Chapter 10

Hawking Radiation

September 28, 2021

Radio frequencies between 20 cm and 1 mm show that there is a background whose spectrum (figure 62) very close to that of a black body at 2.7 K (see, for example, Field (1969». This background appears to be isotropic to within 0.2 % (figure 62 (ii); see, for example, Sciama (1971) and references given there for further discussion).

(i) The spectrun of the microwave background radiation. The plotted . points show the observed values of the • excess’ background radiation. The solid line is a Planck spectrwn corresponding to a temperature of 2.7 oK. (ii) The isotropy of the microwave background radiation. The temperature distribution along the celestial equator is shown; more than two years of data have been averaged to obtain these points. From D. W. Sciama, Modern Oosmology, Cambridge University Press, 1971.350

The high degree of isotropy indicates that it cannot come from within our own galaxy (we are not symmetrically placed in the plane of the galaxy) but must be of extragalactic origin. At these frequencies we can see discrete sources some of whose distances are known from other evidence to be of the order of 10 27 cm, so we know that the universe is transparent to this distance at these wavelengths.

Thus radiation which is produced by sources at distances greater than 10 27 cm must have propagated freely towards us for at least that distance.

Possible explanations of the origin of .the radiation are:

(1) the radiation is black body radiation left over from a hot early stage of the universe; (2) the radiation is the result ofsuperposition ofa very large number of very distant unresolved discrete sources; (3) the radiation comes from intergalactic grains which thermalize other forms of radiation (perhaps infra-red).

Of these explanations, (1) seems the most plausible. (2) seems im- probable, as there do not appear to be sufficient sources with the right sort of spectrum to produce an appreciable fraction of the observed radiation in this frequency range. Further, the small scale isotropy of the radiation implies that the number of discrete sources would have to be very large (of the order of the number of galaxies) and most galaxies do not seem to radiate appreciably in this region of the spectrum. (3) also seems unlikely, since the density of interstellar grains which would be needed is very large indeed. Although (1) seems the most probable, we will not base our arguments on it, since to do so would be to presuppose that the universe had a hot early stage.

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At centimetre wavelengths, the largest ratio of opacity to density for matter at reasonable densities is that given by Thomson scattering off free electrons in ionized hydrogen. Thus the optical depth to a distance v will be less than where K is the Thomson scattering opacity per unit mass, p is the density of the matter, and Y,. is the local velocity of the gas.

The redshift z of the matter is given by z = KaY,. - 1. Since no matter has been seen with significant blue-shifts, we shall assume KaY,. is always greater than one on our past light cone, out to an optical depth unity.

As galaxies are observed at these wavelengths with redshifts of 0.3, most of the scattering must occur at redshifts greater than this. (In fact if quasars really are cosmological, the scattering must occur at redshifts greater than two.) With a Hubble constant of 100Km/secl Mpc (,.., 10 10 years- I ), a redshift of 0.3 corresponds to a distance of about 3 x 10 27 cm. Taking this value for VOl the contribution to the integral (9.9) of the matter causing the scattering is 3.7 x 10 28 ftJ, p(Ka V a)2dv, 1l. while the optical depth ofthe matter between V o and VI is less than 6.6 x 10 27 f 1l 1 p(Kay")dv. 1l. Since KaY,. ~ 1, it can be seen that the inequality (10.6) will be satisfied at an optical depth ofless than 0.2.

If the optical depth of the universe was less than 1, one would not expect either an almost black body spectrum or such a high degree ofsmall scale isotropy, unless there was a very large number of discrete sources which covered only a small fraction of the sky and each of which had a spectrum roughly the same as a 3 oK black body but with much higher intensity. This seems rather unlikely. Thus we believe that the condition (4)(iii) of theorem 2 is satisfied, and so there should be a singularity somewhere in the universe provided the other conditions hold.

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