BICEP2: Primordial Gravitational Waves!

Guest blog by Liam McAllister, Cornell University.

The BICEP2 team has just announced a remarkable discovery (FAQ): they argue that they have detected, at very high significance, the imprint of primordial gravitational waves on the polarization of the cosmic microwave background.  Moreover, the signal they see is very strong.  If they are right, this is The Big One.

BICEP on site at the South Pole

BICEP's map of the CMB:

Vorticity in the CMB according to BICEP

How should we interpret this result, and what are its implications?

If the BICEP measurement really is a detection of primordial gravitational waves, and if we interpret this finding in the context of the overwhelmingly favored theory for producing primordial gravitational waves — namely, inflation — then the implications of this finding are staggering.  I find it hard to imagine a more powerful, more transformative experimental result anywhere in fundamental physics, short of a discovery of extra dimensions or of a violation of quantum mechanics.

Let me now temper this excitement with a number of cautionary remarks, and then explain why a detection of inflationary gravitational waves would be so important.


Is it real?

The first caution involves the experiment itself.  Before the community has a chance to pore over the BICEP analysis, and before this result is confirmed or refined by another experiment, it is difficult to be certain that the BICEP team has detected primordial gravitational waves, let alone primordial gravitational waves from inflation.  First of all, what they actually see are polarized photons.  They have made a beautiful map of the polarization of a small patch of the CMB, and have then carefully teased out the B-mode component of the polarization.  (The B-mode is the curl — more precisely, divergenceless — component of the polarization field— see the earlier post by Luboš for more details and references.)

At this stage, the natural questions that a naive theorist like myself will ask about the experiment include: have they correctly measured a B-mode on the sky, or could the observed B-mode be the result of a systematic effect in the instrument?  For example, could there be an overlooked coupling that converts E-modes, which are bright and were seen long ago, into a trace of B-modes?  Next, if there really is a B-mode in the CMB, is it primordial, or could it come from polarized foreground sources?   Foregrounds should look different at different frequencies and in different parts of the sky, so given enough measurements one should be able to remove this ambiguity.  BICEP claims to have done so, and I have no specific concern to share.

Suppose now that BICEP has indeed seen primordial B-modes.  How does this become a measurement of gravitational waves?  The connection is that gravitational waves propagating at the time that the CMB decoupled from protons and electrons ($$$t \approx 380,000$$$ yr) leave a distinctive imprint, inducing vorticity in the polarization field.  Gravitational waves are by far the most plausible explanation for primordial B-mode polarization, so unless the BICEP signal is due to an instrumental problem or a foreground, it is very likely a consequence of primordial gravitational waves.  For the purpose of the rest of this discussion, I will suppose that BICEP really has detected primordial gravitational waves, even though some sort of confirmation will be essential.

What do we learn?

Direct detection of gravitational waves
On its own, even without any framing in the context of inflation (see below), the detection of primordial gravitational waves is a spectacular event of historic significance.  Gravitational waves are a central prediction of general relativity, and we have had indirect experimental evidence for their existence for many years: most notably, the Hulse-Taylor binary pulsar PSR B1913+16, discovered in 1974, is gradually losing orbital energy at exactly the rate predicted based on losses to gravitational radiation:



But seeing gravitational waves directly is quite another matter!  Seeing them imprinted in the universe on the largest scales is icing on the cake: while a pulsar counts as an exotic source with strong gravitational fields by the standards of terrestrial life, the gravitational waves seen by BICEP were generated on an absolutely epic scale, a fraction of a second after the Big Bang.

Implications for inflation
The impact of the BICEP discovery is greatly heightened if we interpret primordial gravitational waves in the context of the leading model of the early universe, inflation.  Inflation provides a compelling framework for understanding the origin of the anisotropies in the CMB, and the theory makes a number of clear predictions:

The universe on large scales should be approximately

1. Homogeneous,
2. Isotropic, and
3. Flat.

The primordial scalar (density) perturbations should be

4. correlated on super-horizon scales,
and should be approximately
5. Gaussian,
6. Adiabatic,
   as well as approximately, but not exactly,
7. Scale-invariant.

The meaning of condition (6) is that to good approximation the perturbations should come from collective, synchronous excitations of all species (such as dark matter, baryons, and photons), rather than from relative excitations in which increases in the density of one species are compensated by decreases in the density of another.

Every one of these predictions agrees beautifully with observations.  Here is the Planck 2013 CMB power spectrum: the curve corresponds to the maximum-likelihood model.



A beautiful fit, to be sure.

