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Cosmic constraint on the unified model of dark sectors with or without a cosmic string fluid in the varying gravitational constant theory

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 $w_{0dm}=0.0072^{+0.0170}_{-0.0170}$ with assuming $w_{0de}=-1$ in unified model, and dark energy is $w_{0de}=-0.9986^{+0.0011}_{-0.0011}$ with assuming $w_{0dm}=0$ at prior. Restriction on UM parameters are $B_{s}=0.7442^{+0.0137+0.0262}_{-0.0132-0.0292}$ and $α=0.0002^{+0.0206+0.0441}_{-0.0209-0.0422}$ with $1σ$ and $2σ$ 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 $Λ$CDM ($Ω_{s}=0$, $β=0$ and $α=0$) is included in this VG-UM model at $1σ$ confidence level, and the larger errors are given: $Ω_{s}=-0.0106^{+0.0312+0.0582}_{-0.0305-0.0509}$ (dimensionless energy density of cosmic string), $(\frac{G}{G_{0}})_{z=3.5}\simeq 1.0008^{+0.0620}_{-0.0584}$ and $(\frac{\dot{G}}{G})_{today}\simeq -0.3496^{+26.3135}_{-26.3135}\times 10^{-13}yr^{-1}$.