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Relic radiolines of hydrogen at decimeter
and meter wavelength band
By V l a d i s l a v S t o l ya r o v AND V i k t o r Dub r o v i c h
e­mail: vlad@ratan.sao.stavropol.su
Special Astrophysical Observatory of Russian Academy of Sciences, Nizhnij Arkhyz,
Karachaevo­Cherkessia Republic, 357147, RUSSIA
The formation of spectral disturbances of Cosmic Background Radiation (CBR) during the
period of hydrogen recombination at redshift z ' 1500 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.
1. Introduction
A recombination of hydrogen in a standard model of hot Universe takes place with
redshifts z ' 1500. This process can be investigated both with the purpose of determin­
ation 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 Tm and radiation T r 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 temper­
ature 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 prim­
ordial matter, these stages take place within an interval from z ' 6000 to z ' 1500.
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 non­equilibrium of some kind, for instance with essential changes of ionization degree.
Exactly therefore the disturbances are not appeared in the period of full ionization, be­
cause 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 z. In first approximation this process may be
considered as quasi­equilibrium, 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 quasi­equilibrium. We shall describe this fact below. And now let us note
1

2 V. Stolyraov & V.Dubrovich: Relic hydrogen radio lines
another special characteristics of spectral disturbances. As one can see from calculations
(Bernshtein et al. (1977)), a value Ÿ ij is diminishing with the increase of level number
i; j = i \Gamma 1, approximately as i \Gamma3=2 or ¸ – \Gamma1=2 . In the same time a background intensity
is I – ¸ – \Gamma2 . This means that relative value of spectral disturbances \DeltaT =T must increase
approximately as \DeltaT =T ¸ – 3=2 . From another side during the growth of – the distance
between the lines \Deltaš 0 =š ¸ 3=i becomes smaller than a width \Deltaš z =š of a single line
(independent from i). 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.
2. Mathematical model
A synthesis of recombination radiolines spectra is divided into several stages. Firstly,
the mathematical modelling of electron's transitions processes in hydrogen atom is con­
ducted and efficiency matrix Ÿ ij is calculated. Secondly, it is necessary to solve differential
equation describing ionization degree of gas by numerical methods and to obtain dx=dz
for different parameters
h;\Omega ; !; fi. And the last stage will be a construction of synthetic
spectrum.
The techniques of efficiency matrix Ÿ ij calculating (average number of quanta on fre­
quency š ij , which are given off by one recombination electron), is based on mathematical
model of electron's level­to­level 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 W ij . 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 (! 10 \Gamma5 of the initial amount). The number of escaped
quanta of frequency š ij , equal to (š i W ij \Gamma š j W ji ), where š i is the density of atoms on the
level i, is calculated on every step; a sum through all iterations, normalized on the num­
ber of particles that reach the second level, is the element of matrix Ÿ ij to be found. In
the present model the system of 50 levels plus continuum level was considered, therefore
the matrix size was (51 \Theta 51).
In order to obtain the synthetic spectrum of hydrogen recombination radiolines, it
is necessary to calculate the element of Ÿ ij matrix for every transition i ! j, then to
obtain a dependence of \DeltaT =T from z and to add the received profiles (\DeltaT =T )(z) 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)):
x 2
1 \Gamma x =
k mT
¯h 2 ú
3
2
2
p
2
1
e I
k T nH
(2.1)
where x = n e =nH is the ionization degree, k is Boltzmann constant, m is the mass of
electron, T is the CBR temperature, ¯h is Plank constant, nH = n p + n 0 is the hydrogen
density in the Universe and I is ionization potential of hydrogen.
If we consider the recombination as quasi­equilibrium process, we must solve this equa­
tion in order to receive the profile to be found. One can see that an area of essential
change of the function lies quite near z of recombination, because before and after re­
combination period the number of free electrons is practically constant.

