Initial perturbations in the Universe

Fluctuations power spectrum: non-relativistic approach

Problem 1

problem id: per16

Construct the correlation function of the Fourier components of the relative density fluctuations, which satisfies the cosmological principle.

Problem 2

problem id: per17

Express the correlation function of the relative density fluctuations through the power spectrum of these fluctuations.

Quantum fluctuations of fields in inflationary Universe

Problem 3

problem id: infl_vac_fluc0

Estimate the amplitude of vacuum fluctuations of the free massless scalar quantum field $\varphi(\vec{x},t)$ with characteristic momenta $q$ and frequencies $w_q=q$ with background Minkowski metrics.

solution

First, note that the energy of the vacuum fluctuations in the characteristic spatial volume $\lambda \sim q^{-1}$ coincides by the order of magnitude with the energy of zero--point oscillations with frequency $w_q$, which is $\hbar w_q/2$. The action for such oscillator is \begin{equation}\nonumber S_\varphi =\frac{1}{2}\int d^4 x [(\partial_t \varphi)^2 - (\partial_i \varphi)^2], \end{equation} and energy functional is \begin{equation}\label{E_phi} E_\varphi =\frac{1}{2}\int d^3 x [(\partial_t \varphi)^2 + (\partial_i \varphi)^2]. \end{equation} Given $\delta \varphi (q)$ is an amplitude of considered fluctuations, according to the \eqref{E_phi} the estimate for corresponding energy density is $$\rho \sim w_q^2 (\delta \varphi)^2.$$ Hence, $$\frac{w_q}{2}\sim \rho\lambda^3 \sim w_q^2 (\delta \varphi)^2\lambda^3,$$ where $\hbar =1$. Accounting for $w_q=q, \lambda \sim q^{-1}$ one obtains \begin{equation}\label{delta_varphi} \delta \varphi (q)=q. \end{equation} Thus the amplitudes of vacuum fluctuations are large at large $q$ and, respectively, at short wavelengths, while conversely, they are small at long wavelengths. Consequently, the effects of the vacuum fluctuations are negligible on a macroscopic scale.

Problem 4

problem id: infl_vac_fluc1

Considering the free massless scalar quantum field as a set of quantum harmonic oscillators, refine the estimate obtained in the problem #infl_vac_fluc0.

solution

To improve the estimate, we use the standard expression for the field operator: \begin{equation}\label{phi_qun} \varphi(\vec{x},t)=\int \frac{d^3q}{(2\pi)^{3/2}\sqrt{2w_q}}\left(e^{iw_qt-i\vec{q}\vec{x}}A_{\vec{p}}^\dag+e^{-iw_qt+i\vec{q}\vec{x}}A_{\vec{p}}\right) \end{equation} Note that creation $A_{\vec{p}}^\dag$ and annihilation $A_{\vec{p}}$ operators satisfy the standard commutation relations \begin{equation}\label{A_comm} \left[ A_{\vec{p}},A_{\vec{p}'}^\dag\right]=\delta(\vec{p}-\vec{p}'). \end{equation} Vacuum fluctuation amplitude $\delta \varphi (q)$ is understood as an rMS fluctuation of the field with haracteristic momentum $q$: \begin{equation}\label{delta_varphiQ} \delta \varphi (q)\equiv\Delta_{\varphi}=\sqrt{\mathcal{P}_{\varphi}(q)}, \end{equation} where the power spectrum $\mathcal{P}_{\varphi}(q)$ of vacuum fluctuations is determined in the same way as in the theory of ransom Gaussian fields: \begin{equation}\label{correl} \langle 0|\varphi(\vec{x},t)\varphi(\vec{x},t)|0\rangle \equiv \langle \varphi(\vec{x},t)^2\rangle = \int_0^{\infty} \mathcal{P}_{\varphi}(q) \frac{dq}{q}. \end{equation} The integration is over the module of momentum $q$ and $|0\rangle$ is a vacuum state normalized as $\langle 0|0\rangle=1$. As usual, the power spectrum represents the contribution of the logarithmic momentum interval in the mean square fluctuation, particularly $\langle \varphi(\vec{x},t)^2\rangle$. To find this value, we substitute the expansion of the field operator \eqref{phi_qun} into the left--hand side of \eqref{correl}. Note, that annihilation operator acts as $A_{\vec{p}}|0\rangle=0$, since the vacuum already has no particles. Hence, \begin{equation}\label{varphi_ket} \varphi(\vec{x},t)|0\rangle=\int \frac{d^3q}{(2\pi)^{3/2}\sqrt{2w_q}}\left(e^{iw_qt-i\vec{q}\vec{x}}A_{\vec{p}}^\dag|0\rangle\right). \end{equation} The easiest way to determine the vector $\langle 0|\varphi(\vec{x},t)$ is by using the relation \eqref{varphi_ket} and definition $\langle 0|\equiv (|0\rangle)^{\dag}$: \begin{equation}\label{bra_varphi} \langle 0|\varphi(\vec{x},t)\equiv (\varphi(\vec{x},t)^{\dag}|0\rangle)^{\dag} =\int \frac{d^3q}{(2\pi)^{3/2}\sqrt{2w_q}}e^{-iw_qt+i\vec{q}\vec{x}}\langle 0|A_{\vec{p}}. \end{equation} Using the expressions \eqref{varphi_ket} and \eqref{bra_varphi} one obtains \begin{equation}\label{correl_comm} \langle \varphi(\vec{x},t)^2\rangle = \int \frac{d^3q d^3q'}{(2\pi)^{3}\sqrt{2w_q 2w_{q'}}}e^{i(q-q')}\langle 0|A_{\vec{p}}A_{\vec{p}'}^{\dag}|0\rangle. \end{equation} From commutation relation \eqref{A_comm}: \begin{equation}\label{corr_phi2} \langle \varphi(\vec{x},t)^2\rangle =\int \frac{d^3q}{(2\pi)^{3}2w_q}=\int_0^{\infty} \frac{4\pi q^2dq}{(2\pi)^{3}2q}=\int_0^{\infty} \frac{ q^2}{(2\pi)^{2}}\frac{dq}{q}, \end{equation} and hence, \begin{equation}\label{power_spectrum} \mathcal{P}_{\varphi}(q)=\frac{ q^2}{(2\pi)^{2}}. \end{equation} Thus, we see that vacuum fluctuations amplitude for the free massless quantum of the field is given by \begin{equation}\label{vac_fluc} \delta \varphi (q)=\frac{ q}{2\pi} \end{equation} Generally speaking, integral \eqref{correl} has ultraviolet divergence, since operators \eqref{phi_qun} in the left--hand side of \eqref{correl} act at the same point. This, however, does not prevent the power spectrum to be calculated using the formula \eqref{correl}. Physical quantities, such as correlators of fields in different points are finite in the theory of free quantum field \cite{Rubakov2}.

