The infinite inflation and the end of time

by @ulaulaman about #cosmology #mathematics #inflation #Hawking #AlanGuth

I published this post some years ago (archived version), but for unilateral decision of the online publisher, it is deleted, so I decide to recover it.

In the early years of the 3rd millennium there was a discussion about eternal inflation. This theoric ipothesis was introduce by Alan Guth and other physicists. In particular you can read Guth's paper Eternal Inflation⁽¹⁾:

The basic workings of inflationary models are summarized, along with the arguments that strongly suggest that our universe is the product of inflation. It is argued that essentially all inflationary models lead to (future-)eternal inflation, which implies that an infinite number of pocket universes are produced. Although the other pocket universes are unobservable, their existence nonetheless has consequences for the way that we evaluate theories and extract consequences from them. The question of whether the universe had a beginning is discussed but not definitively answered. It appears likely, however, that eternally inflating universes do require a beginning.

We have a lot of observations that confirms not only the big bang theory, but also the inflation period: in some time after the first expansion of the universe, there is a faster expansion of space time. The most important observation that supports inflation is the anisotropy of the cosmic background radiation (we could add also the absence of magnetic monopole...).
The background of eternal inflation ipothesis is the existence of repulsive-gravity material, that is unstable and decay with an exponential law (like any radiactive atom). In every decay process the volume of repulsive-gravity material grow instead decrease and prodece a never ending series of pocket universes^{(1, 2)}:

In Cosmology from the Top Down, a talk presented at Davis Inflation Meeting in 2003, Stephen Hawking speak about some criticism on eternal inflation:

(...) In the case of inflation, the idea is that the exponential expansion, obliterates the dependence on the initial conditions, so we wouldn't need to know exactly how the universe began, just that it was inflating. To lose all memory of the initial state, would require an infinite amount of exponential expansion.
This leads to the notion of ever lasting or eternal inflation. The original argument for eternal inflation, went as follows. Consider a massive scalar field in a spatially infinite expanding universe. Suppose the field is nearly constant over several horizon regions, on a space like surface. In an infinite universe, there will always be such regions. The scalar field will have quantum fluctuations. In half the region, the fluctuations will increase the field, and in half, they will decrease it. In the half where the field jumps up, the extra energy density will cause the universe to expand faster, than in the half where the field jumps down. After a certain proper time, more than half the region will have the higher value of the field, because the high field regions will expand faster than the low. Thus the volume averaged value of the field will rise. There will always be regions of the universe in which the scalar field is high, so inflation is eternal. The regions in which the scalar field fluctuates downwards, will branch off from the eternally inflating region, and will exit inflation.
Because there will be an infinite number of such exiting regions, advocates of eternalinflation get themselves tied in knots, on what a typical observer would see. So even if eternal inflation worked, it would not explain why the universe is the way it is. But in fact, the argument for eternal inflation that I have outlined, has serious flaws.
First, it is not gauge invariant. If one takes the time surfaces to be surfaces of constant volume increase, rather than surfaces of constant proper time, the volume averaged scalar field does not increase.
Second, it is not consistent. The equation relating the expansion rate to the energy density, is an integral of motion. Neither side of the equation can fluctuate, because energy is conserved.
Third, it is not covariant. It is based on a 3+1 split. From a four-dimensional view, eternal inflation can only be de Sitter space with bubbles. The energy momentum tensor of the fluctuations of a single scalar field, is not large enough to support a de Sitter space, except possibly at the Planck scale, where everything breaks down.
For these reasons, not gauge invariant, not consistent, and not covariant, I do not believe the usual argument for eternal inflation...⁽³⁾

In particular, following Hawking, the not covariant nature of the theory it could simply explain the results recently obtained by Raphael Bousso, Ben Freivogel, Stefan Leichenauer, Vladimir Rosenhaus in Eternal inflation predicts that time will end. In the preprint the theorists discuss the possibility that the time could end. They start by a metric \[\mathrm{d} s^2 = - \mathrm{d} \tau^2 + a(\tau)^2 \left ( \mathrm{d} \chi^2 + \chi^2 \mathrm{d} \Omega^2 \right )\] that it's like Friedmann-Lemaître-Robertson-Walker metric
\[-c^2 \mathrm{d} \tau^2 = - c^2 \mathrm{d} t^2 + a(t)^2 \mathrm{d} \Sigma\] Bousso et al. used for $d \Sigma$ the hypersferical coordinates in the hypotesis of $k=0$, where $k$ is the curvature of the universe: \[\mathrm{d} \Sigma^2 = \mathrm{d} r^2 + S_k (r)^2 \mathrm{d} \Omega^2\] where \[\mathrm{d} \Omega^2 = \mathrm{d} \theta^2 + \sin ^2 \theta \mathrm{d} \varphi^2\] and \[S_k = \begin{cases}\sqrt{k}^{-1} \sin \left (r \sqrt{k} \right ), & k>0\\ r, & k=0\\ \sqrt{|k|}^{-1} \sinh \left ( r \sqrt{|k|} \right ), & k<0\end{cases}\] The evaluation of $a (\tau)$ (or $a(t)$) is given using WMAP data combined with SN and BAO.
Let $\tau_0$ the life of the universe (about 13,7 billion of years), they find that, after 5,3 billion of year an observer located near the boundary of the de Sitter event horizon reaches the end of time. There is also a 50% chance that this observer reaches the end of time in 3,7 billion of years.
However, the idea of the end of time is the consequence of the isometric of our spacetime manifold to a proper subset of another spacetime. In other words: we live in a subset of another manifold, and in certain point of our spacetime we arrive at the conclusion of our travel. And so I quest: this conclusion is coincide to our beginning? Or to another?

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(1) Alan H. Guth (2001). Eternal Inflation, arXiv: astro-ph/0101507v1
(2) Guth A.H. (2007). Eternal inflation and its implications, Journal of Physics A: Mathematical and Theoretical, 40 (25) 6811-6826. DOI: 10.1088/1751-8113/40/25/S25 (arXiv)
(3) Alan Guth and Stephen Hawking on Eternal Inflation