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Gennaro Miele » 9.Theory of Quantum Cosmological Perturbations

Theory of Quantum Cosmological Perturbations

Quantum theory of cosmological perturbations consists in the quantization of the metric perturbations and of the matter fields. The action is obtained by a first order expansion around a homogeneous and isotropic background, which solves the EoM in the unperturbed case. In order to obtain the perturbation action, one must consider the generic action for matter and gravity

$S=-\frac{1}{16 \pi G}\int d^4x \sqrt{-g}R + \int d^4x \sqrt{-g} \mathcal{L}_m(g)$

and expand it up to the second order in the perturbation variables, being the EoM of the first order. It is worth noticing that the first order term of the expansion vanishes because it is proportional to the Euler-Lagrange EoM for the background. For the gravitational part of the action we have

$\delta_2 S_{gr} = &\frac{1}{16 \pi G} \int d^4x \ a^2 [ -6\psi^{'2}-12\mathcal{H}(\phi+\psi)\psi^{'} - 9 \mathcal{H}^2 (\phi+\psi)^2$

$- 2\psi_{,i}(2\phi_{,i}-\psi_{,i}) - 4\mathcal{H} (\phi+\psi)(B-E^{'})_{,ii} + 4\mathcal{H}\psi^{'}E_{,ii}$

$- 4\psi^{'}(B-E^{'})_{,ii} -4\mathcal{H}\psi_{,i}B_{,i} + 6\mathcal{H}^2 (\phi+\psi)E_{,ii}$

$- 4\mathcal{H}E_{,ii}(B-E^{'})_{,jj} + 4\mathcal{H}E_{,ii}B_{,jj} + 3\mathcal{H}^2E^2_{,ii} + 3\mathcal{H}^2 B_{,i}B_{,i}] ,$

where we have cancelled terms corresponding to total derivatives. The matter dynamics is specified once is given the Lagrangian density, that for a real scalar field reads

$\mathcal{L}(\phi)= \frac{1}{2}\phi_{,\mu}\phi^{,\mu}-V(\phi)$

Theory of Quantum Cosmological Perturbations

Hence we have

$\delta_2 S_m = \int d^4 x \ \sqrt{-g_0} \left( \frac{\delta_2 \sqrt{-g_0}}{\sqrt{-g_0}}\mathcal{L}_0 + \frac{2\delta_1\sqrt{-g_0}\delta_1 \mathcal{L}}{\sqrt{-g_0}} + \delta_2 \mathcal{L}\right) ,$

where the subscript “0″ stands for quantities evaluated in the background. The quantities $\delta_1 \mathcal{L}$ and $\delta_2 \mathcal{L}$ can be obtained expanding the lagrangian density of scalar field around $\phi_0$ . The total action up to the second order in the perturbations will be given by

$\delta_2 S = \delta_2 S_{gr} + \delta_2 S_{m} =$

$= \frac{1}{6\ell^2} \int d^4 x \ a^2 [-6\psi{'}^2 - 12\mathcal{H}\phi\psi^{'} - 2\psi_{,i}(2\phi_{,i}-\psi_{,i}) - 2(\mathcal{H}^{'}+2\mathcal{H}^2)\phi^2$

$+3 \ell^2(\delta\phi^{'2}- \delta\phi_{,i}\delta\phi_{,i} - V_{,\phi\phi}a^2 \delta\phi^2) + 6\ell^2 [\phi_0^{'}(\phi + 3\psi)^{'}\delta\phi - 2V_{,\phi}a^2\phi\delta\phi]$

$+4(B-E^{'})_{,ii}(3/2\ell^2\phi_0^{'}\delta\phi-\psi^{'}-\mathcal{H}\phi)] ,$

where once again we have obmitted the total derivative terms.

Theory of Quantum Cosmological Perturbations

Varying the previous expression with respect to B-E’ we get the constraint

$\psi^{'}+\mathcal{H}\phi= \frac{3}{2}\ell^2\phi_0^{'}\delta\phi,$

and introducing the gauge-invariant potential

$v = a \left[\delta\phi + \left(\frac{\phi_0^{'}\psi}{\mathcal{H}}\right)\right] = a \left[ \overline{\delta\phi} + \left(\frac{\phi_0^{'}\Psi}{\mathcal{H}}\right)\right] ,$

$\overline{\delta\phi} = \delta\phi + \phi_0^{'}(B-E^{'}),$

Theory of Quantum Cosmological Perturbations

The previous involved expression for the action takes now the simpler form

$\delta_2 S = \frac{1}{2} \int d^4 x \ \left( v^{'2}-v_{,i}v_{,i}+\frac{z^{''}}{z}v^2 \right) , \qquad z= \frac{a\phi^{'}_0}{\mathcal{H}}.$

Observing this expression it appears evident that it is the action of a scalar field with a time dependent mass term.

