BICEP2 results: inflation and the tensor modes

It’s not every day that you go into work and hear how the Universe began.  Today I was in the audience as the BICEP2 team presented observations of gravitational waves coming to us from the first 10^{-35} seconds of the Universe’s history.   In the time it took the text of this sentence to go from your eye to your brain, more than a billion billion billion intervals this long have elapsed.  To say this signal is from the first blinks of the nascent cosmos is to vastly understate how primordial it is. (Note: a few seats away from me in the audience was friend and fellow grad student Kirit Karkare, a former astrobiter who is one of the authors of the paper—read a series about his time working on BICEP2 at the South Pole here through here!)

BICEP2 observed a pattern imprinted on the cosmic microwave background (CMB) due to the passage of gravitational waves most likely produced by inflation (see this astrobite for more, and this one). Gravitational waves are propagating ripples in the fabric of space-time itself, and the best way to imagine them is to consider a cross of four spherical masses, initially symmetrically placed away from a common center.  A gravitational wave passing through them would cause them to move in and out from this center, distorting the cross.

As the waves passed through the early Universe, they set charged particles in motion, just like these spheres of mass. The periodic, regular motion of these particles imprints a characteristic pattern on the light of the CMB by polarizing it: making the electromagnetic waves (i.e. light) oscillate in particular directions.  If you could see the CMB to high, high precision, this pattern would look like tiny vortexes of hot and cold, much like a weather map of wind circulation.  But it’s not a wind: rather, these vortices are due to electromagnetic waves oscillating in different directions on the sky.  These directions are known as polarizations, and inflation’s signature is to polarize the CMB.

This shows the tensor modes, also called B modes because they are like a magnetic field (which has curl only).  Note the swirling of the vectors, like little vortexes.  It is particularly impressive that one can see this tiny signal in this raw map of one patch of sky.

1.  This shows the tensor modes in one particular patch of sky, also called B modes because they are like a magnetic field (which has curl only). Note the swirling of the vectors, like little vortics. It is particularly impressive that one can see this signal directly without needing to add up many regions of the sky, as an experiment like Planck will do. BICEP2 paper.

Inflation was first proposed in the early 1980s by Alan Guth, Andrei Linde, Alexei Starobinsky, Andreas Albrecht, and Paul Steinhardt, among others. Guth’s paper couched inflation as a response to two problems.  First, the Universe must have been highly homogeneous at the beginning, since it is still relatively so now, but different regions of it could not have been in the causal contact necessary for this level of homogeneity.  Second, the expansion rate at the beginning must have had a very particular, perhaps unlikely value (out to 59 decimal places!) for the Universe to be geometrically flat today.  Inflation resolved both of these problems. If you expand a tiny, causally connected (and hence relatively homogeneous) initial region exponentially fast, it can become the whole Universe we see today.  And more, take a small enough patch on any curved surface (like the Earth), blow it up, and it will seem flat.

Since 1980, inflation has fit in well with many subsequent observations, such as the result that the CMB is uniform on scales that could not have been in causal contact without some new mechanism (first observed by the COBE satellite in 1991).  However, there has been no smoking gun: an observation of a unique prediction of inflation. The tensor mode imprint now observed is a relatively unique prediction of inflation that is unlikely to come from any other scenario for the Universe’s beginning.

The key plot.  The black points are BICEP2 data, the red dashed curves are an r=0.2 model (lower) and r=0.2 plus lensing of the CMB (upper).  The solid red curve is the polarization expected from lensing of the CMB by intervening mass on the light's path to us.

2.  The key plot. The vertical axis measures the strength of the signal while the horizontal is the multipole moment l, which describes the physical length scale on which the maps of the polarization are correlated with themselves (higher l means smaller scales, and a typical scale is ~a degree on the sky).  The black points are BICEP2 data. The solid red curve is the polarization expected from lensing of the CMB by intervening mass on the light’s path to us. The red dashed curves are an r = 0.2 model (lower) and r = 0.2 plus lensing of the CMB (upper) (i.e. adding the dashed and solid red curves). The number $1.3 \times 10^{-7}$ is the Probability To Exceed by chance: how likely is it to get these results by chance, as a statistical fluke, if the real Universe does not have inflationary tensor modes? BICEP2 paper.

The tensor-to-scalar ratio is the ratio of gravitational waves to density fluctuations in the Universe.  Confusingly, the gravitational waves are also called tensor modes because they have directions as well as magnitudes*, while the density fluctuations are also called scalar perturbations because they are just numbers (scalars). That’s why r is the tensor-to-scalar ratio.  BICEP2’s major result is a 5.3 sigma detection of tensor modes. The best fit model is one with r=0.2, and fitting this model permits the team to rule out r=0 at 7 sigma. See the Figure above, from the paper, which clearly shows that an r = 0.2 model (the two dashed red curves) fits the data well.

