Can dark energy be decaying?

Abstract

We explore the fate of the universe given the possibility that the density associated with `dark energy' may decay slowly with time. Decaying dark energy is modeled by a homogeneous scalar field which couples minimally to gravity and whose potential has {\em at least one} local quadratic maximum. Dark energy decays as the scalar field rolls down its potential, consequently the current acceleration epoch is a transient. We examine two models of decaying dark energy. In the first, the dark energy potential is modeled by an analytical form which is generic close to the potential maximum. The second potential is the cosine, which can become negative as the field evolves, ensuring that a spatially flat universe collapses in the future. We examine the feasibility of both models using observations of high redshift type Ia supernovae. A maximum likelihood analysis is used to find allowed regions in the $\lbrace m, φ_0\rbrace$ plane ($m$ is the tachyon mass modulus and $φ_0$ the initial scalar field value; $m\sim H_0$ and $φ_0\sim M_P$ by order of magnitude). For the first model, the time for the potential to drop to half its maximum value is larger than $\sim 8$ Gyrs. In the case of the cosine potential, the time left until the universe collapses is always greater than $\sim 18$ Gyrs (both estimates are presented for $\om = 0.3$, $m/H_0 \sim 1$, $H_0 \simeq 70$ km/sec/Mpc, and at the 95.4% confidence level).

Figures (5)

• Figure 1: Confidence levels at $68.3\%$ (light grey inner contour) $95.4\%$ (medium grey contour) and $99.73\%$ (dark grey outer contour) are shown in the $m$-$\phi_0$ plane for the potential $V(\phi)=V_0-\frac{1}{2}m^2\phi^2$. In the left panel, the present value of the matter density is $\Omega_{0 {\rm m}}=0.3$, and in the right panel it is $\Omega_{0 {\rm m}}=0.4$. The mass $m$ is measured in units of $\sqrt{8 \pi G V_0/3}$, and the initial field value, $\phi_0$, is in units of the reduced Planck mass $\tilde{M}_P=\sqrt{3/8 \pi G}$. The best-fit point in each plot is marked by a star. The $\chi^2$ per degree of freedom at the best-fit is $1.053$ for $\Omega_{0 {\rm m}}=0.3$ and $1.049$ for $\Omega_{0 {\rm m}}=0.4$. For both figures, in the region to the right of the thick solid line, parameter values are such that the matter density never reaches its present value, hence this region is disallowed by observations.
• Figure 2: A magnified part of the Fig. \ref{['fig:msq_full']} with (dashed) lines of constant $\Delta T_{1/2}$ added. $\Delta T_{1/2}$ is the time, measured from the present epoch, to when the DDE potential has dropped to half its maximum value: $V(\phi)=V_0/2$. The values of $\Delta T_{1/2}$ for the dashed curves (from top to bottom) are listed in Table \ref{['tab:half']} (from left to right). For both $\Omega_{0 {\rm m}}=0.3$ and $\Omega_{0 {\rm m}}=0.4$, the minimum time elapsed before the potential drops to half its maximum value is $\Delta T_{1/2} \simeq 0.6 H_0^{-1} \simeq 8 \ {\rm Gyrs}$($H_0=70 \ {\rm km/s/Mpc}$) at the $95.4 \%$ confidence level. In the region to the right of the thick solid curve, parameter values are such that the matter density never reaches its present value. This region is therefore disallowed by observations.
• Figure 3: Confidence levels at $68.3\%$ (light grey inner contour) $95.4\%$ (medium grey contour) and $99.73\%$ (dark grey outer contour) are shown in the $m$-$\phi_0$ plane for the potential $V(\phi)=V_0 {\rm cos}(m \phi/\sqrt{V_0})$. Here $m$ is in units of $\sqrt{8 \pi G V_0/3}$, and $\phi_0$ is in units of the reduced Planck mass $\tilde{M}_P=\sqrt{3/8 \pi G}$. The best-fit point in each plot is marked as a star. The $\chi^2$ per degree of freedom at the best-fit is $1.050$ for $\Omega_{0 {\rm m}}=0.3$ and $1.047$ for $\Omega_{0 {\rm m}}=0.4$. For both figures, in the region to the right of the thick solid line, parameter values are such that the matter density never reaches the present value, hence this region is disallowed by observations.
• Figure 4: A magnified part of the Fig. \ref{['fig:cos_full']} with (dashed) lines of constant $\Delta T_{\rm end}$ (upper panel) and constant $\Delta T_{\rm coll}$ (lower panel) added. In the upper panels, the time $\Delta T_{\rm end}$ is measured from the present epoch to when the universe stops accelerating: $q(t_0+\Delta T_{\rm end})=0$. The values of $\Delta T_{\rm end}$ for the dashed curves (from top to bottom) are listed in Table \ref{['tab:collapse']} (from left to right). For both $\Omega_{0 {\rm m}}=0.3$ and $\Omega_{0 {\rm m}}=0.4$, the minimum time taken for the deceleration parameter to rise to zero is $\Delta T_{\rm end} \simeq 0.7 H_0^{-1} \simeq 10 \ {\rm Gyrs}$ (at the $95.4 \%$ confidence level). For the lower panel, the values of $\Delta T_{\rm coll}$ for the dashed curves (from top to bottom) are listed in Table \ref{['tab:collapse']} (from left to right). For both $\Omega_{0 {\rm m}}=0.3$ and $\Omega_{0 {\rm m}}=0.4$, the minimum time to collapse is $\Delta T_{\rm coll} \simeq 1.3 H_0^{-1} \simeq 18 \ {\rm Gyrs}$ at the $95.4 \%$ confidence level (we assume $H_0=70 \ {\rm km/s/Mpc}$). In the region to the right of the thick solid curve the matter density never reaches its present value of $\Omega_{0 {\rm m}} = 0.3$ (left panel) and $\Omega_{0 {\rm m}} = 0.4$ (right panel), therefore this region is disallowed by observations.
• Figure 5: The evolution of the deceleration parameter $q$ and the matter density $\Omega_{\rm m}$ is shown for four different DDE models corresponding to different choices of $m$ and $\phi_0$ in the DDE potential $V(\phi)=V_0 {\rm cos}(m \phi/\sqrt{V_0})$ ($\Omega_{0 {\rm m}} = 0.3$). Time $t$ is in units of $\sqrt{3/8\pi G V_0}$. The models have parameter values: $m=1.0, \phi_0=0.6$ (A), $m=1.0, \phi_0=0.2$ (B), $m=0.74, \phi_0=0.23$ (C), $m=1.0, \phi_0=1.2$ (D). Models A,B,C are allowed by supernova observations at the $95.4\%$ confidence level. The dashed line D in both panels shows the time evolution of $q$ and $\Omega_{\rm m}$ for a DDE model with $m=1.0, \phi_0=1.2$. This model is disallowed by observations since the matter density always remains larger than $0.3$ (see figure \ref{['fig:cos_coll']}). The horizontal dotted line in the left panel ($q=0$) divides this panel into two regions. In the upper region $q > 0$ and the universe decelerates, whereas $q < 0$ in the lower region in which the universe accelerates. The points of intersection of $q=0$ with A,B,C show the commencement and end of the acceleration epoch in these models. The horizontal dotted line in the right panel marks the present epoch when $\Omega_{0 {\rm m}}=0.3$. The solid circles in both left and right panels show the epoch when the potential energy of the scalar field falls to zero. Note that this occurs after the universe stops accelerating.