#### How Much Can We Learn about the Physics of Inflation?

#### Inflationary Tensor Perturbations after BICEP2

*Editor’s Note: Below, following this article, we provide short explanations of selected BICEP2-related theory papers that are not discussed in the main text.*

The BICEP2 team’s detection of polarization features in the cosmic microwave background, reported in *Physical Review Letters*, represents a fundamentally new piece of evidence for inflation, which has been a staple of cosmological theorizing for decades (see 19 June 2014 Viewpoint). Despite some concerns among experts about whether the polarization is truly of cosmological origin, theorists have been considering the next steps. In several other reports in PRL, theorists explain how further observations can test whether BICEP2’s reported polarization truly represents a lingering trace of inflation. Their analyses also discuss how such observations can probe differences among the various inflationary scenarios that cosmologists have devised.

Inflation was developed over $30$ years ago to solve several major problems in classic big-bang cosmology, including the overall isotropy of the Universe and the origin of galaxies. The basic idea is that an unusual kind of potential energy existed everywhere in space and fueled a brief but rapid expansion of the Universe shortly after the big bang, starting as early as ${10}^{-36}$ seconds in some models. This energy had the property that it was not diluted even as space expanded. According to general relativity, the presence of this kind of energy caused the Universe to grow at an exponential pace. When the period of inflation came to an end, expansion continued at a slower pace.

Theorists associate a field, the inflaton, with this potential energy. To picture how inflation proceeds, they think of a point rolling downhill into a valley: the horizontal axis is the value or strength of the inflaton field, and the vertical axis represents the potential energy. When the inflaton reached the bottom of the valley, the energy fell to zero, and inflation ended. Exactly how inflation proceeded, in any given version of the theory, depends on the shape of the hill and valley.

Quantum fluctuations during inflation—slight variations in the strength of the field from one place to another—seeded density variations that ultimately grew into galaxies and clusters of galaxies. Those density fluctuations also resulted in small temperature variations in the $3$-kelvin cosmic microwave background (CMB) that cosmologists have been mapping since the early 1990s.

Those CMB temperature variations could have other explanations besides inflation. But the pattern and magnitude of CMB polarization variations that BICEP2 detected—the so-called $B$-mode—are difficult to reproduce theoretically by other mechanisms without contradicting previous cosmological observations (see first three papers in the list below). The idea is that during inflation, there were quantum fluctuations in the gravitational field. These fluctuations gave rise to large-scale gravitational waves that stretched and squeezed spacetime, polarizing the microwaves and creating the $B$-mode signal.

The parameter reported by the BICEP2 team is the so-called tensor-to-scalar ratio $r$, which measures the strength of the $B$-mode signal relative to the magnitude of CMB temperature variations. After their best effort at subtracting interfering foreground emissions from galactic dust and other sources, they estimated a value $r\approx 0.20$ of cosmological origin. From basic principles of inflationary theory, this value implies an inflation energy scale of ${10}^{16}$ giga-electron-volts ($\text{GeV}$), essentially the height at which the inflaton began its roll. If this value of $r$ holds up, it would rule out many versions of inflation that assume lower energies. But there are other details of inflation that theorists hope to explore with improved observations.

In the simplest models of inflation, the inflaton rolls very slowly along a very gentle incline, so that the potential energy driving inflation also changes slowly. “Slow roll” models became popular in the early years of inflation as a way of providing a sufficiently long period of exponential expansion followed by a smooth return to normal cosmic expansion. With this assumption, the spectrum of the B-mode polarization signal (intensity versus wavelength) is approximately a power law, decreasing with increasing wavelength. Specifically, the power-law exponent $n$ is equal to $-r/8$, so that the BICEP2 estimate implies $n\approx -0.025$.

In one recent theoretical paper commenting on the BICEP2 results, Jerod Caligiuri and Arthur Kosowsky of the University of Pittsburgh explore whether future experiments could measure $n$ directly from the $B$-mode data and thus test the simple inflationary prediction $n=-r/8$. But the fact that $n$ is so close to zero creates observational difficulties. The wavelength of gravitational fluctuations translates, in observational terms, into an angular distance across the sky. Measuring the $B$-mode power at a specific angular distance involves taking every pair of points having that angular separation, multiplying their polarization signals, and then finding the average of those products. To estimate $n$ directly, observers would need to measure the $B$-mode signal over a wide range of angular distances. However, BICEP2 observed only a small patch of sky—$380$ square degrees, or about $1\%$ of the total.

