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8.2. Infrared Background from VMOs

We now compare these limits with the background expected from a population of VMOs. This can be predicted very precisely since all VMOs have a surface temperature Ts of about 105 K and generate radiation with efficiency epsilon approx 0.004. We normalize the VMO density parameter to the value Omega* approx 0.1 required to explain galactic halos and assume that they produce black-body radiation with temperature Ts. If the radiation is affected only by cosmological redshift, its density and peak wavelength at the present epoch should be

Equation 8.1        (8.1)

where z* is the redshift at which the VMOs burn. We can place an upper limit on z* by noting that the main-sequence time of a VMO is tMS approx 2 × 106y (independent of mass), so that z* cannot exceed the redshift when the age of the Universe is tMS. This implies z* < 240h-2/3 and so lambdapeak < 15 µ and OmegaR > 10-6 for h > 0.5. We can place a lower limit on z* from UV/optical background light limits. As discussed by McDowell (1986) and Negroponte (1986), these imply a constraint on the density of VMOs burning at any redshift z*. If one requires Omega* approx 0.1, this places a lower limit on z*, mainly because one needs the radiation to be redshifted into the near-IR band, where the background light limits arc weaker. In the absence of neutral hydrogen absorption, one requires z* > 30, which implies lambdapeak > 1 µ and OmegaR < 10-5. The peak of the VMO background must then lie somewhere on the heavy line in Figure 6 and the spectrum must lie within the region bounded by the broken line. Note that the observational limits are only just beginning to constrain the VMO scenario and they may never be able to exclude it if z* is so large (> 200) that most of the VMO light is pushed beyond 10 µ, where it would be hidden by interstellar dust.

The constraints on the VMO scenario would be much stronger if the light was reprocessed by dust as discussed by many workers (McDowell 1986, Negroponte 1986, Bond et al 1986, Wright & Malkan 1987, Lacey & Field 1988, Adams et al 1989, Draine & Shapiro 1989). Such dust could either be pregalactic in origin or confined to galaxies themselves if galaxies cover the sky. If the dust cross-section for photons of wavelength lambda is assumed to be geometric (pi r2d for a grain radius rd) for lambda >> rd, but to fall off as lambda-1 for lambda >> rd then the spectrum should peak at a present wavelength (Bond et al 1986)

Equation 8.2        (8.2)

where zd is the epoch of dust production and we have used Equation (8.1) with Omega approx 0.1 to express OmegaR in terms of z*. The crucial point is that the wavelength is very insensitive to the various parameters appearing in Equation (8.2) because the exponents are so small. At one time the Nagoya-Berkeley experiment (Matsumoto et al 1984 appeared to indicate a submillimeter excess peaking at almost exactly the wavelength predicted. However, the Nagoya-Berkeley excess has now been disproved by FIRAS and the question arises of whether the VMO-plus-dust scenario is still compatible with COBE results.

It should be stressed that one does not necessarily expect dust reprocessing anyway. Pregalactic dust with density Omegad would only absorb UV photons for

Equation 8.3        (8.3)

where Omegad is normalized to the sort of value appropriate for galaxies. It is not clear whether this condition can be satisfied. One has no direct evidence for pregalactic dust but in any hierarchical clustering picture one would expect at least some pregalactic dust production (Najita et al 1990). For example, one could envisage the dust produced by the first dwarf galaxies being blown into intergalactic space because the gravitational potential of the dwarfs would be so small. The dust in galaxies themselves would suffice to reprocess the VMO background only if galaxies cover the sky which - for galaxies like our own - requires the redshift of galaxy formation to exceed about 10 (Ostriker & Heisler 1984, Heisler & Ostriker 1988, Ostriker et al 1990). Even if galaxies do cover the sky, the analysis of Fall et al (1989) indicates that the dust-to-gas ratio in primordial galaxies may only be 5-20% that of the Milky Way for 2 < z < 3, which makes the opaqueness condition difficult to satisfy.

In general, one would expect there to be both a far-IR dust background and a near-IR attenuated starlight background, with the relative intensity reflecting the efficiency of dust reprocessing. By changing the amount of dust, one can redistribute the light between the near-IR and far-IR in an attempt to obviate the constraints. In order to examine the issue in more detail, Bond et al (1991) have carried out a more sophisticated analysis, in which the dust cross section is assumed to scale as lambda-alpha at infrared wavelengths. They also introduce a more realistic model for the source luminosity history, allowing for both "burst" and "continuous" models. Comparison with the far-IR and COBE constraints is shown in Figure 6 for two of their models with alpha = 1.5 and z* = 100. One has Omegad = 10-5 and zd = 50 (which is above the FIRAS constraint); the other has Omegad = 10-6 and zd = 10 (which is below it). This shows that the VMO-plus-dust scenario is only viable for models with a high redshift of energy release (z* = 100) and small amounts of dust (Omegad = 10-6). Of the models considered by Bond et al (1991), Wright et al (1994) claim that only their model 12 still survives.

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