The field 9 can take any value in the range —V2nf to V2nf. A natural estimate for the initial value of the axion field would therefore be 9 = 0(f), and the initial value of V(d) should be of order m%. An investigation of the rate at which the energy of the axion field p$ falls off as the Universe expands shows that, for f > 1012 GeV, most of the energy density would presently be contributed by axions, while the baryon energy density would be considerably lower than its presently observed value of pB ~ 0.05p0. This information was used to derive the constraint f < 1012 GeV [57-59].

This is a very strong constraint, especially since the astrophysical considerations lead to a constraint f > 1011 GeV. Note also that the standard scale for f in string theory is f ~ Mp ~ 1018 — 1019 GeV. This means that one should construct theories with an unnaturally small value of f, and even this may not help unless the parameter f is in the very narrow 'axion window' 1011 <f < 1012 GeV.

Let us now take a somewhat closer look at whether one can actually obtain the constraint f < 1012 GeV in the context of inflationary cosmology. Long-wave fluctuations of the axion field 9 are generated during inflation if Peccei-Quinn symmetry-breaking, resulting in the potential given by Eq. (8.8), takes place before the end of inflation. By the end of inflation, therefore, a quasi-homogeneous distribution of the field 9 will have appeared in the Universe, with the field taking on all values from —\[2nf to at different points in space with a probability that is almost in dependent of 9. This means that one can always find exponentially large regions of space within which 9 ^ f. The energy of the axion field always remains relatively low in such regions and there is no conflict with the observational data.

This feature does not itself remove the constraint f < 1012 GeV. Indeed, when f ^ 1012 GeV, only within a very small fraction of the volume of the Universe is the axion field energy density small enough by comparison with the baryon density. It might therefore seem extremely improbable that we live in one of these particular regions.

Consider, for example, those regions initially containing a field 90 ^ f, for which the present ratio of the energy density of the axion field to the baryon density is consistent with the observational data (i.e. where the density of dark matter is about five times greater than the baryon density). It can be shown that the total number of baryons in regions with 9 ~ 1090 should be ten times the number in regions with 9 ~ 90. One might therefore expect the probability of randomly ending up in a region with 9 ~ 1090 (incompatible with the observational data) to be ten times that of ending up in a region with 9 ~ 90.

However, closer examination of this problem indicates that the properties of galaxies formed in such a region should be very different from the properties of our galaxy. This makes it unclear whether life can exist in the regions with 9 > 1090 [29]. Let us compare the domains with 9 = d0 and 9 = Nd0 at the same cosmological time t in an early universe dominated by hot matter. Since the Universe after inflation becomes flat, the total density during this post-inflationary stage is proportional to t-2, practically independent of the relative fraction of matter in axions and in baryons. This has two interesting implications. The first is that at t ~ 1010 y the total density in both domains will be the same but the baryon density will be N2 times smaller in the domain with 9 = Nd0. In other words, in a domain with 9 ~ 1090, the observable region after 1010 y will contain one hundred times fewer baryons than a domain with 9 ~ d0. As discussed below, this alone may reduce the probability of the emergence of life.

The second implication is related to the properties of galaxies in domains with 9 ~ N90. The ratio nB/nY ~ 10-10 is fixed by some processes in the early Universe, which are not expected to depend on the axion abundance. The main difference between the two domains discussed above is that the relative energy density of non-relativistic particles is N2 times higher in the second domain. Also, at the same time t, the ratio of the energy density of photons and cold dark matter will be N2 smaller, i.e. this domain is colder. The cold dark matter energy density decreases as t-3/2, whereas the energy density of photons decreases as t-2, i.e. t-1/2 times faster. Therefore the period of cold dark matter dominance occurred N4 times earlier in the second domain. The energy density of cold dark matter at that moment was ~ N8 times higher than in the first domain.

Note that the beginning of cold dark matter dominance is the time when density perturbations 5p/p ~ 10-4 start growing. Since they start growing earlier, the moment when they reach 0(1) - i.e. the stage when overdense regions separate into galaxies - also occurs earlier. The density of matter inside galaxies in the future remains of the same order as the density of the Universe at the time of the galaxy formation. This means that the density of matter in the first (smallest) galaxies to be formed in the second domain will be N8 higher than in the first domain, and the density of baryons there will be N6 times higher.

The matter density in large galaxies, which formed later in the evolution of the Universe, should be less sensitive to 9. However, if most of the matter is packed into superdense dwarf galaxies formed in the very early Universe, the total amount of remaining matter - which would be distributed more smoothly like in our own galaxy - may be relatively small. Also, any galaxy of a given mass M will contain N2 times fewer baryons than our galaxy.

Naively, it would seem ten times more probable to live in domains with d = 10 d0 because the total volume of such domains is ten times bigger. However, since the properties of galaxies in a universe with d = 10 d0 are very different from those of our galaxy, it well may happen that domains with d ~ d0 provide much better conditions for the emergence of life than domains with d = 10^o [29]; see also ref. [60].

In order to study this situation quantitatively, one should perform a detailed investigation of galaxy formation in a model with pM » pB, similar to the investigation of galaxy formation in a model dominated by a cosmo-logical constant (pa ^ pM), as discussed in Section 8.5.1. This investigation has been performed very recently [61]. The results obtained confirmed the expectations of ref. [29]: if dark matter is represented by axions with f ^ 1012 GeV, then one is most likely to live in a model where the density of dark matter is about one or two orders of magnitude greater than the density of ordinary matter. This is quite consistent with the observed value PDM ~ 5pm.

This result has two interesting implications. First, it will not be too surprising to find that the standard constraints 1011 < f < 1012 GeV are violated. Second, in the context of the axion cosmology with f » 1012 GeV, it is not surprising that we live in a universe with pDM ~ 5pM. In this respect, such a theory has an important advantage with respect to many other dark matter theories, where one must fine-tune the parameters to obtain pdm ~ 5pM.

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