Constraints on the anisotropy of dark energy

If the equation of state of dark energy is anisotropic there will be additional quadrupole anisotropy in the cosmic microwave background induced by the time-dependent anisotropic stress quantified in terms of $\Delta w$. Assuming that the entire amplitude of the observed quadrupole is due to this anisotropy, we conservatively impose a limit of $|\Delta w|<2.1\times {10}^{-4}$ for any value of $w\ge -1$ assuming that ${\Omega }_{\mathrm{m}}<0.5$. This is considerably tighter than that which comes from supernovae. Stronger limits, up to a factor of 10, are possible for specific values of ${\Omega }_{\mathrm{m}}$ and $w$. Since we assume this component is uncorrelated with the stochastic component from inflation, we find that both the expectation value and the sample variance are increased. There is no improvement in the likelihood of an anomalously low quadrupole as suggested by previous work on an elliptical universe.

Figure 1

The power law slope of the function $F(a)$ for $w=-1$ (solid line), $-2/3$ (dotted line), $-1/3$ (dashed line). In each case we have used ${\Omega }_{\mathrm{m}}=0.3$.

Figure 2

The function $J({\Omega }_{\mathrm{m}},w)$ against $w$ for ${\Omega }_{\mathrm{m}}=0.1$ (dashed line), 0.2 (dotted line), 0.3 (solid line), 0.4 (short-dashed line), and 0.5 (dotted-short-dashed line).

Figure 3

Quadrupole likelihood function for $\alpha =0$ (solid line), 0.1 (dotted line), 0.5 (short-dashed line), and 1.0 (long-dashed line). The value of $C_2^{\mathrm{I}}$ has been arbitrarily set to unity.

Figure 4

The $2\sigma$ upper limit on the anisotropy, $\Delta w_{\lim }$, against $w$ for ${\Omega }_{\mathrm{m}}=0.1$ (dashed line), 0.2 (dotted line), 0.3 (solid line), 0.4 (short-dashed line), and 0.5 (dotted-short-dashed line).