Model-independent reconstruction of the linear anisotropic stress $\eta$

In this work, we use recent data on the Hubble expansion rate $H(z)$, the quantity $f\sigma_8(z)$ 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 $\eta$. 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 $\eta_{{\rm obs}}$. If $\eta_{{\rm obs}}$ 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 $\eta_{{\rm obs}}$ at three redshift values, $z=0.29, 0.58, 0.86$. Using the method of polynomial regression, we find $\eta_{{\rm obs}}=0.57\pm1.05$, $\eta_{{\rm obs}}=0.48\pm0.96$, and $\eta_{{\rm obs}}=-0.11\pm3.21$, respectively. Assuming a constant $\eta_{{\rm obs}}$ in this range, we find $\eta_{{\rm obs}}=0.49\pm0.69$. We consider this method as our fiducial result, for reasons clarified in the text. The other two methods give for a constant anisotropic stress $\eta_{{\rm obs}}=0.15\pm0.27$ (binning) and $\eta_{{\rm obs}}=0.53 \pm 0.19$ (Gaussian Process). We find that all three estimates are compatible with each other within their $1\sigma$ error bars. While the polynomial regression method is compatible with standard gravity, the other two methods are in tension with it.