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53. Eternal inflation: stochastic approach 1 (Inflationary perturbations 7)

Posted by Dmitry
at June 3, 2008

This post is the next in the series devoted to the discussion of physics of cosmological perturbations. Today I will start the physical regime which is ofthe most interest for me - regime where super-Hubble cosmological perturbations are of the order 1.

As we have found, inflation typically predicts slightly red power spectrum of primordial perturbations due to the necessity to have graceful exit from inflation. “Red” tilt of the power spectrum means that the amplitude of primordial perturbations is larger at smaller k or longer length scales. Why the tilt of inflationary perturbations should be read, can also be easily understood if we recall that longer length scales correspond to earlier moments of time in the evilution of the Unverse.

Let us take for simplicity a chaotic inflationary model with a monotonic potential V(\phi) having a single minimum \phi=0. Earlier moments of time correspond to larger values of \phi, because inflaton field slowly rolls down towards its minimum \phi=0 during inflation. At the moment of time t=t_{0} the modes with k=a(t_{0})H(t_{0}) cross the horizon and their amplitude freezes at the level

\sqrt{\langle\delta\phi_{k}^{2}\rangle}\approx\frac{H(t_{0})}{2\pi}. (1)

Clearly, the earlier is the the moment t_{0}, the larger is H(t_{0}), since, according to the Friedmann equation

H^{2}(t_{0})=\frac{8\pi}{3M_{P}^{2}}V(\phi(t_{0})). (2)

Therefore, the amplitude of generated primordial perturbations is larger at earlier times than at later times.

If we follow this logic, we can conclude that there may exist wavelength

\lambda_{EI}=\frac{2\pi}{k_{EI}} (3)

such that the relative amplitude of primordial perturbations \delta_{k} will become of order of 1, and our perturbative treatment of primordial perturbations (i.e., description in terms of background field \phi and small fluctuations \delta\phi near it) may break down. This is indeed the case for many inflationary models and another typical prediction of inflation. Again, considering chaotic inflatioary model with monotonic potential V(\phi) let us find the moment of time t_{EI} when the inflationary perturbation theory breaks down. During one Hubble time \Delta t\sim H^{-1} the value of the inflaton field decreases by

\Delta\phi\sim\dot{\phi}\Delta t\sim\frac{1}{3H^{2}}\frac{\partial V}{\partial\phi}\sim\frac{M_{P}^{2}}{8\pi V}\frac{\partial V}{\partial\phi} (4)

due to the action of the classical force (derivative of the potential w.r.t. the scalar field) and, as we said, there will be fluctuations \delta\phi which for l\sim k^{-1}\sim H^{-1} have the amplitude

|\delta\phi|\sim\frac{H}{2\pi}\sim\sqrt{\frac{2V}{3\pi M_{P}^{2}}}. (5)

When

(M_{P}^{2}/V)(\partial V/\partial\phi)\lesssim\sqrt{V/M_{P}^{2}} (6)

or, in other words, when V\gtrsim\epsilon M_{P}^{4}, where \epsilon is the usual slow roll parameter, \Delta\phi\sim\delta\phi, and at the corresponding time scale inflationary perturbation theory breaks down. Typically, at a similar time scale (and associated wave length) non-gaussianity parameter NG defined at the previous lecture also becomes of the order of 1.

Let us find the critical value of field \phi_{EI} for the chaotic inflationary model with the potential \lambda\phi^{4}. Using our estimations we find that

\phi_{EI}\sim\lambda^{-1/6}M_{P}. (7)

Now, if we set initial condition for inflation at planckian energy density which corresponds to \phi_{I}\sim\lambda^{-1/4}M_{P}, we see that the inflation can proceed in the non-perturbative regime for quite a long time if \lambda is small enough.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Lectures on de Sitter spacetime, Quantum field theory

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52. Introduction to non-gaussianities (Inflationary perturbations 6)

Posted by Dmitry
at June 2, 2008

This post is the next in the series devoted to study of inflationary perturbations. The last time we discussed quantization of inflationary perturbations, i.e., we constructed the quadratic effective action for the cosmological perturbations, diagonilized the corresponding hamiltonian and quantized it. Now the time has come to discuss interacting quantum field theory of the cosmological perturbations.