Problem 5

problem id:

Representing the inflanton field as a superposition of uniform scalar field $\varphi_b(t)$ and small perturbation $\psi(\vec{x},t)$ on the background of unperturbed FRW metrics, obtain the equations of motion of small perturbation $\psi(\vec{x},t),$, assuming, that action for perturbation is quadratic.

solution

The inflanton field has the form: \begin{equation}\label{varphi_sum} \varphi(\vec{x},t)=\varphi_b(t)+\psi(\vec{x},t), \end{equation} where $\varphi_b(t)$ is a background field, satisfying usual Klein--Gordon equation. Quadratic action for $\psi(\vec{x},t)$ at the background of spatially flat metrics can be obtained by substituting the expansion \eqref{varphi_sum} into general expression for action: \begin{equation}\label{act_Sphi} S_\varphi = \int {d^4 x\sqrt { - g} } \left( {{1 \over 2}\nabla _\mu \varphi \nabla ^\mu \varphi - V(\varphi )} \right) \end{equation} As a result: \begin{equation}\label{act_Sphi1} S_\psi = \frac{1}{2}\int {d^4 x\sqrt { - g} } \left(\nabla _\mu \psi \nabla ^\mu \psi - V''(\varphi_b)\psi^2\right) \end{equation} Variation of action is $$ \delta S_\psi = -\int {d^4 x\sqrt { - g} } \left(\nabla _\mu \nabla ^\mu \psi + V''(\varphi_b)\psi\right)\delta \psi, $$ and equation of motion reads $$\nabla _\mu \nabla ^\mu \psi + V''(\varphi_b)\psi.$$ Proceeding as in problem #inf5: \begin{equation}\label{inf_eq_fp} \ddot\psi+ 3H\dot\psi-\frac{1}{a^2}\partial_i\partial_i\psi + V''(\varphi_b)\psi = 0 \end{equation}

Problem 6

problem id:

Perform a qualitative analysis of the equation for small perturbations of the inflaton from previous problemin the different modes of inflation.

solution

Due to the spatial uniformity of background fields $ \varphi_b $ solutions of this equation can be found in the form of plane waves $$\psi(\vec{x},t)\propto e^{\pm i\vec{k}\vec{x}}\psi_{\vec{k}}(t)$$ where $k$ is a constant in time conformal momentum. Equation for Fourier components has the form: \begin{equation}\label{ddot_psi_per} \ddot\psi_{\vec{k}}+ 3H\dot\psi_{\vec{k}}-\frac{k^2}{a^2}\psi_{\vec{k}} + V''(\varphi_b)\psi_{\vec{k}} = 0 \end{equation} Recalling the features of the inflationary regime, this equation can be simplified for qualitative analysis. one of the main features of inflationary phase is the slow evolution of Hubble parameter $H$ and fast growth of scale factor. Physical momentum $q(t) = k/ a(t)$ is large compared to Hubble parameter and, moreover, $q^2 \gg V''(\varphi_b)$. The largest terms in \eqref{ddot_psi_per}, thus, are first and third terms; field $\psi_{\vec{k}}(t)$ oscillates rapidly and behaves mostly like in Minkowski space-time. In other words, mode $\psi_{\vec{k}}(t)$ is below the horizon, space--time curvature is effectively small and has little effect on its evolution. The results of problem #infl_vac_fluc1 can be used in this regime and amplitudes of vacuum fluctuations decrease in correspondence with cosmological "reddening" of the momentum. At later times the relation $q(t)\ll H$ holds and term $3H\dot\psi_{\vec{k}}$ dominates the equation \eqref{ddot_psi_per}. Effective wavelength of the mode in this regime is much greater than cosmological horizon and doesn't depend on time, so that its amplitude "freezes", although its wavelength increases and physical momentum decreases.

