You may be asking yourself "How does the space time curvature impact the probability of the universe?" And on seeing the title of this article, you may feel your anxieties have been put to rest, as they will be addressed herein. However, you are mistaken, for this article is dedicated to considering the evaluation of the probability of the universe with regards to the space time curvature, as opposed to the impact of the space time curvature on the probability itself, which is worthy of its own article.

So to begin, we must consider the fundamental assumptions about physical reality, and explore those assumptions with regards to the probability of the universe just in case our current theoretical models of space and time turn out to be incomplete.

For canonical probability, the probability is considered as the chances of one thing being selected from a larger, finite set of things.

Probability theory is enumerative combinatorial analysis applied to finite sets. Thus, we need to establish the finiteness of the set of possible configurations the universe can reside within. If the universe (or a part of it) can not be described as a finite set, then finite probability mathematics will not apply.

One can consider many types of probability, such as the probability of finding particular sets from among a larger finite set, or the probability of finding a particular instance as belonging to a larger set (as the thermodynamics equations tell us) but here we are considering the choice of one particular thing, a set unto itself, which represents the specific microstate that specifies every aspect of physical reality at a particular point in time. In thermodynamics, we consider the probability that a particular microstate belongs to a macrostate with a set of observables (temperature, volume) with a priori equal weights between those microstates. However, the probability of the universe is concerned with the probability of a particular microstate irregardless of macrostate ensembles.

A microstate specifies the exact location and momentum of every particle within the system, including every atom, photon, and quark.

A first question which must be overcome is the finiteness, or graininess, of tiny sizes. We may ask: in a universe with a single atom, what are the chances it is at a specific position at a specific point in time? For we could imagine this single atom, and specifying its position to higher and higher degrees. For example, we might say the atom is located at (34.55, 36.12, 82.33). Then we might ask among all the possible microstates this atom could be located- what are the odds of this? But then, what about the atom located at (34.5512, 36.1242, 82.3367)? Is this located in the same place, or a different place? If we change the last significant digit, is this atom in a new location or not?

Classical thermodynamics had this same problem, and solved it by saying "everything within these bounds are considered the same thing". So it would specify a box in phase space, a very small one, for which all things within the box would be considered to be the same thing. This is fine and handy, but begs the question- is this now reality, or our representation of reality? for if we are putting approximations on things, is that not moving away from reality and towards a human conceptual framework which approximates, but does not equal, reality? This question bothered Gibbs greatly, especially because his equations did not match up with experimental reality with regards to heat. Both of these problems were solved with the introduction of quantum theory, which both specified that there is a finite size (planks length) by which the momentum and position of an atom is fundamentally indistinguishable beneath a certain discreteness. In other words- reality is like a bunch of tiny boxes (but with momentum as part of the box axis), where the particle can exist in one, or another box, but does not exist in between these boxes. This is like pixels on a computer screen.

The universe has a graininess through space with a distance of 1.5 10e-35 meters, and a graininess in time of 10e-43 seconds, which are called the Planck length, and Planck time, respectively.

This particular question though is worth exploring a bit more, but we will merely comment on it here and leave it open for discussion. If the "future reality" of the universe can not distinguish beneath these momentum and positions, is it possible that it is "self aware" of these differences? That is, even if the difference makes no difference in terms of its interaction with the environment, and never will, could it be possible that the atom itself is aware of these sub-plank length distances? This is a worthy question, and I hope it will be addressed in these forums.

Since we are concerned here with the future trajectory of the universe with considerations of its past, it is truly "planck grained" and does not have any sort of physical reality at positions beneath these sizes (with respect to its future evolution). Therefore, this puts a cap on considering the set size of a particular microstate bounded within a specific volume. That is to say, we can apply normal probability theory because we have a finite volume and finite number of denumerable positions in the phase space. If space were classically defined with rational (and infinitely fine precision fractions) then we would be left with the dilemna of statistical mechanics, and resort to mathematics which certainly are only approximations.

But now let us consider a finite bounded volume of space. The atoms move about within its environs, changing positions, and hence, their probability function (as each microstate has its own, unique, probability at that specific point in time). Is it possible to consider this finite bounded volume completely within a mathematical framework?

Yes and no. The problem is, other parts of space may be impacting it, such as a particle which flies in from the side. This then changes the internal configuration of the space which we were considering, and thus changes the probability. But can we not model this "side particle" as a possible probability, thus leaving us with an intact probability equation for the finite volume?

We can, but we have moved away from "stochastic behavior unto itself" and into "stochastic behavior from the outside world". As we know, for a quantum system it can be completely described with stochastic equations (including decoherence), and thus a quantum ensemble can be completely understood with regards to its probability. The mathematics may be intractable, but they are complete unto themself. Now if we introduce this "lack of knowledge of the outside world that we represent with a statistical representation" the nature of the probability has changed completely- it is no longer fundamentally reality, but merely of our representation of reality. It is similar to the dilemna proposed above about the infinitely fined grained nature of a classical universe that is solved with an approximation box. It is a useful model still, that may provide us with insights about reality, but no longer can be considered complete.

