Reconstructing Inflation

All sorts of responsibilities have been sadly neglected, as I’ve been zooming around the continent — stops in Illinois, Arizona, New York, Ontario, New York again, and next Tennessee, all within a matter of two weeks. How is one to blog under such trying conditions? (Airplanes and laptops are involved, if you must know.)

But the good news is that I’ve been listening to some very interesting physics talks, the kind that actually put ideas into your head and set off long and convoluted bouts of thinking. Possibly conducive to blogging, but only if one pauses for a moment to stop thinking and actually write something. Which is probably a good idea in its own right.

One of the talks was a tag-team performance by Dick Bond and Lev Kofman, both cosmologists at the Canadian Institute for Theoretical Astrophysics at the University of Toronto. The talk was part of a brief workshop at the Perimeter Institute on “Strings, Inflation, and Cosmology.” It was just the right kind of meeting, with only about twenty people, fairly narrowly focused on an area of common interest (although the talks themselves spanned quite a range, from a typically imaginative propsoal by Gia Dvali about quantum hair on black holes to a detailed discussion of density fluctuations in inflation by Alan Guth).

Dick and Lev were interested in what we should expect inflationary models to predict, and what data might ultimately teach us about the inflationary era. The primary observables connected with inflation are primordial perturbations — the tiny deviations from a perfectly smooth universe that were imprinted at early times. These deviations come in two forms: “scalar” perturbations, which are fluctuations in the energy density from place to place, and which eventually grow via gravitational instability into galaxies and clusters; and the “tensor” perturbations in the curvature of spacetime itself, which are just long-wavelength gravitational waves. Both arise from the zero-point vacuum fluctuations of quantum fields in the very early universe — for scalar fluctuations, the relevant field is the “inflaton” φ that actually drives inflation, while for tensor fluctuations it’s the spacetime metric itself.

The same basic mechanism works in both cases — quantum fluctuations (due ultimately to Heisenberg’s uncertainty principle) at very small wavelengths are amplified by the process of inflation to macroscopic scales, where they are temporarily frozen-in until the expansion of the universe relaxes sufficiently to allow them to dynamically evolve. But there is a crucial distinction when it comes to the amount of such fluctuations that we would ultimately see. In the case of gravity waves, the field we hope to observe is precisely the one that was doing the fluctuating early on; the amplitude of such fluctuation is related directly to the rate of inflation when they were created, which is in turn related to the energy density, which is given simply by the potential energy V(φ) of the scalar field. But scalar perturbations arise from quantum fluctuations in φ, and we aren’t going to be observing φ directly; instead, we observe perturbations in the energy density ρ. A fluctuation in φ leads to a different value of the potential V(φ), and consequently the energy density; the perturbation in ρ therefore depends on the slope of the potential, V’ = dV/dφ, as well as the potential itself. Once one cranks through the calculation, we find (somewhat counterintuitively) that a smaller slope yields a larger density perturbation. Long story short, the amplitude of tensor perturbations looks like

T ² ~ V ,

while that of the scalar perturbations looks like

S ² ~ V ³/(V’ )² .

Of course, such fluctuations are generated at every scale; for any particular wavelength, you are supposed to evaluate these quantities at the moment when the mode is stretched to be larger than the Hubble radius during inflation.

To date, we are quite sure that we have detected the influence of scalar perturbations; they are responsible for most, if not all, of the temperature fluctuations we observe in the Cosmic Microwave Background. We’re still looking for the gravity-wave/tensor perturbations. It may someday be possible to detect them directly as gravitational waves, with an ultra-sensitive dedicated satellite; at the moment, though, that’s still pie-in-the-sky (as it were). More optimistically, the stretching caused by the gravity waves can leave a distinctive imprint on the polarization of the CMB — in particular, in the type of polarization known as the B-modes. These haven’t been detected yet, but we’re trying.

Problem is, even if the tensor modes are there, they are probably quite tiny. Whether or not they are substantial enough to produce observable B-mode polarization in the CMB is a huge question, and one that theorists are presently unable to answer with any confidence. (See papers by Lyth and Knox and Song on some of the difficulties.) It’s important to get our theoretical expectations straight, if we’re going to encourage observers to spend millions of dollars and years of their time building satellites to go look for the tensor modes. (Which we are.)

So Dick and Lev have been trying to figure out what we should expect in a fairly model-independent way, given our meager knowledge of what was going on during inflation. They’ve come up with a class of models and possible behaviors for the scalar and tensor modes as a function of wavelength, and asked which of them could fit the data as we presently understand it, and then what they would predict for future experiments. And they hit upon something interesting. There is a well-known puzzle in the anisotropies of the CMB: on very large angular scales (small l, in the graph below), the observed anisotropy is much smaller than we expect. The red line is the prediction of the standard cosmology, and the data come from the WMAP satellite. (The gray error bars arise from the fact that there are only a finite number of observations of each mode at large scales, while the predictions are purely statistical — a phenomenon known as “cosmic variance.”)

It’s hard to tell how seriously we should take that little glitch, especially since it is at one end of what we can observe. But the computers don’t care, so when Dick and Leve fit models to the data, the models like to do their best to fit that point. If you have a perfectly flat primordial spectrum, or even one that is tilted but still a straight line, there’s not much you can do to fit it. But if you allow some more interesting behavior for the inflaton field, you have a chance.

Let’s ask ourselves, what would it take for the inflaton to be generating smaller perturbations at earlier times? (Larger wavelengths are produced earlier, as they are the first to get stretched outside the Hubble radius during inflation.) We expect the value of the inflaton potential V to monotonically decrease during inflation, as the scalar field rolls down. So, from the second equation above, the only way to get a smaller scalar amplitude S at early times is to have a substantially larger value of the slope V’. So the inflaton potential might look something like this.

Maybe it’s a little contrived, but it seems to fit the data, and that’s always nice. And the good news is that a large slope at early times implies that the actual value of the potential V was also large at early times (because the field was higher up a steep slope). Which means, from the equation for T above, that we expect (relatively) large tensor modes at large scales! Which in turn is exactly where we have some hope to look for them.

This is all a hand-waving reconstruction of the talk that Dick and Lev gave, which involved a lot more equations and Monte Carlo simulations. The real lesson, to me, is that we are still a long way from having a solid handle on what to expect in terms of the inflationary perturbations, and shouldn’t fool ourselves into thinking that our momentary theoretical prejudices are accurate reflections of the true space of possibilities. If it’s true that we have a decent shot at detecting the tensor modes at large scales, it would represent an incredible triumph of our ability to extend our thinking about the universe back to its earliest moments.