There’s no question that quantum fluctuations play a crucial role in modern cosmology, as the recent BICEP2 observations have reminded us. According to inflation, all of the structures we see in the universe, from galaxies up to superclusters and beyond, originated as tiny quantum fluctuations in the very early universe, as did the gravitational waves seen by BICEP2. But quantum fluctuations are a bit of a mixed blessing: in addition to providing an origin for density perturbations and gravitational waves (good!), they are also supposed to give rise to Boltzmann brains (bad) and eternal inflation (good or bad, depending on taste). Nobody would deny that it behooves cosmologists to understand quantum fluctuations as well as they can, especially since our theories involve mysterious aspects of physics operating at absurdly high energies.
Kim Boddy, Jason Pollack and I have been re-examining how quantum fluctuations work in cosmology, and in a new paper we’ve come to a surprising conclusion: cosmologists have been getting it wrong for decades now. In an expanding universe that has nothing in it but vacuum energy, there simply aren’t any quantum fluctuations at all. Our approach shows that the conventional understanding of inflationary perturbations gets the right answer, although the perturbations aren’t due to “fluctuations”; they’re due to an effective measurement of the quantum state of the inflaton field when the universe reheats at the end of inflation. In contrast, less empirically-grounded ideas such as Boltzmann brains and eternal inflation both rely crucially on treating fluctuations as true dynamical events, occurring in real time — and we say that’s just wrong.
All very dramatically at odds with the conventional wisdom, if we’re right. Which means, of course, that there’s always a chance we’re wrong (although we don’t think it’s a big chance). This paper is pretty conceptual, which a skeptic might take as a euphemism for “hand-waving”; we’re planning on digging into some of the mathematical details in future work, but for the time being our paper should be mostly understandable to anyone who knows undergraduate quantum mechanics. Here’s the abstract:
De Sitter Space Without Quantum Fluctuations
Kimberly K. Boddy, Sean M. Carroll, and Jason Pollack
We argue that, under certain plausible assumptions, de Sitter space settles into a quiescent vacuum in which there are no quantum fluctuations. Quantum fluctuations require time-dependent histories of out-of-equilibrium recording devices, which are absent in stationary states. For a massive scalar field in a fixed de Sitter background, the cosmic no-hair theorem implies that the state of the patch approaches the vacuum, where there are no fluctuations. We argue that an analogous conclusion holds whenever a patch of de Sitter is embedded in a larger theory with an infinite-dimensional Hilbert space, including semiclassical quantum gravity with false vacua or complementarity in theories with at least one Minkowski vacuum. This reasoning provides an escape from the Boltzmann brain problem in such theories. It also implies that vacuum states do not uptunnel to higher-energy vacua and that perturbations do not decohere while slow-roll inflation occurs, suggesting that eternal inflation is much less common than often supposed. On the other hand, if a de Sitter patch is a closed system with a finite-dimensional Hilbert space, there will be Poincaré recurrences and Boltzmann fluctuations into lower-entropy states. Our analysis does not alter the conventional understanding of the origin of density fluctuations from primordial inflation, since reheating naturally generates a high-entropy environment and leads to decoherence.
The basic idea is simple: what we call “quantum fluctuations” aren’t true, dynamical events that occur in isolated quantum systems. Rather, they are a poetic way of describing the fact that when we observe such systems, the outcomes are randomly distributed rather than deterministically predictable. But when we’re not looking, a system in its ground state (like an electron in its lowest-energy orbital around an atomic nucleus) isn’t fluctuating at all; it’s just sitting there. And in de Sitter space — empty space with a positive cosmological constant — all of the fields are in their ground states. If we were to probe empty de Sitter space with a particle detector, it would certainly detect particles — but there are no particle detectors around, so in fact the quantum fields are sitting there quietly in a stationary state with no definite particle number. Therefore, these kinds of fluctuations aren’t “really happening.”
To get into a bit more detail, there are two things going on here: a certain interpretation on the meaning of “quantum fluctuations,” and some claims about de Sitter space. As far as quantum fluctuations are concerned, we readily admit that our analysis relies heavily on the Everett/Many-Worlds formulation of quantum theory. In that view, there is nothing truly random and unpredictable about quantum dynamics. There is only the smooth, unitary evolution of the wave function according to the Schrödinger equation. Apparent unpredictability arises because that smooth evolution can take a quantum state from a single connected “world” into several distinct “branches,” each of which features certain entanglements between subsystems (like the spin of a particle and the readout of a measuring apparatus that just measured that spin). But such branching doesn’t happen willy-nilly; it’s crucial that the system undergoes decoherence. Roughly speaking, that’s when a macroscopic quantum system becomes entangled with an unobserved environment. Macroscopically different states of the system (like different readouts on a measuring apparatus, or alive/dead states of a cat in a box) become entangled with different environment states. Once that happens, the two states of the macroscopic system can never talk to each other again, and in particular cannot experience mutual quantum interference. It’s as if they have become part of two different worlds.
So in the Everett picture, a quantum system in its lowest-energy state (or in any state of precisely-defined energy) isn’t fluctuating at all. It’s just sitting there, until some nosy measuring device comes poking at it. From the point of view of any given observer, the outcome of those pokes is intrinsically random. Because our brains are wired for classical physics, we therefore sometimes speak as if the system is fluctuating around even when we’re not looking at it — as if an electron is actually bouncing around in the vicinity of the nucleus of an atom, and its orbital represents the likelihood of it being in one place or another. But that’s not right: the orbital (the wave function) is the electron, it doesn’t represent our knowledge of it. And when nobody is observing it, literally nothing is fluctuating.
