44. Cosmological perturbations in the presence of scalar field (Inflationary perturbations 5)

Posted by Dmitry
May 15, 2008

This is the next post in the series devoted to study of cosmological perturbations. Last time we have discussed IR quasi-classical dynamics of the inflaton and gravitational fields, so today we are ready to perform perturbation theory analysis at the level of linear perturbations.

Let us consider the universe, where the energy density is dominated by a scalar field \varphi with potential V=V(\varphi). We will be especially interested in the physics of slow roll regime

M_{P}^{2}\left(\frac{V'}{V}\right)^{2}, M_{P}^{2}\frac{V'{}'}{V}\ll1.

Constructing linear perturbation theory in the same way as we did it for the universe filled with ideal fluid, i.e., choosing the longitudnal gauge and perturbing Einstein equations, we find (again, \phi=\psi)

\phi_{;i}^{;i}-3{\cal H}\phi'-({\cal H}'+2{\cal H}^{2})\phi=\frac{4\pi}{M_{P}^{2}}\left(\varphi_{0}'\delta\varphi'+\frac{\partial V}{\partial\varphi}a^{2}\delta\varphi\right), (1)

\phi'+{\cal H}\phi=\frac{4\pi}{M_{P}^{2}}\varphi_{0}'\delta\varphi, (2)

\phi'{}'+3{\cal H}\phi'+({\cal H}'+2{\cal H}^{2})\phi=\frac{4\pi}{M_{P}^{2}}\left(\varphi_{0}'\delta\varphi'-\frac{\partial V}{\partial\varphi}a^{2}\delta\varphi\right). (3)

We can combine these equations to construct a single equation for the gravitational potential

\phi'{}'+2\left({\cal H}-\frac{\varphi_{0}'{}'}{\varphi_{0}'}\right)\phi'-\phi_{;i}^{;i}+2\left({\cal H}'-{\cal H}\frac{\varphi_{0}'{}'}{\varphi_{0}'}\right)\phi=0, (4)

analagous to the equation for gravitational potential in the universe filled with ideal fluid (we have to take \delta S=0). Clearly, the physical picture realized in the presence of the scalar field is not very different from this one : there is again a Jeans scale, the phase of short wavelength modes oscillates while their amplitude decays due to the expansion of the Universe, amplitude of long wavelength modes freezes.

Since we are also interested to know detailed behavior of the fluctuations of \delta\varphi, instead of the Eq. (4) we will deal with the perturbaed equation of motion for the scalar field. The latter has the form

\delta\varphi'{}'+2{\cal H}\delta\varphi'-\delta\varphi_{;i}^{;i}+a^{2}V'{}'\delta\varphi-4\varphi_{0}'\phi'+2a^{2}V'\phi=0 (5)

or

\ddot{\delta\varphi}+3H\dot{\delta\varphi}-\delta\varphi_{;i}^{;i}+V'{}'\delta\varphi-4\dot{\varphi_{0}}\dot{\phi}+2V'\phi=0 (6)

(we rewrote it in terms of physical time dt=ad\eta).

In the UV limit k\to\infty only first three terms in (5) are important, so that we effectively have

\delta\varphi_{k}(\eta)\approx\frac{c_{1\, k}}{a}e^{ik\eta}+\frac{c_{2\, k}}{a}e^{-ik\eta}. (7)

The IR limit is much more interesting. At k\to0, using the slow roll approximation, we can rewrite Eqs. (2) and (6) as

3H\dot{\delta\varphi}_{k=0}+V'{}'\delta\varphi_{k=0}+2V'\phi_{k=0}\approx0, (8)

H\phi_{k=0}\approx\frac{4\pi}{M_{P}^{2}}\dot{\varphi_{0}}\delta\varphi_{k=0}. (9)

After introducing new variable u=\frac{\delta\varphi_{k=0}}{V'} and using the Friedmann equation H^{2}\approx\frac{8\pi}{3M_{P}^{2}}V we finally find that \frac{d(uV)}{dt}=0, so that the solution for the non-decaying adiabatic mode of
\varphi and gravitational potential is given by

\delta\varphi_{k\to0}=C_{1\, k}\frac{V'}{V},\phi_{k\to0}=-\frac{1}{2}C_{1\, k}\left(\frac{V'}{V}\right)^{2}.

If we suppose that at the moment of time when a given mode crosses the Hubble scale its amplitude is minimal, we see that at later times its amplitude is slowly growing, since the slow roll parameter M_{P}^{2}(V'/V)^{2} grows towards the end of inflation. Therefore, inflation generally predicts a spectrum of primordial perturbations very close to the flat (Harrison-Zeldovish) one.

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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft

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