Problem 7

problem id: infl_vac_per_app

Estimate the amplitude of vacuum fluctuations at the moment of exit of cosmological perturbations beyond the on the horizon.

solution

The transition from the regime below the horizon to the regime beyond the horizon (exit beyond horizon) happens, when the wavelength of a given mode is the order of the horizon scale (hence the name), or equivalently, the momentum is of the order of the Hubble parameter $$q(t)\sim H$$. the amplitude of the fluctuation at this moment can be estimated according to \eqref{vac_fluc} as $$\delta \varphi(k) \sim\frac{ H}{2\pi}.$$

Problem 8

problem id:

Demonstrate that the inflationary stage provides amplification of vacuum fluctuations of inflaton field.

solution

As it was already shown, after exit beyond the horizon vacuum fluctuation amplitude becomes constant. This means that the fluctuations of the inflaton field with large wavelengths (far below the horizon), have increased as compared to the same fluctuations with wavelengths in the Minkowski space by the \begin{equation}\nonumber \frac{\delta \varphi}{\delta \varphi_{_M}}\sim \frac{H}{q}. \end{equation} As is known, in realistic models of inflation, which do not conflict with the observational data, the period between the exit of considered modes beyond the horizon (see e.g. problem \ref{?????}) and the end of inflation is about $N_e \sim 60$ $e-$foldings. Thus, momentum had decreased $e^{N_e}-$fold, while fluctuations increase factor, compared to the Minkowski space, amounts a giant value of \begin{equation}\nonumber \frac{H}{q}\sim e^{N_e}\sim e^{60}. \end{equation} This is the mechanism of inflationary amplification of vacuum fluctuations of inflaton field, which ultimately provides the generation of primary scalar perturbations in the Universe.

Problem 9

problem id:

Consider the difference in the sequence of events during the evolution of cosmological perturbations in the radiation--dominated stage, or during the stage of domination nonrelativistic matter and inflationary cosmology.

solution

For each mode of cosmological perturbations the sequence of events on the radiation-dominated stage, or stage of domination of nonrelativistic matter, is reverse of that which takes place at the inflationary stage. Particularly, at inflationary stage mode is first below the horizon $(q(t)\gg H)$, and then -- beyond the horizon $(q(t) \ll Í)$. At the radiation-dominated stage or the stage of domination of nonrelativistic matter the situation is reversed. This sequence of events is directly related to the resolution of problems in the hot Big Bang in the inflationary theory, which requires scale factor to grow faster with time than $H^{-1}$ (or, more precisely, than particles horizon), so that physical momentum $q(t) = k/a(t)$ decreased faster than $H(t)$. The dependence of the physical momentum and the Hubble parameter on time at the inflationary and subsequent stages is shown at Figure \ref{inf-horiz-H}. \begin{figure}[ht] \begin{center} \includegraphics[width=.7\textwidth]{inf-horiz-H.eps} \caption{Dependence of the wave vector UNIQ-MathJax67-QINU and the Hubble parameter UNIQ-MathJax68-QINU on time at the inflationary stage and later.} \end{center} \label{inf-horiz-H} \end{figure}

Problem 10

problem id: field_inf_Mink

Demonstrate that in the slow-roll regime at the beginning of inflation the inflaton field behaves like a massless scalar field in the Minkowski space.