What does this have to do with cosmology? We often contemplate situations in which space is completely empty other than for vacuum energy — perhaps during inflation in the very early universe, or perhaps in our own future once all the matter and radiation has been dispersed by the expansion of the universe. We’re left with de Sitter space. Back in the 70′s, Gibbons and Hawking showed that de Sitter space, just like a black hole, has a temperature. That’s because, just like a black hole, de Sitter space comes with an horizon. That horizon cuts off the degrees of freedom to which any observer has access, leaving them in a thermal state at a well-defined temperature. It’s as if — but, we are claiming, only as if! — the cosmological horizon is radiating into the interior, just as the black hole horizon radiates to the outside world.
But this quantum-mechanical “thermal state” is different from our intuition, once again trained by classical mechanics, of a bunch of particles randomly bouncing around inside a box. Globally (including outside the horizon), the quantum state is static. It only appears thermal to an observer because the horizon cuts them off from the rest of the world. This gives us a mixed state, in which the local observer doesn’t know exactly what state they’re actually in — but all of the allowed possibilities are completely stationary. So once again, nothing is actually fluctuating.
My confidence in this story about quantum fluctuations and de Sitter space is extremely high, even though it does conflict with the way many cosmologists think about the situation. The less secure part of our story is when we move from the idealization of pure de Sitter space to the messy real world. In the real world, you might think you’re in de Sitter space once and for all, but you could actually be in a temporary false-vacuum state. If there is only one vacuum, we can appeal to a “cosmic no-hair theorem” (analogous to similar theorems for black holes) that says a universe with a cosmological constant will eventually dissipate all of its excitations and turn into de Sitter space. But when there are false vacua, the situation is admittedly tricker. We’ve thought about it, and decided that the story we told above for de Sitter space is the one that is usually right, even if you’re in a false vacuum. (There are some subtleties dealing with complementarity and the dimensionality of Hilbert space, but that’s the typical situation.)
The ramifications are very interesting. The idea that Boltzmann brains fluctuate into existence and should count as “observers” in a multiverse cosmology has been a troubling one, and now we’re saying it might not be nearly as severe as people have thought. Whereas before Boltzmann brains were hard to avoid if your cosmological model ever entered a de Sitter phase, now we think it’s quite hard to get them to appear in any appreciable abundance. This might mean that the last paper by Kim and me, asking whether the Higgs field could provide an escape from the BB problem in our actual universe, is addressing a non-problem (in at least some models).
You might worry that our dismissal of quantum fluctuations is too sweeping — after all, don’t we see their effects in the cosmic microwave background? Fortunately, no. The standard story says that the inflaton field undergoes quantum fluctuations, which then get imprinted as fluctuations in density. What we’re saying is that the inflaton doesn’t actually “fluctuate,” it’s just in some calculable quantum state. But there’s nothing “observing” it, causing decoherence and branching of the wave function. At least, not while inflation is going on. But when inflation ends, the universe reheats into a hot plasma of matter and radiation. That actually does lead to decoherence and branching — the microscopic states of the plasma provide an environment that becomes entangled with the large-scale fluctuations of the inflaton, effectively measuring it and collapsing the wave function. So in our picture, all of the textbook predictions for inflation perturbations remain unchanged.
Eternal inflation is a different story. The idea there is that the inflaton field slowly rolls down its potential during inflation, except that quantum fluctuations will occasionally poke the field to go higher rather than lower. When that happens, space expands faster and inflation continues forever. Like Boltzmann brains — and unlike density perturbations — this story relies on the idea that the “fluctuations” are actual events happening in real time, even in the absence of measurement and decoherence. And we’re saying that none of that is true. The field is essentially in a pure state, and simply rolls down its potential. Clearly a lot more careful analysis has to be done here, and we’ve started thinking about it. The stakes are substantial: the fact that inflation is eternal is a key part of its motivation in the minds of many cosmologists. (Note that we’re not saying eternal inflation is impossible; if you are stuck in a false vacuum with a very tiny decay rate, you can stay there for an arbitrarily long time. But the set of models in which inflation is eternal might be much tinier than was previously believed.) As with the Higgs and Boltzmann brains, this might be another case where I am undermining one of my own previous papers. So be it — in science you have to be willing to change your mind when faced with new data or better ideas. (I think that both the Higgs paper and the out-of-equilibrium paper are perfectly correct, given their working assumptions; I just think that the assumptions are much less likely to apply than I used to.)
Finally, it’s interesting to note the role of “interpretations of quantum mechanics” in this story. (I don’t like that term, since we’re not discussing “interpretations,” we’re comparing manifestly different physical theories.) In the Everett formulation, the wave function is a direct reflection of reality; when it is stationary, so is the quantum system. Other approaches take a very different tack. There are formulations of quantum mechanics where collapse of the wave function is truly random and unpredictable; there are others with hidden variables, in which the true state of the universe isn’t defined by the wave function. In any of those cases, our analysis is completely beside the point. It’s interesting to think — but perhaps unsurprising in retrospect — that the correct formulation of quantum mechanics might have crucial implications for the evolution of the universe.