Model-independent reconstruction of the linear anisotropic stress $η$

TL;DR

The paper develops a model-independent estimator for the linear anisotropic stress eta, named eta_obs, based on H(z), E_g, and fσ8 data to test gravity without strong model assumptions. It introduces three reconstruction methods—binning, Gaussian Process, and polynomial regression—to recover E(z), P2, and P3 from sparse data and hence eta_obs. Across redshifts around 0.3–0.86, the methods yield eta_obs values that are broadly compatible within 1σ; the fiducial polynomial regression result is 0.49±0.69, while the other methods show mild tensions. The authors advocate polynomial regression as the most robust approach and anticipate much tighter tests with upcoming surveys such as Euclid.

Abstract

In this work, we use recent data on the Hubble expansion rate $H(z)$, the quantity $fσ_8(z)$ from redshift space distortions and the statistic $E_g$ from clustering and lensing observables to constrain in a model-independent way the linear anisotropic stress parameter $η$. This estimate is free of assumptions about initial conditions, bias, the abundance of dark matter and the background expansion. We denote this observable estimator as $η_{\rm obs}$. If $η_{\rm obs}$ turns out to be different from unity, it would imply either a modification of gravity or a non-perfect fluid form of dark energy clustering at sub-horizon scales. Using three different methods to reconstruct the underlying model from data, we report the value of $η_{\rm obs}$ at three redshift values, $z=0.29, 0.58, 0.86$. Using the method of polynomial regression, we find $η_{\rm obs}=0.57\pm1.05$, $η_{\rm obs}=0.48\pm0.96$, and $η_{\rm obs}=-0.11\pm3.21$, respectively. Assuming a constant $η_{\rm obs}$ in this range, we find $η_{\rm obs}=0.49\pm0.69$. We consider this method as our fiducial result, for reasons clarified in the text. The other two methods give for a constant anisotropic stress $η_{\rm obs}=0.15\pm0.27$ (binning) and $η_{\rm obs}=0.53 \pm 0.19$ (Gaussian Process). We find that all three estimates are compatible with each other within their $1σ$ error bars. While the polynomial regression method is compatible with standard gravity, the other two methods are in tension with it.

Figures (4)

• Figure 1: Data sets used in this work (black dots with error bars), plotted together with the corresponding reference $\Lambda\mathrm{CDM}$ prediction as a function of redshift (solid red line), using a Planck 2018 cosmology as reported on Table \ref{['tab.fiducial']}. Left panel:$E(z)$ data from Table Yu2017. We used the value of $H_{0}$ from the HST collaboration to rescale part of the data points from $H(z)$ to $E(z)$ (see main text). Central panel: Plot of the natural logarithm of the $f\sigma_{8}$ data points from Table \ref{['tab.fs8']}. Right panel: Data set for $P_{2}$, obtained using $E_{g}$ data from Table \ref{['tab.egdata']}. For $z>0.5$ we see a large discrepancy between $\Lambda\mathrm{CDM}$ and the data points, which was also noted in Amon2017.
• Figure 2: Comparison of the three reconstruction methods for each of the model-independent variables. The binning method in blue squares with error bars, Gaussian Process as a green dotted line with green bands, polynomial regression as a solid yellow line with yellow bands. All of them depicting the $1\sigma$ uncertainty. Left panel: Plot of the reconstructed $E(z)$ function on the top and its derivative $E'(z)$ on the bottom. Right panel: Plot of the reconstructed $P_{2}(z)$ function on the top and the reconstructed $P_3(z)$ function on the bottom. For each case, we show the theoretical prediction of our reference $\Lambda\mathrm{CDM}$ model as a red dashed curve.
• Figure 3: Plot of the reconstructed $\eta_{\rm obs}$ as a function of redshift, using the binning method (blue squares), Gaussian Process (green dotted line) and polynomial regression (yellow solid line). The corresponding error bands (error bars for the binning method), represent the $1\sigma$ estimated error on the reconstruction. As a reference, we show in a dashed red line the value in standard gravity.
• Figure 4: Reconstrunction of the $\ln (f \sigma_{8}(z))$ data by the Gaussian Process method using the hyperparameters obtained by the GaPP code (left panel) and using a grid in hyperparameter space with a prior on $\ell_f$ (right panel).