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Relic radiolines of hydrogen at decimeter
and meter wavelength band
Vladislav
Stolyarov
and Viktor
Dubrovich
e-mail:
vlad@ratan.sao.stavropol.su
Special Astrophysical Observatory of Russian Academy of Sciences,
Nizhnij Arkhyz, Karachaevo-Cherkessia Republic, 357147, RUSSIA
Abstract:
The formation of spectral disturbances of Cosmic Background Radiation (CBR)
during the period of hydrogen recombination at redshift 
is
considered. Special attention is paid to the decimeter and meter wavelength
band. The most distinctive properties of spectral disturbances are noted: an
amplitude increase with the following decreasing, growing asymmetry of the
profiles of separate lines. It is shown that according to the observational
data it will be possible to determine an average baryon and total density of
matter in the Universe with great accuracy.
A recombination of hydrogen in a standard model of hot Universe takes place
with redshifts 
. This process can be investigated both with the
purpose of determination of its influence on smoothing of initial spatial
fluctuations of microwave background temperature, and for estimation of the
values of isotropic spectral disturbances in the cosmic background radiation
(CBR) (Dubrovich (1975),Bernshtein et al. (1977)). In this paper we will return to the question on the
spectral lines caused by recombination of hydrogen discussed by Dubrovich (1975),
in the longer wavelength part of spectrum of the CBR.
Let us remember the main ideas of Dubrovich (1975). When the Universe expands, the
temperatures of matter 
and radiation 
decrease. These temperatures
are equal with great accuracy because interaction between the radiation and
matter on the early stages of evolution is strong, and the radiation has Planck
spectra. When the temperature becomes lower than a certain value, the matter
turns from ionized state into a neutral state. For basic elements such as
hydrogen and helium which compose primordial matter, these stages take place
within an interval from 
to 
. The changes of
temperatures of radiation and matter occur very slowly in comparison with the
average times of radiation transitions in atoms. Therefore, one can expect the
extremely small value for spectral disturbances of the CBR. However, it is easy
to show (Dubrovich (1975)), that the main part of transitions caused by interaction
with the CBR, i.e. by absorption and emission of the CBR quanta without any
changes of its spectrum (owing to LTE principle). The disturbance may appears
only with the presence of non-equilibrium of some kind, for instance with
essential changes of ionization degree. Exactly therefore the disturbances are
not appeared in the period of full ionization, because all recombination acts
are compensated by appropriate number of ionizations. The first important
conclusion which is followed from this fact: the profile of recombination line
(or its width) will be determined by dynamics of recombination, i.e. by the
rate of ionization degree change as a function of 
. In first approximation
this process may be considered as quasi-equilibrium, i.e. ionization degree is
determined from Saha equation. However, as it have shown in Zeldovich et al. (1968), a
real recombination rate is quite different from quasi-equilibrium. We shall
describe this fact below. And now let us note another special characteristics
of spectral disturbances. As one can see from calculations (Bernshtein et al. (1977)), a
value 
is diminishing with the increase of level number 
, approximately as 
or 
. In the same time
a background intensity is 
. This means that
relative value of spectral disturbances 
must increase
approximately as 
. From another side during the
growth of 
the distance between the lines 
becomes smaller than a width 
of a single line (independent
from 
). In a low frequency band lines contrast becomes decrease rapidly in
result, but a total intensity of them increases relative to the equilibrium the
CBR. The consideration of free-free absorption in recombination plasma imposes
a restriction on this growth. The energy absorbed by free electrons is
redistributed rapidly between all particles but it is small so it does not lead
to visible influence on the matter and radiation temperature. The concrete
calculation method and main results is given below.
A synthesis of recombination radiolines spectra is divided into several stages.
Firstly, the mathematical modelling of electron's transitions processes in
hydrogen atom is conducted and efficiency matrix 
is calculated.
Secondly, it is necessary to solve differential equation describing ionization
degree of gas by numerical methods and to obtain 
for different
parameters 
. And the last stage will be a construction
of synthetic spectrum.
The techniques of efficiency matrix 
calculating (average number
of quanta on frequency 
, which are given off by one recombination
electron), is based on mathematical model of electron's level-to-level moving
and is taken from Bernshtein et al. (1977). The electrons distribution through levels with
given initial distribution, at every new iteration step is determined with the
help of relative transitions probabilities matrix 
. The particles which
are come to be in ionization and electrons which are get onto the second level,
exit the system. The calculation is finished when comparatively small amount of
particles remains in the system (
of the initial amount). The number
of escaped quanta of frequency 
, equal to (
), where 
is the density of atoms on the level 
, is calculated
on every step; a sum through all iterations, normalized on the number of
particles that reach the second level, is the element of matrix 
to be found. In the present model the system of 50 levels plus continuum level
was considered, therefore the matrix size was (
).
In order to obtain the synthetic spectrum of hydrogen recombination radiolines,
it is necessary to calculate the element of 
matrix for every
transition 
, then to obtain a dependence of 
from
and to add the received profiles 
on the common scale of
wavelengths.
In the presence of thermodynamical equilibrium we can describe ionization state
of gas by Saha equation (Zeldovich et al. (1968),Bernshtein et al. (1977)):
where 
is the ionization degree, 
is Boltzmann constant, 
is
the mass of electron, 
is the CBR temperature, 
is Plank constant,
is the hydrogen density in the Universe and 
is ionization
potential of hydrogen.
If we consider the recombination as quasi-equilibrium process, we must solve
this equation in order to receive the profile to be found. One can see that an
area of essential change of the function lies quite near 
of recombination,
because before and after recombination period the number of free electrons is
practically constant.
