We show how to improve constraints on \Omega _ { m } , \sigma _ { 8 } , and the dark-energy equation-of-state parameter , w , obtained by Mantz et al .
( 2008 ) from measurements of the X-ray luminosity function of galaxy clusters , namely MACS , the local BCS and the REFLEX galaxy cluster samples with luminosities L > 3 \times 10 ^ { 44 } erg/s in the 0.1–2.4 keV band .
To this aim , we use Tinker et al .
( 2008 ) mass function instead of Jenkins et al .
( 2001 ) and the M-L relationship obtained from Del Popolo ( 2002 ) and Del Popolo et al .
( 2005 ) .
Using the same methods and priors of Mantz et al .
( 2008 ) , we find , for a \Lambda CDM universe , \Omega _ { m } = 0.28 ^ { +0.05 } _ { -0.04 } and \sigma _ { 8 } = 0.78 ^ { +0.04 } _ { -0.05 } while the result of Mantz et al .
( 2008 ) gives less tight constraints \Omega _ { m } = 0.28 ^ { +0.11 } _ { -0.07 } and \sigma _ { 8 } = 0.78 ^ { +0.11 } _ { -0.13 } .
In the case of a w CDM model , we find \Omega _ { m } = 0.27 ^ { +0.07 } _ { -0.06 } , \sigma _ { 8 } = 0.81 ^ { +0.05 } _ { -0.06 } and w = -1.3 ^ { +0.3 } _ { -0.4 } , while in Mantz et al .
( 2008 ) they are again less tight \Omega _ { m } = 0.24 ^ { +0.15 } _ { -0.07 } , \sigma _ { 8 } = 0.85 ^ { +0.13 } _ { -0.20 } and w = -1.4 ^ { +0.4 } _ { -0.7 } .
Combining the XLF analysis with the f _ { gas } +CMB+SNIa data set results in the constraint \Omega _ { m } = 0.269 \pm 0.012 , \sigma _ { 8 } = 0.81 \pm 0.021 and w = -1.02 \pm 0.04 , to be compared with Mantz et al .
( 2008 ) , \Omega _ { m } = 0.269 \pm 0.016 , \sigma _ { 8 } = 0.82 \pm 0.03 and w = -1.02 \pm 0.06 .
The tightness of the last constraints obtained by Mantz et al .
( 2008 ) , are fundamentally due to the tightness of the f _ { gas } +CMB+SNIa constraints and not to their XLF analysis .
Our findings , consistent with w = -1 , lend additional support to the cosmological-constant model .