COBE, WMAP, and now PLANCK, have or are in the process of measuring anisotropies in the cosmic microwave background. WMAP has dectected E-mode polarization from electron Thomson scattering, but not B-mode polarization. One of PLANCK's objectives is to detect B-mode polarization as a means to determine the scale of inflation. The B-mode is expected to originate from large scale gravitational waves created by inflation, but gravtational lensing is also a B-mode source. My question is, how is B-mode polarization generated by gravity waves, and how is it used to determine the scale of inflation? Also, can B-mode detection be a verification of the inflation theory?


1 Answer 1


I'll try to hit the main points, but for details you need to read something longer than this post. I think I'd start with the various tutorials at Wayne Hu's web site. The Weiss report, which lays out the case for search for $B$ modes, might also be a good place to look.

The general picture: (You may already know this, in which case skip ahead.)

Both temperature anisotropy and polarization are sourced by inhomogeneities in either the matter distribution or the spacetime geometry. These inhomogeneities can be categorized into scalar, vector, and tensor perturbations. This categorization essentially refers to the way the perturbation transforms under rotations: scalars are invariant, vectors transform with spin 1, tensors transform with spin 2.

The dominant kind of perturbations are scalars, especially "ordinary" density perturbations and the gravitational potential perturbations arising from them. These contribute to both temperature anisotropy and $E$-type polarization, but not to $B$-type polarization. The reason is essentially a symmetry argument. $B$-type polarization can be sourced by pseudoscalars but not scalars -- that is, any process that's invariant under reflections can't act as a source of $B$ polarization.

To be precise, the above is true to linear order in perturbation theory. At higher order, you can get $B$-type polarization from scalar perturbations. That's the source of the gravitational lensing signal you refer to. The $B$ modes produced by gravitational lensing result from $E$ modes (produced by the "usual" process) interacting with other inhomogeneities along the line of sight.

Tensor perturbations (perhaps more accurately called spin-2 perturbations) can and do cause temperature anisotropy, $E$-type polarization, and $B$-type polarization. The type of tensor perturbations that are most likely to be found in our Universe are those arising from a background of gravitational waves. Such a background must produce a smaller contribution than scalar perturbations -- if it produced a signal that was comparably large, we'd have seen it by now. That means that searching for this signal in either temperature anisotropy or $E$-type polarization will probably be impossible: there'll be no good way to separate the small contribution from the large one. But since scalars don't produce $B$-type polarization, the $B$ modes are a "clean" channel to use in searching for these gravitational waves.

Why do gravitational waves cause B modes?

About the best exposition I've seen of this is Wayne Hu's. The big idea is that anything that transforms like a scalar can't couple to a pseudoscalar. Since gravitational waves are spin-2, they can. One way to think about this is that gravitational waves can be right- or left-circularly polarized. On average, you expect both polarizations to be equally strong, but at any given point, you might see one dominating over the other, just by chance. That is, you might see "handedness" at any given place. That handedness is what you need to get $B$ modes. To be more precise, you can think of everything in Fourier space and consider one Fourier mode at a time. For scalar perturbations, each Fourier mode will have azimuthal symmetry about the direction of the wave vector, but for tensor modes they won't necessarily. That extra degree of freedom makes it mathematically possible to excite $B$ modes.

If that's confusing, check out Wayne's pictures.

Gravitational waves from inflation:

During inflation, the energy in the Universe is dominated by a scalar field known as the inflaton. At the end of inflation, the inflaton decays into "ordinary" matter. The density perturbations in the Universe are due to quantum fluctuations of the inflaton laid down during inflation.

At the same time, according to the theory, the spacetime metric itself should be undergoing quantum fluctuations. These fluctuations also get "frozen in" as classical perturbations after the end of inflation. They propagate (in part, anyway) as gravitational waves. These gravitational waves are the source of the $B$-type polarization that people are searching for.

It turns out that, when you solve the equations for the quantum fluctuations of the inflaton, the amplitude depends on things like the rate at which the field is changing, but the amplitude of the metric perturbations (which give rise to the gravitational waves) depends only on the energy scale at the time. So measuring the amplitude of density perturbations tells you something complicated about the dynamics of the inflaton field, but measuring the amplitude of gravitational waves tells you something relatively simple about the energy scale of inflation.

The chief prediction of any specific inflation scenario is a set of angular power spectra, which essentially means the amplitudes of temperature, $E$, and $B$ modes as a function of wavenumber in Fourier space (multipole of spherical harmonics, to be a bit more precise). See, for instance, Figure 2.2 of the Weiss report. The predicted amplitude of the $B$ signal, especially on large angular scales, essentially depends only on the energy scale of inflation. On small scales (high wavenumbers / multipoles), the dominant cause of $B$ modes is the gravitational lensing signal, which is interesting in itself but which doesn't tell you about inflation.

Can $B$-modes be used to verify inflation?

I'll say yes on this. The mere presence of $B$ modes isn't necessarily proof of inflation: you can imagine other mechanisms that might produce them. But inflationary scenarios predict the shape of the $B$ mode power spectrum. If we observe $B$ modes whose power spectrum matches this prediction, that's strong evidence in support of the theory.

Unfortunately, a failure to detect $B$ modes does not falsify inflation. The amplitude of the signal depends on the energy scale of inflation, and it's easy to construct inflationary models in which that amplitude is undetectably low. So a failure to detect $B$ modes would rule out some of the parameter space for inflationary models, but it wouldn't rule out the entire parameter space.

  • $\begingroup$ Thank you Ted for a lucid set of answers. The Wayne Hu and Weiss tutorials have perhaps somewhat more information than my current Stokes parameter polarization knowledge can readily absorb, but I'll give it a try. $\endgroup$ Jun 8, 2011 at 20:53
  • $\begingroup$ Wayne Hu's tutorials are at different levels. I think he's labeled some of them with things like "easy" or "intermediate," but you have to watch out: Wayne is so smart that what he thinks is easy is not in fact easy at all. $\endgroup$
    – Ted Bunn
    Jun 8, 2011 at 21:31
  • $\begingroup$ What does that "parameter space" mean? $\endgroup$
    – Wang Yun
    May 12, 2017 at 8:12

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