As we have found, the power spectrum \delta_{\Phi}^{2}(k) of the newtonian potential does not carry too much information about the physics of inflation: in the leading approximation it is flat, in the NLO approximation it has the form of power law k^{n_{s}-1}, where the spectral index n_{s} is the linear function of slow roll parameters \epsilon and \eta. The same can be said about the power spectrum of curvature perturbation or any other two-point correlation function.

It is relatively hard to measure n_{s} with really good precision, and it is much harder to find the scale dependence of n_{s}(k). Therefore, what we have at hand given the results of current observations, is just a single parameter characterizing inflationary potential. If we suppose that the inflation potential has the form

V(\phi)\approx\frac{1}{2}m^{2}(\phi-\phi_{0})^{2}+\cdots (1)

in the vicinity of its minimum \phi=\phi_{0}, the only parameter we know relatively well is the mass of the inflaton m.

Clearly, more information about inflationary potential can be gained from higher order correlation functions. In this respect, the situation in cosmology (where observables are correlation functions of \frac{\delta\rho}{\rho}, gravitational potential or curvature perturbation \zeta) is somewhat similar to the situation we have in QFT (where observables are various cross-sections related to S-matrix elements): if one is able to measure multi-point correlation functions, much more information about underlying QFT lagrangian is gained. While in QFT the leading order (LO) corresponds to free field theory, the leading order in inflationary perturbation theory corresponds to exact de Sitter universe; in both cases, the only non-trivial observables are two-point correlation functions. The NLO (next-to-leading order) of perturbation theory and multi-point correlation functions give us information about interaction terms in the QFT largrangian, while NLO in inflationary perturation theory (and higher order correlation functions) - more information about inflationary potential allowing us to descriminate between different inflationary models.

Another thing regarding this analogy that is worth mentioning is that in QFT we calculate multipoint correlation functions and construct perturbation theory using the Wick theorem, since the functional integral for the free field theory is gaussian. Analogously, the LO in the iflationary perturbation theory corresponds to the gaussian approximation, while next orders - to the deviations from gaussianity.

To describe (at least at the very naive level) the behavior of these non-gausianities is the subject of this lecture. We will be especially interested in the correlation functions

\langle\zeta_{k}\zeta_{k'}\rangle,\langle\zeta_{k}\zeta_{k'}\zeta_{k'{}'}\rangle,\ldots (2)

of the curvature perturbation \zeta. The reason is that, as we already know, \zeta is conserved at supercurvature scales and therefore carries information about inflationary stage untouched directly to us. As we have said, multi-point correlation functions of \zeta determine non-gaussianities. To determine their relative strength, it is convenient to represent the overall curvature perturbation as expansion

\zeta(x)=\zeta_{g}(x)+\frac{3}{5}f_{NL}\zeta_{g}^{2}(x)+\cdots,

where \zeta_{g} is purely gaussian variable, so that odd power correlators like \langle\zeta_{g}^{3}\rangle are all zero. For example, one has

\langle\zeta^{3}\rangle\sim f_{NL}\langle\zeta_{g}^{2}\rangle^{2}+\cdots (3)

etc. The relative strength of non-gausianity can be estimated as the ratio

\frac{\langle\zeta^{3}\rangle}{\langle\zeta_{g}^{2}\rangle^{3/2}}\sim f_{NL}\langle\zeta_{g}^{2}\rangle^{1/2}\sim f_{NL}\cdot10^{-5}, (4)

where we used the fact that the average amplitude of the curvature perturbation at the present horizon scale is of the order of 10^{-5}. One can say that non-gaussianities are large (of the same order as gaussian cotribution) if f_{NL}\sim10^{5}. As it turns out, it is rather hard to construct a model which would predict so large value of f_{NL}. In particular, typical prediction of single fied inflationary models is f_{NL}\sim1 (so that standard inflationary scenario generically predicts a very low value of non-gaussianity).

