Ever since we all heard the exciting news that the BICEP2 experiment had detected “B-mode” polarization in the cosmic microwave background — just the kind we would expect to be produced by cosmic inflation at a high energy scale — the scientific community has been waiting on pins and needles for some kind of independent confirmation, so that we could stop adding “if it holds up” every time we waxed enthusiastic about the result. And we all knew that there was just such an independent check looming, from the Planck satellite. The need for some kind of check became especially pressing when some cosmologists made a good case that the BICEP2 signal may very well have been dust in our galaxy, rather than gravitational waves from inflation (Mortonson and Seljak; Flauger, Hill, and Spergel).

Now some initial results from Planck are in … and it doesn’t look good for gravitational waves. (Warning: I am not a CMB experimentalist or data analyst, so take the below with a grain of salt, though I tried to stick close to the paper itself.)

Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes
Planck Collaboration: R. Adam, et al.

The polarized thermal emission from Galactic dust is the main foreground present in measurements of the polarization of the cosmic microwave background (CMB) at frequencies above 100GHz. We exploit the Planck HFI polarization data from 100 to 353GHz to measure the dust angular power spectra CEE,BBℓ over the range 40<ℓ<600. These will bring new insights into interstellar dust physics and a precise determination of the level of contamination for CMB polarization experiments. We show that statistical properties of the emission can be characterized over large fractions of the sky using Cℓ. For the dust, they are well described by power laws in ℓ with exponents αEE,BB=−2.42±0.02. The amplitudes of the polarization Cℓ vary with the average brightness in a way similar to the intensity ones. The dust polarization frequency dependence is consistent with modified blackbody emission with βd=1.59 and Td=19.6K. We find a systematic ratio between the amplitudes of the Galactic B- and E-modes of 0.5. We show that even in the faintest dust-emitting regions there are no "clean" windows where primordial CMB B-mode polarization could be measured without subtraction of dust emission. Finally, we investigate the level of dust polarization in the BICEP2 experiment field. Extrapolation of the Planck 353GHz data to 150GHz gives a dust power ℓ(ℓ+1)CBBℓ/(2π) of 1.32×10−2μK2CMB over the 40<ℓ<120 range; the statistical uncertainty is ±0.29 and there is an additional uncertainty (+0.28,-0.24) from the extrapolation, both in the same units. This is the same magnitude as reported by BICEP2 over this ℓ range, which highlights the need for assessment of the polarized dust signal. The present uncertainties will be reduced through an ongoing, joint analysis of the Planck and BICEP2 data sets.

We can unpack that a bit, but the upshot is pretty simple: Planck has observed the whole sky, including the BICEP2 region, although not in precisely the same wavelengths. With a bit of extrapolation, however, they can use their data to estimate how big a signal should be generated by dust in our galaxy. The result fits very well with what BICEP2 actually measured. It’s not completely definitive — the Planck paper stresses over and over the need to do more analysis, especially in collaboration with the BICEP2 team — but the simplest interpretation is that BICEP2’s B-modes were caused by local contamination, not by early-universe inflation.

Here’s the Planck sky, color-coded by amount of B-mode polarization generated by dust, with the BICEP2 field indicated at bottom left of the right-hand circle:

Every experiment is different, so the Planck team had to do some work to take their measurements and turn them into a prediction for what BICEP2 should have seen. Here is the sobering result, expressed (roughtly) as the expected amount of B-mode polarization as a function of angular size, with large angles on the left. (Really, the BB correlation function as a function of multipole moment.)

The light-blue rectangles are what Planck actually sees and attributes to dust. The black line is the theoretical prediction for what you would see from gravitational waves with the amplitude claimed by BICEP2. As you see, they match very well. That is: the BICEP2 signal is apparently well-explained by dust.

Of course, just because it could be dust doesn’t mean that it is. As one last check, the Planck team looked at how the amount of signal they saw varied as a function of the frequency of the microwaves they were observing. (BICEP2 was only able to observe at one frequency, 150 GHz.) Here’s the result, compared to a theoretical prediction for what dust should look like:

Again, the data seem to be lining right up with what you would expect from dust.

It’s not completely definitive — but it’s pretty powerful. BICEP2 did indeed observe the signal that they said they observed; but the smart money right now is betting that the signal didn’t come from the early universe. There’s still work to be done, and the universe has plenty of capacity for surprising us, but for the moment we can’t claim to have gathered information from quite as early in the history of the universe as we had hoped.

Nothing focuses the mind like a hanging, and nothing focuses the science like an unexpected experimental result. The BICEP2 claimed discovery of gravitational waves in the cosmic microwave background — although we still don’t know whether it will hold up — has prompted cosmologists to think hard about the possibility that inflation happened at a very high energy scale. The BICEP2 paper has over 600 citations already, or more than 3/day since it was released. And hey, I’m a cosmologist! (At times.) So I am as susceptible to being prompted as anyone.

