The nature of cosmological metric perturbations in presence of gravitational particle production

Abstract

The present paper tries to answer the question: Can a de Sitter phase in presence of radiation be a competitor of the standard inflationary paradigm for the early universe? This kind of a de Sitter phase can exist in cosmological models where gravitational particle production takes place. To address the issue the metric perturbations in the de Sitter phase in presence of radiation must be known. The evolution of metric perturbations are explicitly calculated in the paper. It is seen that the evolution of scalar and vector perturbations are considerably different from standard inflationary models. These differences arise due to the particle production mechanism. The scalar perturbation power spectrum grows exponentially at small length scales. However, one cannot uniquely specify the scale at which this exponential growth starts because of the dependence of the power spectrum on the initial perturbation value. The time slice on which the initial perturbation starts is arbitrary. The arbitrariness of the initial time slice is related to the problem of setting the initial condition on the perturbations in the present model. The paper briefly opines on vector and tensor perturbations in the de Sitter space filled with radiation. It is seen that quantization of the perturbation modes in the present model is considerably difficult. One has to modify the present model to provide a consistent scheme of quantization of the perturbation modes.

Appendices

Appendix I

In an expanding universe the energy density naturally decreases with time. In presence of particle production the decrement in energy density, with time, slows down (compared to the corresponding rate in standard cosmological models) due to the production of new particles which contributes to the energy density. This decrement in the rate of energy density, in models with particle production, can be explained if we take into account the action of the creation pressure $P_c$. To understand this point we must note that the effective pressure of the system is the sum of the fluid pressure and the creation pressure,

$$\begin{aligned} P_T=P+P_c\,, \end{aligned}$$

where $P=\omega \rho$ is the fluid pressure. The total pressure appears in the total energy momentum tensor appearing in Eq. (6). As total energy momentum tensor is conserved, we must have

$$\begin{aligned} \frac{d(\rho \Delta V)}{dt}=-P_T\frac{d(\Delta V)}{dt}\,. \end{aligned}$$

Assuming $d(\Delta V)/dt>0$ for an expanding universe, we can decrease the time rate of change of the energy (compared to standard cosmological models where there is no particle production and $P_T=P$) by making $P_T<P$. For positive values of $P_T$ and P one can get $P_T<P$ when the creation pressure $P_c$ is negative. As because by introducing the negative creation pressure we can decrease the rate of decrement of the of energy (compared to the case where no particles are created) we do not again require to alter the energy density. This is the reason why we do not alter the energy density for the cosmological scenarios where there is particle production.

Appendix II

Assuming ${\bar{T}},{\bar{n}}$ to be the basic thermodynamic variables we have ${\bar{\rho }}={\bar{\rho }}({\bar{n}},{\bar{T}})$. As a consequence

$$\begin{aligned} \dot{{\bar{\rho }}}=\left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T \dot{{\bar{n}}} + \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{T}}}\right) _n \dot{{\bar{T}}}\,. \end{aligned}$$

Using the above relation in $\dot{{\bar{\rho }}}+3H({\bar{\rho }}+{\bar{P}})={\Gamma }({\bar{\rho }}+{\bar{P}})$, which comes from Eq. (13), one gets

$$\begin{aligned} \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T \dot{{\bar{n}}} + \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{T}}}\right) _n \dot{{\bar{T}}}=-3H({\bar{\rho }} + {\bar{P}}) - 3H{\bar{P}}_c\,. \end{aligned}$$

Using the relation of $\dot{{\bar{n}}}/{\bar{n}}=\Gamma -3H$, the above equation can also be written as

$$\begin{aligned} \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{T}}}\right) _n \dot{{\bar{T}}}=-3H({\bar{\rho }} + {\bar{P}}) - 3H{\bar{P}}_c -\left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T({\bar{n}}{\bar{\Gamma }}-3H{\bar{n}})\,. \end{aligned}$$

The above equation can also be written as:

$$\begin{aligned} \dot{{\bar{T}}} = -\frac{1}{({\partial {\bar{\rho }}}/{\partial {\bar{T}}})_n}\left\{ \left[ ({\bar{\rho }} + {\bar{P}})-{\bar{n}}\left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T \right] 3H + 3H{\bar{P}}_c + \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T {\bar{n}}{\Gamma } \right\} \,. \qquad \end{aligned}$$

(72)

To proceed further we have to spend some time on thermodynamic relations. From Eq. (12) (for the background cosmological model) we can write

$$\begin{aligned} d{\bar{s}} = \frac{1}{{\bar{n}}{\bar{T}}}\left[ d{\bar{\rho }} - \left( \frac{{\bar{\rho }} + {\bar{P}}}{{\bar{n}}}\right) d{\bar{n}}\right] \,, \end{aligned}$$

which gives,

$$\begin{aligned} \left( \frac{\partial {\bar{s}}}{\partial {\bar{n}}}\right) _T= & {} \frac{1}{{\bar{n}}{\bar{T}}}\left[ \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T - \left( \frac{{\bar{\rho }} + {\bar{P}}}{{\bar{n}}}\right) \right] \,, \end{aligned}$$

