Crossing the phantom divide

We consider fluid perturbations close to the “phantom divide” characterized by $p=-\rho$ and discuss the conditions under which divergencies in the perturbations can be avoided. We find that the behavior of the perturbations depends crucially on the prescription for the pressure perturbation $\delta p$. The pressure perturbation is usually defined using the dark energy rest-frame, but we show that this frame becomes unphysical at the divide. If the pressure perturbation is kept finite in any other frame, then the phantom divide can be crossed. Our findings are important for generalized fluid dark energy used in data analysis (since current cosmological data sets indicate that the dark energy is characterized by $p\approx -\rho$ so that $p<-\rho$ cannot be excluded) as well as for any models crossing the phantom divide, like some modified gravity, coupled dark energy, and braneworld models. We also illustrate the results by an explicit calculation for the “Quintom” case with two scalar fields.

Figure 1

Energy density (black solid line) and pressure (red dashed line) as functions of the scale factor. $w$ crosses $-1$ at $a_{\times }=0.5$, so that $\rho$ is minimal at this point while $p$ decreases monotonically there (but has a maximum earlier).

Figure 2

The pressure as a function of the energy density. The graph of $p(\rho )$ shows that $p$ is not a single valued function of the energy density, and that there is a point with an infinite slope (corresponding to the minimum of $\rho$ and the divergence of $c_a^2$). The point where $p(\rho )$ has a zero slope corresponds to the maximum of $p$ and a vanishing $c_a^2$.

Figure 3

This figure shows the logarithmic divergence of $\delta /(a-a_{\times })$ near $a_{\times }=0.5$. The black curve is the numerical solution while the red dashed line shows the approximation given by Eq. (35) for $\alpha =1$. The approximate formula describes the structure of the divergence quite well. The pressure perturbation $\delta p$ exhibits a very similar divergence in this case.

Figure 4

This figure shows the density contrast ${\delta }_{\mathrm{DE}}$ (red and blue lines, negative for small scale factor $a$) and the velocity perturbation $V_{\mathrm{DE}}$ (cyan and magenta lines, positive for small $a$) for a dark-energy component with a pressure perturbation given in the conformal Newtonian gauge through $\gamma$ [as defined in Eq. (45)]. The solid lines show the results for $\gamma =0.2$ and the dashed lines for $\gamma =1$. The crossing of the phantom divide at $a_{\times }=0.5$ (dotted green vertical line) is not apparent in these figures, everything is finite.

Figure 5

The gravitational potential $\psi$ in the same scenario as Fig. 4 (with $\gamma =0.2$ for the black solid line and $\gamma =1$ for the red dashed line). Again, the crossing of the phantom divide at $a_{\times }=0.5$ (dotted green vertical line) is not apparent. The gravitational potential is constant during matter domination (small $a$) and starts to decay as the dark energy begins to dominate and the expansion rate of the universe accelerates. The contribution of the dark-energy perturbations to $\psi$ is very small ($\lesssim 0.1\%$) so that there is no visible difference for the two choices of $\gamma$.

Figure 6

This figure shows the different divergent contributions to the pressure perturbation, Eq. (54), multiplied by ${10}^9$. The relative pressure perturbation is shown as red dotted lines, the nonadiabatic pressure perturbation as green dash-dotted lines and the contribution from the gauge transformation to the conformal Newtonian frame as blue dashed lines. Each of the contributions diverges at the phantom crossing, but their sum (shown as black solid line), and so $\delta p$, stays finite.

Figure 7

We plot the apparent sound speed $c_x^2$ defined by Eq. (58) for three different wave vectors, $k=1/H_0$ (black solid line), $k=10/H_0$ (red dashed line), and $100/H_0$ (blue dotted line). Although the real sound speed is just ${\widehat{c}}_s^2=1$, the apparent sound speed diverges at $w=-1$ and can even become negative.

Figure 8

The phaxion: The sign in front of the kinetic term of a normal scalar field is flipped every time $\dot{\varphi }$ passes through zero (so that $\dot{\varphi }\ge 0$). As potential we use $V(\varphi )=1+\cos (\varphi /m_0)$ which is bounded both from above and below. The field then moves to large values (top left graph), while $w$ oscillates around $-1$ (top right). The energy density $\rho$ also oscillates so that the field mimics an effective cosmological constant (bottom left). As the phantom divide is crossed with zero slope, there is no divergence in the pressure perturbation (bottom right), but the time-dependent effective mass can lead to strong growth of the perturbations in spite of ${\widehat{c}}_s^2=1$.

Figure 9

The pressure perturbation $\delta p$ for three fluids with different rest-frame sound speed, ${\widehat{c}}_s^2=0.1$ (top curve, red dash-dotted line), ${\widehat{c}}_s^2=0.01$ (middle curve, blue dashed line), and ${\widehat{c}}_s^2=0$ (lowest, solid line). Only the last case does not have a logarithmic divergence in the pressure perturbation at the phantom divide.

Figure 10

The pressure perturbation for a fluid example with ${\widehat{c}}_s^2=0.1$. The red dash-dotted curve shows the standard case with the logarithmic divergence at $w=-1$. The blue (dashed), black (solid), and green (dotted) curves use a regularized adiabatic sound speed ${\tilde{c}}_a^2$ which does not diverge. The black curve with $\lambda =1/1000$ provides a reasonable fit. Larger values smooth too much, while smaller values start to follow the divergence and exhibit a temporary instability in the solution for $V$.