Quantization of classical action

The first step to quantize a theory consists in the determination of the canonical conjugate momentum to the variable v, namely

$\pi(\eta, x) = \frac{\partial\mathcal{L}}{\partial v^{'}} = v^{'}(\eta, x),$

By using this expression one can construct the Hamiltonian

$H = \frac{1}{2} \int d^4 x \ \left( \pi^2 + v_{,i}v_{,i}-\frac{z^{''}}{z}v^2 \right).$

In the quantum theory the corresponding rules associate to the variables v and π the operators $\hat{v}$ and $\hat{\pi}$ , which satisfy the commutation relations

$\left[\hat{v}(\eta, x), \hat{v}(\eta, x') \right] = \left[\hat{\pi}(\eta, x), \hat{\pi}(\eta, x') \right] = 0 , \qquad \left[\hat{v}(\eta, x), \hat{\pi}(\eta, x') \right] = i \delta(x-x').$

By varing the action with respect to v we get its EoM that provide the field equations for $\hat{v}$ , namely

$\hat{v}'' - \nabla^2 \hat{v} - \frac{z''}{z}\hat{v}= 0.$

which is equivalent to the Heisenberg relations

$i\hat{v}'= \left[\hat{v}, \hat{H} \right], \qquad i\hat{\pi}' = \left[\hat{\pi}, \hat{H} \right]$

Quantization of classical action

Where the Hamiltonian operator $\hat{H}$ has been obtained from H replacing v and π with the operators $\hat{v}$ and $\hat{\pi}$ . The operator $\hat{v}$ can be expanded in terms of an orthonormal basis of solutions of the field equation, which are denoted by

$\psi_k(x) v^*_k(x)$

and that can be obtained by separation of variables. For a spatially flat universe, the function $\psi_k(x)$ is

$\psi_k(x)= \frac{1}{(2 \pi)^{3/2}} \exp(i x \cdot k),$

whereas $v_k$ is the solution of the following equation

$v_k''(\eta) + E^2_k v_k(\eta) = 0, \qquad E^2_k = k^2 - \frac{z''}{z}.$

Theory of Quantum Cosmological Perturbations

In terms of modes $\psi_k(x) v^*_k(x)$ the operator $\hat{v}$ is

$\hat{v}= \frac{1}{\sqrt{2}} \int d^3k \left(\psi_k(x) v^*_k(x)a_k + \psi^*_k(x) v_k(x)a^\dagger_k \right),$

where creation and annihilation operators $a^\dagger_k$ and $a_k$ satisfy the commutation relations for bosonic variables

$\left[a^\dagger_k,a^\dagger_{k'} \right]=\left[a_k,a_{k'} \right]=0, \qquad \left[a_k,a^\dagger_{k'} \right]= \delta(k-k').$

The commutation relations between $\hat{v}$ and $\hat{\pi}$ ., and the previous ones between $a^\dagger_k$ and $a_k$ are consistent in case the functions $v_k$ satisfy the condition

$v'_k(\eta) v^*_k(\eta) -v'^{*}_k(\eta) v_k(\eta) = 2i.$

Quantization of classical action

The next step in the quantization concerns the definition of a Fock representation of Hilbert space of states on which the operators $\hat{v}$ and $\hat{\pi}$ act.

For a scalar field on a flat space-time the representation is built by defining a vacuum state $\left\vert 0 \right\rangle$ such that

$a_k \left\vert 0 \right\rangle = 0 \quad \forall k,$

where the annihilation operators $a_k$ are the coefficients of modes with positive frequencies in the $\hat{v}$ expansion. All states of Fock basis can be obtained by acting on the vacuum state with combinations of creation operators. Such prescription is well defined since the notion of positive and negative frequencies is time invariant. This is not the case if one tries to quantize a scalar field in a FRW universe where we do not have an unique notion of time. However, even providing such a definition of time, solution with positive frequency at a time t₁ would not have the same property at a sequent time t₂, hence there is not a univocal definition of ground state.

Quantization of classical action

Let us fix a generic instant of time η₀, it is possible to construct a linear combination of two solutions of the equation

$v_k''(\eta) + E^2_k v_k(\eta) = 0, \qquad E^2_k = k^2 - \frac{z''}{z}.$

which has positive frequency at η₀. This can be obtained by choosing as initial conditions the following ones

$v_k(\eta_0)= E^{-1/2}_k(\eta_0), \qquad v'_k(\eta_0)= i E^{1/2}_k(\eta_0),$

provided that E_k is positive for each mode. In analogy with the Minkowsky case, it is possible to define a ground state at η₀denoted with $\left\vert 0_{\eta_0} \right\rangle \equiv \left\vert \psi_0 \right\rangle$ .