(73)

$$\begin{aligned} \left( \frac{\partial {\bar{s}}}{\partial {\bar{T}}}\right) _n= & {} \frac{1}{{\bar{n}}{\bar{T}}} \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{T}}}\right) _n\,. \end{aligned}$$

(74)

For partial derivatives we know

$$\begin{aligned} \frac{\partial }{ \partial {\bar{T}}}\left\{ \left( \frac{\partial {\bar{s}}}{\partial {\bar{n}}}\right) _T\right\} _n =\frac{\partial }{ \partial {\bar{n}}}\left\{ \left( \frac{\partial {\bar{s}}}{\partial {\bar{T}}}\right) _n\right\} _T\,, \end{aligned}$$

which gives

$$\begin{aligned} \frac{\partial }{ \partial {\bar{T}}}\left\{ \frac{1}{{\bar{n}}{\bar{T}}}\left[ \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T - \left( \frac{{\bar{\rho }} + {\bar{P}}}{{\bar{n}}}\right) \right] \right\} _n =\frac{\partial }{\partial {\bar{n}}}\left\{ \frac{1}{{\bar{n}}{\bar{T}}} \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{T}}}\right) _n \right\} _T\,. \end{aligned}$$

Calculating the derivatives in both sides and cancelling appropriate terms the above equation yields [26]

$$\begin{aligned} {\bar{T}}\left( \frac{\partial {\bar{P}}}{\partial {\bar{T}}}\right) _n=({\bar{\rho }} + {\bar{P}}) -{\bar{n}}\left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T\,. \end{aligned}$$

(75)

Using this relation in Eq. (72) we get

$$\begin{aligned} \dot{{\bar{T}}} = -\frac{1}{({\partial {\bar{\rho }}}/{\partial {\bar{T}}})_n}\left\{ 3H{\bar{T}}\left( \frac{\partial {\bar{P}}}{\partial {\bar{T}}}\right) _n + 3H{\bar{P}}_c + \left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T {\bar{n}}{\Gamma } \right\} \,, \end{aligned}$$

which can also be written as

$$\begin{aligned} \frac{\dot{{\bar{T}}}}{{\bar{T}}}=-3H \left( \frac{\partial {\bar{P}}}{\partial {\bar{\rho }}}\right) _n -\frac{3H{\bar{P}}_c + ({\partial {\bar{\rho }}}/{\partial {\bar{n}}})_T \,\,{\bar{n}}{\Gamma }}{{\bar{T}}({\partial {\bar{\rho }}}/{\partial {\bar{T}}})_n}\,, \end{aligned}$$

(76)

where we have used

$$\begin{aligned} \frac{({\partial {\bar{P}}}/{\partial {\bar{T}}})_n}{({\partial {\bar{\rho }}}/{\partial {\bar{T}}})_n}=\left( \frac{\partial {\bar{P}}}{\partial {\bar{\rho }}}\right) _n\,. \end{aligned}$$

If specific entropy is conserved then the expression of ${\bar{P}}_c$ is given as in Eq. (14) and we know the expression of ${\bar{n}}({\partial {\bar{\rho }}}/{\partial {\bar{n}}})_T$ from Eq. (75). Using these expressions we can write

$$\begin{aligned} 3H{\bar{P}}_c + {\bar{n}}\left( \frac{\partial {\bar{\rho }}}{\partial {\bar{n}}}\right) _T {\Gamma } = -{\Gamma }({\bar{\rho }} + {\bar{P}}) +\left[ ({\bar{\rho }} + {\bar{P}}) -{\bar{T}}\left( \frac{\partial {\bar{P}}}{\partial {\bar{T}}}\right) _n\right] \Gamma =-{\bar{T}}\,\,{\Gamma }\left( \frac{\partial {\bar{P}}}{\partial {\bar{T}}}\right) _n\,. \end{aligned}$$

Using this result in Eq. (76) for cosmological evolution conserving specific entropy we get

$$\begin{aligned} \frac{\dot{{\bar{T}}}}{{\bar{T}}}=-3H \left( \frac{\partial {\bar{P}}}{\partial {\bar{\rho }}} \right) _n + {\Gamma } \left( \frac{\partial {\bar{P}}}{\partial {\bar{\rho }}}\right) _n\,, \end{aligned}$$

and using the expression of ${\dot{n}}/n$ the above equation becomes

$$\begin{aligned} \frac{\dot{{\bar{T}}}}{{\bar{T}}}=\left( \frac{\partial {\bar{P}}}{\partial {\bar{\rho }}}\right) _n \frac{\dot{{\bar{n}}}}{{\bar{n}}}\,. \end{aligned}$$

(77)