Typicality. Cosmological Constant and Boltzmann Brains. Note Quote.

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In a multiverse we would expect there to be relatively many universe domains with large values of the cosmological constant, but none of these allow gravitationally bound structures (such as our galaxy) to occur, so the likelihood of observing ourselves to be in one is essentially zero.

The cosmological constant has negative pressure, but positive energy.  The negative pressure ensures that as the volume expands then matter loses energy (photons get red shifted, particles slow down); this loss of energy by matter causes the expansion to slow down – but the increase in energy of the increased volume is more important .  The increase of energy associated with the extra space the cosmological constant fills has to be balanced by a decrease in the gravitational energy of the expansion – and this expansion energy is negative, allowing the universe to carry on expanding.  If you put all the terms on one side in the Friedmann equation – which is just an energy balancing equation – (with the other side equal to zero) you will see that the expansion energy is negative, whereas the cosmological constant and matter (including dark matter) all have positive energy.

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However, as the cosmological constant is decreased, we eventually reach a transition point where it becomes just small enough for gravitational structures to occur. Reduce it a bit further still, and you now get universes resembling ours. Given the increased likelihood of observing such a universe, the chances of our universe being one of these will be near its peak. Theoretical physicist Steven Weinberg used this reasoning to correctly predict the order of magnitude of the cosmological constant well before the acceleration of our universe was even measured.

Unfortunately this argument runs into conceptually murky water. The multiverse is infinite and it is not clear whether we can calculate the odds for anything to happen in an infinite volume of space- time. All we have is the single case of our apparently small but positive value of the cosmological constant, so it’s hard to see how we could ever test whether or not Weinberg’s prediction was a lucky coincidence. Such questions concerning infinity, and what one can reasonably infer from a single data point, are just the tip of the philosophical iceberg that cosmologists face.

Another conundrum is where the laws of physics come from. Even if these laws vary across the multiverse, there must be, so it seems, meta-laws that dictate the manner in which they are distributed. How can we, inhabitants on a planet in a solar system in a galaxy, meaningfully debate the origin of the laws of physics as well as the origins of something, the very universe, that we are part of? What about the parts of space-time we can never see? These regions could infinitely outnumber our visible patch. The laws of physics could differ there, for all we know.

We cannot settle any of these questions by experiment, and this is where philosophers enter the debate. Central to this is the so-called observational-selection effect, whereby an observation is influenced by the observer’s “telescope”, whatever form that may take. But what exactly is it to be an observer, or more specifically a “typical” observer, in a system where every possible sort of observer will come about infinitely many times? The same basic question, centred on the role of observers, is as fundamental to the science of the indefinitely large (cosmology) as it is to that of the infinitesimally small (quantum theory).

This key issue of typicality also confronted Austrian physicist and philosopher Ludwig Boltzmann. In 1897 he posited an infinite space-time as a means to explain how extraordinarily well-ordered the universe is compared with the state of high entropy (or disorder) predicted by thermodynamics. Given such an arena, where every conceivable combination of particle position and momenta would exist somewhere, he suggested that the orderliness around us might be that of an incredibly rare fluctuation within an infinite space-time.

But Boltzmann’s reasoning was undermined by another, more absurd, conclusion. Rare fluctuations could also give rise to single momentary brains – self aware entities that spontaneously arises through random collisions of particles. Such “Boltzmann brains”, the argument goes, are far more likely to arise than the entire visible universe or even the solar system. Ludwig Boltzmann reasoned that brains and other complex, orderly objects on Earth were the result of random fluctuations. But why, then, do we see billions of other complex, orderly objects all around us? Why aren’t we like the lone being in the sea of nonsense?Boltzmann theorized that if random fluctuations create brains like ours, there should be Boltzmann brains floating around in space or sitting alone on uninhabited planets untold lightyears away. And in fact, those Boltzmann brains should be incredibly more common than the herds of complex, orderly objects we see here on Earth. So we have another paradox. If the only requirement of consciousness is a brain like the one in your head, why aren’t you a Boltzmann brain? If you were assigned to experience a random consciousness, you should almost certainly find yourself alone in the depths of space rather than surrounded by similar consciousnesses. The easy answers seem to all require a touch of magic. Perhaps consciousness doesn’t arise naturally from a brain like yours but requires some metaphysical endowment. Or maybe we’re not random fluctuations in a thermodynamic soup, and we were put here by an intelligent being. An infinity of space would therefore contain an infinitude of such disembodied brains, which would then be the “typical observer”, not us. OR. Starting at the very beginning: entropy must always stay the same or increase over time, according to the second law of thermodynamics. However, Boltzmann (the Ludwig one, not the brain one) formulated a version of the law of entropy that was statistical. What this means for what you’re asking is that while entropy almost always increases or stays the same, over billions of billions of billions of billions of billions…you get the idea years, entropy might go down a bit. This is called a fluctuation. So backing up a tad, if entropy always increases/stays the same, what is surprising for cosmologists is that the universe started in such a low-entropy state. So to (try) to explain this, Boltzmann said, hey, what if there’s a bigger universe that our universe is in, and it is in a state of the most possible entropy, or thermal equilibrium. Then, let’s say it exists for a long long time, those billions we talked about earlier. There’ll be statistical fluctuations, right? And those statistical fluctuations might be represented by the birth of universes. Ahem, our universe is one of them. So now, we get into the brains. Our universe must be a HUGE statistical fluctuation comparatively to other fluctuations. I mean, think about it. If it is so nuts for entropy to decrease by just a little tiny bit, how nuts would it be for it to decrease enough for the birth of a universe to happen!? So the question is, why aren’t we just brains? That is, why aren’t we a statistical fluctuation just big enough for intelligent life to develop, look around, see it exists, and melt back into goop. And it is this goopy-not-long-existing intelligent life that is a Boltzmann brain. This is a huge challenge to the Boltzmann (Ludwig) theory.

