Disclaimer: The following is poorly thought out. For this thesis to be correct there would need to be 3-4 major assumptions that must be correct, and these assumptions stand on weak grounds. The following should be treated as an fun piece of thinking, not as fact.
Nobel Prize winner Roger Penrose seems to be convinced that our universe is cyclical. Under his theory Big Bangs are the beginning of a new eon and there are many cyclical Big Bangs. After an eon begins inflation expands matter spreading it outwards. Eventually black holes swallow that matter and later hawking radiation dissipates all of the black holes leaving nothing but photons bouncing around. Once the universe is nothing but photons then there begins a new eon and another big bang. I am skipping many steps but this is the short hand version.
This is a very controversial idea. But let us assume it is correct. If an intelligent society recognized this pattern of the universe and it was in fact verified that this occurs, then you could have a way of one way communication to the next eon. Let me explain.
The way we identified the Big Bang was the origin of the universe was through Cosmic Microwave Background Radiation. We looked up at the stars and saw that it was formed around a specific area, the universe was spreading outwards from a point in the sky. Astrophysicists have studied the spread of that material and are able to get an idea of what the universe was like just seconds after the big bang by studying the motion of the spreading material.
If there were cyclical big bangs it would be a very chaotic and energetic moment for the universe when a new eon was beginning. It would be very difficult to control the matter. But like a pyrotechnic who places the explosive, colorant, and oxidizing agent into a firework to ignite a perfect explosion in the sky, previous civilizations could have figured out how to construct a big bang to convey information. Just bind up all the material in the right way before the next big bang and you could spell out concepts in the stars.
Now I know what you are thinking, the initial conditions of the universe are too chaotic. You couldn't expect to calculate out how the material would spread and structure all of the interactions of the matter once the big bang occurred without unreasonably powerful computers. Plus, isn’t everyone dead at this point? This would require an enormous amount of calculating power at the end of the last eon in order to get some message transmitted to the next eon.
Well, everything might be swallowed up by black holes in the very final moments of the universe, but you might be able to organize matter before the end. There could be a period in which intelligent life forms are able to direct the collapse of the material and control the way it was organized before the very last moments, allowing a bit of information to be conveyed into the next big bang
The computation needed to organize material to spell out concepts into the stars following a Big Bang would be astronomically high. Interestingly, the perfect time to calculate such a difficult problem might just be at the end of the universe before the next Big Bang.
The type of computer fit to calculate such a problem would need to be able to predict the interactions of many chaotic collisions, advanced supercomputers would have to calculate hundreds of quadrillions more calculations than we are capable of now. However, at the end of the universe the computational power present in the final moments would be way more than the the computing output we have now. This relates to Landauer’s Principle in computer science. The fact that computing gets easier in low entropy environments leads some researchers to believe there would be incentives for a civilization narrowly focused on computing to arbitrage this fact of the universe when the universe is in its final moments.
Anders Sandberg has been popularizing the Aestivation Hypothesis, the idea that intelligent life lays dormant for trillions of years and sparks up back to life in the final moments of the universe to conduct many more computations than they otherwise could to take advantage of the good computational conditions at the end of the universe.
You could get very “woo woo” with this and jump to crazy conclusions. Imagine there are many intelligent creatures that come online when the music begins winding down. They could realize the next party is starting, a new Big Bang is going to occur. Before the next party starts they could be scrambling to encode the ”genetic information” necessary to bring them back online in the next Big Bang into the astral fallout that spreads from the next Big Bang. They could try to increase the odds that someone like us would bring them back into existence in this eon by writing an instruction manual into the spreading star dust that immediately follows a Big Bang. This could even be an evolved mechanism that increases the fitness of an information system, life form, or replicating AI.
Evolving for replication is core to many organisms, inorganic or organic. Controlling the spread of the Big Bang to then allow future generations to read the tea leaves and piece together the ways to bring you back online would be an evolved mechanism for inter-eon evolution.
But there would likely be a competition at the end to get ideas to spread or to write the formula to bring your alien organization back online. After all, someone might have the same idea and want their own genetic code written into the stars for an organization like us to read . The agents writing messages into big bangs might not be alone. This creates several issues.
Other issues might be that, due to constraints in your position in the universe, you might not be positioned well enough in the universe to get your portion of the canvas painted fast enough. The part of the Big Bang that you might influence may be only a small corner. Or more likely, the canvas could be painted by different hands simultaneously in the final battle to get a message through to the next eon.
