Rick's Cosmology Tutorial: Chapter 20 Abstract

The Entropy Of The Universe

According to the Big Bang theory, the universe is supposed to have started in a highly dense and extremely hot state consisting of radiation and particles in random motion, devoid of structure. And yet here we are, 13.7 billion years later. We inhabit a planet which is replete with natural order, not least its flora and fauna, including ourselves. It is not immediately obvious how this apparent transition from disorder to order can be reconciled with the second law of thermodynamics, namely that the entropy of the universe should never decrease.

In Appendix B1 we show that the entropy of matter collapsing under gravity does indeed decrease. In collapsing, matter moves to a state of lower potential energy, and, it turns out, to a state of lower total energy. Hence, in order to collapse, there must be some mechanism for removing energy from the collapsing matter. This is provided by some cooling mechanism, and the energy is lost in the form of radiation. This radiation carries away at least as much entropy as the matter looses. Consequently the second law of thermodynamics is respected, and entropy overall does not decrease. Nevertheless, the entropy of the matter involved does decrease, and this is the origin of orderly structure within the universe.

In this Chapter we show that 99.9% of the photons in the observable universe originated from the Big Bang, and there are 10^89 of them. The entropy of the Big Bang photons and neutrinos are about the same, and total 7 x 10^89 in units of Boltzmann's constant. The entropy of ordinary matter, after recombination but before star formation, would have been about 3 x 10^81 in these same units. If all the ordinary matter were now in the form of stars of roughly solar mass, its total entropy would be ~10^81 in these units. Thus, the formation of structure in the universe has been accompanied by a roughly 3:1 reduction in the entropy of the ordinary matter involved.

The above results imply that the photon entropy has increased by ~3.5 x 10^86 since recombination, whereas that of ordinary matter has decreased by ~2 x 10^81. Hence, the entropy of the universe as a whole has increased, as it must.

The contribution of dark matter to the universe's entropy depends upon the mass of the individual particles involved. If dark matter consists of particles lighter than ~2eV then their entropy would dominate that of the Big Bang photons. If dark matter consists of particles comparable in mass to the electron, their entropy would be negligible compared with that of the Big Bang photons and neutrinos, but still far larger than that of ordinary matter. If dark matter consists of particles more massive than the nucleons, then its entropy is smaller than that of ordinary matter.

We have avoided discussing the entropy of dark energy. In the absence of any certainty about the nature of dark energy this would be premature.

The entropy of ordinary matter is, roughly speaking, proportional to the number of particles, and hence proportional to the total mass. In contrast, the entropy of a black hole is proportional to its surface area, and hence to the square of its mass. For sufficiently large masses it follows that the entropy of a black hole will far exceed that of the matter from which it formed. Thus, the entropy of a solar mass black hole is ~10^18 times greater than the entropy of the Sun. We can ask three interesting questions:-

Most intriguingly it turns out that the answer to all three questions is roughly the same, namely of order of the Planck mass squared divided by the mass of the nucleon, which is of order 10^11 kg.

The entropy inventory of the universe changes dramatically if the entropy of black holes is included. Thus, just one solar mass black hole per galaxy is enough to increase the entropy of 'matter' (including the black holes) by a factor of ~10^7, although the CMB would still (just) dominate. However, galaxies may contain, say, a million solar mass black holes, as well as a single, central supermassive black hole of 10^6 to 10^9 solar masses. The entropy of the former, though huge compared with that of the equivalent mass of stars, is negligible compared with the latter. The entropy of 'matter' (including black holes) might be increased by a factor of ~10^25 if every galaxy contains a single supermassive black hole of 10^9 solar masses. These black holes would then dominate over the entropy of the CMB by a factor of ~10^17.

These observations have a bearing on the question, "by how much has the entropy of the universe increased since the Big Bang?" If the entropy of black holes is included, the answer might be, "by a factor of ~10^17". However, if black holes are excluded then the entropy of the universe, dominated by the CMB, has increased by a paltry fraction ~0.001 - simply due to stellar production of photons. The amount by which the entropy of ordinary matter has decreased is five orders of magnitude smaller still, and yet it is this entropy decrease which has given rise to all the structure in the universe. It would seem that all structured matter in the universe is but a thin scum floating on an enormous ocean of chaos.

If the entropy increases so dramatically when a black hole is formed, why is the second law not violated when the black hole evaporates? The answer lies in the fact that the bulk of the mass-energy which results from black hole evaporation is in the form of extremely large numbers of very low energy photons, neutrinos, etc - i.e. with a strong bias towards the lightest particles or quanta. It is shown, albeit via a rather heuristic argument, that the entropy of this radiation is comparable to, but slightly larger than, that of the black hole.

Finally, we have avoided the question of whether information is preserved in the process of forming and then evaporating a black hole. At one time it was common to believe that information could not be preserved, on the grounds that the thermal nature of the Hawking radiation would not provide a vehicle for the conveyance of information. However, Hawking has recently changed his mind on this matter, and many others have long believed that information would be preserved. The issue hinges on whether the black hole can be regarded as a quantum mechanical pure state, and upon the unitarity of the S-matrix. These matters are beyond our scope.

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The Bullet cluster (1E 0657-56) consists of two colliding clusters of galaxies. Whilst the stars in the clusters generally just pass through each other like ghosts, the ordinary matter in the form of gas clouds interact quite strongly and end up lagging behind the stars. But the bulk of a cluster's mass is in the gas clouds, not the stars. The result is that the greatest concentration of ordinary matter is in the middle, between the two lobes representing the two clusters. This can be deduced from electromagnetic emissions, including X-rays. Dark matter only interacts weakly, via gravity only, and would therefore be expected to stay with the stars. Gravitational lensing effects imply that the total mass is concentrated in the two separated lobes. This is currently one of the most compelling arguments in favour of cold dark matter (and against a modified gravitational force law explanation of galactic rotation curves).