It might be observed that there would have been many more primordial black holes (PBHs) if the spectrum of primordial fluctuations, f (n), were tilted to increase their amplitude on small scales . This does not refute S directly unless the inflationary model has a parameter that can be varied to achieve this tilt. The Standard Model does not, but it is reasonable to examine whether some plausible extension of it, E, might do so. One plausible extension is to add a field that could serve as the inflaton. The spectrum of primordial fluctuations may then be predicted as a function of the parameters of E. Thus S is refuted if (a) some model E of inflation is ob-servationally confirmed and (b) that particular model has some parameter, pinf, that can be modified to increase the total number of PBHs produced. Given the accuracy expected for observations of the CMB from the WMAP and PLANCK satellites, there is a realistic possibility that these will distinguish between different hypotheses E and measure the values of their parameters.
In the standard 'new' inflationary scenario , there is no parameter that fulfils the function required of pinf. There is the inflaton coupling, A, and it is true that the amplitude of f (n) is proportional to A, so that the number of PBHs can be increased by increasing this. However, the size of the region that inflates, R, scales as eA 1/2. This means that A should be at the lower limit of the range of values for which galaxy formation occurs. An exponentially larger universe, which produces black holes only through supernova remnants, still has vastly more black holes than an exponentially smaller universe with many primordial black holes. In fact, if the observations confirm the new inflationary scenario, S is refuted if A is not tuned to the value that maximizes the total production of black holes in the inflated region . Because of the exponential decrease in R with increasing A, this is likely to be close to the smallest value that leads to appreciable black hole production. This should correspond to the smallest A that allows prolific formation of galaxies .
This seems consistent with the actual situation, in which there appears to have been little PBH production. Therefore, given that Q = 5p/p & 10_5, the primary mode of production of black holes seems to be through massive star production in galaxies that do not form until rather late. However, there are non-standard models of inflation that have parameters pinf that can be varied in a manner that tilts f (n) so that more PBHs are created without decreasing R [104-109]. If future CMB observations show that standard new inflation is ruled out, so that only models with such a parameter pinf are allowed, then S will be refuted. This is a weaker argument than the first one, but - given the scope for increased accuracy of the CMB measurements -such a refutation is plausible.
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