How should I respond to an atheist who denies creation,
claims there has been an infinite number of universes before this one, and believes that the universe was made not by God but through the "Big Bang," part of the universe's endless cycle of expansion and contraction?
There are several points you could make. First note that in order for the universe to fall back in on itself there would have to be enough mass for gravity to overcome the outward force of the expansion. But according to the best estimates scientists have, there isn't enough matter to cause such a contraction. Consequently, if the universe cannot contract, there could have been no previous big bangs and no endless cycles. At most you could posit one big bang.
There is nothing that would cause the universe to expand again after it had collapsed. If such a collapse occurred - a big crunch, so to speak - the universe would remain in an unimaginably compressed state, never capable of expanding again. The tremendous gravity exerted by such a great mass would prevent any expansion. Black holes, which are most probably collapsed stars, are an example of this phenomenon on a much smaller scale. A black hole's gravitational pull is so strong not even light is able to escape from it.
If all the matter in the universe were compressed into a single black hole or something like a black hole, the gravitational forces would be incalculable, and it is hard to imagine anything that could overcome them. If they couldn't be overcome, nothing could escape in the form of another big bang.
Even if there were a mechanism to re-expand the universe, each cycle of expansion and contraction would lose energy because of entropy, the tendency to of matter run down, much as a spring-driven clock runs down. The extent of the universe's expansion would diminish with each cycle - consider how swings of a pendulum slowly diminish - and eventually the universe would cease expanding entirely, its mass remaining collapsed. There could never be an infinite number of successive expansions and contractions.
Keep in mind that the idea that the universe came into existence as a result of a cataclysmic explosion of highly compressed matter is not inconsistent with the Catholic teaching that God created the universe. A big bang could have been part of his method of creation.
But an atheist has a problem here. If there really was a big bang, and if there could not have been an infinite series of big bangs before the present one, then there are only two possibilities: Either God created matter out of nothing and (arguably) set things going through a big bang - this alternative destroys atheism - or matter existed for an infinite amount of time in a primordial black hole state. But if it existed that way for an infinite amount of time, it never could have exploded in the big bang.
If an infinite amount of time passed without a big bang, then every combination of protoplasmic matter and energy would have existed at one time or another within that black hole, without any one combination leading to the big bang. All the combinations would have been tried, and none of them would have produced the explosion. (Remember, this presumes an infinite amount of time.) If none of the combinations could have produced a big bang, and if a big bang occurred anyway, it could have arisen only from outside intervention, not from anything inside the black hole.
Since the whole of the universe - all matter and energy, even space Itself - was compressed into the black hole, "outside" must imply a non-natural force, a force above nature, and that is the definition of supernatural. No matter which alternative an atheist takes, he ends up with God.