Inflation takes many forms, but the most plain vanilla version is to have a  field or fields (just some amount of energy at every point in space), which might describe particles, losing potential energy and driving exponential expansion.  A good analogy is a ball rolling down a hill: as the ball rolls down, it loses height but gains speed.  To generate exponential expansion, the ball has to be rolling on a very shallow hill. Otherwise, it would end up at the bottom too quickly and the Universe would fail to inflate enough to solve the problems inflation was proposed to address.  Inflationary models where the ball is rolling down a shallow hill are called “slow-roll” inflation.

In slow roll inflation, r has a simple physical interpretation: it measures the slope of the hill the ball is rolling down.** The hill is shallow, so the slope is small.  That’s why r was expected to be small, and consequently hard to detect.

And it was.  The BICEP2 team’s telescope, about 30 cm across, stared at the same patch of sky for approximately 3 years, more or less continuously, to get the noise down sufficiently.  They also sliced their data in many different ways to try to eliminate any other sources for the signal.  In particular, they divided the data into smaller sub-samples and subtracted them from each other in a strategy called “jackknifing”.  If the signal was roughly the same in all sub-samples, these subtractions should result in random noise.  And they did, compared across stretches of time, across different detectors (the telescope has 512 separate ones), and many other categories.

The left hand column is the temperature map and then, moving down, two different directions of polarization, called Q and U. The jacknifing here has been to subtract the maps from one temporal half of the observation cycle from the other. As should be the case if there are no systematic errors, the jacknifed results look like noise.  (Technical point: Light has two polarizations, so Q and U completely describe it).

3.  The left hand column is the temperature map and then, moving down, two different directions of polarization, called Q and U. The units are all in microKelvin.  The jackknifing here has been to subtract the maps from one temporal half of the observation cycle from the other. As should be the case if there are no systematic errors, the jackknifed results look like noise. (Technical point: Light has two polarizations, so Q and U completely describe it). BICEP2 paper.

The team also estimated whether contamination from other astrophysics could create tensor modes, such as dust in our own galaxy, which can conceivably mimic a signal.  Using the latest publicly-available data from the Planck mission, the team found dust cannot come close to explaining the signal.  Another possible contaminant is lensing of light coming from the CMB to us by galaxies in between (detected by South Pole Telescope in 2013 and by Polarbear in 2014; see also this astrobite and this one for more discussion).  Light experiences gravity, and mass can therefore bend it so as to create a tensor-mode like signal.  But this signal is not large enough to be confused with the BICEP2 detection, and moreover has a quite different shape from the inflationary tensor modes. Lensing tensor modes are dominant at higher multipoles than those from inflation, and in fact BICEP2 detects these lensing modes separately at 2.7 sigma. (They are in solid red in 2. above).


Planck constraints on r without (red) and with running of the scalar spectral tilt. BICEP2 measurements have been added into the blue confidence ellipses.  These curves represent 68%, and 95% confidence contours; the larger ones within each color are 95%. Allowing this running is one way to resolve the tension between BICEP2's value of r and Planck's constraint.

Planck constraints on r in red when the scalar spectral tilt is allowed to run. These curves represent 68%, and 95% confidence contours; the larger ones within each color are 95%. Note both easily cover the r value BICEP2 measures.  The blue is a constraint from combining Planck data with running scalar spectral tilt  with the BICEP2 results. The vertical axis is r, and the horizontal axis is the scalar spectral tilt $n_s$. Allowing this running is one way to resolve the tension between BICEP2’s value of r and Planck’s constraint. BICEP2 paper.

A surprising aspect of the result is that the value of r BICEP2 found is actually above current constraints from other experiments.  The Planck mission found that r < 0.11 at 95% confidence, about two BICEP2 error bars below the BICEP2 result.  More certainly remains to be done to understand this tension, but the BICEP2 team notes that one possibility for resolving this tension is if running of the scalar spectral tilt is allowed. The scalar spectral tilt measures whether there was a preferred lengthscale at which structures correlated with each other at the beginning of the Universe.  Inflation predicts some correlation, which has been observed (e.g. by WMAP). Running of the scalar spectral tilt incorporates a dependence of the slope on physical scale: it says that the Universe’s density perturbations are correlated, but different physical scales are more or less strongly correlated.  The BICEP results in combination with the Planck results suggest that this may be so, but there are many other possibilities for new physics that could also be responsible.

What’s certain is that these results open a new era of precision early Universe cosmology.  Humans have been telling themselves stories about their origins at least since Hesiod’s Theogony in the 8th century BC.  Hesiod began that work with  a hopeful question: “These things declare to me from the beginning, ye Muses who dwell in the house of Olympus, and tell me which of them first came to be.”  Stretching back through the unfathomable vastness of cosmic time and distant space, the BICEP2 results bring us nearer a final answer.


*Note I am not being fully detailed here: there is actually a matrix (really, a tensor, but just think “table of numbers, with two rows and two columns” that describes the stretching and squeezing of space-time itself, that is the tensor perturbation.  This is an important subtlety since vectors also have magnitudes and directions, but we are not interested here in vector perturbations! Hartle’s “Gravity: An Introduction to Einstein’s General Relativity”, chapter 23, provides a first-timer-suitable discussion of this.

