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.