Thus, to model probability on a fundamental level we need to have a finite space, otherwise we are moving away from fundamentals and towards an approximation. A real world example of this is considering the chances of life existing on earth. If we consider the mathematics of the life evolving here on its own, from scratch, it is completely different than if an asteroid came from thousands of years ago carrying some DNA on board. Considering life evolving here on earth is like considering just the system of the earth and ignoring the possible asteroid.

So in order to consider the asteroid, we must move our solution space to a larger domain, namely all of the stars which could have sent an asteroid our way during the evolution time of the universe.
To make things worse, what if a particular beam of light were fired into the earths atmosphere from far away, caused by a quasar or other anomoly, which caused life to begin its formation? The arrival of life could have arrived at the speed of light.

As you can see, we must reach backwards on the light cone from the present to reach all parts of the universe which can reach the present. This is still, for every time slice we go backards, a finite countable set. Or is it?

It depends on how far back we need to go. If we need to go back forever, then we are exceeding the finite set size, and moving to an infinite set size. Thus, our probability model would need to become a local approximation, both bounding the possible starting conditions at some arbitrary time in the past, and bounding the edges of the spatial volume under consideration. These would be the conditions for a steady state universe, and this was considered to be the actual model for the cosmos by many leading thinkers.

If the universe is expanding, does this impact our calculations? It removes from consideration some parts of the universe- they are moving away from us faster than the speed of light, and thus would not figure into the countability of the phase space for probability measures. But this is not particularly relevant since we are looking backwards along the light cone, not horizontally through space, unless there were some form of inflation/deflation which bent the light cone to either make the past have bounds (in the infinite universe) or reach infinity (as the starting universe perhaps is infinite in future extent but bounded at the moment of creation).

So it appears that the present space time continuum, with the universe expanding away from us at the speed of light, is only relevant in that we do NOT have to consider points beyond this boundary. If the universe were to begin collapsing, then light from distant places would arrive, and in fact it would start heating up rapidly and burn everything up. This would impact the probability of the local region, and necessicate reaching further and further out for mathematical considerations to reach the furthest points of light and their histories which were now coming in. Fortunately, this does not appear to be the case in the present, but could in the future if the universe is closed.

The space time curvature thus impacts the enumerability of the phase space for considering the probability. The possible things that should be considered are:

graininess of universe
steady state or not
infinite or finite universe
the speed of light
closed (like a sphere that wraps back onto itself)
open ( consantly accelerating outwards)
flat (slows in acceleration)

As we have established, it appears the universe is grainy with regards to plancks length and plancks time. For a different mathematical universe, it may not adhere to these graininess measures and finite probability analysis would not be applicable.

For a steady state universe, the backwards propagation of information goes to infinity, so that we can consider the probability of a particular configuration, but only with regards to specific earlier times which are fixed arbitrarily relative to the present time. Furthermore, we must put bounds on the spatial volume if the universe is infinite in extent, for similar purposes. A steady state finite universe would need to have a temporal bound, but not a spatial bound, for finite probability theory to be applicable.

For a closed universe with a starting time, the constraint on time is provided by the universe itself, and the constraint on space is limited by the speed of light reaching back on the light cone to the start of time. If there were not a speed of light, then the entire universe would have to be considered with regards to probability measures, and it would not longer be tractable with a finite probability analysis. As the universe begins to close back in on itself though, the consideration of probability needs to take into account areas which had previously been moving away faster than the speed of light, and the countability would be contingent on the finite or infinite size of such a closed universe. At specific points in time before it began to accelerate back inwards, this would not be a concern.

For an open universe, it is subject to the constraints of the closed universe but without the problem of the universe collapsing back on itself raising the enumerability from finite to infinite.

For a flat universe, it too is subject to finite probability analysis because of the speed of light constraining the backwards influence area to a finite size. It is arguable though for the starting instants of an infinite universe how the backwards light cone should be considered (does it reach to infinity, or not?)

There is a concept I shall call "feeding the void" for preserving low probability structures in an open universe. For considerations of the probability, what about light escaping from here and going into space? This certainly must be included, for there may be important messages being sent out into space. If these light beams move out into interstellar space, and in fact do not impact any other object, they retain information, and hence retain probability as they move out. Let us suppose some of these beams move into space where they are now headed towards objects which are accelerating away from the tip of the beam faster than the speed of light. That is to say- these beams will NEVER reach anything. It seems that the universe has "preserved" probability here, but it is difficult to say, for it is also possible that the light changes character under such extreme deformations of space time, and is perhaps "collapsed to a flat wave function" and thus does not preserve information. It appears though, that it does preserve information at first glance. Thus, it would be useful for a civilization interested in leaving some residue in the universe to fire coherent beams of light out into interstellar space where it would not impact onto any physical object before being "trapped in the void" where it will never impact a solid object again yet retain a legacy of the civilization which created it.

Probability theory explained:

http://www.toequest.com/forum/showthread.php?t=1020

References

http://www.physlink.com/Education/AskExperts/ae281.cfm?CFID=5302737&CFTOKEN=91188874

http://www.efunda.com/math/probability/probability.cfm

http://www.dartmouth.edu/~chance/teaching_aids/books_articles/probability_book/book.pdf