Observations indicate that most of the universal matter are invisible and the gravitational constant G ( t ) maybe depends on the time .
A theory of the variational G ( VG ) is explored in this paper , with naturally producing the useful dark components in universe .
We utilize the observational data : lookback time data , model-independent gamma ray bursts , growth function of matter linear perturbations , type Ia supernovae data with systematic errors , CMB and BAO to restrict the unified model ( UM ) of dark components in VG theory .
Using the best-fit values of parameters with the covariance matrix , constraints on the variation of G are ( \frac { G } { G _ { 0 } } ) _ { z = 3.5 } \simeq 1.0015 ^ { +0.0071 } _ { -0.0075 } and ( \frac { \dot { G } } { G } ) _ { today } \simeq - 0.7252 ^ { +2.3645 } _ { -2.3645 } \times 10 ^ { -13 } yr ^ % { -1 } , the small uncertainties around constants .
Limit on the equation of state of dark matter is w _ { 0 dm } = 0.0072 ^ { +0.0170 } _ { -0.0170 } with assuming w _ { 0 de } = -1 in unified model , and dark energy is w _ { 0 de } = -0.9986 ^ { +0.0011 } _ { -0.0011 } with assuming w _ { 0 dm } = 0 at prior .
Restriction on UM parameters are B _ { s } = 0.7442 ^ { +0.0137 + 0.0262 } _ { -0.0132 - 0.0292 } and \alpha = 0.0002 ^ { +0.0206 + 0.0441 } _ { -0.0209 - 0.0422 } with 1 \sigma and 2 \sigma confidence level .
In addition , the effect of a cosmic string fluid on unified model in VG theory are investigated .
In this case it is found that the \Lambda CDM ( \Omega _ { s } = 0 , \beta = 0 and \alpha = 0 ) is included in this VG-UM model at 1 \sigma confidence level , and the larger errors are given : \Omega _ { s } = -0.0106 ^ { +0.0312 + 0.0582 } _ { -0.0305 - 0.0509 } ( dimensionless energy density of cosmic string ) , ( \frac { G } { G _ { 0 } } ) _ { z = 3.5 } \simeq 1.0008 ^ { +0.0620 } _ { -0.0584 } and ( \frac { \dot { G } } { G } ) _ { today } \simeq - 0.3496 ^ { +26.3135 } _ { -26.3135 } \times 10 ^ { -13 } % yr ^ { -1 } .