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 $\alpha=0.0002^{+0.0206+0.0441}_{-0.0209-0.0422}$ 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 $(\frac{\dot{G}}{G})_{today}\simeq -0.3496^{+26.3135}_{-26.3135}\times 10^{-13}yr^{-1}$.


I. Introduction
Gravity theories are usually studied with an assumption that Newton gravity constant G is a constant. But some observations hint that G maybe depends on the time [1], such as observations from white dwarf star [2,3], pulsar [4], supernovae [5] and neutron star [6]. In addition, cosmic observations predict that about 95% of the universal matter is invisible, including dark matter (DM) and dark energy (DE). The unified models of two unknown dark sectors (DM and DE) have been studied in several theories, e.g. in the standard cosmology [7,8], in the Hořava-Lifshitz gravity [9], in the RS [10] and the KK higher-dimension gravity [11]. In this paper, we study the unified model of dark components in theory of varying gravitational constant (VG). The attractive point in this model is that the variation of G could result to the invisible components in universe, by relating the Lagrangian quantity from the generalized Born-Infeld theory to the VG theory. One source of DM and DE is explored. Using the Markov Chain Monte Carlo * Electronic address: lvjianbo819@163.com (MCMC) method [12], the cosmic constraints on unified model of DM and DE are performed in the framework of time-varying gravitational constant, with flat and non-flat geometry. The used cosmic data include the lookback time (LT) data [13,14], the model-independent gamma ray bursts (GRBs) data [15], the growth function (GF) of matter linear perturbations [16][17][18][19][20][21][22][23], the type Ia supernovae (SNIa) data with systematic errors [24], the cosmic microwave background (CMB) [25], and the baryon acoustic oscillation (BAO) data including the radial BAO scale measurement [26] and the peak-positions measurement [27][28][29].

II. A time-varying gravitational constant theory with unified dark sectors in a non-flat geometry
We adopt the Lagrangian quantity of system with a parameterized time-varying gravitational constant G = G 0 a(t) −β . t is the cosmic time, a = (1 + z) −1 is the cosmic scale factor, and z denotes the cosmic redshift. g is the determinant of metric, R is the Ricci scalar, and L u = L b + L r + L d corresponds to the Lagrangian density of universal matter including the visible ingredients: baryon L b and radiation L r and the invisible dark ingredients: L d . Utilizing the variational principle, the gravitational field equation can be derived [30], in which R µν is the Ricci tensor, T µν is the energy-momentum tensor of universal matter that comprise the pressureless baryon (w b = p b ρ b = 0), positive-pressure photon (w r = pr ρr = 1 3 ) and the unknown dark components (w d = p d ρ d ). w is the equation of state (EoS), p is the pressure and ρ denotes the energy density, respectively.
In a non-flat geometry, one gets the evolutional equation of universe in VG theory The effective energy conservation equationρ Integrating Eq. (6) can gain the energy density of baryon , the energy density of radiation ρ r ∝ a −β 2 −4β−8 2+β and the energy density of curvature ρ k ∝ a −β 2 −4 2+β . Relative to the constant-G theory, the evolutional equations of energy densities are obviously modified in VG theory, for the existence of VG parameter β.
We concentrate on the Lagrangian density of dark components with the form from the generalized Born-Infeld theory [31], in which V (ϕ) is the potential. Relating this scalar field ϕ with the time-varying gravitational constant by ϕ(t) = G(t) −1 , it is then found that the dark ingredients can be induced by the variation of G. The energy density of dark fluid in V G frame complies with here parameter β reflects the variation of G, α and B s = 6+6β 0V G−GCG are model parameters. Eq. (7) shows that the behavior of ρ d is like cold DM at early time 1 (for a ≪ 1, with Hubble constant H 0 and dimensionless energy densities Ω b = 8πG0ρ 0b , and Ω 0d + Ω b + Ω r + Ω k = 1 + β. Above equations are reduced to the standard forms in the constant-G theory, for β = 0.

A. Lookback Time
Refs. [32,33] define the LT as the difference between the current age t 0 of universe at z = 0 and the age t z of a light ray emitted at z, Then the age t(z i ) of an object can be expressed by the difference between the age of universe at z i and the age at z F (object was born) [13], For an object born at z i , the observed LT subjects to One defines with σ 2 is the uncertainty of the total universal age, and σ i is the uncertainty of the LT of galaxy i. Marginalizing the 'nuisance' parameter df results to [34] . p s denotes the theoretical model parameters. erfc(x) = 1-erf(x) is the complementary error function of x. The observed universal age at today t obs 0 = 13.75 ± 0.13 Gyr [35] is used, and the observed data on the galaxies age are listed in table I.   [13,14]. The first 6 data are from Ref. [13].