Can this bizarre vision possibly be real, or does it indicate something fundamentally wrong with our notion of “typicality”? Or is our notion of “the observer” flawed – can thermodynamic fluctuations that give rise to Boltzmann’s brains really suffice? Or could a futuristic supercomputer even play the Matrix-like role of a multitude of observers?

Speculative String-Cosmologies as Geodesically Incomplete.

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Penrose diagrams of de Sitter space in the flat (left) and static (right) slicings that each cover only part of the whole de Sitter space, and that are both geodesically incomplete.

Alan Guth, Alvin Borde and Alexander Vilenkin have argued that within the framework of a future-eternal inflationary multiverse, as well as some more speculative string-cosmologies, all worldlines are geodesically incomplete and, thus, the multiverse has to have a beginning. Unfortunately, if future-eternal inflation is true, all “hypotheses about the ultimate beginning of the universe would become totally divorced from any observable consequences. Since our own pocket universe would be equally likely to lie anywhere on the infinite tree of universes produced by eternal inflation, we would expect to find ourselves arbitrarily far from the beginning. The infinite inflating network would presumably approach some kind of steady state, losing all memory of how it started […] Thus, there would be no way of relating the properties of the ultimate origin to anything that we might observe in today’s universe.” (Guth).

On the other hand, Andrei Linde has argued that the multiverse could be past-eternal, because either all single world lines might have to start somewhere, but not the whole bundle of them (Linde), or there could even exist some (albeit strange) space-times with single past-eternal world lines.

This issue is not settled, and even in those scenarios a global arrow of time may not necessarily exist. However, there are other frameworks possible – and they have even already been developed to some extent, where a future-eternal inflationary multiverse is both not past-eternal and beginningless but arise from some primordial vacuum which is macroscopically time-less. Thus, again, the beginning of some classical space-times is not equivalent with the beginning of everything.

We can even imagine that there is no multiverse, but that the whole (perhaps finite) universe – our universe – once was in a steady state without any macroscopic arrows of time but, due to a statistical fluctuation above a certain threshold value, started to expand  – or to contract, bounce and expand – as a whole and acquired an arrow of time. In such a case the above-mentioned reply, which was based on the spatial distinction of a beginning of some parts of the world and the eternity of the world as a whole, would collapse.

Nevertheless it is necessary to distinguish between the different notions and extensions of the term “universe”. In the simplest case, Kant’s antinomy might be based on an ambiguity of the term “world” (i.e. the difference between “universe” and “multiverse”), but it does not need to; and it was not assumed here that it necessarily does. The temporal part of Kant’s first antinomy was purely about the question whether the macroscopic arrow of time is past-eternal or not. And if it is not past-eternal this does not mean that time and hence the world has an absolute beginning in every respect – it is still possible that there was or is a world with some underlying microscopic time. (By the way, one can also imagine that, even if our arrow of time is past- and/or future-eternal, there might exist “timeless islands” someday: for instance isolated black holes if they would not ultimately radiate away due to quantum effects, or empty static universes if they could split off of our space-time.)

Of course it is possible that firstly a natural principle of plentitude is realized and different multiverses (sets of universes) exist totally independent from each other, and secondly that some of them are truly past-eternal while others have an absolute beginning and others have only local starting points of local arrows of time as it was suggested here. If so, we might not be able to tell in what kind we live in. And this would be irrelevant in the end, because then every possible world is actual and probably exists infinitely often. But we do not know whether such an extreme principle of plentitude does apply or if cosmology is ultimately just and only a matter of pure logical consistency, allowing us finally to calculate the complete architecture of the world by armchair-reasoning.