If one applies the second law of thermodynamics to the Universe[1] as a whole then one might expect that the entropy of the Universe always increases as time goes forward (or more accurately that the entropy never decreases).
But the second law of thermodynamics is based on time-reversal invariant physics so that one would also expect that the entropy of the Universe should always increase as one goes backwards in time as well.
Both these conditions are met simultaneously by a linearly expanding cosmology such that the scale factor $a(t)$ is given by:
$$a(t) = \frac{t}{t_0},\tag{1}$$
where $t_0$ is the present time.
According to Raphael Bousso's Covariant Entropy Conjecture, the future entropy of the Universe is bounded by the area of the event horizon at a distance $d_e(t)$ given by:
$$d_e(t) = \int_{t_0}^\infty \frac{c\ dt}{a(t)}.$$
Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the event horizon, $d_e(t)$, diverges logarithmically as $t \rightarrow \infty$. Thus in a linear cosmology the future entropy of the Universe is not bounded.
Similarly the past entropy of the Universe is bounded by the area of the particle horizon at a distance $d_p(t)$ given by:
$$d_p(t) = \int_0^{t_0} \frac{c\ dt}{a(t)}.$$
Substituting the equation (1) for a linear cosmology into the above expression we find that the distance to the particle horizon, $d_p(t)$, diverges logarithmically as $t \rightarrow 0$. Thus in a linear cosmology the past entropy of the Universe is not bounded either.
Is this reasoning correct?
Does this show that such a linear cosmology is interesting even though the current evidence seems to point to an accelerating Universe with a finite future entropy?
A universe with a finite future entropy leads to strange hypotheses like Boltzmann brains so it seems reasonable to consider other alternatives (see Leonard Susskind's latest talk for example).
1.Instead of "Universe" I should have stated my argument in terms of the causal Universe defined by the volume inside the event/particle horizons of a particular observer.
