# 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.

### 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.

### 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.

### 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.

### 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.

### Problem 8

problem id:

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

### 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.

### 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.

### Problem 11

problem id:

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

### 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.

### 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.

### 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.)

### Problem 15

problem id: eq_infPhi

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

### 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?

### 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.

### 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.

### 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.

### Problem 20

problem id:

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

### Problem 21

problem id:

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