solution

The following conditions hold in the slow--roll approzimation, as is well known: \begin{equation}\label{slow-row} \varepsilon = \frac{1}{2}M_{Pl}^{2} \left( {\frac{V'}{V}} \right)^2\ll 0,~~\eta = \frac{1}{2}M_{Pl}^{*2} \frac{V''}{V}\ll 0. \end{equation} Condition on $\eta$ can be rewritten as \begin{equation} V'' = \frac{8\pi V}{M_{Pl}^2}\eta=3H^2\eta\ll H^2 \end{equation} where $\eta$ is a slow--roll parameter. Last term in the left--hand side of \eqref{inf_eq_fp}, thus, can be neglected. Transition to conformal time transforms $FRW$ metrics into the Minkowski metrics $$ds^2=dt^2-a^2(t)\delta_{ik}(dx_idx_k)=a^2(\eta)\eta_{ik}(dx_idx_k)$$, while action for scalar field perturbations obtains the form: \begin{equation}\label{field_NoMass} S_\psi = \frac{1}{2}\int {d^4 x\sqrt { - g} } \left(\nabla _\mu \psi \nabla ^\mu\psi\right)= \frac{1}{2} \int d^4 x a^2(\eta) \left[(\partial_\eta\psi)^2- (\partial_i\psi)^2\right], \end{equation} In order to geobtain the equation of motion of this field, one can just rewrite the equation \eqref{inf_eq_fp} in terms of conformal time, resulting in \begin{equation}\label{chi_psi_a0} \psi''+2\frac{a'}{a}\psi'-\Delta \psi=0, \end{equation} where dash denotes derivative with the respect of conformal time$\eta$. It is convenient to introduce a new field $\chi$ (analogously to problem \ref{inf52}), which is not sensitive to the Hubble friction: \begin{equation}\label{chi_psi_a} \psi=\frac{1}{a(\eta)}\chi \end{equation} Equation, thus, obtains the form \begin{equation}\label{chi_psi_a} \chi'' - 2\frac{a''}{a^2}\chi -\Delta \chi=0, \end{equation} or, in momentum representation: \begin{equation}\label{chi_k} \chi'' - 2\frac{a''}{a^2}\chi +k^2 \chi=0. \end{equation} At small times terms $\frac{a''}{a^2}$ is small sompared to $k$, so that corresponding term in \eqref{inf_eq_fp} can be neglected. Technically, when $\eta\to -\infty$ this term tends to zero (e.g., for exponential expansion one have $a(\eta)=-1/(H\eta)$, so that $\frac{a''}{a^2}\to 2/\eta^2$). Field $\chi$, thus, being the function of conformal time, behaves in the same manner as massless scalar field in Minkowski space: \begin{equation}\label{phi_qun_chi} \chi(\vec{x},\eta)=\int \frac{d^3k}{(2\pi)^{3/2}\sqrt{2k}}\left(e^{ik\eta-i\vec{k}\vec{x}}A_{\vec{k}}^\dag+e^{-ik\eta+i\vec{k}\vec{x}}A_{\vec{k}}\right),~\eta \to -\infty. \end{equation}

Problem 11

problem id:

What is the initial state of the inflaton quantum field towards the creation and annihilation operators?

solution

Physically reasonable assumption about the initial state of the quantum inflaton field $ \psi $ is that it coincides with the vacuum state with respect to the operators $A_{\vec{k}\dag}$ è $A_{\vec{k}}$ for momenta of interest $\vec{k}$. Indeed, modes with considered momenta exit beyond the horizon during $N_e(k)\sim 60$ $e-$foldings before the end of inflation (see problem \ref{?????}), sothat at the beginning of inflation their physical momenta are $$q(t_i)\sim He^{N^{tot}_e-N_e(k)}$$, which is very large value, generally speaking. Thus, the difference of the initial state from the vacuum one would mean that in the very early stages of inflation in the Universe, there were particle with very large physical momenta. Such an assumption does not seem physically reasonable.

Problem 12

problem id:

For modes beyond the horizon at the inflationary stage, obtain a qualitative solution to the equation, obtained in problem #field_inf_Mink.

solution

We will use momentum representation and equation \eqref{chi_k}. In this regime $\chi$ has increasing mode $$\chi = {\rm const}\, a(t),$$ which corresponds to the time independent mode of field $\varphi.$ This equation also has decreasing solution $$\chi(\eta)=a(\eta)\int d\eta \frac{\rm{const}}{a^2}.$$ From the cosmological point of view the decreasing solution is not of interest, since corresponding field $\varphi$ rapidly decreases and becomes negligible just after the exit beyond the horizon. Thus, at large times field operator $\chi$ increases as scale factor, while field operator $\varphi$ remains constant.

Problem 13

problem id: inf_chi1

Obtain the exact solution to the equation from problem \ref{field_inf_Mink} at the inflationary stage. Consider the case of modes below and beyondthe horizon.