**More precisely, the initial value of the logarithmic derivative of the potential, (\partial V/\partial \phi)/V, scales as the square root of r; see a superb review by Lesgourgues (2006), equations (61) and (148), with notation defined above equation (143). Note Lesgourgues is taking the Hubble parameter and the velocity of the field \dot{\phi} to be constant in this calculation.

About Zachary Slepian

I’m a 2nd year grad student in Astronomy at Harvard, working with Daniel Eisenstein on the effect of relative velocities between regular and dark matter on the baryon acoustic oscillations. I did my undergrad at Princeton, where I worked with Rich Gott on dark energy, Jeremy Goodman on dark matter, and Roman Rafikov on planetesimals. I also spent a year at Oxford getting a master’s in philosophy of physics, which remains an interest.


  1. “The tensor mode imprint now observed is a relatively unique prediction of inflation that is unlikely to come from any other scenario for the Universe’s beginning.”

    You say it’s unlikely, but this seems like hand waving. I would really appreciate some description of the unique connection of inflation to the presence of these gravity waves.

    A paper specifically considering alternate scenarios would be helpful.

    • I put it this way to be conservative. I don’t know of any other theories besides inflation that do produce gravitational waves that would imprint on the CMB. While some of my research is in this area and I have published in related ones, I can’t be sure someone somewhere doesn’t have some theory other than inflation that has this effect.

      Inflation is an extremely rapid stretching of space-time. Imagine two people holding a blanket between them close enough to each other that the blanket hangs down. Now imagine they leap away from each other, snapping the blanket tight and horizontal between them. There will be ripples in the blanket created by this. I think that’s the rough idea behind gravitational waves produced by inflation, if the blanket is space-time.

      This paper (‎) is the best single review of all of inflationary theory I’ve found: section 2.5, page 21, derives the tensor mode signature, if you are prepared for relatively intense math. Another good review on inflation, though not very informative on gravitational waves, is the Planck paper linked where I mention the r<0.11 constraint in the astrobite. Finally, here is, direct from the horse's mouth, what the BICEP2 paper says on how inflation generates these waves.

      “Inflation predicts that the quantization of the gravitational field coupled to exponential expansion produces a primordial background of stochastic gravitational waves with a characteristic spectral shape (Grishchuk 1975; Starobinsky 1979; Rubakov et al. 1982; Fabbri & Pollock 1983; Abbott & Wise 1984; also see Krauss & Wilczek 2013). Though unlikely to be directly detectable in modern instruments, these gravitational waves would have imprinted a unique signature upon the CMB. Gravitational waves induce local quadrupole anisotropies in the radiation field within the last-scattering surface, inducing polarization in the scattered light (Polnarev 1985). This polarization pat- tern will include a “curl” or B-mode component at degree an- gular scales that cannot be generated primordially by density perturbations. The amplitude of this signal depends upon the tensor-to-scalar ratio, r, which itself is a function of the energy scale of inflation. The detection of B-mode polarization of the CMB at large angular scales would provide a unique confirmation of inflation and a probe of its energy scale (Seljak 1997; Kamionkowski et al. 1997; Seljak & Zaldarriaga 1997).”

      Nick Hand’s astrobite also discusses how these waves are evidence gravity is quantized (

  2. Thanks Zach, the paper looks interesting.

    I like section 4.6 on page 52: it seems that inflation theory is compatible with r in the range 10^-4 … 10^-2 or with r ~ 0.1, or other values as far as I can tell. Perhaps the measured value helps us address self consistency, but self consistency does not constitute evidence for inflation.

    I don’t see the case where inflation would have been excluded, and if there is no such case, then the whole discussion around this discovery seems pseudo-scientific to me.

    • I think the self-consistency relation’s being satisfied would constitute evidence. If it’s not inflation causing the gravitational waves or the tensor tilt (n_t in the consistency relation), then it would be I think surprising if the two just happened to satisfy this relation.

      Inflation would have not necessarily have been excluded if no r>0 was detected, but there are other competing models that have r=0 that would still be allowed in this case.

      I think it’s important to be careful in calling things pseudo-scientific: there has been significant debate in the philosophy of science literature on what demarcates science from non-science. The criterion of falsifiability, which is what you seem to allude to, was first developed by Karl Popper in the 1930’s. But that is only one approach to demarcation. In practice, with many theories with a number of unconstrained parameters, they are not falsifiable in that you can often find some set of parameters to fit any data. But then simplicity becomes an important additional criterion. To me, it is persuasive, if these results are borne out, that inflation is a simple explanation for r>0 (which we don’t have any other explanations for right now, as far as I know).

  3. Thanks Zach
    I’m just wonder about the limits of the r, according to the graph maximum order of r is 10^-5, while when we deal with chaotic inflation we find the values order 0.1, and the graph of r,n_s is steel the same and but the r axis now doesn’t multiplied by in the paper , how they can do this ?



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