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
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
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
Hence we have
where the subscript “0″ stands for quantities evaluated in the background. The quantities and can be obtained expanding the lagrangian density of scalar field around . The total action up to the second order in the perturbations will be given by
where once again we have obmitted the total derivative terms.
Varying the previous expression with respect to B-E’ we get the constraint
and introducing the gauge-invariant potential
The previous involved expression for the action takes now the simpler form
Observing this expression it appears evident that it is the action of a scalar field with a time dependent mass term.
The first step to quantize a theory consists in the determination of the canonical conjugate momentum to the variable v, namely
By using this expression one can construct the Hamiltonian
In the quantum theory the corresponding rules associate to the variables v and π the operators and , which satisfy the commutation relations
By varing the action with respect to v we get its EoM that provide the field equations for , namely
which is equivalent to the Heisenberg relations
Where the Hamiltonian operator has been obtained from H replacing v and π with the operators and . The operator can be expanded in terms of an orthonormal basis of solutions of the field equation, which are denoted by
and that can be obtained by separation of variables. For a spatially flat universe, the function is
whereas is the solution of the following equation
In terms of modes the operator is
where creation and annihilation operators and satisfy the commutation relations for bosonic variables
The commutation relations between and ., and the previous ones between and are consistent in case the functions satisfy the condition
The next step in the quantization concerns the definition of a Fock representation of Hilbert space of states on which the operators and act.
For a scalar field on a flat space-time the representation is built by defining a vacuum state such that
where the annihilation operators are the coefficients of modes with positive frequencies in the 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.
Let us fix a generic instant of time η0, it is possible to construct a linear combination of two solutions of the equation
which has positive frequency at η0. This can be obtained by choosing as initial conditions the following ones
provided that Ek is positive for each mode. In analogy with the Minkowsky case, it is possible to define a ground state at η0 denoted with .
The time dependence of the notion of positive frequency has an important consequence: an observer at a time η0 defines as vacuum state , whereas for an observer at a sequent time η1 the state 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 E2k is positive for all modes k. In particular during the inflationary stage for a scalar field one has
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.
Therefore, to define the vacuum for η = η0 will be applied the following initial conditions
where the functions M and N satisfy the normalization conditions
The starting point for the calculus of the density fluctuation spectrum is the EoM for the gauge-invariant potential
For a spatially flat universe, it is possible to represent the operator as a Fourier integral in terms of creation and operators, namely
The power spectrum of the metric perturbations is a measure of the two points correlation function of the operator
Substituting the Fourier integral in the left hand side of previous expression, and by using the canonical commutation relations we get
where describes the modulus squared of the perturbation amplitude for a comoving scale. From the EoM for the gauge-invariant potential it follows that the functions satisfy the EoM
The relation between and the operator , used inside the general scheme of quantization previously discussed, is given by
whereas for the functions and the relation is
The initial conditions
yield for to the following relations
The solution of the EoM of , in case of small wavelengths , with the previous initial conditions is
whereas in the limit of large wavelengths it results
If we consider inflationary models that solve the horizon problem, all scales smaller than the Hubble radius at present time are inside the Hubble radius at the beginning of de Sitter phase . 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 and are satisfied.
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 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 can be written as
where the subscript denotes that the expression has to be evaluated at the time , 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 , . Its expression can be obtained from (*) and in particular it reads
where t is the physical time and is the wave number in physical coordinates.
For scales that were inside the Hubble radius at the beginning of inflation, but outside at the time t, by using (**) one gets
Denoted with the time in which inflation ends, for scales that are inside the Hubble radius at the beginning of inflation , but outside at the end of it , namely
the power spectrum is given by (**). Hence for each we have
where we have used the expression for .
Let us summarize the main results
1) Perturbations of the metric always exist, and their amplitude growths during inflation
2) Given the exponential expansion, the final power spectrum has an amplitude that is almost the same for all scales of cosmologcal interest.
6. Dark Matter
11. Cosmic Rays - I
12. Cosmic Rays - II