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Astron. Astrophys. 324, 27-31 (1997)

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3. Consequences of CBR fluctuations

It has been shown previously (see Sunyaev and Zel'dovich, 1970, Dubrovich, 1977, Maoli et al., 1994) what the magnitude is of the temperature fluctuations, [FORMULA] of the CBR due to the pure reflection of photons by the moving object. In this case the effect depends on its peculiar velocity [FORMULA] and optical depth [FORMULA],

[EQUATION]

here [FORMULA] is the component of the peculiar velocity along the line of sight and c is the speed of light. It should be pointed out, that if this fluctuation is caused by the interaction with a resonant system, it must occur only at the corresponding wavelength and, what is very important, there should be no influences from the one resonance to any other. In our case it means that this effect could be at the wavelength corresponding to the rotational and the rovibration transitions, but the amplitudes of the [FORMULA] from the separate transitions are fully independent. It is one of the fundamental properties of pure reflection.

On the contrary, the luminescence process causes the appearance of some photons at one wavelength due to the absorption of the appropriate photons at another wavelength. This new property of the process of interaction of matter and radiation leads us to new possibilities for SSF formation.

In order to see what consequences might follow from this new effect let us take into account that there is no luminescence due to the transitions between different rotational levels only (these are forbidden in the dipole order by the law of momentum conservation). So, the first allowed mechanism which leads to luminescence is: the absorption of a rovibration photon followed by the emission of a new rovibration photon and two new rotational quanta (Fig1).

[FIGURE] Fig. 1. The scheme of some rovibration transitions in diatomic molecules

In terms of the quantum numbers: J, which corresponds to the number of rotational levels (J =0,1,2,...) and v, which corresponds to the number of the rovibration levels (v =0,1,2,..) the simplest circle of the transitions can be written: (J =0,v =0)-(J =1,v =1)-(J =2,v =0)- (J =1,v =0)-(J =0,v =0). Here we have used the law for the allowed dipole transitions: [FORMULA] =([FORMULA])1, [FORMULA] =([FORMULA])1. The frequencies of these transitions are: [FORMULA] and [FORMULA] respectively, and they must satisfy the equation (1 is the absorbing photon and 2,3,4 are emitted photons):

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] -the line's width and [FORMULA] -the photon's number. The energy levels of the diatomic molecules could be described by:

[EQUATION]

where E is the total energy of the level, [FORMULA] and [FORMULA] are the specific molecule constants. For several molecules the ratio [FORMULA] / [FORMULA] is rather similar - about 50-100. But this is the ratio of the frequencies of the first rovibration and the first rotational photon, as follows from expression (5). It is very important for our estimations. The key point is that the number of photons which are reflected by pure elastic scattering in the rotational lines at the frequencies [FORMULA] and [FORMULA] corresponds to the number of the CBR photons at these frequencies, but according to (3), the number of these photons due to the luminescence will correspond to the number of the CBR photons at [FORMULA]. Dubrovich (1977,1994) estimated the redshift, when several molecules could appear using Saha recombination rates as z = 300-100. At these redshifts, [FORMULA] and [FORMULA] lie in the extreme Rayleigh-Jeans wing of the CBR and [FORMULA] - near it's maximum! So, the luminescence causes much more intense disturbances of the CBR.

Let us calculate its quantitative value. The amplitude of the [FORMULA] in the Rayleigh-Jeans wing of the CBR could be defined as the ratio of number of the new photons ([FORMULA] or [FORMULA]) to the number of the CBR photons [FORMULA] at the same frequency ([FORMULA] or [FORMULA]) and in the same spectral interval [FORMULA]:

[EQUATION]

According to the previous investigations by Dubrovich (1977) and Maoli et al. (1994), and our new considerations, we obtain:

[EQUATION]