But inflation makes one more prediction: there should be a spectrum of primordial tensor (gravitational wave) perturbations, with amplitude determined by the energy scale at which inflation occurred:\[

V^{1/4} = 2.2\times 10^{16}\GeV \times \zav{\frac{r}{0.2}}^{1/4}

\] Here V is the energy density at the time of inflation, and $$$r$$$, the tensor-to-scalar ratio, measures gravitational wave perturbations normalized to the (well-measured) size of the scalar perturbations.

But why do theorists care so much about this final prediction — why isn't this just one more successful prediction of a framework that many of us already think is broadly correct?  I will go through the implications of this result in roughly increasing order of subtlety/ambiguity.

Implication 1: inflation at the GUT scale
The first reason is the energy scale.  The BICEP measurement, taken at face value, gives us a direct window on processes with energy around $$$ 10^{16}\GeV $$$, which is 12 orders of magnitude higher than the center of mass energy of the LHC. (Go on, read that again.)  Zel'dovich said that the universe is the poor man's accelerator, but we now see that it far surpasses the most expensive devices built on Earth.

Moreover, this is (obviously) a positive measurement, where gravitational waves, rather than their absence, is what has been detected.  All previous experimental probes of physics above the TeV scale have been negative: most notably, limits on proton decay place bounds on processes around the GUT scale, but do not directly reveal phenomena at that scale.  This distinction is important: despite decades of theoretical arguments about the nature of physics at extremely high energy scales (meaning, well above the TeV scale), on the experimental front we have had to settle for knowing what that high-energy physics is not, rather than what it is.  Until now.

So, granting the context of inflation, the BICEP measurement tells us that inflation occurred around the GUT scale, just two orders of magnitude below the Planck scale.  This is on the doorstep of quantum gravity.  I will say more about this below.

Implication 2: exclusion of most of parameter space
The next reason for caring about this result is that it cuts a terribly impressive swath through the space of inflationary models, leaving a tiny fraction of the parameter space that appeared possible a priori.  The energy scale of inflation is an essentially free parameter, and this measurement restricts it to the extreme upper limit of the previously allowable range.

Planck 2013 constraints on r

BICEP2 constraints

Andrei Linde's model of chaotic inflation with potential $$$ V = \frac{1}{2} m^2 \phi^2$$$ is a famous example that is allowed by BICEP2.

Implication 3: quantization of the gravitational field
Another reason that this result is significant is that the primordial gravitational waves from inflation are quantum-mechanical in origin.  Just as the temperature anisotropies — and the distribution of large-scale structures, including our own galaxy — originate from quantum fluctuations of the inflaton, the tensor perturbations responsible for primordial B-mode polarization are the result of quantum fluctuations of the two polarization modes of the graviton.  To belabor the point: the inflationary prediction is derived by promoting the fluctuations of the gravitational field to operators, imposing canonical commutation relations, specifying the vacuum state, and computing the correlation functions.  The tensor fluctuations write quantum gravity on the sky.

Does this mean that the BICEP measurement, if confirmed by other experiments, provides direct evidence of quantum gravity?  Perhaps, with more work.  Within the context of inflation, quantum fluctuations of the gravitational field are indeed the simplest, best-established explanation for this signal.  But one should bear in mind that even in inflation (to say nothing of other models), there could be alternative sources for a strong primordial gravitational wave signal, and these alternative sources could well be classical (or at least, not involve vacuum fluctuations of the graviton): see e.g. this paper.  We do not presently have a sharp analogue of Bell's inequality through which one could really prove the quantum-mechanical origin of the observed perturbations.  Instead, the argument is that our best theory at present for how the perturbations could have arisen is quantum-mechanical, and until a more compelling classical theory is presented, we may take the primordial tensor signal as preliminary evidence for quantum gravity.

So we now have the first piece of experimental evidence that gravity is quantized.  But what quantum gravity theory is realized in Nature?  This brings us to a final and most fascinating implication of the BICEP measurement.

Implication 4: symmetry properties of quantum gravity
The BICEP measurement has one more major implication, with the potential to connect experimental cosmology to string theory.  Let us continue to work in the context of inflation; this is a reasonable assumption because all existing alternatives are now under extreme pressure in light of BICEP2.

Here is the key fact, known as the Lyth bound:

In an inflationary model producing detectably large primordial gravitational waves, the inflaton field $$$\phi$$$ moves over a distance $$$ \Delta \phi > M_{p}$$$ during inflation.

