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45. Quantization of cosmological perturbations. Mukhanov-Sasaki variable (Inflationary perturbations 5)

Posted by Dmitry
at May 16, 2008

Classical primordial fluctuations of the gravitational potential which are imprinted into CMB fluctuations on the sky originate from quantum fluctuations of the scalar field and gravitational potential in the inflationary Universe. Therefore, to determine the correlation properties of classical fluctuations of the gravitational potential, we have to quantize the Einstein-Hilbert action plus the effective action for the scalar field

S=\int d\eta d^{3}x\left(-\frac{M_{P}^{2}}{16\pi}R+\frac{1}{2}(\partial\varphi)^{2}-V(\varphi)\right) (1)

taking only linear fluctuations (i.e., quadratic terms in the action (1)) into account and determine their quantum correlation properties. The complication is that fluctuations of the scalar field \delta\varphi and gravitational potential \phi are coupled to each other already at the linear level. Thus, one has to construct a linear combination u(\eta,x) of \delta\varphi and \phi such that its quadratic effective action is canonically normalized (i.e., to introduce a rotation in the field space of \delta\varphi and \phi). After, that we will be able to correctly introduce the vacuum of the theory, the Fock space, etc.

From the equations of motion for the fluctuations \delta\varphi(\eta,x) and \phi(t,x) one can see that the proper linear combination is

v(t,x)=a\left(\delta\varphi+\frac{\varphi_{0}'{}'}{{\cal H}}\phi\right), (2)

and the corresponding effective action is

S^{(2)}=\int d\eta d^{3}x\left((v')^{2}-v_{,i}v^{,i}+\frac{z'{}'}{z}v^{2}\right), (3)

where z=a\frac{\varphi_{0}'}{{\cal H}}.The variable v(\eta,x) is known as Mukhanov-Sasaki variable, it is closely related to the curvature perturbation: namely, v=z{\cal R}.

Quantization of the theory (3) is straightforward (it is the theory of harmonic oscillaor with variable frequency). The corresponding equation of motion is

v'{}'_{k}+k^{2}v_{k}-\frac{z'{}'}{z}v_{k}=0 (4)

for a given Fourier mode k of the field v (as usual, we can expand it into Fourier series due to translation invariance in 3-dim space). Note that at long walengths the amplitude of v_{k} behaves as v_{k}\sim z.

The effective frequency \omega_{k}^{2}=k^{2}-\frac{z'{}'}{z} depends on conformal time (note that it is of tachyonic type, so that long wave length modes are tachyonically unstable; this is another face of the Jeans instability). If this dependence is slow enough - namely,

\frac{\omega_{k}'}{\omega_{k}}\ll1, (5)

one can define “adiabatic”modes

v_{k}=\frac{a_{k}}{\sqrt{2\omega_{k}}}e^{i\int\omega_{k}d\eta}+\frac{a_{k}^{\dagger}}{\sqrt{2\omega_{k}}}e^{-i\int\omega_{k}d\eta} (6)

and “adiabatic” vacuum, since in the classical theory adiabatic invariant

n_{k}=\frac{E_{k}}{\omega_{k}}-\frac{1}{2}=\frac{1}{2}\left(\frac{(v_{k}')^{2}}{\omega_{k}}+\omega_{k}v_{k}^{2}\right)-\frac{1}{2} (7)

is conserved when the effective frequency \omega_{k} is changing slowly with t. In the corresponding quantum picture the adiabatic invariant (7) can be associated to the number of particles in a given mode with momentum k. On the other hand, when the condition (5) is no longer valid, adiabatic invariant (7) is changing rapidly, and we can interpret this fact as particle creation at the quantum level.

To canonically quantize the theory, we need to define canonical momentum \pi=v' and promote Poisson brackets to commutators. Decomposition into modes will automatically promote the constants a_{k} and a_{k}^{\dagger} into Fock operators with appropriate commutation relations; we are also able to define the Fock vacuum for a givnen mode k according to the prescription a_{k}|0\rangle=0.This quantum state describes the absence of excitations. If the mode starts in such a physical state, then after crossing the horizon adiabaticity condition is broken, the quick particle creation happens, after which the amplitude of the given mode freezes.

We can now easily estimate the power spectrum of the generated curvature perturbations. First, we notice that z(\eta)\sim a(\eta), since {\cal H} and \varphi_{0}' are proportional to each other. For the power spectrum of the curvature perturbation one has

P_{{\cal R}}\sim k^{3}|{\cal R}_{k}|^{2}=k^{3}z^{-2}|v_{k}|^{2}

according to the definition of v_{k}. Than,

P_{{\cal R}}\sim k^{3}z^{-2}\left(\frac{z}{z_{H}}\right)^{2}|v_{kH}|^{2}=k^{3}z_{H}^{-2}|v_{kH}|^{2}=
=k^{3}a_{H}^{-2}|v_{kH}|^{2},

where z_{H} and v_{kH} are z and the mode amplitude at the moment of Hubble scale crossing, and we used the fact that after the crossing v_{k}\sim z. Finally, we get

P_{{\cal R}}\sim k^{3}k^{-2}H^{2}k^{-1}=H^{2},

where we used vacuum initial conditions for the mode v_{k}\sim1/\sqrt{w_{k}}\sim k^{-1/2}, i.e., we again find that inflation predicts flat power spectrum of the primordial perturbations.

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

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

Posted by Dmitry
at 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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