How isotropic is dark energy?

TL;DR

The paper asks how isotropic dark energy must be and tests time-dependent anisotropic stress in a Bianchi I background. It introduces a constrained-basis parameterization that enforces $Q_T \equiv -\int s(a)/(a\mathcal{H}(a))\,da \approx 0$ to respect the CMB ISW quadrupole bound on $C_2^{\rm aniso}$, and applies this to Pantheon+SH0ES SNe and DESI BAO. Unconstrained anisotropic models improve the isotropic $w$CDM fit but violate the quadrupole, while a five-bin constrained model yields a substantial improvement and remains quadrupole-compatible with $\log_{10}(C_2^{\rm aniso}) \approx -10$. The results show that late-time anisotropic stress can mimic dynamical dark energy to some extent, but only with finely tuned amplitudes and careful handling of polarization constraints; future work could extend to polarization and directional BAO analyses.

Abstract

Tensions in late-time expansion data have renewed interest in models beyond $Λ$CDM. We ask: \emph{how isotropic must dark energy be?} Working in Bianchi~I, we allow time-dependent anisotropic stress and introduce a parameterisation that enforces a vanishing line-of-sight integral of the shear, thereby satisfying the CMB ISW quadrupole bound by construction. Using Pantheon+SH0ES SNe together with DESI BAO distances, single-bin (constant) and five-bin anisotropic models improve the fit over $w$CDM by $Δ(-2\ln L_{\rm iso})=14.8$ and $26.6$ respectively, but both violate the quadrupole constraint. In contrast, a five-bin constrained model achieves $Δ(-2\ln L_{\rm iso})=15.4$ while remaining compatible with the quadrupole limit. The fit improvement arises from two sources: capturing directional structure in the Pantheon+ SNe data, and partially alleviating the tension between the SH0ES $H_0$ value and DESI BAO distances.

Figures (4)

• Figure 1: Effective equation of state $w_{\rm eff}(a)$ as a function of scale factor. Comparison of a dynamical dark energy model (solid red) with $w(a)=w_0+w_a(1-a)$ and the single-bin anisotropic model (dashed blue).
• Figure 2: Best-fit evolution of $w_{\rm eff}(a)$, shear $s(a)$, and ellipticity $e^2(a)$ for the constant anisotropic model (solid red), 5-bin unconstrained model (dashed blue), and 5-bin SVD model (dot-dashed green).
• Figure 3: Anisotropy sky maps for the 5-bin unconstrained model. Top panel: Directional variation in distance modulus at redshift $z=0.5$, shown in Galactic coordinates with the preferred anisotropy axis marked by a black star. White dots indicate Pantheon+SH0ES SNe positions. Bottom panel: Predicted CMB temperature quadrupole pattern. The best-fit anisotropy axis points toward Galactic coordinates $(l,b) = (288.8^{\circ}, 6.5^{\circ})$, consistent with the preferred direction found in Verma:2024lex.
• Figure 4: Demonstration of the SVD continuation method for a 2-bin parameterization with $\alpha_1 = 0.3$. Upper panel: Evolution of the bin angle $\tan^{-1}(\Delta w_1/\Delta w_0)$ as a function of the continuation parameter $\lambda$. The blue line shows how the direction evolves during continuation, while the red dashed line indicates the fixed direction from the standard SVD method. Lower panel: Evolution of $C_2$ during continuation. The continuation method (blue) maintains much better constraint satisfaction than the standard SVD approach (red dashed line).