The derivation is quite simple, but I will not reproduce it now; you can find a more complete presentation here or here.  And here is some fine print:

1. When multiple fields evolve, it is the collective excitation that obeys the bound.
2. Slightly smaller $$$\Delta\phi$$$ can be achieved by contrived dynamics: make the model live at high energies only while the observed CMB is produced, then very quickly reduce the energy scale.  I'd expect this feature to be discernible, in most cases, from the tilt and running of the scalar spectrum. 
3. The Lyth bound refers to the arc length of the inflaton trajectory. The distance from the start to the end could be much smaller, as in this paper.
4. In models with nontrivial kinetic terms, the form of the bound is superficially modified.  The bound formulated by Baumann and Green is always stronger than the naive Lyth bound.

So what?  Well, the hallowed principle of naturalness tells us that in an effective quantum field theory with cutoff scale $$$\Lambda$$$, displacements of a field $$$\phi$$$ over distances $$$\gtrsim \Lambda$$$ generally leave the regime of validity of the original theory.  A simplified version of the idea is that around one point in field space, all the couplings in the action, both kinetic and potential terms, can be expanded in powers of $$$\phi/\Lambda$$$, and such series have a finite radius of convergence.  The higher-dimensional terms that result are interpreted as couplings of the light fields induced by integrating out the degrees of freedom of the ultraviolet completion.  When the light fields have generic (namely, order-unity) couplings to the heavy degrees of freedom, then the theory stops making sense at distances $$$\gtrsim \Lambda$$$ from the initial origin.

This line of thinking would not constrain inflationary models if we were simply free to take the cutoff $$$\Lambda \gg M_p $$$.    But we cannot: graviton-graviton scattering violates unitarity around the Planck scale, so general relativity + quantum field theory must change somehow at this scale.  There could be new degrees of freedom (e.g., massive string states),  and/or the theory could become strongly coupled,  but it cannot remain unmodified.   So it is extremely implausible that the cutoff scale for an  inflationary effective field theory will be $$$\gg M_p $$$.  At first sight, this seems to imply that super-Planckian field displacements cannot be described in a consistent effective theory.
 
There is a critical loophole!   The light field $$$\phi$$$ might have suppressed couplings to the heavy degrees of freedom, if the ultraviolet theory respects an approximate shift symmetry\[

\phi \to \phi + {\rm const}.

\] In a Wilsonian effective field theory, such a shift symmetry is a totally legitimate possibility, and is not spoiled by loops of the light fields.

Now, in the case of an inflationary model producing detectable $$$r$$$ (a.k.a. a 'large-field model'), the shift symmetry must protect the inflaton $$$\phi$$$ over distances $$$\gtrsim M_p$$$, so the corresponding cutoff scale $$$\Lambda$$$ (i.e. the scale up to which $\phi$  is sequestered from interacting with massive degrees of freedom) is at least the Planck scale.

Here is the summary:

An inflationary model producing detectably large primordial gravitational waves can be natural in the Wilsonian sense if an approximate shift symmetry protects the inflaton $$$\phi$$$ from coupling to massive degrees of freedom at the Planck scale.

Because the cutoff scale must be at or beyond the Planck scale, to assume such a shift symmetry is to make an assumption about global symmetries in the ultraviolet completion of gravity.  In this sense, every inflationary model producing detectable gravitational waves rests on assumptions (or, sometimes, knowledge) about quantum gravity.

Several approaches to the Lyth bound in large-field inflation are evident in the literature:

1. Reject/ignore naturalness arguments, write down desired potential energy function.
2. Agree that there is an issue, write down shift symmetry in low-energy theory, propose that quantum gravity will not spoil this symmetry.
3. Work in an actual quantum gravity theory in order to see whether suitable symmetries arise.

To their great credit, adherents of (1) with very good taste in simple polynomial/trigonometric functions were able to make broadly correct predictions!  But I think that the approach (1) itself remains questionable, and more importantly by not grappling with the issues of ultraviolet completion, one quite plainly gives up the opportunity to learn something about quantum gravity from this connection.

The shadow hanging over approach (2) is that very general arguments in quantum gravity suggest that all (exact continuous internal) global symmetries are broken at some level in quantum gravity.  (At least, in a quantum gravity theory with reasonable black hole thermodynamics.  This need not be string theory.)  So I personally find it hard to accept as an axiom that any desired low-energy global symmetry — one protecting Planck-scale displacements! — can be UV-completed without problem.  Rather, the conservative position is that obstructions could arise, and one needs to work directly in a quantum gravity theory in order to establish the existence of the necessary symmetries.

