Simplest models of inflation involve a scalar field with a potential. During inflation, the value of the scalar field is such that the potential is large and positive, effectively acting as a cosmological constant that supports a faster-than-light expansion of the universe. The potential should be almost but not exactly flat, so that the scalar field slowly creeps down the potential slope; once it falls into the minimum inflation ends and the modern history begins. Clearly, that sounds like a spherical cow model rather than a fundamental picture. However, the single-field slow-roll inflation works surprisingly well at the quantitative level. There is no sign of isocurvature perturbations that would point to a more complicated inflaton sector. There is no sign of running of the spectral index that would point to departures from the slow-roll conditions. There is no sign of non-gaussianities, that would point to large self-interactions of the inflaton field. There is no sign of wiggles in the CMB spectrum that would point to some violent events happening during inflation. One can say that the slow-roll inflation is like a spherical cow model that correctly predicts not only the milk yield, but also the density, hue, creaminess, and even the timbre of moo the cow makes when it's being milked.

Let's look into more details of the slow-roll inflation. Assuming the standard kinetic term for the inflaton field φ, the model is completely characterized by the scalar potential V(φ). The important parameters are the first and second derivatives of the potential at the time when the observable density fluctuations are generated. Up to normalization, these derivatives are the slow-roll parameters ε and η (see the equation box for a precise definition). Both have to be much smaller than 1, otherwise the inflaton field evolves too quickly to support inflation. Several observables measured by Planck depend primarily on ε and η. In particular, the spectral index, which measures the departure of the primordial density fluctuation spectrum from scale invariance, is given by ns - 1=2η-6ε. Since Planck measured ns=0.9603±0.0073, we know the order of magnitude of the slow-roll parameters: either ε or η or both have to be of order 0.01.

Another important observable that depends on the slow roll parameters is the tensor-to-scalar ratio

*r*. The system of an inflaton coupled to gravity has 3 physical degrees of freedom: the scalar mode linked to curvature perturbations, and the tensor mode corresponding to gravitational waves. The scalar mode was detected in a distant past by the COBE satellite and its amplitude

*As*is of order 10^-10. The tensor mode has not been detected so far. From the box you see that the amplitude

*At*of the tensor mode is directly sensitive to the value of the inflaton potential, and for the slow-roll inflation it is expected to be somewhat smaller than

*As*. In fact, the relative amplitude of tensor and scalar fluctuations is a direct measure of the parameter ε:

*r=At/As = 16ε.*Now, the latest limit from Planck is r≲0.11 at 95% confidence level and, given we expect ε∼0.01 to fit the spectral index, it is already a non-trivial constraint on the shape of the inflaton potential. That's why in the plot of the best-fit area in the

*ns*vs.

*r*plane many inflationary models fall into the excluded region. Basically, power-law potentials V(φ)∼φ^n that are too steep, n≳2, are excluded. The quadratic potential V(φ) = m^2 φ^2, perhaps the most popular one, is on the verge of being excluded. What survives are power-law potentials with n≲2, or hilltop models where inflation happens near a maximum of the potential. The latter is predicted e.g. in the so-called

*natural inflation*where the inflaton is a Goldstone boson with a periodic cosine potential.

So, the current situation is interesting but unsettled. However, the limit

*r*≲0.11 may not be the last word, if the Planck collaboration manages to fix their polarization data. The tensor fluctuations can be better probed via the B-mode of the CMB polarization spectrum, with the sensitivity of Planck often quoted around r∼0.05. If indeed the parameter ε is not much smaller than 0.01, as hinted by the spectral index, Planck may be able to pinpoint the B-mode and measure a non-zero tensor-to-scalar ratio

*.*That would be a huge achievement because we would learn the absolute scale of inflation, and get a glimpse into fundamental physics at 10^16 GeV!. Observing no signal and setting stronger limits would also be interesting, as it would completely exclude power-law potentials. We'll see in 1 year.

See the original Planck paper for more details.