What an observer essentially measures is the correlator \langle\zeta^{3}(x)\rangle (or \langle\zeta(x_{1})\zeta(x_{2})\zeta(x_{3})\rangle with x_{1,2,3} within the given Hubble patch). In the momentum space it is naturally reduced to

\langle\zeta_{k_{1}}\zeta_{k_{2}}\zeta_{k_{3}}\rangle=(2\pi)^{3}\delta(\sum_{i}k_{i})\frac{3}{5}f_{NL}(k_{1},k_{2},k_{3})\times
\times(P_{\zeta}(k_{1})P_{\zeta}(k_{2})+{\rm permutations}). (5)

Any possible configuration of k_{i} contributes into the overall correlation function \langle\zeta^{3}(x)\rangle. Configurations with one of the k much smaller than the two others are called squeezed limit configurations. These configurations test correlations which exist at supercurvature scales, and typically the main contribution into \langle\zeta^{3}\rangle comes from these configurations. Configurations with all k of the same order of magnitude are called equivilateral configurations, and they test physics near the Hubble crossing scales. In some specific models they may give the main contribution into \langle\zeta^{3}(x)\rangle. More precisely,

\langle\zeta^{3}\rangle=\int_{k\gg{}aH}+\int_{k\sim{}aH}+\int_{k\ll{}aH}. (6)

The first integral in the r.h.s. is always zero (subhubble fluctuations are in the vacuum state), second integral corresponds to equilateral contribution, while the third - to squezzed limit contribution into the three-point function.

Let us now show how it is possible to estimate the relative order of magnitude for non-gaussinities in squeezed and equivilateral limits. We will first consider a test scalar field in de Sitter background

S=\int d^{4}x\left((\partial\phi)^{2}-m^{2}\phi^{2}\right). (7)

From the previous lecure we know how to construct the corresponding QFT. At supercurvature scales k\ll aH the corresponding modes have the form

u_{k}(t)\approx\frac{H_{*}}{\sqrt{2k^{3}}}, (8)

(where * corresponds to the moment of curvature scale crossing as usual), so that the amplitude is frozen after Hubble crossing, and the two point correlation function is

\langle\phi_{k}\phi_{k'}\rangle\approx\delta(k+k')\frac{H_{*}^{2}}{2k^{3}}. (9)

We now want to add an interaction term \frac{V'{}'{}'}{3!}\phi^{3} to the action (7) and estimate the time-dependent correlation function \langle\phi^{3}\rangle. Let us first estmate the equivilateral contribution (k\sim{}aH): taking into account that

\phi\sim(\phi^{2})^{1/2}\sim H_{*} (10)

at Hubble scale crossing crossing we can estimate from the effective action that

\frac{\langle\phi^{3}\rangle^{1/3}}{\langle\phi^{2}\rangle^{1/2}}\sim\frac{V'{}'{}'}{H_{*}}\ll1. (11)

It is a bit more non-trivial task to estimate the squeezed limit contribution (k\ll{}aH). For that, we will use the equation of motion for \phi at supercurvature scales:

3H\dot{\phi}+m^{2}\phi\approx-\frac{V'{}'{}'}{2}\phi^{2}. (12)

We get

\phi_{NG}\sim\frac{V'{}'{}'}{H_{*}^{2}}\phi_{g}^{2}\Delta N, (13)

where \Delta N is the number of efolds passed between the given moment of time and the beginning of inflation. Therefore, the squeezed limit contribution is generally much larger than the equilateral contribution.

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Cosmological perturbations and large scale structure, Cosmology, Entered Apprentice, Quantum field theory

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