Caligiuri and Kosowsky ran computer simulations of future observational scenarios covering half of the sky and assuming instruments with three different levels of sensitivity, all significantly better than BICEP2’s current sensitivity. They acknowledge that increasing the sky coverage complicates the problem of subtracting out foreground contamination from galactic dust and other sources. But Kosowsky says that because dust has a different emission spectrum than the CMB, measurements over a wide range of frequencies should make it possible to subtract dust emission with greater confidence than is now possible.

An additional problem is that gravitational lensing—distortion of light as it passes through gravitational fields—also alters the polarization of CMB photons and creates a $B$-mode signal that must be separated from the one due to inflation. Citing previous studies, Caligiuri and Kosowsky say that CMB observations can be “delensed” through statistical methods that distinguish between the different spatial patterns for lensing and inflation.

Caligiuri and Kosowsky conclude that, provided delensing can be performed with sufficient accuracy, future observations with greater sky coverage and with sensitivity about 20 times better than BICEP2’s can not only reduce the uncertainty in $r$ but can also yield a measurement of $n$ with enough accuracy to distinguish it from zero. Caligiuri says there is no scientific obstacle to achieving the required sensitivity, although it is beyond what is currently available. He and Kosowsky caution, however, that if the true value of $r$ is closer to 0.1 than 0.2, $n$ becomes so small that even ideal observations would not be able to distinguish it from zero.

In another recent paper, Scott Dodelson of the Fermi National Accelerator Laboratory in Illinois offers a similar conclusion about the need to conduct sensitive, large-area observations to have a chance of pinning down the value of $n$, but he proposes a different strategy to more accurately determine $r$. Whole-sky observations with modest angular resolution but very low noise can, in principle, yield a value of $r$ with accuracy of around $2\%$, Dodelson shows.

Dodelson also discusses a theoretical puzzle that arises if the BICEP2 result holds up. If the inflaton began its slow roll at a height of ${10}^{16}\phantom{\rule{0.333em}{0ex}}\text{GeV}$, as suggested by BICEP2, standard calculations indicate that the horizontal distance (field strength) it traveled must have exceeded ${10}^{19}\phantom{\rule{0.333em}{0ex}}\text{GeV}$, the so-called Planck energy (at which quantum gravitational influences on particle physics are expected to set in). As Dodelson points out, slow roll models generally assume that the valley can be approximated by the simplest possible well shape—a parabola. However, this assumption is only valid if the inflaton field value remained significantly smaller than the Planck energy. Another reason theorists need more CMB polarization data, he says, is to better understand just what the shape of the potential is.

This puzzle is the jumping-off point for Paolo Creminelli of the Institute for Advanced Study in Princeton, New Jersey, and the Abdus Salam International Center for Theoretical Physics in Trieste, Italy, and his colleagues. They define a quantity that combines $r$ with the power-law exponent of the density fluctuations, as measured by CMB temperature variations. This quantity is zero for a potential that is purely quadratic (a parabola), and the BICEP2 data indicate that it is indeed zero, within experimental error. But if future observations were to find a nonzero value, the result could be used as an empirical test of more complex inflationary models, the team finds. “There are many models on the market,” says Creminelli, with different inflationary potentials derived from a variety of particle physics mechanisms. However, in the absence of observational constraints, judging “what is nice or not nice depends on your theoretical prejudice.”

Future CMB measurements, Creminelli and his colleagues say, could provide empirical evidence that rules out some of these models and favors others. Caligiuri agrees: If the expected relationship between $r$ and $n$ doesn’t hold, he says, “we can pretty much rule out simple inflation models,” but others will remain viable.

However, many experts are skeptical about the procedure the BICEP2 team used to subtract the $B$-mode signal attributable to dust in our galaxy. David Spergel of Princeton University in New Jersey says there is a “reasonable probability” that the results from another CMB-measuring team known as Planck will show the BICEP2 signal to be largely due to galactic dust. But even if the value of $r$ attributable to cosmological effects turns out to be much smaller, “some inflationary models can produce $r$ as small as you like,” says Dodelson.

The availability of so many different models of inflation is a problem, says Paul Steinhardt, also of Princeton, a pioneer of inflationary theory who has become a critic. “Inflation can accommodate any parameters you can measure,” he says; “there is not a test or combination of tests that can ever disprove it.” Spergel agrees that inflation does not uniquely predict the magnitude of gravitational waves, but it is the only theory he knows that predicts their existence: “The detection of gravitational waves…would be widely seen as a confirmation of the inflationary theory.” And Steinhardt adds that “chasing after $r$ is worthwhile nevertheless…because it is important for sorting out the correct scientifically meaningful [cosmological] theory.”