solution

Hubble parameter can be considered as constant at inflationary stage, so that relation $$a(\eta)=-\frac{1}{H\eta},$$ holds. Equation \eqref{chi_k} besomes \begin{equation}\label{chi_k_inf} \chi'' -\frac{ 2}{\eta^2}\chi +k^2 \chi=0. \end{equation} This is the modified Bessel equation, which has the solution in form of superposition of Bessel function of first $J_{\nu}(k\eta)$ and second $Y_{\nu}(k\eta)$ kinds: \begin{equation}\label{chi_k_inf} \chi(\eta)= (k\eta)^{1/2}\left[C_1J_{3/2}(k\eta)+C_2Y_{3/2}(k\eta)\right], \end{equation} where $C_1$ and $C_2$ are integration constants. Note the asymptotics of Bessel function, in order to determine the behavior of these modes beyond and below the horizon: $$J_\nu(k\eta) \sim \frac{1}{\Gamma(\nu+1)} \left( \frac{k\eta}{2} \right) ^\nu, ~ Y_\nu(k\eta)\sim \frac{1}{\pi}\Gamma(\nu) \left( \frac{k\eta}{2} \right) ^{-\nu},$$ ïðè $k|\eta| \ll 1,$ è $$J_\nu(k\eta) \sim \sqrt{\frac{2}{\pi k\eta}} \cos \left( k\eta-\frac{\nu\pi}{2} - \frac{\pi}{4}\right) ,~Y_\nu(k\eta) \sim \sqrt{\frac{2}{\pi k\eta}} \sin \left( k\eta-\frac{\nu\pi}{2} - \frac{\pi}{4} \right) $$ ïðè $k|\eta| \gg 1.$ Since in our case $\nu =3/2$, we obtain $$ \chi(\eta)\simeq\frac{C}{2\sqrt{\pi k^3} \eta},~\mbox{ïðè}~ k|\eta| \ll 1, ~\mbox{è} ~\chi(\eta)\simeq\sqrt{\frac{2}{\pi k }}C \cos(k\eta),~\mbox{ïðè}~ k|\eta| \gg 1. $$ Thus, at large times field operator $\chi$ increases as scale factor, while field operator $\varphi$ remains constant, which coincide our previous quantitative consideration. Note, that the solution with positive frequency asymptotics in the formal limit of $\eta \to -\infty $ can be expressed as \begin{equation}\label{infchi_k} \chi_k^{(+)}(\eta)=e^{ik\eta} \left(1+\frac{i}{k\eta}\right), \end{equation} while the solution with negative frequency asymptotics can be obtained by conjugation: $$\chi_k^{(-)}=\left[\chi_k^{(+)}\right]^\dag.$$

Problem 14

problem id: infl_vac_fluc_hor-p

Demonstrate, that power spectrum $\mathcal{P}_k(\varphi)$ of the modes, which cross the horizon is the same as for free massless scalar field (see problem #infl_vac_fluc1.)

solution

using the result of previous problem, one can deduce, that these solutions oscillate with frequency $k$ below the horizon (early times, $k\eta\gg 1$), while they behave as $\eta^{-1}\propto a(\eta)$ beyond the horizon ($k\eta\ll 1$, later periods). Requiring that long before crossing the horizon field has the form of \eqref{field_inf_Mink}, one can obtain the contribution of modes with momenta of order of $k$ to the field operator $\chi$ near the crossing of the horizon in more general form: \begin{equation}\label{phi_qun_chi-hor1} \chi(\vec{x},\eta)=\int \frac{d^3k}{(2\pi)^{3/2}\sqrt{2k}}\left(e^{ik\eta}\chi_k^{(+)}(\eta)A_{\vec{k}}^\dag+e^{-ik\eta}\chi_k^{(-)}(\eta)A_{\vec{k}}\right),~\eta \to -\infty, \end{equation} where functions $\chi_k^{(\pm)}$ are determined according to \eqref{infchi_k} from previous problem. After quite long time after horizon crossing, i.e. when $|\eta|\ll k^{-1}$, expression \eqref{phi_qun_chi-hor} can be simplified by substitution of $\chi_k^{(\pm)}$ for this evolution regime. Hence, \begin{equation}\label{phi_qun_chi-hor2} \chi(\vec{x},\eta)=\int \frac{d^3k}{(2\pi)^{3/2}\sqrt{2k}}\left(-\frac{1}{k\eta}\right)\left(e^{ik\eta}\widetilde{A}_{\vec{k}}^\dag+e^{-ik\eta}\widetilde{A}_{\vec{k}}\right),~\eta \to -\infty, \end{equation} where operators $\widetilde{A}_{\vec{k}}^\dag$ and $\widetilde{A}_{\vec{k}}$ differ from standard ones ${A}_{\vec{k}}^\dag$ and ${A}_{\vec{k}}$ by irrelevant phase factors and, thus, satisfy the standard commutation relations \eqref{A_comm}, considered in problem \ref{infl_vac_fluc1}. Using the replacement relation $\varphi=a(\eta)^{-1}\chi$ and dependence $a(\eta)=-1/(H\eta)$ for inflanton field $\varphi$ after crossing the horizon, one obtains \begin{equation}\label{phi_qun_chi-hor3} \chi(\vec{x},\eta)=\int \frac{d^3k}{(2\pi)^{3/2}\sqrt{2k}}\left(\frac{H}{k}\right)\left(e^{ik\eta}\widetilde{A}_{\vec{k}}^\dag+e^{-ik\eta}\widetilde{A}_{\vec{k}}\right),~\eta \to -\infty, \end{equation} Thus, beyond the horizon the inflaton field is time independent and is a Gaussian random field: \begin{equation}\label{corr_phi-hor} \langle \varphi(\vec{x},t)^2\rangle =\int \frac{d^3k}{(2\pi)^{3}}\frac{H^2}{k^3}=\int \frac{ H^2}{(2\pi)^{2}}\frac{dk}{k}, \end{equation} and, hence \begin{equation}\label{power_spectrum-hor} \mathcal{P}_{\varphi}(k)=\frac{ H_k^2}{(2\pi)^{2}}. \end{equation} Note, that previous consideration referred to the period of time near the exit of mode with momentum $k$ beyond the horizon, which we denote as $\eta_k$, After the exit beyond the horizon the field $\varphi$ remains constant. Thus, equations \eqref{phi_qun_chi-hor3}, \eqref{corr_phi-hor} and \eqref{power_spectrum-hor} contain in fact contain Hubble parameter at this moment, $H_k=H(\eta_k)$, which is determined from the condition of equality of physical momentum to Hubble parameter $$H(\eta_k)=\frac{k}{a(\eta_k)}.$$ The right--hand side of \eqref{power_spectrum-hor}, hence, has weak dependence on $k$, i.e. the mode's wavelength. If neglecting this dependence, and setting $H_k=H=const,$ we obtain the power spectrum of inflanton beyond the horizon at inflationary stage according \eqref{power_spectrum-hor} to be independent on momentum. Such spectrum is called flat or Harrison--Zeldovich spectrum.