B. Gamma Ray Bursts
In GRBs observation, the famous Amati's correlation can be expressed [36,37], log Eiso are the isotropic energy and the cosmological rest-frame spectral peak energy, respectively. d L is the luminosity distance and S bolo is the bolometric fluence of GRBs. Ref. [38] introduced a model-independent quantity of distance measurement, with z 0 being the lowest GRBs redshift. For GRBs constraint, χ 2 GRBs has a form in which ∆r p (z i ) = r data p (z i ) − r p (z i ), and (Cov −1 GRBs ) ij is the covariance matrix. Using 109 GRBs data, Ref.
[15] obtained 5 model-independent datapoints listed in table II, where σ(r p (z i )) + and σ(r p (z i )) − are the 1σ errors. The with the covariance matrix

C. Growth Function of Matter Linear Perturbations
The χ 2 GF from growth function of matter linear perturbations f can be constructed where the used observational values of f obs are listed in table δρ ρ (a=1) . ′ denotes derivative with respect to a. So in theory, f can be gained by numerically solving the following differential equation in VG theory with D(a) ≃ a as the initial condition for a ≃ 0. Comparing with the most popular ΛCDM model, the effective current matter density can be written,
Here r s (z) is the comoving sound horizon size r s = c t 0 csdt a . c s is the sound speed of the photon−baryon fluid, c −2 The measurement of BAO peak positions can be performed by the WiggleZ Dark Energy Survey [27], the Two Degree Field Galaxy Redshift Survey [28] and the Sloan Digitial Sky Survey [29]. Introducing D V (z) = [(1 + z) 2 D 2 A (z) cz H(z;ps) ] 1/3 , the observational data from BAO peak positions can be exhibited by with the inverse covariance matrix V −1 shown in Ref. [57].
Thus, the χ 2 BAO can be constructed in a form, X t denotes the transpose of X.

IV. Cosmic constraints on unified model of dark sectors in VG theory with flat and non-flat geometry
Taking a joint likelihood analysis L ∝ e −χ 2 /2 as the products of the separate likelihoods, the total χ 2 is adopted A. Flat-universe case In order to obtain the stringent constraint on VG theory, we utilize the cosmic data different from Ref. [30], to calculate the total likelihood. Concretely, the LT data, the GRBs data, the GF data, the SNIa data with systematic error and the BAO data from radial measurement are not used in Ref. [30]. After calculation, the 1-dimension  distribution and the 2-dimension contours of VG-UM model parameters are illustrated in Fig. 2 for a flat universe.
A stringent constraint on VG parameter is β = 0.0008 +0.0034+0.0063 −0.0032−0.0064 . Obviously, a small uncertainty around zero for parameter β is obtained, at 1σ and 2σ regions. In VG theory, the constraint on UM model parameters are For a non-flat geometry, restriction on dimensionless curvature density is Ω k = −0.0311 +0.0259+0.0517 −0.0248−0.0501 in the varying-G theory with containing unified dark sectors. Obviously, at 1σ error a closed universe Ω k < 0 is predicted. Furthermore, the uncertainty of Ω k is more enlarged than other results. For example, using the same data to constrain    the best-fit evolutions ofĠ G with their confidence level (the shadow region) are illustrated in Figure 3. "Dot" denotes the derivative with respect to t. Fig. 3 provides prediction that the today's value (Ġ G ) today ≃ 0.7977 +2.3566 −2.3566 ×10 −13 yr −1 in a flat geometry. This restriction on (Ġ G ) today is more stringent than other results seen in table VII. Also, using the best-fit value of parameter β with error the shapes of G G0 = (1 + z) β are exhibited. Taking high redshift z = 3.5 as another reference, we find ( G G0 ) z=3.5 ≃ 1.0003 +0.0014 −0.0016 and (Ġ G ) z=3.5 ≃ −0.4012 +1.1797 −1.1797 × 10 −12 yr −1 in the flat VG-UM model. The upper panel of Fig. 3 reveal that the behaviors of G and its derivative are around the constant-G theory in the flat geometry. But the behavior of G has the more departure from constant-G theory for a non-flat case. In the non-flat geometry, limits on the variation of G at today is (Ġ G ) today ≃ 19.3678 +21.8262 −21.8262 × 10 −13 yr −1 , and at z = 3.5 they are ( G G0 ) z=3.5 ≃ 0.9917 +0.0104 −0.0131 and (Ġ G ) z=3.5 ≃ 101.828 +119.524 −119.524 × 10 −13 yr −1 . It tends to hint,Ġ > 0. Finally, from the best-fit evolutions ofĠ G , they give the different evolutional tendency in flat and non-flat universe, respectively. Since the monotonicity ofĠ G = −βH depends on the symbol of β, it is important to sternly constrain the value of β.

VI. Behaviors of EoS with the confidence level in flat and non-flat VG-UM theory
The EoS of UM in VG theory is demonstrated For both flat and non-flat universe, we point out at early time w V G−UM ∼ 0 (DM), and in the future w V G−UM ∼ −1 (DE), as illustrated in Figure 4 (left). If the dark sectors are thought to be separable, it is interested to investigate the properties of both dark components in VG-UM model. Considering that the behavior of dark matter is known i.e. its EoS w dm = 0 (ρ dm = ρ 0dm a −β 2 −2β−6 2+β ), the EoS of dark energy in VG-UM model subjects to, Using the best-fit values of model parameters and the covariance matrix, the evolution of w de with confidence level in VG-UM is plotted in Fig. 4 (middle), for flat and non-flat universe. If one deems the behavior of dark energy is the cosmological constant i.e. w Λ = −1 (p Λ = −ρ Λ ), the EoS of dark matter in VG-UM model obeys which is drawn in Fig. 4 (right) with the confidence level, for flat and non-flat universe.   −0.0011 ) and non-flat (w 0de = −1.0092 +0.0104 −0.0104 ) universe. In addition, at high redshift the uncertainty of w de (or w dm ) is enlarged (or narrowed). For the best-fit evolution, w de and w dm are variable with the time, and at the early time of universe they tend to have the small deviation from zero pressure.