Cosmologically viable generalized Einstein-aether theories

We investigate generalized Einstein-aether theories that are compatible with the Planck cosmic microwave background (CMB) temperature anisotropy, polarization, and lensing data. For a given dark energy equation of state, $w_{\mathrm{de}}$, we formulate a designer approach and we investigate their impact on the CMB temperature anisotropy and matter power spectra. We use the equation of state approach to parametrize the perturbations and find that this approach is particularly useful in identifying the most suitable and numerically efficient parameters to explore in a Markov chain Monte Carlo analysis. We find the data constrains models with $w_{\mathrm{de}}=-1$ to be compatible with $\mathrm{\Lambda }$ cold dark matter ($\mathrm{\Lambda }\mathrm{CDM}$). For $w_{\mathrm{de}}\ne -1$ models, which avoid the gravitational waves constraint through the entropy perturbation, we constrain $w_{\mathrm{de}}$ to be $w_{\mathrm{de}}=-1.06_{-0.03}^{+0.08}$ (CMB) and $w_{\mathrm{de}}=-1.04_{-0.02}^{+0.05}$ ($\mathrm{CMB}+\mathrm{Lensing}$) at 68% C.L., and find that these models can be different from $\mathrm{\Lambda }\mathrm{CDM}$ and still be compatible with the data. We also find that these models can ameliorate some anomalies in $\mathrm{\Lambda }\mathrm{CDM}$ when confronted with data, such as the $\mathrm{low}\text{−}\ell$ and $\mathrm{high}\text{−}k$ power in the CMB temperature anisotropy and matter power spectra respectively, but not simultaneously. We also investigate the anomalous lensing amplitude, quantified by $A_{\mathrm{lens}}$, and find that for $w_{\mathrm{de}}=-1$ models, $A_{\mathrm{lens}}=1.15_{-0.08}^{+0.07}$ (CMB) and $A_{\mathrm{lens}}=1.12\pm 0.05$ ($\mathrm{CMB}+\mathrm{Lensing}$) at 68% C.L. $\sim 2\sigma$ larger than expected, similar to previous analyses of $\mathrm{\Lambda }\mathrm{CDM}$.

Figure 1

The evolution of ${\mathrm{\Omega }}_{\mathrm{de}}{\mathrm{\Delta }}_{\mathrm{de}}$ is shown for $k=0.001{\mathrm{Mpc}}^{-1}$, $0.01{\mathrm{Mpc}}^{-1}$, and $0.1{\mathrm{Mpc}}^{-1}$, for $\log {\mathcal{P}}_1=1.0$ and ${\mathcal{P}}_2=1.0$ fixed, compared with the attractor solution for each scale (black dashed lines).

Figure 2

The Weyl potential, $2\mathrm{\Phi }=\psi +\varphi$, (solid lines) and its derivative (dashed lines) as a function of scale factor for different values of $G_{\mathrm{eff},0}$, corresponding to ${\mathcal{P}}_2=1$, 0, and $-1$, with $\log {\mathcal{P}}_1=0.1$, $w_{\mathrm{de}}=-1$, and ${\mathrm{K}}_0=1$ fixed. The $\mathrm{\Lambda }\mathrm{CDM}$ potential is denoted by the black dotted line.

Figure 3

Left panels: The matter power spectrum relative to $\mathrm{\Lambda }\mathrm{CDM}$ (top panel) and CMB temperature angular anisotropy power spectrum (bottom panel) for ${\mathcal{P}}_2=1$ and $w_{\mathrm{de}}=-1$ fixed. The late-time ISW component is shown by the dashed lines. The black dotted line denotes $\mathrm{\Lambda }\mathrm{CDM}$. Right panels: As the left panels but with $\log {\mathcal{P}}_1=0.1$ fixed.

Figure 4

Left panels: The matter power spectrum relative to $\mathrm{\Lambda }\mathrm{CDM}$ (top panel) and CMB temperature angular anisotropy power spectrum (bottom panel) in $w\mathrm{CDM}$ models, compatible with the gravitational waves constraint $c_{13}=0$, for different $w_{\mathrm{de}}$ and ${\mathcal{P}}_3=1$ and ${\mathcal{P}}_4=-1$ fixed. The late-time ISW component is shown by the dashed lines. The black dotted line denotes $\mathrm{\Lambda }\mathrm{CDM}$. Right panels: As with the left panels, but for models with different ${\mathcal{P}}_4$ and $w_{\mathrm{de}}=-1.1$ and ${\mathcal{P}}_3=1$ fixed. All other cosmological parameters are fixed to $\mathrm{\Lambda }\mathrm{CDM}$ values given previously.

