Constraining the Parameters of Modified Chaplygin Gas in Einstein-Aether Gravity

We have assumed FRW model of the universe in Einstein-Aether gravity filled with dark matter and Modified Chaplygin gas (MCG) type dark energy. We present the Hubble parameter in terms of some unknown parameters and observational parameters with the redshift z. From observed Hubble data (OHD) set (12 points), we have obtained the bounds of the arbitrary parameters (A,B) of MCG by minimizing the \chi^{2} test. Next due to joint analysis of BAO and CMB observations, we have also obtained the best fit values and the bounds of the parameters (A,B) by fixing some other parameters. We have also taken type Ia supernovae data set (union 2 data set with 557 data points). Next due to joint analysis with SNe, we have obtained the best fit values of parameters. The best-fit values and bounds of the parameters are obtained by 66%, 90% and 99% confidence levels for OHD, OHD+BAO, OHD+BAO+CMB and OHD+BAO+CMB+SNe joint analysis. The distance modulus \mu(z) against redshift z for our theoretical MCG model in Einstein-Aether gravity have been tested for the best fit values of the parameters and the observed SNe Ia union2 data sample.


I. INTRODUCTION
Observational evidence strongly points to an accelerated expansion of the Universe, but the physical origin of this acceleration is unknown. The observations include type Ia Supernovae and Cosmic Microwave Background (CMB) [1][2][3][4][5] radiation. The standard explanation invokes an unknown "dark energy" component which has the property that positive energy density and negative pressure. Observations indicate that dark energy occupies about 70% of the total energy of the universe, and the contribution of dark matter is ∼ 26%. This accelerated expansion of the universe has also been strongly confirmed by some other independent experiments like Sloan Digital Sky Survey (SDSS) [6], Baryonic Acoustic Oscillation * ujjaldebnath@yahoo.com , ujjal@iucaa.ernet.in (BAO) [7], WMAP data analysis [8,9] etc. Over the past decade there have been many theoretical models for mimicking the dark energy behaviors, such as the simplest (just) cosmological constant in which the equation of state is independent of the cosmic time and which can fit the observations well. This model is the so-called ΛCDM, containing a mixture of cosmological constant Λ and cold dark matter (CDM). However, two problems arise from this scenario, namely "fine-tuning" and the "cosmic coincidence" problems. In order to solve these two problems, many dynamical dark energy models were suggested, whose equation of state evolves with cosmic time. The scalar field or quintessence [10,11] is one of the most favored candidate of dark energy which produce sufficient negative pressure to drive acceleration. In order to alleviate the cosmological-constant problems and explain the acceleration expansion, many dynamical dark energy models have been proposed, such as K-essence, Tachyon, Phantom, quintom, Chaplygin gas model, etc [12][13][14][15][16].
Also the interacting dark energy models including Modified Chaplygin gas [17], holographic dark energy model [18], and braneworld model [19] have been proposed. Recently, based on principle of quantum gravity, the agegraphic dark energy (ADE) and the new agegraphic dark energy (NADE) models were proposed by Cai [20] and Wei et al [21] respectively. The theoretical models have been tally with the observations with different data sets say TORNY, Gold sample data sets [3,[22][23][24]. In Einstein's gravity, the modified Chaplygin gas [17] best fits with the 3 year WMAP and the SDSS data with the choice of parameters A = 0.085 and α = 1.724 [25] which are improved constraints than the previous ones −0.35 < A < 0.025 [26].
Another possibility is that general relativity is only accurate on small scales and has to be modified on cosmological distances. One of these is a modified gravity theories. In this case cosmic acceleration would arise not from dark energy as a substance but rather from the dynamics of modified gravity. Modified gravity constitutes an interesting dynamical alternative to ΛCDM cosmology in that it is also able to describe the current acceleration in the expansion of our universe. The simplest modified gravity is DGP brane-world model [27]. The other alternative approach dealing with the acceleration problem of the Universe is changing the gravity law through the modification of action of gravity by means of using f (R) gravity [28] instead of the Einstein-Hilbert action. Some of these models, such as 1/R and logarithmic models, provide an acceleration for the Universe at the present time [29]. Other modified gravity includes f (T ) gravity, f (G) gravity, Gauss-Bonnet gravity, Horava-Lifshitz gravity, Brans-Dicke gravity, etc [30][31][32][33][34].
In the present work, we concentrate on the generalized Einstein-Aether theories as proposed by Zlosnik et al [35,36], which is a generalization of the Einstein-Aether theory developed by Jacobson et al [37,38].
These years a lot of work has been done in generalized Einstein-aether theories [39][40][41][42][43][44][45]. In the generalized Einstein-Aether theories by taking a special form of the Lagrangian density of Aether field, the possibility of Einstein-Aether theory as an alternative to dark energy model is discussed in detail, that is, taking a special Aether field as a dark energy candidate and it has been found the constraints from observational data [46,47]. Since modified gravity theory may be treated as alternative to dark energy, so Meng et al [46,47] have not taken by hand any types of dark energy in Einstein-Aether gravity and shown that the gravity may be generates dark energy. Here if we exempt this assumption, so we need to consider the dark energy from outside. So we assume the FRW universe in Einstein-Aether gravity model filled with the dark matter