In the case of recombination, caused by slow ``decay'' of Lyman quanta at the
expense of two-photon processes, one cannot use Saha equation. The dynamic of
recombination is determined now by the underequilibrium Lyman quanta conversion
rate. The equation describing the ionization degree change (Bernshtein et al. (1977)), is
rewritten as
where 
is a coefficient, determined from Saha
equation for 
, 
is the coefficient of two-photons decays from
high levels (Dubrovich (1987)), 
is total density to critical
density ratio, 
is the hydrogen density to critical density
ratio, 
is the Hubble constant normalized on 
,
is redshift, normalized on 
(for 
ionization
degree is 0.5 according to Saha), and 
(
is a number of free
electrons), with 
.
It is solved numerically by Runge-Kutta-Merson method of 4th order with
automatic step choosing and with accuracy up to 
.
The constructing of synthetic spectrum demands of profile 
calculating according to the equation from Bernshtein et al. (1977) for transitions 
with corresponding laboratory wavelength.
Then we must allocate rightly the central frequencies of profiles on appropriate
places of spectrum where we are observing them today with 
. A
summarizing of different 
we must conduct, taking into account that
. Certainly it is important to take
into consideration an optical depth due to dissipation processes 
, and to
multiplicate initial profiles on 
. This question will be discussed
below.
Figure 1: Continuum for non-equilibrium case.
Figure 2: Lines contrast for non-equlibrium case.
We received the set of profiles for different parameters 
. The width of obtained profiles vary from 
to 
in
relation to parameters. It is easy to calculate what the distance between the
neighbouring lines is ought to be equal. It is clear that the neighbouring
profiles will be put one onto another from a certain 
, and synthetic spectrum
will be smooth in that wavelength band. Moreover, we must take into account
that in forming of spectra take part not only main lines with 
but
secondary 
, which have an amplitude only in 2-4 times less and fill up
the frequency axis very closely. In the present model we consider only main
lines and secondary lines with 
. The synthetic spectrum for profile
obtained from Saha equation for parameters 
and 
(
) has 
and allows us to see the single
line profiles. The spectrum calculated with consideration of non-equilibrium
with the help of the equation for non-equilibrium case for the width 
and 0.2 with the same parameters has the same value 
but it
is very smooth and we can not distinguish the single lines (see
Figure 1). The lines after continuum subtracting for equilibrium case
have an amplitude 
in decimeters and meters
wavelength band, but non-equilibrium cases give us only 
lines
amplitude (see Figure 2).
And now let us consider the dissipation processes due to the interaction
between the radiation and matter. The main mechanisms are: free-free (f-f),
bound-free (b-f), bound-bound (b-b) absorption, and lines broadening by
electron scattering (Bernshtein et al. (1977)).
- Optical depth due to free-free transitions. For transitions with
this optical depth is about 
.
- Optical depth due to bound-free transitions. For the same values of
then 
.
- For bound-bound transitions The lines absorption is not eliminate
quantum by itself. It is occurred only in the case if the inverse transition
transfers electron on the other level, or if the capture of another quantum
from background is occurred and the system returns to the initial state not by
strictly back way. Besides, in both cases some quanta disappear from the CBR
and another quanta arise. As follows from condition of entropy growth this
process will lead for quanta forming that lies as nearer as possible to the
maximum of Planck spectra (Dubrovich (1985)).
- The broadening of line by electron scattering is not depended on
wavelength. It is clear from the quoted estimates that the main contribution in
absorption will give the absorption on free-free electrons. Let us consider
its role in spectra formation in more detail. Firstly, the strong dependence
from 
lead to the decreasing of spectra in the
range of the long waves from certain value 
. The value
is determined numerically according to the place of maximum
of spectra, formed after multiplication of the initial profile by
. It was found that 
for
different set of parameters. However, there is one more feature of free-free
absorption. The fact is that an integral in expression for 
must be
taken from current moment 
, i.e. from the moment when the given quantum has
escaped. This means that emission in the ``red'' wing of the line will be
absorbed more powerfully than in ``blue'' wing. This distinction will be very
large, beginning from the certain 
, because the function under
integral changes in 20-30 times through the width of line. In result the
profile will be asymmetrical. In synthetic spectrum in this case the contrast
must rise again but finally it will fall, because with 
becomes large than 1 almost for all points of profile. From
expression for 
it is easy to see how 
depends on
parameters. It is clear that mainly this is dependence from 
:
. An influence of other parameters is more
weak and intrinsically non-linear.
In conclusion let us once more pay attention to the exceptional amount of
information in the recombination lines of hydrogen. The wavelength range that
was considered has some advantages from the side of the equipment
possibilities for lines detection. The main advantage is a simplicity of
receivers and spectrometers in this band. This may allow to create a system of
large number 
of simultaneously and independently working receivers not
demanding complex antennas. It may give us a gain in sensitivity about square
root from 
when the spectra are summed.
We shall not dwell upon the detailed description of observation techniques and
analysis of advantages and shortcomings of this method. Let us only note this
variant as one of the ways to solve the problem of search for and study
spectral disturbances of the CBR describing above.
The authors are grateful to Dr N.S.Kardashev and Dr Y.N. Parijskij for
stimulated discussions. This work is supported in part by grant from
Scientific-Educational Center ``Kosmion'' (Moscow).
- Bernshtein I.N., Bernshtein D.N., Dubrovich V.K., 1977,
SvA, 54, 727.
- Dubrovich V.K., 1975,
SvAL, 1, 10.
- Dubrovich V.K., 1985,
Izv. Spec. Astroph. Observ., 20, 63.
- Dubrovich V.K., 1987,
Sov. Optics and Spectr., 63, 439.
- Zeldovich I.B., Kurt V.G., Sunaev R.A., 1968,
Sov. Zh. Exp. Theor. Phys., 55, 287.
YERAC 94 Account
Wed Feb 22 22:22:29 GMT 1995