Problem 15

problem id: eq_infPhi

Obtain the equation, connecting the gravitational potential $\Phi$, background inflanton scalar field ôîíîâîå $\varphi(t)$ and its perturbation $\psi(\vec{x},t).$

solution

Using the expressions for the energy-momentum tensor obtained in the problem \ref{Tmunu_inf} and $00$ -- and $0i$--components of linearized Einstein equations \eqref{Ein_1}-\eqref{Ein_2} from problem \ref{per27nnn}, one obtains \begin{eqnarray} \Delta \Phi - 3\frac{a'}{a}\Phi ' - 3\frac{a'^2}{a^2}\Phi =4\pi G \varphi'^2 \Phi + 4\pi G \left[\varphi'\psi'- \left(\varphi''+2\frac{a'}{a}\varphi' \right)\psi\right] ;\label{Ein_1phi}\\ \Phi ' + - 3\frac{a'}{a}\Phi = 4\pi G \varphi'\psi. \label{Ein_2phi} \end{eqnarray} These two equations are sufficient to find the evolution of the two unknown time functions $ \Phi $ and $ \psi$. The convenience of choosing this pair of solutions is that it does not include the second order time derivatives of these functions. So far we have not used the fact that $ a (\eta) $ and $ \varphi (t) $ satisfy Einstein equations. Using this fact, or particularly, using the Friedman equations in terms of conformal time (see problem \ref{equ66} of Chapter 2): \begin{equation}\label{pre_fri1} \frac{a''}{a}- 2\frac{a'^2}{a^2}=-4\pi G(\rho+p)=-4\pi G\varphi'^2 \end{equation} Using this relation, the first term in right--hand side of \eqref{Ein_1phi} can be rewritten and, hence: \begin{equation}\label{pre_fri2} \Delta \Phi - 3\frac{a'}{a}\Phi ' - \left(\frac{a''}{a}+\frac{a'}{a}\right)\Phi = 4\pi G \left[\varphi'\psi' - \left(\varphi''+2\frac{a'}{a}\varphi' \right)\psi\right] \end{equation} System \eqref{Ein_2phi}, \eqref{pre_fri2} can be transformed to more convenient form. We now express $\Phi$ through $\Phi'$ and $\varphi'$ using \eqref{Ein_2phi} and substitute obtained relation into \eqref{pre_fri2}. Using \eqref{pre_fri1}again, we rewrite the equation in the form: \begin{equation}\label{Phi_inf} \Delta \Phi = 4\pi G \frac{a'}{a}\varphi'^2 \frac{d}{d\eta} \left(\Phi+\frac{a'}{a\varphi'}\psi' \right). \end{equation} According to definition \eqref{calR_def} and taking into account the expression of velocities potentialof inflanton field (see problem \ref{v_inf}), the quantity in the right--hand side is exactly $-\mathcal{R}$, since \begin{equation}\label{mathcalR_pre} \mathcal{R}=-\Phi+\frac{a'}{a}v_{tot}=-\left(\Phi+\frac{a'}{a\varphi'}\psi' \right) \end{equation} In accordance with the above, the value of $ \mathcal {R} $ is a spatial curvature of hypersurfaces of constant inflaton field in the comoving reference frame.

