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Special Astrophysical Observatory of Russian Academy of Sciences, Nizhnij Arkhyz, Karachaevo-Cherkessia Republic, 357147, RUSSIA
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)).
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.