Based on the latest MLCS17 SNe Ia data provided by Hicken et al .
( 2009 ) , together with the Baryon Acoustic Oscillation ( BAO ) and strong gravitational lenses ( SGL ) , we investigate the dark energy equation-of-state parameter for both constant w and time-varying w ( z ) = w _ { 0 } + w _ { a } z / ( 1 + z ) in the flat universe , and its correlation with the matter density \Omega _ { M } and Hubble constant h .
The constraints from SNe data alone arrive at : ( a ) the best-fit results are ( \Omega _ { M } ,w,h ) = ( 0.358 , -1.09 , 0.647 ) , while both \Omega _ { M } and w are very sensitive to the difference \Delta h = h - \tilde { h } of the Hubble constant deviating to the prior input \tilde { h } = 0.65 ; ( b ) the likelihoods of parameters are found to be : w = -0.88 ^ { +0.31 } _ { -0.39 } and \Omega _ { M } = 0.36 ^ { +0.09 } _ { -0.15 } , which is consistent with the \Lambda CDM at 95 \% C.L .
; ( c ) the two parameters in the time-varying case are found to be ( w _ { 0 } ,w _ { a } ) = ( -0.73 ^ { +0.23 } _ { -0.97 } , 0.84 ^ { +1.66 } _ { -10.34 } ) after marginalizing other parameters ; ( d ) there is a clear degeneracy between constant w and \Omega _ { M } , which depresses the power of SNe Ia to constrain both of them ; ( e ) the likelihood of parameter w _ { a } has a high non-Gaussian distribution ; ( f ) an extra restriction on \Omega _ { M } is necessary to improve the constraint of the SNe Ia data on ( w _ { 0 } , w _ { a } ) .
A joint analysis of SNe Ia data and BAO is made to break the degeneracy between w and \Omega _ { M } , and it provides a stringent constrain with the likelihoods : w = -0.88 ^ { +0.07 } _ { -0.09 } and \Omega _ { M } = 0.29 ^ { +0.02 } _ { -0.03 } .
For the time-varying w ( z ) , it leads to the interesting maximum likelihoods w _ { 0 } = -0.94 and w _ { a } = 0 .
When marginalizing the parameters \Omega _ { M } and h , the fitting results are found to be ( w _ { 0 } ,w _ { a } ) = ( -0.95 ^ { +0.45 } _ { -0.18 } , 0.41 ^ { +0.79 } _ { -0.96 } ) .
After adding the splitting angle statistic of SGL data , a consistent constraint is obtained ( \Omega _ { M } ,w ) = ( 0.298 , -0.907 ) and ( w _ { 0 } ,w _ { a } ) is further improved to be ( w _ { 0 } ,w _ { a } ) = ( -0.92 ^ { +0.14 } _ { -0.10 } , 0.35 ^ { +0.47 } _ { -0.54 } ) , which indicates that the phantom type models are disfavored .