Unification of inflation, dark energy, and dark matter within the Salam-Sezgin cosmological model

We investigate a cosmological model, based on the Salam-Sezgin six-dimensional supergravity theory and on previous work by Anchordoqui, Goldberg, Nawata, and Nuñez. Assuming a period of warm inflation, we show that it is possible to extend the evolution of the model back in time, to include the inflationary period, thus unifying inflation, dark matter, and dark energy within a single framework. Like the previous authors, we were not able to obtain the full dark matter content of the universe from the Salam-Sezgin scalar fields. However, even if only partially successful, this work shows that present-day theories, based on superstrings and supergravity, may eventually lead to a comprehensive modeling of the evolution of the universe. We find that the gravitational-wave spectrum of the model has a nonconstant negative slope in the frequency range $({10}^{-15}–{10}^6)\mathrm{rad}/\mathrm{s}$, and that, unlike standard (cold) inflation models, it shows no structure in the $\mathrm{MHz}/\mathrm{GHz}$ range of frequencies.

Figure 1

Time evolution of the energy densities of radiation ${\rho }_r$ (dashed curve) and of the scalar fields ${\rho }_{xy}={\dot{x}}^2/2+{\dot{y}}^2/2+V(x,y)$ during the first stage of evolution. The smooth transition from inflation to a radiation-dominated universe occurs at about $1.1\times {10}^8{t}_{\mathrm{p}}$.

Figure 2

Time evolution of the scalar fields $x$ (dashed curve) and $y$ during the first stage of evolution. After the inflationary period $y$ decreases very slowly, while the $|x|$ decreases exponentially.

Figure 3

Time evolution of the density parameters ${\Omega }_r$ (thin curve), ${\Omega }_{\mathrm{M}}$ (dashed curve), and ${\Omega }_y$ (thick curve) during the second stage of evolution. In this case, which corresponds to a short duration of the first stage of evolution, $8.00\times {10}^8{t}_{\mathrm{p}}$, the transition from a radiation-dominated to a matter-dominated universe occurs earlier in the history of the universe (but well after nucleosynthesis) and the density parameter for dark energy becomes non-negligible already at a redshift $z\approx {10}^6$. The $x$-field dark matter amounts to about 10% of the total matter content of the universe at the present time $u_0=0$.

Figure 4

Time evolution of the $y$-field dark energy. This field, which is practically constant during most of the second stage of evolution, became dominant only in the recent past.

Figure 5

Time evolution of the equation-of-state parameter for dark energy, $w_y$, during the second stage of evolution. At the present time, $u_0=0$, the value of this parameter is about $-0.62$. In the future, it will approach the value $-1/3$.

Figure 6

Time evolution of the deceleration parameter $q$ during the second stage of evolution. This parameter has its minimum (negative) value around the present time and approaches zero in the future.

Figure 7

Time evolution of the density parameters ${\Omega }_r$ (thin curve), ${\Omega }_{\mathrm{M}}$ (dashed curve), and ${\Omega }_y$ (thick curve) during the second stage of evolution. In this case, which corresponds to a duration of the first stage of evolution of $8.90\times {10}^8{t}_{\mathrm{p}}$, the density parameter for dark energy became non-negligible only recently. At the present time, $u_0=0$, the density parameters for radiation, ${\Omega }_{r,0}=4.8\times {10}^{-5}$, matter, ${\Omega }_{\mathrm{M},0}=0.27$, and dark energy, ${\Omega }_{y,0}=0.73$, are in good agreement with observational data. The $x$-field dark matter contributes only about 6% to the total matter content of the universe.

Figure 8

Time evolution of the scale factor $a$ (thick curve). To guide the eye, a linear growth is also represented (dashed straight line). The transition to an accelerated expansion takes place at about $(2–3)\times {10}^{17}\mathrm{s}$. At the present time, $t\approx 4\times {10}^{17}\mathrm{s}$, the expansion of the universe is slightly accelerated. In the future, the scale factor growth approaches a linear behavior, a consequence of the fact that the equation-of-state parameter for dark energy approaches $-1/3$.

Figure 9

Gravitational-wave spectrum of the Salam-Sezgin cosmological model.

Figure 10

Time evolution of the equation-of-state parameter, $w=p/\rho$, during the first stage of evolution.