Problem 16

problem id: inf-z-u

Obtain the equations of evolution of scalar perturbations generated by the inflaton field perturbation in the case, when there is no other fields of matter in the Universe. What form does its solution has in the case for the mode under the horizon and what its implies?

solution

In order to obtain the desired equation, we note that natural variable in equation \eqref {mathcalR_pre} is not the field $ \psi $ itself, nut a linear combination \begin{equation}\label{psitilda} \widetilde{\psi} = \Phi+\frac{a'}{a\varphi'}\psi' \end{equation} In accordance with previously considered cases, it is convenient to introduce a new field by relation $u=\widetilde{\psi}a$, and rewrite $u=-z \mathcal{R},$ $z= \frac{a'}{a^2\varphi'}$. Note, that function $u$ is called the Mukhanov--Sasaki variable. In terms of this variable equation \eqref{Phi_inf} from problem \ref{eq_infPhi} transforms to \begin{equation}\label{Phi_inf1} \Delta \Phi = 4\pi G \varphi'\frac{z}{a} \frac{d}{d\eta} \left(\frac{u}{z} \right). \end{equation} Equation \eqref{Ein_2phi} from problem \ref{eq_infPhi} in this variables takes on the form: \begin{equation}\label{Phi_inf2} \frac{a'}{a^2}\frac{d}{d\eta} \left(\frac{a^3}{a'} \Phi\right)=4\pi G \varphi'u \end{equation} System \eqref{Phi_inf1}, \eqref{Phi_inf2} reduced to the second order equation for $u$: acting by operator $ \Delta$ íà on equation \eqref{Phi_inf2} and using \eqref{Phi_inf1}, one can obtain \begin{equation}\label{Phi_inf3} \frac{d}{d\eta} \left[z^2\frac{d}{d\eta} \frac{u}{z} \right]=z\Delta u, \end{equation} and, hence: \begin{equation}\label{Phi_inf3} u'' - \frac{z''}{z}-\Delta u=0. \end{equation} As it was already noted, over the horizon $ \Delta u \simeq 0 $, and the solution has the form (?) and since $ u =-z \mathcal {R}$, the curvature of $ \mathcal {R} $ for this solution is independent of time. Thus, as mentioned above, in the assumption of the absence of the decreasing mode perturbations beyond the horizon weakly depend on time.

Problem 17

problem id: inf-z-u1

Construct the solution of the equation obtained in the previous problem for the case of inflation in the slow--roll approximation. Find the spatial curvature of hypersurfaces of constant inflaton field $\mathcal{R}$ in this regime.

solution

Take into account that although the field $ \varphi (t) $ in the inflationary stage changes slowly with time $ t$, the value of $ \varphi '$ is not small and is rapidly changing due to the rapid changes in the scale factor. it is convenient to rewrite the quantity $z$ (see previous problem) in the form $$z=\frac{a\dot{\varphi}}{H}.$$ In the considered stage of evolution all quantities, except of scale factor, can be considered constant, hence the equation \eqref{Phi_inf3} in momentum representation obtains the known form (compare with problem #field_inf_Mink) \begin{equation}\label{Phi_inf3} u'' - \frac{z''}{z}+k^2u=0. \end{equation} Since this equation is identical to the equation \eqref {chi_k}, then obviously their solutions must also match. Thus, when the mode with fixed conformal momentum $ k $ is deep below the horizon, the field $ u $ coincides with the field $ \chi = a \psi$. This implies that the quantum field has the form of right--hand side of field operator \eqref{phi_qun_chi} in problem #field_inf_Mink also for large negative $ \eta $.

Problem 18

problem id:

Find the power spectrum of $ \mathcal {P}_\varphi (k) $ of spatial curvature of hypersurfaces generated by fluctuations of the inflaton field in the comoving reference frame.

solution

In the slow-roll approximation, the expression for the $ u $ is exactly the same as for $ \chi $ in problem #infl_vac_fluc_hor-p, while expression $\widetilde{\psi} $ coincides with the expression for $\chi/a$, which plays as perturbations operator of inflanton field in simplified analysis in problem #infl_vac_fluc_hor-p. Beyond the horizon operator $\widetilde{\psi} $ is constant and equals the right--hand side of \eqref{phi_qun_chi-hor3} and corresponding Gaussian random field has a power spectrum \eqref{power_spectrum-hor}: \begin{equation}\nonumber \mathcal{P}_{\varphi}(k)=\frac{ H_k^2}{(2\pi)^{2}}. \end{equation} Relations $$u=-z \mathcal{R},~~z=\frac{a\dot{\varphi}}{H}$$ from problems #inf-z-u and #inf-z-u1 give the connection between$\widetilde{\psi}$ and $\widetilde{R}$: \begin{equation}\label{mathcal_R_psi} \mathcal{R}=-\frac{H}{\dot{\varphi}} \widetilde{\psi}. \end{equation} Using the results of the problem #inf-z-u1, one can deduce that field $ \widetilde {\psi} $ is the same as the inflaton field perturbations in Minkowski space, and hence is Gaussian, and its power spectrum is calculated in the same way as in the problem \ref {infl_vac_fluc_hor-p}. Thus the power spectrum of the spatial curvature of hypersurfaces generated by fluctuations of the inflaton field in the comoving reference frame is given by \begin{equation}\label{mathcal_P_varphi} \mathcal{P}_{\varphi}(k)=\left.\left(\frac{ H^2}{2\pi\dot{\varphi}}\right)^2\right|_{\eta_k}, \end{equation} where $\eta_k$ denotes, that corresponding quantities should be taken at the moment when the relation of equality of physical momentum and Hubble parameter holds: $$H(\eta_k)=\frac{k}{a(\eta_k)}.$$

