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In their article Disturbing Implications of a Cosmological Constant, Leonard Susskind and others write:

Certainly, given enough time and a suitable inflaton, recurrences will eventually bring the box to a configuration that could serve as an initial starting point for a standard inflationary theory. The entropy of such configurations is very low. In a typical high-scale inflationary theory, the entropy of the initial de Sitter-like space can be as low as 10^10 . The entropy of the final de Sitter space (assuming the observed value for the dark energy) is of order 10^120 . Inflationary starting points are very rare in time. That in itself is not a problem. Most of the rest of time during which nothing interesting is happening can, from an anthropic point of view, be thrown away. We can also throw away large fluctuations which lead to un–livable conditions. The danger is that there are too many possibilities which are anthropically acceptable, but not like our universe. Let us consider the entropy in observable matter in today’s universe. It is of order 10^100 . This means that the number of microstates that are macroscopically indistinguishable from our world is exp (10^100). But only exp (10^10) of these states could have evolved from the low entropy initial state characterizing the usual inflationary starting point.

I do not understand some points.

  1. What does "macroscopically indistinguishable from our world" mean? On what scale are they indistinguishable? On the scale of galaxy clusters, individual galaxies, stars, planets, individual grains of sand on each planet? That is, there are exp (10 ^ 100) microstates, which at the macroscopic level (starting from grains of sand the size of a micron) have an absolutely identical appearance?

  2. Why among exp (10 ^ 100) possible microstates only exp (10 ^ 10) come from inflation? Entropy should grow over time, right? Does this mean that only a small number of configurations are possible after the exit from inflation? But how is it? Indeed, for example, the density perturbations that caused the large-scale structure of the Universe appeared as a result of quantum fluctuations. And the possible distributions of such fluctuations in space are much larger than exp (10 ^ 10).

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When dealing with this kind of magnitudes, the difference between a grain of sand and the observable universe is just 60 orders of magnitude in terms of mass (and hence roughly entropy). That is $10^{-40}$ of $10^{100}$. So the macroscopically indistinguishable states could indeed be entire observable universes, although of course perfectly replicated solar system with a different outside universe will vastly outnumber these.

The second question is less clear to me. I assume Susskind just means that there are few microstates in an inflation event, which means that a lot of the current microstates are independent of the microstates in the inflation ("new" randomness due to quantum fluctuations). I think a lot hinges on the model here.

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  • $\begingroup$ The entropy of the inflationary state may be small, I agree. This would correspond to a small number of microstates that look the same macroscopically. But starting with inflation, the entropy is constantly growing: a hot soup of elementary particles and radiation is already a state with much higher entropy. There are many more microstates that look the same at the macroscopic level.A state with an entropy of 10 ^ 10 (inflation), which corresponds to exp (10^10) microstates, may well evolve into a state with an entropy of 10^100 (modern universe), which corresponds to exp (10^100) microstates. $\endgroup$ Sep 28, 2021 at 11:26
  • $\begingroup$ Just like ice (a system with low entropy) turns into water (a system with high entropy). What is the point of claiming that only exp (10 ^ 10) out of exp (10 ^ 100) states started with inflation, if all exp (10 ^ 100) states could well start with inflation? $\endgroup$ Sep 28, 2021 at 11:26

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