# Vacuum Decay into Cold Dark Matter

Let us consider the Einstein field equations $R_{\mu\nu}-frac12Rg_{\mu\nu}=8\pi G\left(T_{\mu\nu}+\frac\Lambda{8\pi G}g_{\mu\nu}\right).$ According to the Bianchi identities, (i) vacuum decay is possible only from a previous existence of some sort of non-vanishing matter and/or radiation, and (ii) the presence of a time-varying cosmological term results in a coupling between $T_{\mu\nu}$ and $\Lambda$. We will assume (unless stated otherwise) coupling only between vacuum and CDM particles, so that $u_\mu,T_{;\nu}^{(CDM)\mu\nu}=-u_\mu\left(\frac{\Lambda g^{\mu\nu}}{8\pi G}\right)_{;\nu}= -u_\mu\left(\rho_\Lambda g^{\mu\nu}\right)_{;\nu}$ where $T_{\mu\nu}^{(CDM)}=\rho_{dm}u^\mu u^\nu$ is the energy-momentum tensor of the CDM matter and $\rho_\Lambda$ is the vacuum energy density. It immediately follows that $\dot\rho_{dm}+3H\rho_{dm}=-\dot\rho_\Lambda.$ Note that although the vacuum is decaying, $w_\Lambda=-1$ is still constant, the physical equation of state (EoS) of the vacuum $w_\Lambda\equiv=p_\Lambda/\rho_\Lambda$ is still equal to constant $-1$, which follows from the definition of the cosmological constant.

### Problem 1

Since vacuum energy is constantly decaying into CDM, CDM will dilute in a smaller rate compared with the standard relation $\rho_{dm}\propto a^{-3}$. Thus we assume that $\rho_{dm}=\rho_{dm0}a^{-3+\varepsilon}$, where $\varepsilon$ is a small positive small constant. Find the dependence $\rho_\Lambda(a)$ in this model.

### Problem 2

Solve the previous problem for the case when vacuum energy is constantly decaying into radiation.

### Problem 3

Show that existence of a radiation dominated stage is always guaranteed in scenarios, considered in the previous problem.

### Problem 4

Find how the new temperature law scales with redshift in the case of vacuum energy decaying into radiation.

## Vacuum decay into CDM particles

Since the energy density of the cold dark matter is $\rho_{dm}=nm$, there are two possibilities for storage of the energy received from the vacuum decay process:

(i) the equation describing concentration, $n$, has a source term while the proper mass of CDM particles remains constant;

(ii) the mass $m$ of the CDM particles is itself a time-dependent quantity, while the total number of CDM particles, $N=na^3$, remains constant.

Let us consider both the possibilities.

### Problem 5

Find dependence of total particle number on the scale factor in the model considered in problem #IDE_78.

### Problem 6

Find time dependence of CDM particle mass in the case when there is no creation of CDM particles in the model considered in problem #IDE_78.

### Problem 7

Consider a model where the cosmological constant $\Lambda$ depends on time as $\Lambda=\sigma H$. Let a flat Universe be filled by the time-dependent cosmological constant and a component with the state equation $p_\gamma= (\gamma-1)\rho_\gamma$. Find solutions of Friedman equations for this system [1].

### Problem 8

Show that the model considered in the previous problem correctly reproduces the scale factor evolution both in the radiation-dominated and non-relativistic matter (dust) dominated cases.

### Problem 9

Find the dependencies $\rho_\gamma(a)$ and $\Lambda(a)$ both in the radiation-dominated and non-relativistic matter dominated cases in the model considered in problem #IDE_84.

### Problem 10

Show that for the $\Lambda(t)$ models $T\frac{dS}{dt}=-\dot\rho_\Lambda a^3.$

# Time-dependent cosmological "constant"

### Problem 11

Consider a two-component Universe filled by matter with the state equation $p=w\rho$ and cosmological constant and rewrite the second Friedman equation in the following form

     $$\label{mainDE} \frac{\ddot{a}}{a} = \frac{1}{2}\left( 1 + 3w\right) \left( \frac{\dot{a}^2}{a^2} + \frac{k}{a^2} \right) + \frac{1-3w}{6} \Lambda .$$


### Problem 12

Consider a two-component Universe filled by matter with the state equation $p=w\rho$ and cosmological constant with quadratic time dependence $\Lambda(\tau)=\mathcal{A}\tau^2$ and find the time dependence for of the scale factor.

### Problem 13

Consider a flat two-component Universe filled by matter with the state equation $p=w\rho$ and cosmological constant with quadratic time dependence $\Lambda(\tau)=\mathcal{A}\tau^{\ell}$. Obtain the differential equation for Hubble parameter in this model and classify it.

### Problem 14

Find solution of the equation obtained in the previous problem in the case ${\ell =1}$. Analyze the obtained solution.

### Problem 15

Solve the equation obtained in the problem #tauell for ${\ell =2}.$ Consider the following cases
a) $\lambda_0 > -1/(3\gamma\tau_0)^2,$
b) $\lambda_0 = -1/(3\gamma\tau_0)^2,$
c) $\lambda_0 < -1/(3\gamma\tau_0)^2$
(see the previous problem). Analyze the obtained solution.

### Problem 16

Consider a flat two-component Universe filled by matter with the state equation $p=w\rho$ and cosmological constant with the following scale factor dependence $$\Lambda = {\cal B} \, a^{-m}. \label{Bam}$$ Find dependence of energy density of matter on the scale factor in this model.

### Problem 17

Find dependence of deceleration parameter on the scale factor for the model of previous problem.