Cosmic inflation, a period of super-fast accelerated expansion in the early universe, was initially invented to help explain why the universe is so flat and smooth. (Whether this is a good motivation is another issue, one I hope to talk about soon.) In order to address these problems, the universe has to inflate by a sufficiently large amount. In particular, we have to have enough inflation so that the universe expands by a factor of more than 10²², which is about e⁵⁰. Since physicists think in exponentials and logarithms, we usually just say “inflation needs to last for over 50 e-folds” for short.

So Grant Remmen, a grad student here at Caltech, and I have been considering a pretty obvious question to ask: if we assume that there was cosmic inflation, how much inflation do we actually expect to have occurred? In other words, given a certain inflationary model (some set of fields and potential energies), is it most likely that we get lots and lots of inflation, or would it be more likely to get just a little bit? Everyone who invents inflationary models is careful enough to check that their proposals allow for sufficient inflation, but that’s a bit different from asking whether it’s likely.

The result of our cogitations appeared on arxiv recently:

How Many e-Folds Should We Expect from High-Scale Inflation?
Grant N. Remmen, Sean M. Carroll

We address the issue of how many e-folds we would naturally expect if inflation occurred at an energy scale of order 10¹⁶ GeV. We use the canonical measure on trajectories in classical phase space, specialized to the case of flat universes with a single scalar field. While there is no exact analytic expression for the measure, we are able to derive conditions that determine its behavior. For a quadratic potential V(ϕ)=m²ϕ²/2 with m=2×10¹³ GeV and cutoff at M_Pl=2.4×10¹⁸ GeV, we find an expectation value of 2×10¹⁰ e-folds on the set of FRW trajectories. For cosine inflation V(ϕ)=Λ⁴[1−cos(ϕ/f)] with f=1.5×10¹⁹ GeV, we find that the expected total number of e-folds is 50, which would just satisfy the observed requirements of our own Universe; if f is larger, more than 50 e-folds are generically attained. We conclude that one should expect a large amount of inflation in large-field models and more limited inflation in small-field (hilltop) scenarios.

As should be evident, this builds on the previous paper Grant and I wrote about cosmological attractors. We have a technique for finding a measure on the space of cosmological histories, so it is natural to apply that measure to different versions of inflation. The result tells us — at least as far as the classical dynamics of inflation are concerned — how much inflation one would “naturally” expect in a given model.

The results were interesting. For definiteness we looked at two specific simple models: quadratic inflation, where the potential for the inflaton ϕ is simply a parabola, and cosine (or “natural“) inflation, where the potential is — wait for it — a cosine. There are many models one might consider (one recent paper looks at 193 possible versions of inflation), but we weren’t trying to be comprehensive, merely illustrative. And these are two nice examples of two different kinds of potentials: “large-field” models where the potential grows without bound (or at least until you reach the Planck scale), and “small-field” models where the inflaton can sit near the top of a hill.

Think for a moment about how much inflation can occur (rather than “probably does”) in these models. …

How Much Cosmic Inflation Probably Occurred?Read More »

I want to tell you about a paper I recently wrote with grad student Grant Remmen, about how much inflation we should expect to have occurred in the early universe. But that paper leans heavily on an earlier one that Grant and I wrote, about phase space and cosmological attractor solutions — one that I never got around to blogging about. So you’re going to hear about that one first! It’s pretty awesome in its own right. (Sadly “cosmological attractors” has nothing at all to do with the hypothetical notion of attractive cosmologists.)

Attractor Solutions in Scalar-Field Cosmology
Grant N. Remmen, Sean M. Carroll

Models of cosmological scalar fields often feature “attractor solutions” to which the system evolves for a wide range of initial conditions. There is some tension between this well-known fact and another well-known fact: Liouville’s theorem forbids true attractor behavior in a Hamiltonian system. In universes with vanishing spatial curvature, the field variables () specify the system completely, defining an effective phase space. We investigate whether one can define a unique conserved measure on this effective phase space, showing that it exists for m²φ² potentials and deriving conditions for its existence in more general theories. We show that apparent attractors are places where this conserved measure diverges in the () variables and suggest a physical understanding of attractor behavior that is compatible with Liouville’s theorem.

This paper investigates a well-known phenomenon in inflationary cosmology: the existence of purported “attractor” solutions. There is a bit of lore that says that an inflationary scalar field might start off doing all sorts of things, but will quickly settle down to a preferred kind of evolution, known as the attractor. But that lore is nominally at odds with a mathematical theorem: in classical mechanics, closed systems never have attractor solutions! That’s because “attractor” means “many initial conditions are driven to the same condition,” while Liouville’s theorem says “a set of initial conditions maintains its volume as it evolves.” So what’s going on?

