## 3. Consequences of CBR fluctuationsIt has been shown previously (see Sunyaev and Zel'dovich, 1970, Dubrovich, 1977, Maoli et al., 1994) what the magnitude is of the temperature fluctuations, of the CBR due to the pure reflection of photons by the moving object. In this case the effect depends on its peculiar velocity and optical depth , here is the component of the peculiar
velocity along the line of sight and 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).
In terms of the quantum numbers: where -the line's width and -the photon's number. The energy levels of the diatomic molecules could be described by: where Let us calculate its quantitative value. The amplitude of the in the Rayleigh-Jeans wing of the CBR could be defined as the ratio of number of the new photons ( or ) to the number of the CBR photons at the same frequency ( or ) and in the same spectral interval : According to the previous investigations by Dubrovich (1977) and Maoli et al. (1994), and our new considerations, we obtain: where is the optical depth of the elastic scattering at frequency , and is the ratio of the decaying photons to all scattered photons ( 1). So, using Eqs. (1-4) and (6-8) we obtain: where is the redshift of molecule's recombination by Saha, is derived from this equation at , and . Here we have assumed that /kT 1, and /kT 1, /kT 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. where , are the total
and the baryonic average densities of the matter relative to the
critical one, is the abundance of the molecule
relative to the atomic hydrogen, z is the redshift of the
proto-object, In order to estimate the value of , 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 (), the dash line to the emission of
the one or the three photons (), 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 here for : These expressions show that the value of 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
after adding these pathways. Indeed, If we
neglect in (11) the factor , we will obtain an
increase of by a factor of three from the level
with here: is the dissociation potential of the
molecule, is the wavelength of the first
rotational transition ( =c/2
), is the wavelength of
the rovibration transition (),
refers to the highest wavelength where this
molecule could now be seen, is the limit to the
molecule abundance which could be reached if we assume that
/ Now, we can write the expression for in a more simple form: 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 are shown as a function of wavelength for each molecule and correspond to the value of . 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.
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 . Even so, this value of , eqn. 16 and the peak value of from Fig. 2 lead to predicted values as high as about = for = 0.1. HD : 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 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 at redshift z = 200, which might be about 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 : 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 at high redshift. But it might be the most likely molecule to be searched for. H 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 of these lines depends on the object's size (linear - L or angular - ) (Dubrovich, 1982): Here A proto-object with the mass of an ordinary cluster of galaxies, , will have an angular size: The value of the peculiar velocity at the redshift z might be: These parameters would be the most probable for the standard model of the Universe. © European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |