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For an eternal dS universe, the entropy of an observer in the static patch goes as the area of the horizon which for 3+1 D goes as $S_H = \frac{Area_{H}}{4} \sim H^{-2}$, as given by Gibbons & Hawking.

Since we are considering an eternally dS spacetime, the area of the horizon for the observer does not change with time, though more and more 'information / degrees of freedom' keep going out of the horizon, with time (but at a constant rate).

Now, although we have no concrete idea of the microscopic origins of the entropy, the standard notion is that it counts the amount of information inaccessible to the observer because of his cosmological horizon.

1) Is this notion of deSitter entropy (and the formula which calculates it) something that is true for every point in time? Because there is no preferred time slicing / time coming to an end here. If I am not mistaken, Bousso's spacelike projection theorem (Sec 1.1, Fig. 5) arising out of the covariant entropy conjecture agrees with this.

2a) How does the entropy remaining constant in time make sense with us 'not being able to access more and more information' (pertaining to regions beyond our horizon) with the passage of time?

In the same paper as above, the degrees of freedom of the volume inside the horizon on a specific spacelike slicing is linked to the area as $N_\textrm{DOF on V} \leq A/4$. Is this what the horizon entropy counts/bounds?

2b) If it is so, is this not an odd measure of horizon entropy since these are actually the degrees of freedom which are accessible to us? Is this so because of the fundamental difference between cosmological horizons and BH horizons?

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The observable universe is only approximately de Sitter. The de Sitter spacetime is a vacuum solution and is a mathematical idealization of the physical spacetime we observe. As galaxies cross the horizon is expands slightly, very slightly ub fact. The observable universe will asymptote to a pure de Sitter configuration over the next $10^{100}$ years. As this diagram below illustrates the event horizon has been growing since the big bang, but that growth is slowing down.

enter image description here

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    $\begingroup$ Hi Lawrence! Thank you for the response. I do agree with what you have said. However, my question was pertaining to eternal de Sitter spacetime and not to our universe. $\endgroup$ – Raj Kunkolienkar Oct 20 '16 at 17:26

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