V. Stolyraov & V.Dubrovich: Relic hydrogen radio lines 3
0 20 40 60 80 100
-6.0x10 -8
-4.0x10 -8
-2.0x10 -8
0.0
2.0x10 -8
4.0x10 -8
Figure 2. Lines contrast for non-equlibrium case.
DT/T
Wavelength, cm
0 20 40 60 80 100
0.0
2.0x10 -6
4.0x10 -6
6.0x10 -6
8.0x10 -6
1.0x10 -5
Wavelength, cm
W=1, w=0.1, h=1
Figure 1. Continuum for non-equilibrium case.
DT/T
Figure 1. Continuum for non­equilibrium case.
In the case of recombination, caused by slow ``decay'' of Lyman quanta at the expense
of two­photon 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
dx z
dz
= 8 fi ! 3
4
p
h\Omega
e ff (1\Gammaz)
4 z
`
\Gamma 1 \Gamma x
e ff (1\Gammaz)
z
+ 2 x 2
'
(2.2)
where ff = 38:7 \Gamma ln(h 2 !) is a coefficient, determined from Saha equation for x = 0:5, fi
is the coefficient of two­photons decays from high levels (Dubrovich
(1987)),\Omega = n t =n c
is total density to critical density ratio, ! = nH =n c is the hydrogen density to critical
density ratio, h is the Hubble constant normalized on 75 km s \Gamma1 Mpc \Gamma1 , z is redshift,
normalized on z 0 = 158600=2:73ff (for z 0 ionization degree is 0.5 according to Saha), and
x = n e =nH (n e is a number of free electrons), with nH = n p + n 0 .
It is solved numerically by Runge--Kutta--Merson method of 4th order with automatic
step choosing and with accuracy up to 10 \Gamma8 .
The constructing of synthetic spectrum demands of profile K(z) = \DeltaT =T calculating
according to the equation from Bernshtein et al. (1977) for transitions i ! j with
corresponding laboratory wavelength.
K(z) = 0:000127 Ÿ ij – ij 2 !
– 0
2
dx z 3
dz
(2.3)
Then we must allocate rightly the central frequencies of profiles on appropriate places
of spectrum where we are observing them today with z = 0. A summarizing of different
K ij (z) we must conduct, taking into account that \Delta–=– = \Deltaz=(z + 1). Certainly it is
important to take into consideration an optical depth due to dissipation processes Ü , and
to multiplicate initial profiles on exp(\GammaÜ ). This question will be discussed below.
We received the set of profiles for different parameters
h;\Omega ; !; fi. The width of obtained
profiles vary from \Deltaz = 0:14 to 0:26 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 i, and synthetic

4 V. Stolyraov & V.Dubrovich: Relic hydrogen radio lines
0 20 40 60 80 100
-6.0x10 -8
-4.0x10 -8
-2.0x10 -8
0.0
2.0x10 -8
4.0x10 -8
Figure 2. Lines contrast for non-equlibrium case.
DT/T
Wavelength, cm
0 20 40 60 80 100
0.0
2.0x10 -6
4.0x10 -6
6.0x10 -6
8.0x10 -6
1.0x10 -5
Wavelength, cm
W=1, w=0.1, h=1
Figure 1. Continuum for non-equilibrium case.
DT/T
Figure 2. Lines contrast for non­equlibrium case.
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 i \Gamma j = 1 but secondary
(i \Gamma j ? 1), which have an amplitude only in 2­4 times less and fill up the frequency axis
very closely. In the present model we consider only main lines and secondary lines with
i \Gamma j = 2. The synthetic spectrum for profile obtained from Saha equation for parameters
\Omega = 0:3 and 1:0 (! = 0:1; h = 1:0) has \DeltaT =T ' 10 \Gamma4 and allows us to see the single line
profiles. The spectrum calculated with consideration of non­equilibrium with the help
of the equation for non­equilibrium case for the width \Deltaz = 0:14 and 0.2 with the same
parameters has the same value \DeltaT =T 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 \DeltaT =T ' 10 \Gamma6 in decimeters and meters wavelength band, but
non­equilibrium cases give us only ! 10 \Gamma7 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)).
(a) Optical depth due to free--free transitions. For transitions with i = 10 \Gamma 20 this
optical depth is about Ü f \Gammaf = 0:01 \Gamma 0:1.
(b) Optical depth due to bound--free transitions. For the same values of i then Ü b\Gammaf =
0:001 \Gamma 0:005.
(c) 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)).
(d) 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

V. Stolyraov & V.Dubrovich: Relic hydrogen radio lines 5
absorption on free--free electrons. Let us consider its role in spectra formation in more
detail. Firstly, the strong dependence Ü f \Gammaf (= Ü ) from – lead to the decreasing of spectra
in the range of the long waves from certain value –max . The value –max is determined
numerically according to the place of maximum of spectra, formed after multiplication
of the initial profile by exp(\GammaÜ ). It was found that –max ' 10 \Gamma 15m for different set
of parameters. However, there is one more feature of free--free absorption. The fact is
that an integral in expression for Ü must be taken from current moment z, 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 – ? –max Ü becomes large than 1 almost for all points of profile.
From expression for Ü it is easy to see how –max depends on parameters. It is clear that
mainly this is dependence from !: –max ' 1=!. An influence of other parameters is more
weak and intrinsically non­linear.
3. Conclusion
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 N of simultaneously and independently working
receivers not demanding complex antennas. It may give us a gain in sensitivity about
square root from N 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 Scientific­Educational Center
``Kosmion'' (Moscow).
REFERENCES
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.