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

Posted by Dmitry
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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Cosmological perturbations and large scale structure, Cosmology, Fellow Craft, Quantum field theory

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