Anisotropic dark energy and CMB anomalies

We investigate the breaking of global statistical isotropy caused by a dark energy component with an energy-momentum tensor which has point symmetry, that could represent a cubic or hexagonal crystalline lattice. In such models Gaussian, adiabatic initial conditions created during inflation can lead to anisotropies in the cosmic microwave background whose spherical harmonic coefficients are correlated, contrary to the standard assumption. We develop an adaptation of the line of sight integration method that can be applied to models where the background energy-momentum tensor is isotropic, but whose linearized perturbations are anisotropic. We then show how this can be applied to the cases of cubic and hexagonal symmetry. We compute quantities which show that such models are indistinguishable from isotropic models even in the most extreme parameter choices, in stark contrast to models with anisotropic initial conditions based on inflation. The reason for this is that the dark energy based models contribute to the CMB anisotropy via the integrated Sachs-Wolfe effect, which is only relevant when the dark energy is dominant, that is, on the very largest scales. For inflationary models, however, the anisotropy is present on all scales.

Figure 1

Unit cells of two shapes with cubic and hexagonal symmetry.

Figure 2

Angular dependence in $k$-space for the $(0,-1,+1,-2,+2)$ cubic source functions (top-left, top-right, middle-left, middle right and bottom, respectively), with $\Delta \mu$ arbitrarily set to unity. We have used the equal area Aitoff projection.

Figure 3

Angular dependence in $k$-space for the $(0,+1,-2)$ hexagonal source functions (top, middle and bottom), with $\Delta {\mu }_{\mathrm{A}}=1$ and $\Delta {\mu }_{\mathrm{B}}=0$ (left set of three), and $\Delta {\mu }_{\mathrm{A}}=0$ and $\Delta {\mu }_{\mathrm{B}}=1$ (right set of three).

Figure 4

Isotropic contribution to the scalar transfer function ${\Delta }_{\ell }(\mathbf{k})$ for the ISW component (left) and total $\mathrm{ISW}+$ last-scattering surface (LSS) components (middle). Also shown is the anisotropic component (right) which is generated only via the ISW effect. We use a model with $w=-0.4$, ${\overline{\mu }}_{\mathrm{T}}={\overline{\mu }}^{\min }$ and ${\overline{\mu }}_{\mathrm{L}}={\overline{\mu }}^{\min }+0.1$, where ${\overline{\mu }}^{\min }=0.18$.

Figure 5

Representation of the CMB correlation matrix (see text for details) in $(\mu K{)}^2$ for the ISW component (top-left) and the total $\mathrm{ISW}+\mathrm{LSS}$ (top-right), for an AEDE model with $w=-0.2$, ${\overline{\mu }}_{\mathrm{T}}={\overline{\mu }}^{\min }$ and ${\overline{\mu }}_{\mathrm{L}}={\overline{\mu }}^{\min }+0.1$. Also shown is the ACW correlation matrix for $g_{\star }=1$ (bottom-left) and $g_{\star }=0.1$ (bottom-right).

Figure 6

Cylindrical power spectrum $C_{\ell }$ for $w=-0.2$ (top-left), $w=-0.4$ (top-right), $w=-0.6$ (bottom-left) and $w=-0.8$ (bottom-right). The solid curve shows the ISW contribution for ${\overline{\mu }}_{\mathrm{T}}={\overline{\mu }}^{\min }$ and ${\overline{\mu }}_{\mathrm{L}}={\overline{\mu }}^{\min }+0.1$ (the exception is for $w=-0.8$, where $\Delta \overline{\mu }=0.08$ due to causality and stability bounds on the sound speed), and the dotted curve shows the total $\mathrm{ISW}+\mathrm{LSS}$ component. Also shown for comparison is the cylindrical $C_{\ell }$ for the ACW model (dashed curve).

Figure 7

Fractional increase in cosmic variance over an isotropic model for (solid/dotted) the ISW/total contribution to the power spectrum, with $w=-0.2$, ${\overline{\mu }}_{\mathrm{T}}={\overline{\mu }}^{\min }$ and ${\overline{\mu }}_{\mathrm{L}}={\overline{\mu }}^{\min }+0.1$. Also shown is ACW result with $g_{\star }=1$ (short-dashed) and $g_{\star }=0.1$ (long-dashed).

Figure 8

(Left) Covariance between $C_{\ell }$ for the ISW component of the power spectrum, using an AEDE model with $w=-0.2$, ${\overline{\mu }}_{\mathrm{T}}={\overline{\mu }}^{\min }$ and ${\overline{\mu }}_{\mathrm{L}}={\overline{\mu }}^{\min }+0.1$. (Right) Same quantity for the ACW model with $g_{\star }=1$.

Figure 9

Probability of distinguishing between an anisotropic and isotropic model each with the same $C_{\ell }$. The labeling is the same as for Fig. 6, where for the ACW model we use a value of $g_{\star }=0.1$. On the left, we show the contribution from each $\ell$ mode, and on the right the cumulative probability.