The CMB spectrum measured by the Planck satellite points to a perfectly boring universe: the vanilla ΛCDM cosmological model, no hint of new light degrees of freedom beyond the standard model, no hint of larger-than-expected neutrino masses, etc. However at the quantitative level things are a bit more interesting, as Planck has considerably narrowed down the parameter space of inflation. We may not be far from selecting a small class out the huge zoo of inflationary models.
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
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.