[EQUATION]

where [FORMULA] is the optical depth of the elastic scattering at frequency [FORMULA], and [FORMULA] is the ratio of the decaying photons to all scattered photons ([FORMULA] 1). So, using Eqs. (1-4) and (6-8) we obtain:

[EQUATION]

[EQUATION]

where [FORMULA] is the redshift of molecule's recombination by Saha, [FORMULA] is derived from this equation at [FORMULA], and [FORMULA]. Here we have assumed that [FORMULA] /kT [FORMULA] 1, and [FORMULA] /kT [FORMULA] 1, [FORMULA] /kT [FORMULA] 1. The value of the optical depth (for rotovibrational transitions here) we will estimate on the base of the expressions obtained by Dubrovich (1994) for pure rotational transitions. The accuracy of such an estimation maybe not more than one order of magnitude.

[EQUATION]

[EQUATION]

where [FORMULA], [FORMULA] are the total and the baryonic average densities of the matter relative to the critical one, [FORMULA] is the abundance of the molecule relative to the atomic hydrogen, z is the redshift of the proto-object, H is the Hubble constant, normalized to [FORMULA] = 75km/s/Mpc, [FORMULA] is the specific molecule constant, d is the dipole moment of the molecule, [FORMULA] and [FORMULA] are temperature of CBR and the critical density at z = 0.

In order to estimate the value of [FORMULA], several transition pathways should be considered. These pathways for the first three rotational levels are displayed in Figs. 1 a, b, c - respectively. In these Figs. the solid line corresponds to the first absorption ([FORMULA]), the dash line to the emission of the one or the three photons ([FORMULA]), the wave line - to the "secondary" absorptions. The transition to the initial level corresponds to pure reflection. The transition to the level, which lies lower than the initial one (Fig.1c) actually means that we lose two photons from the CBR which must be absorbed for the initial level to be exited. This is what we mean by "secondary" absorptions and it should be compared with the emission which is due to the process in Fig.1a. The total intensity of each line should be calculated as the sum of all these parts, taking into account the optical depth dependence on J. The dependence of [FORMULA] on J, can be found from the expression of the Einstein coefficient in appropriate form:

[EQUATION]

here J is the number of the rotational level at v =1, and [FORMULA] is the number of the rotational level at [FORMULA] =0, and [FORMULA] very slow depends on J, [FORMULA]. It is easy to see that [FORMULA] depends slowly on the [FORMULA]. So, for the levels with small J and [FORMULA], the main factor is [FORMULA]. In this case, it is easy to calculate the value of the [FORMULA] for several pathways (the first and the second arguments of the [FORMULA] correspond to the initial and the final rotational numbers at v =0, respectively):

[EQUATION]

for [FORMULA]:

[EQUATION]

These expressions show that the value of [FORMULA] is about 0.5 for the different cases. Concerning the comparison of the 1a and 1c pathways, we need to have a more precise value. Using (11) and (14, 15) we can get a lower limit to the effective [FORMULA] after adding these pathways. Indeed, If we neglect in (11) the factor [FORMULA], we will obtain an increase of [FORMULA] by a factor of three from the level with J =2 relative to that with J =0. This means that by the first pathway there will be three times more photons captured than by the second one. But only 1/12 of the first photons give rise to "secondary" absorptions. On the other hand, due to the 1a process, 3/4 of the captured photons give emission. The result will be: [FORMULA] =3/4-3(1/12)=1/2. So, finally for all future estimations we should take [FORMULA] =0.5. In Table 1 we present the main information about the most probable and highly interacting primordial molecules and some estimate of the fundamental parameters that could be measured.

here: [FORMULA] is the dissociation potential of the molecule, [FORMULA] is the wavelength of the first rotational transition ([FORMULA] =c/2 [FORMULA]), [FORMULA] is the wavelength of the rovibration transition ([FORMULA]), [FORMULA] refers to the highest wavelength where this molecule could now be seen, [FORMULA] is the limit to the molecule abundance which could be reached if we assume that [FORMULA] /c = [FORMULA] and that the observational limit which can be achieved is [FORMULA], [FORMULA] is the lower limit which could be placed on the peculiar velocity if we assume [FORMULA] 1 and an observational limit of the [FORMULA]. The triatomic molecule H2 D [FORMULA] is more complicated than the other molecules in Table 1