Quantization of classical action

The time dependence of the notion of positive frequency has an important consequence: an observer at a time η₀ defines as vacuum state $\left\vert \psi_0 \right\rangle$ , whereas for an observer at a sequent time η₁ the state $\left\vert \psi_0 \right\rangle$ would not appear as empty of particles. In the particular case under study, due to the time dependence of the background, quantum flactuations will be produced by an initial vacuum state. Such process is responsible, for example, for the generation of perturbations in the inflationary models.

The previous definition of vacuum is applicable only if E²_k is positive for all modes k. In particular during the inflationary stage for a scalar field one has

$\frac{z''}{z} \simeq \frac{a''}{a} > 0,$

and hence the initial conditions previously defined are not applicable. However, for the calculus of the spectrum of perturbations in inflationary models the results depend on the restriction of the initial vacuum spectrum to small wavelengths only, which is independent of the vacuum choice.

Quantization of classical action

Therefore, to define the vacuum for η = η₀ will be applied the following initial conditions

$v_k(\eta_0)= k^{-1/2}M(k\eta_0), \qquad v'_k(\eta_0)= i k^{1/2}N(k\eta_0),$

where the functions M and N satisfy the normalization conditions

$NM^* + N^*M=2; \qquad |M(k\eta_0)| \rightarrow 1, \quad |N(k\eta_0)| \rightarrow 1 \quad \text{per} \quad k\eta_0 \gg 1.$

Spectrum of density fluctuations in inflationary models of scalar fields

The starting point for the calculus of the density fluctuation spectrum is the EoM for the gauge-invariant potential $\Phi$

$\Phi^{''} + 2\left(\frac{a}{\phi^{'}_0}\right)^{'}\left(\frac{a}{\phi^{'}_0}\right)^{-1} \Phi^{'} - \nabla^2 \Phi + 2 \phi^{'}_0 \left(\frac{\mathcal{H}}{\phi^{'}_0}\right)^{'} \Phi = 0.$

For a spatially flat universe, it is possible to represent the operator $\hat{\Phi}$ as a Fourier integral in terms of creation and operators, namely

$\hat{\Phi}(x, \eta) = \frac{1}{\sqrt{2}} \frac{\phi'_0}{a} \int \frac{d^3k}{(2 \pi )^{3/2}} \left[ u^*_k(\eta)e^{ik\cdot x}a_k + u_k(\eta)e^{-ik\cdot x} a^\dagger_k \right].$

The power spectrum of the metric perturbations $|\delta_k|^2$ is a measure of the two points correlation function of the operator $\hat{\Phi}$

$\left\langle 0 \right\vert \hat{\Phi}(x,\eta)\hat{\Phi}(x+r,\eta)\left\vert 0 \right\rangle=\int_0^\infty \frac{dk}{k}\frac{\sin(kr)}{kr}|\delta_k|^2$

Spectrum of density fluctuations in inflationary models of scalar fields

Substituting the Fourier integral in the left hand side of previous expression, and by using the canonical commutation relations we get

$|\delta_k(\eta)|^2 = \frac{1}{4 \pi^2} \frac{\phi^{'2}_0}{a^2}|u_k(\eta)|^2 k^3,$

where $|\delta_k(\eta)|^2$ describes the modulus squared of the perturbation amplitude for a comoving scale. From the EoM for the gauge-invariant potential $\Phi$ it follows that the functions $u_k(\eta)$ satisfy the EoM

$u''_k(\eta) + \left[k^2 - \left(\frac{1}{z}\right)''\left(\frac{1}{z}\right)^{-1} \right]u_k(\eta)=0, \qquad z=\frac{a \phi'_0}{\mathcal{H}}.$

Spectrum of density fluctuations in inflationary models of scalar fields

The relation between $\hat{\Phi}$ and the operator $\hat{v}$ , used inside the general scheme of quantization previously discussed, is given by

$\nabla^2 \hat{\Phi} = \frac{3}{2} \ell^2 \left(\frac{\phi^{'2}_0}{\mathcal{H}} \right)\left(\frac{\hat{v}}{z} \right)',$

whereas for the functions $u_k(\eta)$ and $v_k(\eta)$ the relation is

$u_k(\eta) = -\frac{3}{2} \ell^2 \left(\frac{z}{k^2} \right)\left(\frac{v_k}{z} \right)'.$

Spectrum of density fluctuations in inflationary models of scalar fields

The initial conditions

$v_k(\eta_0)= k^{-1/2}M(k\eta_0), \qquad v'_k(\eta_0)= i k^{1/2}N(k\eta_0),$

yield for $u_k(\eta)$ to the following relations

$u_k(\eta_i)=-\frac{3}{2}\ell^2 \left( \frac{i}{k^{3/2}} N(k\eta_i) - \frac{z'(\eta_i)}{z(\eta_i)}\frac{1}{k^{5/2}}M(k\eta_i) \right)$