An influential early work of type (2) is Natural Inflation, by Freese, Frieman, and Olinto.  The idea is that the inflaton can be a pseudo-Nambu-Goldstone boson (axion) with a large axion decay constant, $$$ f \gg M_p $$$.  As a low-energy model, this works beautifully.  The question is whether this effective theory admits an ultraviolet completion (including gravity), in other words whether quantum gravity respects the axion shift symmetry to an adequate degree.

In string theory, achieving $$$ f \gg M_p $$$ is not straightforward.  In all computable regions of parameter space explored to date, the decay constants of individual axions are parametrically small.  But on closer examination, achieving $$$ f \gg M_p $$$ for a single axion is not actually necessary in order to achieve large-field axion inflation in string theory.

Two sorts of mechanisms present themselves: let the inflaton be a combination of multiple axions, or let the inflaton involve multiple circuits of a single axion circle.  The former idea may work with alignment of two axions (Kim, Nilles, Peloso 2004), or collective excitation of many axions, a.k.a. 'N-flation' (Dimopoulos, Kachru, McGreevy, Wacker 2005).  The latter idea is known as axion monodromy inflation (Silverstein and Westphal 2008; L.M., Silverstein and Westphal 2008).

Equipped with these basic symmetry mechanisms, one can ask whether complete and consistent models exist, and what their predictions are.  Let me focus on axion monodromy inflation realized on NS5-branes in a compactification of type IIB string theory, as described here and here.  This model is in my view the most computable of the bunch, and can serve as a benchmark within the space of more general monodromy models.  Like all sufficiently honest models of inflation in string theory, the constructions are technically challenging, with a lot of moving parts, but in my own (biased) view the ingredients are plausible ones.

This specific model (linear inflation from axion monodromy) predicted the following:\[

r \approx 0.07

\] and \[

n_s \approx 0.975.

\]
So if the BICEP2 constraints are taken at face value, this model is under significant pressure, because it predicts too low an amplitude of tensors!   Time will tell:  because the BICEP2 measurements are also in  apparent tension with Planck, some parameter adjustments may be necessary (for example, allowing running of the scalar power spectrum).   I am not prepared to use the BICEP2  results to differentiate among large field models without a bit more analysis, but ---  assuming the measurement is real ---  we can certainly use it to exclude everything except for large field models.

The model also predicts the existence of periodic modulations of the spectrum and bispectrum, but with model-dependent amplitude.  The amplitude can easily be small enough to make detection of these modulations impossible.

The modulations illustrate how confronting the real structure of quantum gravity — approach (3) above — can lead to novel predictions, as compared to purely low-energy reasoning.  The limitations on axion decay constants make a straightforward realization of 'traditional' natural inflation seem difficult in string theory, but one can construct close cousins of natural inflation, with slightly different signatures.  Moreover, by comparing experimental constraints on the tensor and scalar perturbations to the predictions of models of inflation in string theory, we can learn which of the symmetry principles allowed by string theory might be realized in our universe.

Further implications for string theory
What does the BICEP2 measurement tell us about string theory itself, beyond the fact that quantum gravity is essential for interpreting the data?

BICEP2 certainly does not tell us that string theory is correct, or that it is incorrect.  It does not tell us which string theory is (or is not) realized in Nature.  The inflationary models described above are arguably best understood in type II string theory, but that is an artifact of our present time and our present theoretical tools.

We do learn one model-independent thing about string theory: because the inflationary Hubble scale is so large,
\[

H \approx 10^{14}  GeV ,

\]
we can exclude a wide range of models in which quantum fluctuations at this scale would destabilize the compactification.  In particular, if the Kaluza-Klein mass is below $$$H$$$, the same fluctuations that give rise to the scalar and tensor perturbations of the CMB would give rise to perturbations of the extra dimensions.  When fluctuations of this sort are large, a four-dimensional description ceases to make sense, because the whole compactification is dynamical.  By this logic we can exclude models with very low Kaluza-Klein scales, i.e. models of large extra dimensions.  (Perhaps there is a model-building trick that can make large compactification robust against quantum fluctuations during inflation, but I'm not aware of a compelling idea.)




Closing thoughts
The BICEP result, if correct, is a spectacular and historic discovery.  In terms of impact on fundamental physics, particularly as a tool for testing ideas about quantum gravity, the detection of primordial gravitational waves is completely unprecedented.  Inflation evidently occurred just two orders of magnitude below the Planck scale, and we have now seen the quantum fluctuations of the graviton.  For those who want to understand how the universe began, and also for those who want to understand quantum gravity, it just doesn't get any better than this.

In fact, it all seems far too good to be true.  And perhaps it is: check back after another experimental team is able to check the BICEP findings, and then we can really break out the champagne.