The VIMOS Public Extragalactic Redshift Survey (VIPERS): $\Omega_{\rm m_0}$ from the galaxy clustering ratio measured at $z \sim 1$

We use a sample of about 22,000 galaxies at $0.65<z<1.2$ from the VIPERS PDR-1 catalogue, to constrain the cosmological model through a measurement of the galaxy {\it clustering ratio} $\eta_{g,R}$. This statistic has favourable properties, being defined as the ratio of two quantities characterizing the smoothed density field in spheres of given radius $R$: the value of its correlation function on a multiple of this scale, $\xi(nR)$, and its variance $\sigma^2(R)$. For sufficiently large values of $R$, this is a universal number, capturing 2-point clustering information independently of the linear bias and linear redshift-space distortions of the specific galaxy tracers. In this paper we discuss in detail how to extend the application of $\eta_{g,R}$ to quasi-linear scales and how to control and remove observational selection effects which are typical of redshift surveys as VIPERS. We verify the accuracy and efficiency of these procedures using mock catalogues that match the survey selection process. These results evidence the robustness of $\eta_{g,R}$ to non-linearities and observational effects, which is related to its very definition as a ratio of quantities that are similarly affected. We measure $\Omega_{m,0}=0.270_{-0.025}^{+0.029}$. In addition to the great precision achieved on our estimation of $\Omega_m$ using VIPERS PDR-1, this result is remarkable because it appears to be in good agreement with a recent estimate $z\simeq 0.3$, obtained applying the same technique to the SDSS-LRG catalogue. It, therefore, suports the robustness of the present analysis. Moreover, the combination of these two measurements at $z\sim 0.3$ and $z\sim 0.9$ provides us with a very precise estimate $\Omega_{m,0}=0.274\pm0.017$ which highlights the great consistency between our estimation and other cosmological probes such as BAOs, CMB and Supernovae.