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Can gravity travel backwards in time?
It may be helpful to think of the universe as a big machine that generates gravity. The machine has levers and pulleys, gears and wheels, pipes and valves. There are puffs of smoke all around it, which are actually particle-antiparticle pairs being created out of nothing for every quantum fluctuation in spacetime. These fluctuations produce the matter/antimatter asymmetry we observe in our present Universe.
The machine is a large and complex system, but it's not perfect. Sometimes the levers get stuck or break, which is what we think happened in our own universe during inflation.
Eventually, gravity will be strong enough to pull the levers back together again by a process called entropic radiation. This is similar to Hawking radiation, but with the important difference that it's one-way in time: after gravity has rewound and reversed itself in spacetime, there would not be much left of our present universe for us to observe.
The process of entropic radiation is analogous to running a VCR tape in reverse.
The video tape would be destroyed by the process. On a cosmological scale, however, there would be many more copies of the universe than we can see today.
The big bang was in fact a collision of the levers on two cosmic machines that were running in opposite directions. The machines collided and merged, while their tapes unwound, forming the new universe we know today.