Figure 5

Left panels: The matter power spectrum (top panel) and CMB temperature angular anisotropy power spectrum (bottom) relative to $\mathrm{\Lambda }\mathrm{CDM}$ for the best fitting values of $\log {\mathcal{P}}_1$ and ${\mathcal{P}}_2$, with $w_{\mathrm{de}}=-1$, for CMB (red) and $\mathrm{CMB}+\mathrm{Lensing}$ (yellow). All other parameters have been kept fixed at the $\mathrm{\Lambda }\mathrm{CDM}$ values. Right panels: As the left panels, but for the best fitting values of ${\mathcal{P}}_3$, ${\mathcal{P}}_4$, and $w_{\mathrm{de}}\ne -1$, in $w\mathrm{CDM}$ models compatible with the gravitational waves constraint of $c_{13}=0$. The oscillations observed at $\mathrm{high}\text{−}\ell$ are due to a slightly different background cosmology which has shifted the peaks of the CMB spectrum.

Figure 6

Top panel: The lensed CMB temperature angular anisotropy power spectrum, for $\mathrm{high}\text{−}\ell$ peaks, in $\mathrm{\Lambda }\mathrm{CDM}$ is shown. The amount of lensing is modified by $A_{\mathrm{lens}}$. A value of $A_{\mathrm{lens}}=1$ is the expected amount of lensing. Bottom panel: Similar to the top panel, but now in a designer $\mathcal{F}(\mathcal{K})$ model with $w_{\mathrm{de}}=-1$. Here, $A_{\mathrm{lens}}=1$ but the amount of lensing is affected by varying ${\mathcal{P}}_2$, with $\log {\mathcal{P}}_1=-0.5$ fixed.

Figure 7

The 68% and 95% constraint contours for $\log {\mathcal{P}}_1$, ${\mathcal{P}}_2$, and ${\sigma }_8$ are shown for $w_{\mathrm{de}}=-1$ models. Note that the correlation between $\log {\mathcal{P}}_1$ and ${\sigma }_8$ is removed once lensing data is included. We see that for CMB only, there is a preference for ${\mathcal{P}}_2$ to be close to zero, with a slight preference for negative values at larger $\log {\mathcal{P}}_1$. From (46) we see that this corresponds to $G_{\mathrm{eff}}/G_{\mathrm{N}}>1$, but only slightly as it is suppressed by a larger $\log {\mathcal{P}}_1$, allowing larger values of ${\sigma }_8$. Once lensing data is included, ${\sigma }_8$ is more strongly constrained to lower values and pushes these models to be similar to $\mathrm{\Lambda }\mathrm{CDM}$, corresponding to either ${\mathcal{P}}_2\approx 0$, or a larger $\log {\mathcal{P}}_1$ which sets $G_{\mathrm{eff}}\approx G_{\mathrm{N}}$. This in turn allows for larger values of ${\mathcal{P}}_2$, since its effect is suppressed in $G_{\mathrm{eff}}$ by a larger $\log {\mathcal{P}}_1$.

Figure 8

The 68% and 95% constraint contours for ${\mathcal{P}}_3$, ${\mathcal{P}}_4$, ${\sigma }_8$, and $w_{\mathrm{de}}$. Note the anticorrelation between ${\sigma }_8$ and $w_{\mathrm{de}}$ as in $w\mathrm{CDM}$ quintessence models. As expected, there is a degeneracy between ${\mathcal{P}}_3$ and ${\mathcal{P}}_4$. We note a feature of the contours for ${\mathcal{P}}_3$ in that they are symmetric with $w_{\mathrm{de}}$ and ${\sigma }_8$. This is a consequence of treating ${\mathcal{P}}_3$ as a free parameter when in fact it is directly proportional to ${\mathcal{P}}_4$ (51), and so these contours show us the cases for when $\{{\mathcal{P}}_3,{\mathcal{P}}_4\}$ are the same or opposite sign.

Figure 9

The 68% and 95% constraint contours for $\log {\mathcal{P}}_1$, ${\mathcal{P}}_2$, $A_{\mathrm{lens}}$ and ${\sigma }_8$ are shown for $w_{\mathrm{de}}=-1$ models. Note that ${\mathcal{P}}_2$ is now very poorly constrained and also that ${\sigma }_8$ increases with the inclusion of lensing data unlike before. We see that the contour for $\{\log {\mathcal{P}}_1,{\mathcal{P}}_2\}$ is the same as before but extends much further now that ${\mathcal{P}}_2$ is more poorly constrained.