Cosmology with a stiff matter era

We consider the possibility that the Universe is made of a dark fluid described by a quadratic equation of state $P=K{\rho }^2$, where $\rho$ is the rest-mass density and $K$ is a constant. The energy density $\epsilon =\rho c^2+K{\rho }^2$ is the sum of two terms: a rest-mass term $\rho c^2$ that mimics “dark matter” ($P=0$) and an internal energy term $u=K{\rho }^2=P$ that mimics a “stiff fluid” ($P=\epsilon$) in which the speed of sound is equal to the speed of light. In the early universe, the internal energy dominates and the dark fluid behaves as a stiff fluid ($P\sim \epsilon$, $\epsilon \propto a^{-6}$). In the late universe, the rest-mass energy dominates and the dark fluid behaves as pressureless dark matter ($P\simeq 0$, $\epsilon \propto a^{-3}$). We provide a simple analytical solution of the Friedmann equations for a universe undergoing a stiff matter era, a dark matter era, and a dark energy era due to the cosmological constant. This analytical solution generalizes the Einstein–de Sitter solution describing the dark matter era, and the $\mathrm{\Lambda }\mathrm{CDM}$ model describing the dark matter era and the dark energy era. Historically, the possibility of a primordial stiff matter era first appeared in the cosmological model of Zel’dovich where the primordial universe is assumed to be made of a cold gas of baryons. A primordial stiff matter era also occurs in recent cosmological models where dark matter is made of relativistic self-gravitating Bose-Einstein condensates (BECs). When the internal energy of the dark fluid mimicking stiff matter is positive, the primordial universe is singular like in the standard big bang theory. It expands from an initial state with a vanishing scale factor and an infinite density. We consider the possibility that the internal energy of the dark fluid is negative (while, of course, its total energy density is positive), so that it mimics anti-stiff matter. This happens, for example, when the BECs have an attractive self-interaction with a negative scattering length. In that case, the primordial universe is nonsingular and bouncing like in loop quantum cosmology. At $t=0$, the scale factor is finite and the energy density is equal to zero. The universe first has a phantom behavior where the energy density increases with the scale factor, then a normal behavior where the energy density decreases with the scale factor. For the sake of generality, we consider a cosmological constant of arbitrary sign. When the cosmological constant is positive, the Universe asymptotically reaches a de Sitter regime where the scale factor increases exponentially rapidly with time. This can account for the accelerating expansion of the Universe that we observe at present. When the cosmological constant is negative (anti–de Sitter), the evolution of the Universe is cyclic. Therefore, depending on the sign of the internal energy of the dark fluid and on the sign of the cosmological constant, we obtain analytical solutions of the Friedmann equations describing singular and nonsingular expanding, bouncing, or cyclic universes.

Figure 1

Relation between the energy density and the rest-mass density.

Figure 2

Equation of state giving the pressure as a function of the energy density.

Figure 3

Speed of sound as a function of the energy density.

Figure 4

Effective potential $V(R)$ in the case where (1) ${\mathrm{\Omega }}_{s,0}>0$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}>0$, (2) ${\mathrm{\Omega }}_{s,0}<0$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}>0$, (3) ${\mathrm{\Omega }}_{s,0}>0$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}<0$, (4) ${\mathrm{\Omega }}_{s,0}<0$ and ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}<0$. We have taken $|{\mathrm{\Omega }}_{s,0}|=10^{-3}$, ${\mathrm{\Omega }}_{\text{rad},0}=0$, ${\mathrm{\Omega }}_{m,0}=0.237$, and $|{\mathrm{\Omega }}_{\mathrm{\Lambda },0}|=0.763$.

Figure 5

Phase portrait of the Universe in the four cases considered in Fig. 4.

Figure 6

Energy density as a function of the scale factor. We have taken ${\mathrm{\Omega }}_{m,0}=0.237$, ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}=0.763$, and ${\mathrm{\Omega }}_{s,0}=10^{-3}$ (here and in the following figures, we have chosen a relatively large value of the density of stiff matter ${\mathrm{\Omega }}_{s,0}$ for a better illustration of the results).

Figure 7

Evolution of the proportion ${\mathrm{\Omega }}_{\alpha }={\epsilon }_{\alpha }/\epsilon$ of the different components of the universe with the scale factor.

Figure 8

Evolution of the scale factor as a function of time.

Figure 9

Evolution of the energy density as a function of time.

Figure 10

Energy density as a function of the scale factor. We have taken ${\mathrm{\Omega }}_{m,0}=0.237$, ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}=0.763$, and ${\mathrm{\Omega }}_{s,0}=-10^{-3}$.

Figure 11

Evolution of the proportion ${\mathrm{\Omega }}_{\alpha }={\epsilon }_{\alpha }/\epsilon$ of the different components of the Universe with the scale factor.

Figure 12

Evolution of the scale factor as a function of time.

Figure 13

Evolution of the energy density as a function of time.

Figure 14

Energy density as a function of the scale factor. We have taken ${\mathrm{\Omega }}_{m,0}=0.237$, ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}=-0.763$, and ${\mathrm{\Omega }}_{s,0}=10^{-3}$.

Figure 15

Evolution of the proportion ${\mathrm{\Omega }}_{\alpha }={\epsilon }_{\alpha }/\epsilon$ of the different components of the Universe with the scale factor.

Figure 16

Evolution of the scale factor as a function of time.

Figure 17

Evolution of the energy density as a function of time.

Figure 18

Energy density as a function of the scale factor. We have taken ${\mathrm{\Omega }}_{m,0}=0.237$, ${\mathrm{\Omega }}_{\mathrm{\Lambda },0}=-0.763$, and ${\mathrm{\Omega }}_{s,0}=-10^{-3}$.

Figure 19

Evolution of the proportion ${\mathrm{\Omega }}_{\alpha }={\epsilon }_{\alpha }/\epsilon$ of the different components of the Universe with the scale factor.

Figure 20

Evolution of the scale factor as a function of time.

Figure 21

Evolution of the energy density as a function of time.

Figure 22

Evolution of the scale factor and energy density as a function of time. We have taken ${\mathrm{\Omega }}_{s,0}=10^{-6}$ and ${\mathrm{\Omega }}_{\text{rad},0}=8.4810^{-5}$.

Figure 23

Evolution of the scale factor and energy density as a function of time. We have taken ${\mathrm{\Omega }}_{s,0}=-10^{-6}$ and ${\mathrm{\Omega }}_{\text{rad},0}=8.4810^{-5}$.

Figure 24

Evolution of the scale factor as a function of time. For illustration, in this figure and in the following ones, we have taken $\alpha =1$ and $n=1$.

Figure 25

Evolution of the energy density as a function of time.

Figure 26

Potential of the scalar field (inflaton).

Figure 27

Evolution of the scalar field as a function of time.