Even a Cyclical Big Bang Could Have a Beginning in Time

In 1965, the Nobel Prize for physical Roger Penrose published a theorem demonstrating that the equations of the theory of general relativity ofEinstein implied that thecollapse gravity of a star sufficiently massive must necessarily lead to a singularity of thespace-time surrounded by a event horizon and therefore inside a black hole. Before him, this occurrence was thought to be an artifact of idealized solutions of Einstein’s equations. In 1969, he published with stephen hawking an improved version of a theorem produced by the latter which had applied the mathematical methods and ideas of Penrose to relativistic cosmology to demonstrate the occurrence of the same phenomenon with the Big Bang. One could indeed consider that many solutions of Einstein’s equations describing a Universe in expansion were more or less equivalent, by reversing the direction of the flow of time, to that describing a star in gravitational collapse.

Under very general conditions, therefore, Einstein’s equations implied that space and time had a beginning, a beginning during which the density and temperature of its contents, as well as the curvature of space-time, tended towards infinity as one asymptotically approached zero time.

But already, in the minds of Hawking and his colleagues at that time, this must have been just an artefact of a non-quantum treatment of spacetime and perhaps also simply of Einstein’s equations. It is indeed possible to consider different equations governing a curved space-time and when a volume of the cosmos observable was much smaller at the beginning of the expansion, its contents had to behave like a atom quantum. Now, we know well that the laws of Quantum mechanics precisely ensure a finite size to an atom by suppressing any collapse of its layers ofelectrons on its core. In fact, by the end of the 1960s, Bryce DeWitt had laid the foundations of a quantum theory of gravitation applicable in cosmology.

The works of cosmology quantum which were to follow, for example those of stephen hawking and more recently from Carlo Rovelliwould revisit issues already addressed in the 1920s and 1930s by Alexandre Friedmann, Georges Lemaitre and Richard Tolman for the most part.

In the hands of these men, it had become clear that Einstein’s equations contained patterns of universes in which the expansion of space eventually slowed before reversing, returning its contents to infinite density. A new phase of expansion could then begin and one could therefore consider that nature had perhaps “chosen” to manifest itself in the form of a cyclic cosmology without beginning or end, oscillating perpetually between a big Bang and one Big Crunch – to use a terminology which will only appear after the Second World War and which is now well known to the general public.

The thermodynamics of the Big Bang

But, as early as the 1930s, the American Richard Tolman, who had begun his career in chemistry physics before becoming a world authority in statistical mechanicswhether classical or quantum, and in relativityhad laid the foundations for reflections which would show that there was potentially a problem with the thermodynamics relativistic cyclic cosmology. A problem that was to get worse after the discovery of the cosmic radiation in 1965.

Indeed, Tolman had thus succeeded in transposing within the framework of curved space-times of relativistic cosmology the laws of thermodynamics and in particular those closely related toentropy, one of the most fundamental state functions of thermodynamics. It finally resulted thereafter that at each new phase of a cyclic cosmology, the entropy of its content in matter and radiation was to grow (it can be estimated with the measurement of the ratio of the number of photons number of baryons in the observable cosmos as well as with its black hole content). This was difficult to reconcile with the observation that the entropy measured today is not only finite but very far from being maximum, if one believes that there are both an infinite number of cycles in the past and in the future, as explained by the Nobel Prize Steven Weinberg at the end of his famous book The first three minutes of the universe.

The question of what happens with entropy for cyclic universes and what Tolman thought about it is more complex than just explained, but we continue to reflect on the difficulties it raises. A few years ago, the famous cosmologist and physicist theorist Paul Steinhardt revisited these questions with his colleague Anna Ijjas.

The two researchers have published articles on arXiv involving a scalar field, like that of the Higgs boson, and equations governing an interaction between this field and the expansion of the observable cosmos. This scalar field, sometimes called quintessencecan be used to describe the nature of dark energy and it allows the acceleration of the expansion of the cosmos to change into deceleration with contraction.

In itself, this is not new, but in the similar scenarios studied so far followed by a rebound phase, a Big Bounce as we say in English, the contraction phase led to the density of Plank and, shortly before reaching it, at a merger black holes formed during the phase. This fusion could make the rebound impossible and above all, the passage through a quantum phase should cause the next phase to begin with a very high state of the entropy of the observable cosmos, which is not observed.

A cyclic cosmology without quantum bounce

Anna Ijjas and Paul J. Steinhardt then showed that with the scalar field model they introduced, the contraction stops well before reaching the Planck density and the cosmos rebounds. But it rebounds with a larger expansion factor than during the previous phase, whereas this factor oscillates periodically, resuming its values ​​in the previous models of cyclic cosmology.

In doing so, the extra entropy produced by the previous phase is somehow diluted and pushed outside of what is called thecosmological horizon. For an observer below this horizon, there is no longer a continual increase at each phase of the expansion of the observable cosmos and there is no longer a contradiction between the measured entropy and an already infinitely old Universe with an infinite number cycles in the past.

But two other cosmologists from the University of Buffalo in the United States, William Kinney and Nina K. Stein, have just thrown a stone into the pond. According to them, as they explain in an open-access publication on arXiveven the cyclic universe of Ijjas and Steinhardt must have a beginning in time with an initial singularity.

The two researchers took up the reasoning already put forward several years ago by Arvind Borde, Alan H. Guth and Alexander Vilenkininspired by those of Penrose and Hawking, which showed that even the famous theory of inflation which was also supposed to make it possible to avoid an initial singularity and to avoid asking questions about the concept of the beginning of the Universe, could not in fact do without these two ideas.

Technically, it is the demonstration of a theorem relating to what is called the completeness of the geodesics of a space-time. These geodesics are the trajectories that light rays and particles of matter must take under the sole effect of the curvature of space-time. William Kinney and Nina K. Stein, as well as Arvind Borde, Alan H. Guth and Alexander Vilenkin came to the conclusion that geodesics in the cosmology of Ijjas and Steinhardt could not be parametrized by a variable which could reach infinity, which which showed in the jargon of differential geometry that these geodesics are incomplete in the past.

This is why in a press release from the University of Buffalo, Kinney explains that: People have come up with bouncing universes to make the Universe infinite in the past, but what we’re showing is that one of the newer types of those models doesn’t work. In this new type of model, which deals with entropy issues, even though the universe has cycles, it must still have a beginning. »

While specifying that: Unfortunately, it’s been known for nearly 100 years that these cyclical patterns don’t work because disorder, or entropy, builds up in the Universe over time, so each cycle is different from the last. It’s not really cyclical. A recent cyclical model circumvents this problem of entropy accumulation by proposing that the Universe expands with each cycle, diluting entropy. You stretch everything to get rid of cosmic structures such as black holes, which return the Universe to its original homogeneous state before another bounce begins.

But, long story short, we showed that by solving the entropy problem, you create a situation where the Universe had to have a beginning. Our proof shows in general that any cyclic pattern that removes entropy by expansion must have a beginning. »

Kinney concedes, however: Our proof does not apply to a cyclic model proposed by Roger Penrose, in which the Universe expands infinitely with each cycle. We are working on this issue. »