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.

## 18 comments:

What do you think of "Inflationary paradigm in trouble after Planck2013"? It makes the weird criticism that yes, simple inflationary models match the data from Planck ... but these models are "anything but simple", according to the authors' version of how inflation ought to be, if it were true. The paper could be subtitled "concern-trolling the inflationary paradigm".

Yes, i read it, it's a very silly paper. It basically says that inflation is in trouble because it does not fit some moronic multiverse ideas.

@Mitchell

I think one does not even have to read the whole paper, just reading the title is enough to guess that it is rather some kind of trolling instead of giving real serious reasonable physics reasoning ;-)

The title reminds me of trolling news and popular articles about HEP ...

question.You say that V(φ)∼φ^n that are too steep, n≳2, are excluded.Don't you need φ^4 term for spontaneous breaking of symmetry? Is this turned off after symmetry breaking? Thanks.

Kashyap Vasavada

During inflation the scalar field is not at a minimum, so you don't need any phi^4 terms to stabilize the potential (it's different than the Higgs field which has to sit in a minimum with a non-zero vev today, to which end you need the negative |H|^2 and the positive |H|^4 term in the SM). Moreover, it is not said that the inflaton potential cannot have any phi^4 term, but that phi^4 cannot be the dominant one during inflation.

None of the models described here are embedded within Particle physics, or as a matter of fact within visible sector.

Inflation dilutes all matter, so the last 50-60 efoldings of inflation must be driven by fields which can directly decay into the SM degrees of freedom.

In these respects none of these models t shed any light on how perturbations and matter are created in the Universe.

Thanks for answering my question. If you don't mind I have a related question.Ok.I sort of understand that phi^4 term is not as important for inflaton as for higgs. But I have problem in understanding the whole concept. In the Lagrangian derivation of spontaneous breaking of symmetry (required for phase transition?), you change the sign of (mass)^2.This seems to me an uncomfortable mathematical jugglery! Admittedly after getting a new lower vacuum, all the (mass)^2 are positive. Is there any physical understanding of what you are doing? Is there a better derivation which does not do this trick? Thanks. Only recently I became aware of your blog. I plan to read it regularly.

I agree with Jester on "Inflationary paradigm in trouble after Planck2013", the authors are desperately seeking their heads high --- there are already number of flaws with respect to pre-Big Bang scenarios originally suggested, unless one resolves the Big Bang singularity one cannot trust iota of the perturbation calculation. It is really sad to see some senior and well respected people can still push this theory without any theoretical and observational backing. It is really sad and rotten science from Princeton and Harvard.

Jester, could this imply that the scalar field is massless to some degree?

I don't think one can put it this way. The data disfavor the inflaton mass term driving the expansion, but there can be a large mass term in the potential as long as some other terms provide larger contributions to the vacuum energy.

Wow. A succinct and very clear summary, that post clears up several points I was a bit hazy on. Thanks for that, Cormac

given that Planck is selecting inflationary models, is there a plot translating this into constraints on the reheating temperature?

Reheating into what degrees of freedom?

if you just mean -- the equation of state parameter just to be radiation, then a very good estimate will be T \sim V^{1/4} ( where V = inflaton potential ) or T\sqrt{\Gamma M_p}, where \Gamma is the decay width of the inflaton into say for instance light fermions. However this calculation does not take into account what are the real degrees of freedom.

However if you wish to ask you need just the Standard Model degrees of freedom, then you really have to know the micrphysics of the inflaton itself and how inflaton is embedded in a Beyond the Standard Model theory. The latter is a very serious issue and only few papers really bother to go into such details.

The right question is to ask what should be the reheat temperature of the Universe where all the Standard Model degrees of freedom are in thermal equilibrium, this is what the Universe cares as far as the BBN is concerned. Note that there is hardly any room for extra relativistic species.

I suspect the word 'not' is missing from the last sentence of the second paragraph. Tnx again for a great post

Cormac

Jester, have you died? No new post since April. I'm missing your blog updates.

Two months in exile ... but now Woit's away it's time you reclaimed the HEP blogging throne!

Three months later... EPS conference and still nothing... Is particle physics officially over then? :)

Are you still alive? Yes I know I saw your name somewhere, but that might be a zombie.

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