$u'_k(\eta_i)=-\frac{3}{2}\ell^2 \left[ \frac{i}{k^{3/2}} M(k\eta_i) + 3 \frac{z'(\eta_i)}{z(\eta_i)}\left(\frac{1}{k^{3/2}}N(k\eta_i) - \frac{z'(\eta_i)}{z(\eta_i)}\frac{1}{k^{5/2}}M(k\eta_i) \right)\right].$

The solution of the EoM of $u_k(\eta)$ , in case of small wavelengths $(k^2 \gg (1/z)''(1/z)^{-1})$ , with the previous initial conditions is

(*) $u_k(\eta) = u_k(\eta_i)\cos\left[k(\eta - \eta_i)\right] + \frac{u'_k(\eta_i)}{k}\sin\left[k(\eta - \eta_i)\right],$

whereas in the limit of large wavelengths $(k^2 \ll (1/z)''(1/z)^{-1})$ it results

(**) $u_k(\eta)= \frac{A_k}{\phi'_0}\left( \frac{1}{a} \int d\eta \ a^2 \right)'.$

If we consider inflationary models that solve the horizon problem, all scales smaller than the Hubble radius at present time $\eta_0$ are inside the Hubble radius at the beginning of de Sitter phase $(\eta=\eta_i)$ . Has already stated previously, in this case the ambiguity on the choice of the state of the system it is not so important, provided that the asymptotic conditions for the functions $M(k\eta_i)$ and $N(k\eta_i)$ are satisfied.

Spectrum of density fluctuations in inflationary models of scalar fields

The perturbations that we would like to consider are initially inside the Hubble radius and evolve according to the Eq. (*) of previous slide. Sequently, they cross the Hubble radius and their evolution will be given by Eq. (**) of previous slide. To determine the coefficient $A_k$ it is possible, for example, to use some gluing conditions between the expressions (*) and (**) at the time of crossing the Hubble radius. Hence, the solution for $u_k(\eta)$ can be written as

$u_k(\eta)= \frac{u_k(\eta_i)\cos\left[k(\eta_i)\right] + \frac{u'_k(\eta_i)}{k}\sin\left[k(\eta_i)\right]}{\left\vert \frac{1}{\phi'_0}\left( \frac{1}{a} \int d\eta \ a^2 \right)'\right\vert_{\eta_\mathcal{H}(k)}} \frac{1}{\phi'_0}\left( \frac{1}{a} \int d\eta \ a^2 \right)',$

where the subscript $\eta_\mathcal{H}(k)$ denotes that the expression has to be evaluated at the time $t_\mathcal{H}(k)$ , namely when the scale k crosses the Hubble radius. Let us consider the power spectrum during inflation, for scales that at the time t are still inside the Hubble $(k_{ph}> H(t))$ , $u_k(\eta)$ . Its expression can be obtained from (*) and in particular it reads

$|\delta_k| \simeq \frac{3 \ell^2}{4 \pi^2}|\dot{\phi_0}(t)| \simeq \frac{\ell}{4 \pi} \frac{V_{,\phi}}{V^{1/2}}, \qquad k_{ph}> H(t),$

where t is the physical time and $k_{ph}$ is the wave number in physical coordinates.

Spectrum of density fluctuations in inflationary models of scalar fields

For scales that were inside the Hubble radius at the beginning of inflation, but outside at the time t, by using (**) one gets

$|\delta_k| \simeq \frac{3 \ell^2}{4 \pi^2}\left\vert \frac{\dot{\phi_0} H^2}{H}\right\vert_{t_\mathcal{H}(k)} \left(\frac{1}{a} \int d\eta \ a \right)'\simeq$ $\frac{\ell}{4 \pi} \left\vert\frac{V^{3/2}}{V_{,\phi}}\right\vert_{t_\mathcal{H}(k)} \frac{V^2_{,\phi}}{V^2}, \qquad H(t)> k_{ph} > H_i\frac{a(t_i)}{a(t)} .$

Denoted with $t_r$ the time in which inflation ends, for scales that are inside the Hubble radius at the beginning of inflation $(t=t_i)$ , but outside at the end of it $(t=t_r)$ , namely

$H_r\frac{a(t_r)}{a(t)}> k_{ph} > H_i\frac{a(t_i)}{a(t)},$

the power spectrum is given by (**). Hence for each $t>t_r$ we have

$|\delta_k| \simeq \frac{3 \ell^3}{4 \pi (p+1)}\left\vert\frac{\dot{\phi_0} H^2}{H}\right\vert_{t_\mathcal{H}(k)} \simeq \frac{3\ell^2}{2 \pi(p+1)} \left\vert\frac{V^{3/2}}{V_{,\phi}}\right\vert_{t_\mathcal{H}(k)},$