Let’s consider the simplest kind of model: you just have a single scalar field φ, and a potential energy function V(φ), in the context of an expanding universe with no other forms of matter or energy. That fully specifies the model, but then you have to specify the actual trajectory that the field takes as it evolves. Any trajectory is fixed by giving certain initial data in the form of the value of the field φ and its “velocity” . For a very simple potential like V(φ) ~ φ², the trajectories look like this:

This is the “effective phase space” of the model — in a spatially flat universe (and only there), specifying φ and its velocity uniquely determines a trajectory, shown as the lines on the plot. See the dark lines that start horizontally, then spiral toward the origin? Those are the attractor solutions. Other trajectories (dashed lines) basically zoom right to the attractor, then stick nearby for the rest of their evolution. Physically, the expansion of the universe acts as a kind of friction; away from the attractor the friction is too small to matter, but once you get there friction begins to dominate and the the field rolls very slowly. So the idea is that there aren’t really that many different kinds of possible evolution; a “generic” initial condition will just snap onto the attractor and go from there.

This story seems to be in blatant contradiction with Liouville’s Theorem, which roughly says that there cannot be true attractors, because volumes in phase space (the space of initial conditions, i.e. coordinates and momenta) remain constant under time-evolution. Whereas in the picture above, volumes get squeezed to zero because every trajectory flows to the 1-dimensional attractor, and then of course eventually converges to the origin. But we know that the above plot really does show what the trajectories do, and we also know that Liouville’s theorem is correct and does apply to this situation. Our goal for the paper was to show how everything actually fits together.

Obviously (when you think about it, and know a little bit about phase space), the problem is with the coordinates on the above graph. In particular, might be the “velocity” of the field, but it definitely isn’t its “momentum,” in the strict mathematical sense. The canonical momentum is actually , where a is the scale factor that measures the size of the universe. And the scale factor changes with time, so there is no simple translation between the nice plot we saw above and the “true” phase space — which should, after all, also include the scale factor itself as well as its canonical momentum.

So there are good reasons of convenience to draw the plot above, but it doesn’t really correspond to phase space. As a result, it looks like there are attractors, although there really aren’t — at least not by the strict mathematical definition. It’s just a convenient, though possibly misleading, nomenclature used by cosmologists.

Still, there is something physically relevant about these cosmological attractors (which we will still call “attractors” even if they don’t match the technical definition). If it’s not “trajectories in phase space focus onto them,” what is it? To investigate this, Grant and I turned to a formalism for defining the measure on the space of trajectories (rather than just points in phase space), originally studied by Gibbons, Hawking, and Stewart and further investigated by Heywood Tam and me a couple of years ago.

The interesting thing about the “GHS measure” on the space of trajectories is that it diverges — becomes infinitely big — for cosmologies that are spatially flat. That is, almost all universes are spatially flat — if you were to pick a homogeneous and isotropic cosmology out of a hat, it would have zero spatial curvature with probability unity. (Which means that the flatness problem you were taught as a young cosmologist is just a sad misunderstanding — more about that later in another post.) That’s fine, but it makes it mathematically tricky to study those flat universes, since the measure is infinity there. Heywood and I proposed a way to regulate this infinity to get a finite answer, but that was a mistake on our part — upon further review, our regularization was not invariant under time-evolution, as it should have been.

That left an open problem — what is the correct measure on the space of flat universes? This is what Grant and I tackled, and basically solved. Long story short, we studied the necessary and sufficient conditions for there to be the right kind of measure on the effective phase space shown in the plot above, and argued that such a measure (1) exists, and (2) is apparently unique, at least in the simple case of a quadratic potential (and probably more generally). That is, we basically reverse-engineered the measure from the requirement that Liouville’s theorem be obeyed!

So there is such a measure, but it’s very different from the naïve “graph-paper measure” that one is tempted to use for the effective phase space plotted above. (A temptation to which almost everyone in the field gives in.) Unsurprisingly, the measure blows up on the attractor, and near the origin. That is, what looks like an attractor when you plot it in these coordinates is really a sign that the density of trajectories grows very large there — which is the least surprising thing in the world, really.

At the end of the day, despite the fact that we mildly scold fellow cosmologists for their sloppy use of the word “attractor,” the physical insights connected to this idea go through essentially unaltered. The field and its velocity are the variables that are most readily observable (or describable) by us, and in terms of these variables the apparent attractor behavior is definitely there. The real usefulness of our paper would come when we wanted to actually use the measure we constructed, for example to calculate the expected amount of inflation in a given model — which is what we did in our more recent paper, to be described later.

This paper, by the way, was one from which I took equations for the blackboards in an episode of Bones. It was fun to hear Richard Schiff, famous as Toby from The West Wing, play a physicist who explains his alibi by saying “I was constructing an invariant measure on the phase space of cosmological spacetimes.” 

The episode itself is great, you should watch it if you can. But I warn you — you will cry.