## Selected PRL Theory Papers Responding to BICEP2

BICEP2’s finding may have implications for fields beyond cosmology. Here is a selection of theory papers from *Physical Review Letters* that have also been inspired by the BICEP2 result.

**Can topological defects mimic the BICEP2 $B$-mode signal?** Joanes Lizarraga *et al.*, Phys. Rev. Lett. 112, 171301 (2014)

**Did BICEP2 see vector modes? First $B$-mode constraints on cosmic defects** Adam Moss and Levon Pogosian, Phys. Rev. Lett. 112, 171302 (2014)

In many theories, the end of inflation can generate topological defects, such as cosmic strings; these would generate gravitational waves that could in principle lead to a polarization signal such as BICEP2 observed. No plausible scenario, however, can attribute the entire $r\approx 0.2$ observation to such structures while also conforming to the measured CMB temperature variations. On the other hand, if there were some cosmic strings, but not enough to account for the BICEP2 results, they could provide an additional degree of freedom in fitting all of the CMB data to a cosmological model.

**Can primordial magnetic fields be the origin of the BICEP2 data?** Camille Bonvin, Ruth Durrer, and Roy Maartens, Phys. Rev. Lett. 112, 191303 (2014)

If the electromagnetic field has a suitable coupling to the inflaton or the metric, magnetic fields generated during inflation can in principle generate a value of $r\approx 0.2$. For this scenario to work, however, the primordial magnetic fields would have to be generated with an implausible non-Gaussian pattern. Weaker fields with a more likely arrangement could nevertheless contribute an $r$-value up to about 0.09.

**On quantifying and resolving the BICEP2/Planck tension over gravitational waves** Kendrick M. Smith *et al.*, Phys. Rev. Lett. (to be published), arXiv:1404.0373

The claimed detection by BICEP2 of a gravitational wave signature from the inflationary era appears to conflict with an upper limit derived from observations by the Planck satellite. The present disagreement is not so large as to point definitively to a conflict, but if future observations show that both measurements are credible, it is likely that cosmological theories will have to be modified to accommodate them.

**Evidence for bouncing evolution before inflation after BICEP2** Jun-Qing Xia *et al.*, Phys. Rev. Lett. (to be published), arXiv:1403.7623

If both the BICEP2 and Planck measurements hold up after further observational scrutiny, one possible cosmological scenario that could explain both involves a “bounce” that occurs before the inflationary era. A bounce connotes a universe that expands from a finite density state following the collapse of an earlier cosmological phase. The change in effective initial conditions can lead to changes in the fluctuation spectrum remaining after inflation.

**Electroweak vacuum stability in light of BICEP2** Malcolm Fairbairn and Robert Hogan, Phys. Rev. Lett. 112, 201801(2014)

Inflation at the energy scale of $\sim {10}^{16}\phantom{\rule{0.333em}{0ex}}\text{GeV}$ implied by the BICEP2 detection would pump large quantum fluctuations into other fields, including the Higgs field. These fluctuations could push the Higgs beyond the value associated with electroweak symmetry-breaking, into an unstable state. There are some relatively straightforward cures for this instability.

**Higgs inflation is still alive after the results from BICEP2** Yuta Hamada, Hikaru Kawai, Kin-ya Oda, and Seong Chan Park, Phys. Rev. Lett. 112 241301 (2014)

Models in which the form of the Higgs potential at energies around ${10}^{17}\phantom{\rule{0.333em}{0ex}}\text{GeV}$ is flat enough to drive slow-roll inflation cannot produce large enough density fluctuations to account for present-day cosmic structure. But a novel coupling of the Higgs field to the Ricci scalar of general relativity leads to an acceptable inflationary cosmology and also produces $r\approx 0.2$.

**Axion cold dark matter in view of BICEP2 results** Luca Visinelli and Paolo Gondolo, Phys. Rev. Lett. (to be published), arXiv:1403.4594

**BICEP2 Tensor detection severely constrains axion dark matter** David J. E. Marsh *et al.*, Phys. Rev. Lett. (to be published), arXiv:1403.4216

The axion, a hypothetical particle arising from symmetry breaking in models of the strong interaction, has been proposed as the source of cosmic dark matter. The measured value $r\approx 0.2$ rules out models in which the axion is heavy and arises from symmetry breaking that occurs before the end of inflation. Models in which the axion arises after inflation and is lighter remain viable, but only if the axion’s mass is in a restricted range.

David Lindley is a freelance writer in Alexandria, Virginia, and author of *Uncertainty: Einstein, Heisenberg, Bohr and the Struggle for the Soul of Science* (Doubleday, 2007).