24. Inflation: description in terms of hydrodynamics (Inflationary perturbations 4)

Posted by Dmitry
April 23, 2008

This post continues my lecture notes on large scale structure, cosmological perturbations and inflation. The last post in this series was about the problem of initial conditions in FRW cosmology. As we have found out, standard Hot Big Bang model with power law expansion a\sim t^p is unable to explain present degree of homogeneity and isotropy of the Universe, so power law expansion of the Universe cannot be complete picture.

Let us recall that both homogeneity and spatial flatness problems of the initial condition are related to the fact that the ratio \frac{\dot{a}_{0}}{\dot{a}_{i}} is extremely small. Therefore, unnaturalness of the initial condition can be avioded if we suppose that the earlier stage of the evolution of the Universe was characterized by \frac{\dot{a}_{0}}{\dot{a}_{i}}\gg1 or in the other words, \ddot{a}>0, i.e., that the Universe was initially expanding with acceleration. How to construct a model of accelerated Universe?

Let us write down the Friedmann equation

\frac{\ddot{a}}{a}=-\frac{4\pi}{3M_{P}^{2}}(\rho+3p) (1).

As we see, it is only possible to have \ddot{a}>0 if the pressure is negative: p\lesssim-\frac{1}{3}\rho. This should correspond to the Universe filled with very special matter, with a built-in, intrinsic instability - the more pressure we put on the matter, the more it wants to collapse and vise versa. This kind of instability is naively very dangerous: for example, if one considers only the class of hydrodnamic models, where pressure p=w\rho is proportional to the energy density, than negative pressure would mean negative w and imaginary speed of sound c_{s}.

As we will see later, this paradox can be easily resolved if one recalls when description in terms of hydrodynamic degrees of freedom is adequite: it only works at length scales much larger than the mean free path l_{{\rm MFP}} for elementary (or collective) excitations in matter; only in this case excitations in such a matter can be described as sound waves. For matter with negative pressure the mean free path is always larger than the horizon scale l_{H}, so that the description of such matter in terms of hydrodynamics breaks down, and one has to use microscopic language of the quantum field theory instead.

Let us forget about this issue for a moment and try to learn something about accelerated Universe at the level of Eq. (1). First of all, we notice that the domination of positive cosmological constant would automatically provide a stage of accelerated expansion. Indeed, the energy-momentum tensor for the cosmological constant is

T_{\alpha}^{\beta}\sim\Lambda\delta_{\alpha}^{\beta},

so that the cosmological constant term represents a matter with negative pressure p=-\rho=-\Lambda. From the Friedmann equation we immediately see that

H^{2}=\left(\frac{\dot{a}}{a}\right)^{2}=\Lambda,

so that the Universe expands exponentially rapidly:

a(t)=a_{i}\exp(\Lambda^{1/2}t)=a_{i}\exp(Ht)=a_{i}\exp(N),

where N is the number of e-folds (number of steps such that the Universe expands e times during each step). Such an exponentially growing solution of the Einstein equations is known as de Sitter or inflationary universe (we will discuss it in much more details later). If there is an inhomogeneity in the de Sitter universe with characteristic scale \lambda_{i} present at t=t_{i}, it will be exponentially stretched to the size \lambda=\frac{a(t)}{a_{i}}\lambda_{i} at time i, and \lambda will exponentially rapidly cross the length scale H^{-1}\sim\Lambda^{-1/2} for any given \lambda_{i}. On the other hand, the exponentially rapid stretching of scales does not contradict causality since the particle horizon size l_{p} is much larger than the scale H^{-1} in the de Sitter universe -

l_{p}=a(t)\int_{a_{i}}^{a(t)}\frac{da}{\dot{a}a}=H^{-1}(\exp(H(t-t_{i}))-1)

- so that it grows as rapidly as any physical length scale \lambda in dS.

When accelerated stage of expansion ends giving rise to the FRW universe with deccelerated expansion, the particle horizon rapidly shrinks to the scale H^{-1}. It is therefore possible to explain homogeneity of the Universe at superhorizon scales adding a stage of accelerated expansion in the history of early Universe. Similarly, isitropy problem of the standard Big Bang cosmology is solved: the amplitude of vector modes behaves as a^{-1} in the expanding Universe and therefore exponentially quickly decays in a de Sitter-like universe. Spatial flatness of the post-de Sitter universe can be explained by the fact that the r.h.s. of the Eq. (1) decreases exponentially quickly during de Sitter stage so that in the end \Omega\sim1 with exponential precision. Finally, entropy problem is solved by considering the thermalization of the decay products of the effective cosmological constant \Lambda.

In principle, why should accelerated expansion stage end? Clearly, eternally expanding de Sitter universe as it is cannot be suitable for the description of observable Universe simply beause it is necesserily empty and cold: in the de Sitter universe the energy density of non-relativistic matter behaves as \rho_{{\rm dust}}\sim a^{-3}\sim\exp(-3Ht), energy density of ultrarelativistic matter - as \rho_{{\rm ur}}\sim a^{-4}\sim\exp(-4Ht), while temperature of hot plasma - as T\sim a^{-1}\sim\exp(-Ht). Therefore, H cannot be constant permanently: if the de Sitter epoch was realized in the very early Universe, cosmological constant \Lambda should somehow decay. Just from the definition of the Hubble parameter one has

\frac{\ddot{a}}{a}=H^{2}+\dot{H},

so the Sitter stage can be realized if |\dot{H}|\ll H^{2} (this condition is related to the slow roll condition, and it is of extreme importance in cosmology). Since H^{2} is always positive, accelerated expansion will end to give rise to the deccelerated one if and only if the running of the Hubble parameter \dot{H} is negative. When de Sitter stage is getting closer to its end, |\dot{H}| approaches the value of H^{2}, and finally, when |\dot{H}|\sim H^{2}, acceleration \ddot{a} changes its sign.

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Cosmological perturbations and large scale structure, Cosmology, Entered Apprentice

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54. Eternal inflation: stochastic approach 2 (Inflationary perturbations 7)

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