Problem 19

problem id:

Express the amplitude of the scalar perturbations $ \Delta_ {\mathcal {P}} \equiv \sqrt {| \mathcal {P} _ \varphi (k) |} $ generated by fluctuations of the inflaton field through the potential of this field.

solution

As it was shown in the previous problem, in the inflationary theory this amplitude is given by the equation \eqref{mathcal_P_varphi}: \begin{equation}\nonumber \Delta_{\mathcal{P}}\equiv \sqrt{\mathcal{P}_\varphi(k)}=\left.\left|\frac{ H^2}{2\pi\dot{\varphi}}\right|\right|_{\eta_k}. \end{equation} Using the equations of motion of the field and the Friedmann equation in the slow--roll approximation: $$\dot{\varphi}=-\frac{V'(\varphi)}{3H},~ H^2=\frac{V(\varphi)}{3M_{Pl}^2},$$ one can express the amplitude of scalar perturbations $\Delta_{\mathcal{P}}$ in the form \begin{equation}\label{Delta} \Delta_\mathcal{P} = \frac{1}{2\sqrt{3}\pi}\frac{V^{3/2}}{M_{Pl}^3V'}. \end{equation} Moreover, as we saw in the previous problem, the values appearing in this expression should be calculated at the moment of exit of the corresponding mode beyond the horizon.

Problem 20

problem id:

Express the amplitude of the scalar perturbations $\Delta_{\mathcal{P}}$ generated by fluctuations of the inflaton field for the potential $V(\varphi)=g\varphi^n$.

solution

By differentiation: $$\frac{V^{3/2}}{V'}=\frac{\sqrt{g}}{n}\varphi^{\frac{n+2}{2}}$$ The unknown field is expressed through the number $ e-$foldings as \begin{equation}\label{Nephin} N_e = \frac{1}{M_{Pl}^2}\int \frac{V(\varphi)}{V'(\varphi)} = \frac{\varphi^2}{2nM_{Pl}^2}. \end{equation} Expressing field through $N_e$ one obtains \begin{equation}\label{Delta} \Delta_\mathcal{P} = \frac{(2\pi n N_e)^\frac{n+2}{4}}{2\pi n}\left(\frac{g}{M_{Pl}^{4-n}}\right)^{1/2} \end{equation}

Problem 21

problem id:

Find the power spectrum of initial perturbations $ \mathcal {P}_h ^{(T)}$, generated at the inflationary stage.

solution

Mechanism for the generation of tensor perturbations is exactly the same as the mechanism of the inflaton field perturbations. As we saw in problems \ref{gr_wave_eq} -- \ref{gr_wave_eq_action} in Chapter \ref{Per_theory_charter}, tensor mode with fixed polarization satisfies the equation \eqref{chi_psi_a0}, which coincides with \eqref{eq_g_wave_s} for inflanton perturbations. The only difference is that action \eqref{S_GravW} for tensor mode differs from action for perturbations of free massless field \eqref{field_NoMass} in common factor $1/(32\pi G)\equiv M_{Pl}^2/4$. In order to find the power spectrum of the initial perturbations, we can use the already known result. For this we introduce a normalized field of tensor perturbations, so that the action for gravitational waves and the action for scalar field perturbations coincide: \begin{equation*} \psi^{(T)} = \frac{M_{Pl}}{2}h^{(T)} \end{equation*} Its quantum dynamics is exactly the same as dynamics of inflanton perturbations, so that power spectrumis given by equation \eqref{power_spectrum-hor}: \begin{equation*} \mathcal{P}_{\psi^{(T)}}(k)=\frac{ H_k^2}{(2\pi)^{2}}. \end{equation*} A spectrum of tensor perturbations, given the amount of the contributions of all helicities, has the form: \begin{equation*} \mathcal{P}_{T}(k)=\frac{4}{M_{Pl}^2}\sum_T \mathcal{P}_{\psi^{(T)}}(k)=\frac{2}{\pi^2}\frac{ H_k^2}{M_{Pl}^2}, \end{equation*} where we account for two polarizations of tensor modes. Unitless amplitude of tensor perturbations is \begin{equation*} \Delta_{T} = \sqrt{\mathcal{P}_{T}(k)}=\frac{\sqrt{2}}{\pi}\frac{ H_k}{M_{Pl}}, \end{equation*} Here, as before, Hubble parameter is taken at the moment of exit of the mode with momentum $k$ beyond the horizon at inflationary stage.

Problem 22

problem id:

Construct the relation between the power amplitudes of the primary gravitational and scalar perturbations generated at the inflationary stage.

solution

\begin{equation}\label{r_Pg_Ps} r\equiv \frac{ \mathcal{P}_{T}}{\mathcal{P}_\mathcal{R}} = 16 \varepsilon \end{equation}