II. EINSTEIN-AETHER GRAVITY THEORY
In order to include Lorentz symmetry violating terms in gravitation theories, apart from some noncommutative gravity models, one may consider existence of preferred frames. This can be achieved admitting a unit timelike vector field in addition to the metric tensor of spacetime. Such a timelike vector implies a preferred direction at each point of spacetime. Here the unit timelike vector field is called the Aether and the theory coupling the metric and unit timelike vector is called the Einstein-Aether theory [37]. So Einstein-Aether theory is the extension of general relativity (GR) that incorporates a dynamical unit timelike vector field (i.e., Aether). In the last decade there is an increasing interest in the Aether theory.
The action of the Einstein-Aether gravity theory with the normal Einstein-Hilbert part action can be written in the form [35,46] where L EA is the vector field Lagrangian density while L m denotes the Lagrangian density for all other matter fields. The Lagrangian density for the vector part consists of terms quadratic in the field [35,46]: where c i are dimensionless constants, M is the coupling constant which has the dimension of mass, λ is a Lagrange multiplier that enforces the unit constraint for the time-like vector field, A a is a contravariant vector, g ab is metric tensor and F (K) ia an arbitrary function of K. From (1), we get the field equations where Here T m ab is the energy momentum tensor for matter field and T EA ab is the energy momentum tensor for the vector field and they are respectively given as follows: [46] T m ab = (ρ + p)u a u b + pg ab (8) where ρ and p are respectively the energy density and pressure of matter and u a = (1, 0, 0, 0) is the fluid

4-velocity vector and
where the subscript (ab) means symmetric with respect to the indices involved and A a = (1, 0, 0, 0) is non-vanishing time-like unit vector satisfying A a A a = −1.
We consider the Friedmann-Robertson-Walker (FRW) metric of the universe as where k (= 0, ±1) is the curvature scalar and a(t) is the scale factor. From equations (3) and (4), we get where β = c 1 +3c 2 +c 3 is constant. From eq. (5), we get the modified Friedmann equation for Einstein-Aether gravity as in the following [35,46]: and β d dt where H (=ȧ a ) is Hubble parameter. Now we see that if the first expressions of L.H.S. of equations (12) and (13) are zero, we get the usual field equations for Einstein's gravity. So first expressions arise for Einstein-Aether gravity. Also the conservation equation is given bẏ Now, assume that the matter fluid is combination of dark matter and modified Chaplygin gas type dark energy. So ρ = ρ m + ρ ch and p = p m + p ch , where ρ m and p m are respectively the energy density and pressure of dark matter and ρ ch and p ch are respectively the energy density and pressure of modified Chaplygin gas.
Assume that the dark matter follows the barotropic equation of state p m = w m ρ m , where w m is a constant.
The equation of state of modified Chaplygin gas (MCG) is given by [17] where A > 0, B > 0 and 0 ≤ α ