New measurements of $\Omega_m$ from gamma-ray bursts
  Context: Data from cosmic microwave background radiation (CMB), baryon
acoustic oscillations (BAO), and supernovae Ia (SNe-Ia) support a constant dark
energy equation of state with $w_0 \sim -1$. Measuring the evolution of $w$
along the redshift is one of the most demanding challenges for observational
cosmology. Aims: We discuss the existence of a close relation for GRBs, named
Combo-relation, based on characteristic parameters of GRB phenomenology such as
the prompt intrinsic peak energy $E_{p,i}$, the X-ray afterglow, the initial
luminosity of the shallow phase $L_0$, the rest-frame duration $\tau$ of the
shallow phase, and the index of the late power-law decay $\alpha_X$. We use it
to measure $\Omega_m$ and the evolution of the dark energy equation of state.
We also propose a new calibration method for the same relation, which reduces
the dependence on SNe Ia systematics. Methods: We have selected a sample of
GRBs with 1) a measured redshift $z$; 2) a determined intrinsic prompt peak
energy $E_{p,i}$, and 3) a good coverage (0.3-10) keV afterglow light curves.
The fitting technique of the rest.frame (0.3-10) keV luminosity light curves
represents the core of the Combo-relation. We separate the early steep decay,
considered a part of the prompt emission, from the X-ray afterglow additional
component. Data with the largest positive residual, identified as flares, are
automatically eliminated until the p-value of the fit becomes greater than 0.3.
Results: We strongly minimize the dependency of the Combo-GRB calibration on
SNe Ia. We also measure a small extra-Poissonian scatter of the Combo-relation,
which allows us to infer from GRBs alone $\Omega_M =0.29^{+0.23}_{-0.15}$
(1$\sigma$) for the $\Lambda$CDM cosmological model, and $\Omega_M
=0.40^{+0.22}_{-0.16}$, $w_0 = -1.43^{+0.78}_{-0.66}$ for the flat-Universe
variable equation of state case.