Now, we can write the expression for [FORMULA] in a more simple form:

[EQUATION]

In order to search for these molecules, the most auspicious wavelength regions (for the first rotational lines) can be found in Fig.2. The expected values of [FORMULA] are shown as a function of wavelength for each molecule and correspond to the value of [FORMULA]. The red wings of these curves are actually due to the rate of recombination of each molecule, assuming Saha recombination rates. The blue wings are described by expression (8). The second rotational line of each molecule has a factor of two higher frequency and a value of K which is four times lower than the first one.

[FIGURE] Fig. 2. The value of [FORMULA] for several molecules

[TABLE]

Table 1.

Here are some comments to Table 1.

LiH: This is a very important molecule, because it consists of primordial Li. Its abundance is a good test for the epoch of nuclear synthesis in the early Universe. Its large dipole moment and relatively low frequency of the rotational and rovibration transitions lead to the high value of K. But unfortunately, its small abundance and some difficulties with the chemical processes of forming this molecule lead to a non-optimistic prediction for [FORMULA]. Even so, this value of [FORMULA], eqn. 16 and the peak value of [FORMULA] from Fig. 2 lead to predicted values as high as about [FORMULA]  =  [FORMULA] for [FORMULA]  = 0.1.

HD [FORMULA]: This is also an important molecule, due to the presence of primordial deuterium, D. The abundance of D is about 5 orders of magnitude larger, than that of Li. But HD [FORMULA] has a dipole moment about 10 times less than LiH and a cross-section which is 100 times smaller. Another small factor is the abundance of H [FORMULA] at redshift z = 200, which might be about [FORMULA] relative to that of neutral hydrogen. Due to the relatively high frequency of the rotational and rovibration transitions, the resulting value of K is not very large. But, if high sensitivity were reached, this molecule might be seen.

HeH [FORMULA]: This molecule does not have any low abundance constituents. There are only two small factors which lead to a low abundance: a high rate coefficient for destruction (by electron recombination and collisions with the neutral atoms of hydrogen) compared with the rate coefficient of formation, and a small abundance of H [FORMULA] at high redshift. But it might be the most likely molecule to be searched for.

H2 D [FORMULA]: This is the simplest triatomic molecule with a high dipole moment. It contains primordial deuterium. Due to the presence in its spectrum of very low frequency transitions, the value of K can be very high. In Table 1 the value K0 corresponds to [FORMULA]. It is very important that the redshift of the recombination H2 D [FORMULA] be relatively high.

The expected abundances of these molecules in the early Universe are discussed by many authors (Lepp and Shull,1984, Puy et al, 1993, Palla et al, 1995, Maoli et al, 1996, Stancil et al, 1996a) and more recent results by Stancil et al.(1996b).

In order to observe SSF due to all these molecules, let us give some simple estimates of their main parameters. These are diffuse, extended objects, which will have only the narrow emission lines with the low brightness temperature in these lines. The width [FORMULA] of these lines depends on the object's size (linear - L or angular - [FORMULA]) (Dubrovich, 1982):

[EQUATION]

[EQUATION]

Here M is the mass of the proto-object, [FORMULA] is the mass of the Sun. At the redshift z = 100, if [FORMULA] then a protogalaxy with the mass [FORMULA] has

[EQUATION]

[EQUATION]

A proto-object with the mass of an ordinary cluster of galaxies, [FORMULA], will have an angular size:

[EQUATION]

[EQUATION]

The value of the peculiar velocity at the redshift z might be:

[EQUATION]

These parameters would be the most probable for the standard model of the Universe.

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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