.
For the Rayleigh scattering effect modeling, we follow BHMS04.
As discussed above, the scattering cross-section of this process has a
very strong dependence on the frequency,
Figure 13 shows the expected level, prior to any
removal, of the discussed foregrounds when looking for the maximum of
the H line. We considered here the case of an ideal experiment
measuring at 430 GHz and 130 GHz, and with an instrumental bandwidth
of
,
which is comparable to the electron
width of the line under discussion.
It is clear that the main diffuse contamination will come
from dust emission, but there are other two signals which will be
larger than the H
contribution, namely
the thermal SZ emission from filaments and the Rayleigh scattering.
For the first one, the signal has a completely different
frequency and angular behavior from the one we are considering, so
these peculiarities could be used to separate both components.
For the case of Rayleigh scattering, the frequency behavior is also
completely different from the case of resonant scattering.
The angular dependence has some similarities in the high multipole
range (damping tail) if we probe a line in the low redshift tail
of recombination, because here both effects are proportional to
.
However, as we probe higher redshifts, the behaviors become different also in this domain.
In addition to this, there is another difference between the Rayleigh scattering and
the coherent scattering in lines, which is related to the
different evolution of the optical depth of each process during recombination,
and which is superimposed on the previous effect.
To illustrate it, we show in Fig. 14 the redshift dependence of the
population of electrons, neutral hydrogen atoms and
the population of hydrogen atoms in the level 2p, all
computed using our code. These three variables (
,
and
)
are proportional to the differential optical depth
for the Thomson, Rayleigh
and coherent scatterings, respectively. From this, it is clear that
the redshift interval
in which the optical depth for
Rayleigh scattering changes significantly is much larger than the corresponding one of
the H
line scattering. Thus, instruments with broad bandwidths
will dilute the signal from the lines, but add up the signal from
Rayleigh scattering. This also could be used to separate them, because
using a broad-band instrument we can isolate the Rayleigh scattering
component, and then subtract it.
One interesting physical remark is that the evolution of the population of the 2p level in Fig. 14 is practically proportional to
the electron density in this redshift range. This is what we expected, as we can immediately see from Eq. (2) (
)
and from the evolution of the electron density
(
,
see ZKS68).
![]() |
Figure 14:
Multilevel hydrogen recombination computed in this paper. It
is presented the evolution of the relative fraction of
electrons (![]() ![]() ![]() ![]() |
We have studied here the imprint of cosmological hydrogen recombination lines on the power spectrum of the CMB. To this end, we have developed a code that follows the evolution of the population of the levels of the hydrogen atom, separately treating those levels with different angular momentum. From here, we have obtained the optical depths associated with coherent scattering in these lines during the epoch of recombination, and these numbers were used to compute the effect on the angular power spectrum of the CMB.
These changes are small (
-
), but could be
separated from other effects due to their peculiar frequency and
angular dependence.
Unfortunately, the line giving the maximum signal (the H
line) is placed in a frequency domain where the contribution from dust emission, the tSZ effect produced in filaments and Rayleigh scattering are important, so it will be necessary either to look for
regions with low contamination (for the case of dust),
or to use component separation techniques to reach this signal.
The important point here is that the signal under discussion has a
very characteristic behavior, both in frequency and in angular scale,
which is completely different from any of the above contaminants.
One of the most important properties of this signal is that each observing frequency is associated with one redshift, so observations of these signals at different frequencies might give information about the amplitude of the fluctuations and corresponding peculiar velocities of matter at different redshifts during the epoch of recombination.
In this paper, the best-fit cosmological model to
the WMAP 1st-year data (Bennett et al. 2003) was
adopted to make predictions about
the shape and the intensity of the signal we expect to measure.
With the announced sensitivities of future experiments, these signals
should be detected, and thus we could try to infer some information
on the values of the cosmological parameters.
The inferred constraints from the detection of these lines
will be clearly independent of those obtained using the
model-fitting to the power spectrum of the CMB.
This additional information could be used
to break some degeneracies in the parameters in two ways.
First, as pointed out above, observations in narrow spectral bands
of these signals can give us information about specific redshift
slices during the process of recombination in the Universe.
The shape and positions of the peaks and zeros of the signal
presented in Figs. 8 and 9
reveal the details of the baryon transfer function at each
considered epoch, so they could be used as an additional test of the
cosmological model.
For a given experiment with a given bandwidth, it is straight-forward to derive
the prediction for
,
so model fitting could be used here, in the
same way as it is used today to extract parameters from the power spectrum.
Secondly, the determination of the amplitude of this effect
at a given redshift could also be used to set an independent constraint on
.
Using this signal, future narrow-band spectral observations might permit one to study all the details of the transfer function describing the evolution and viscous damping of perturbations in the epoch of recombination.
Acknowledgements
We acknowledge use of the CMBFast software package (Seljak & Zaldarriaga 1996). J.A.R.M. and C.H.M. acknowledge the financial support provided through the European Community's Human Potential Programme under contract HPRN-CT-2002-00124, CMBNET.