# Exponentially accelerating universe, dark energy, the universe's temperature and event horizons.. (sorry in advance!)

Ok.. assume that space-time is accelerating away from itself with a>0 and.. jerk - j⃗ (t)>0 (sorry - don't know how to write "a dot"! :s )..

If this continues on without the rate of acceleration ever stopping, or decreasing, the ambient temperature of the universe will fall due to its expansion in volume, approaching absolute 0 (as ours has been doing up until now). However, as the acceleration of space-time increases, at some point in time the Unruh effect would start to become significant with regards to the background temperature of the observable universe (as we assume the rate of acceleration only increases).

Would this then mean that after some time - at some extremely fast rate of acceleration - the universe would begin to "heat up" to an observer? (given jerk>0, does the background temperature of the universe plot as a convex curve with respect to time?) If so.. as the rate of acceleration keeps on increasing, could the background radiation increase enough to form a singularity?

Finally (assuming the above makes sense!), given that the observed cosmic event horizon shrinks and gets ever closer to the observer as the acceleration of space time increases, and that at some point in time the background temperature of the universe increases to a level at which singularities begin to form, if "a dot">0 still stands, the cosmic event horizon and black hole event horizon would be moving in opposite directions - one towards the observer and one away for the observer. If the two event horizons move in opposite directions, what does this mean? (I get lost here.. would this affect "a dot"?!)

(sorry if I lost you somewhere up above, if I did - could you please tell me where and why? Thank you! :D )

• The site has MathJax active to allow LaTeX-mathmode-alike markup (so $j = \dot{a}$ is written as j = \dot{a} placed inside single dollar-signs. You'll find some links in the sidebar when asking a question. You might start with physics.stackexchange.com/editing-help . – dmckee --- ex-moderator kitten Apr 24 '16 at 3:10

Good question.

Not clear to me it reaches infinity, or high enough for a singularity. It seems that if one is talking about non accelerating, i.e. Inertial, observers in a de Sitter spacetime, it is a constant temperature determined by the cosmological constant. The cosmological constant is constant in de Sitter, and per current observations. Please note that even in that case the universe is expanding exponentially, with a, the scale parameter an exponential. And thus also a dot and a double dot. If it changes the solution is different, and may indeed be worse.

If you also accelerate it gets worse, you increase the temperature you see.

Casadio et all in a 2011 paper in arxiv on the Unruh effect in de Sitter space show that for certain observers, called KODAMA observers, they see a thermal bath with temperature approximately H/2pi. H is proportional to sqrt of lambda, the cosmological constant. It is also affected by the red shift, the so called Tolman factor, and the proper acceleration of the observer (wrt comoving coordinates in de Sitter). There's other papers on the topic.

Those observers maintain the same distance from the horizon at all times. So they are accelerating observers (with respect to the of comoving frame the expansion- similar to an astronaut staying at a fixed distance above a black hole horizon, where he also sees the horizon's radiation). For inertial observers, i.e., not accelerating, in the de Sitter space they will still see a temperature, though less than if accelerating.

Now, keep in mind that these effects are not coordinate or observer invariant, they depend on the observer's frame of reference. But as I understand it although dynamical or apparent horizons are observer dependent, real singularities (not simply horizon coordinate singularities) are invariant properties of a spacetime, not dependent on the observer.

As for the black hole in that spacetime and its horizon, need to find the black hole solution in a de Sitter spacetime. There papers on it, perhaps somebody else can summarize the findings. I'd conjecture (just a guess) that the black hole would simply be following the spacetime geodesic so, i.e. Acts as an inertial observer, for the most part, and it's horizon would stay about the same. You as a roving observer may see radiation from both, eg, if you are a Kadama observer. If the black hole gets too close to you, and you can't stay away from the black hole horizon too bad. If you can, you may be ok till you get too close. As for the Unruh horizon, doesn't look like infinite temperatures, though maybe the Tolman factor, due to the Doppler effect, may blue shift without bound. I was not quite sure from the paper.