We should expect either one group of intelligent creatures entirely wrote the message in the CMBR alone or it was scrambled together at the final moments. Considering the importance of the message you might expect some cooperation if the agents realized that no coordination would mean that the message would be ruined, or be incomprehensible. A zero sum game in which all players lose from no cooperation would put heavy pressure on all players to cooperate, or at least the best evolved strategies in this inter-eon competition would be the ones which have evolved advanced forms of cooperation in the final moments to ensure a message was encoded in the stars. This competition could have been occurring for infinitely many eons.
Next comes the objection: Ok sure, lets say the superaliens of the last eon wrote some messages for us in the stars, how can we verify that they did this from the CMBR we have? How do we even begin interpreting their language?
Well analogy is probably the core route of information transfer. Trapped on an island with a tribe who speaks a different dialect? Point at the ground, utter "coconut" and soon everyone understands this foreigner calls that round hairy fruit a coconut. The tools they would use to communicate would probably be something like logic or math. They might bank on the pre-existing universal constants and use them as a form of letters or medium for information transfer.
The way you might differentiate the noise from the the information they wanted to communicate would probably be by studying abnormalities. If there are strange and unexpected phenomenon that don't add up in how the Big Bang spread or otherwise peculiar unnatural findings that showed the motion of the stars might have been drawn by a hand, this would give some credence to the belief that we have pen pals from the last eon.
The task of identifying the message would be very hard. However, we can read back pretty far into the universe's beginning. There is a chance that one “piece of sand” is positioned in such a way as to make you think someone else was walking on this sand on the island before us. Or to use the fireworks analogy, it might be that there are certain periodic or repeating patterns that seem to not occur at a rate you would expect from a random stochastic system.
Plus the further back we can read the less tainted the information would have been. The clearest window into the message the previous eon might have sent would be the time nearest to the beginning of the Big Bang. The less time the message spent getting corrupted through random chaotic interactions that are difficult to calculate, the more clear the potential information would be that was transmitted to our eon. Seasoned players in the inter-eon information transmission game would probably recognize that someone would figure this out and they would only need to look as far back as 0.0000000001s after the Big Bang to see the message written as the message is still written there even billions of years later.
It would be really hard to tell but massive computing projects might get us to a point where we can read the tea leaves. If I were an intelligent super civilization looking to get reincarnated into the next eon, I might try using geometric patterns as a way to convey information. I might spell out certain concepts using math or geometry by inserting simple abnormalities in the way the big bang spread to indicate the existence of the big party that occurs at the end of the eon. If we ran a computer and found strings of logical patterns spelled out in strange ways to communicate a concept or idea from our brothers from another mother, the most important task of our species at the moment would be deciphering the message.
This is very speculative but an interesting way of figuring out what the expected conditions present before the Big Bang would have been, would be to try and model the collapse of the universe. Whatever shape, state, or position the universe will come to in its final moments if we predict the future from now may be the exact way that the universe would have collapsed in the last eon before us. If we notice irregularities between our modeled prediction of how our universe would collapse and the observed state prior to the creation of the Big Bang, this may indicate intelligent intervention into the state which existed prior to the Big Bang.
There could be core ideas that are shared throughout time from one eon to another. Such as, "y=mx+b" or basic formulas of physics, truths about the universe, or whatever big things we haven't thought of yet. You could imagine agreements between the players at the end of the eon where all players agree, "Ok we got a big jump ahead when the last eonites put this formula in here, so we are going to keep this formula in the next big bang. But lets also add this little bit about dark matter. That was a big discovery that could help them in the next eon."
I would imagine that if you got your stuff in a pile enough to do the ridiculous amount of computing necessary to encode messages into big bangs, you probably are an entity which is able to cooperate with other corners or embodiments of the universe because you guys are all playing for the same eon team. Just different representations of the same underlying spacetime matter shaped in different ways, an extension of the same field. But who knows.
Useful ideas:
The Arecibo message was our attempt at sending a message from a radio telescope in the 20th century to tell aliens about our existence. The idea of communicating using a universal language is represented here. Humans attempted to propagate a message and use universal symbols or a type of universal language to communicate an idea.
This is an image of what a team of redditors made when collaborating together to try to make a drawing. Interestingly while essentially an anarchic system, individual players began protecting and maintaining certain portions of the drawing so that a specific message could be included in the finished project. The effective altruists got a portion in the image and kept it by protecting their portion of the drawing and filling in portions of the drawing that were vandalized by defecting agents in the game.
Whereas this image had an audience on the other end that could communicate back to the creators after the image was completed, the big bang message would not have a similar set up. This would alter the incentives of the agents cooperating to write the message. It is hard but not inconceivable to imagine a group of nihilists working to disrupt the image entirely so that no message from the previous eon would have been encoded into our big bang.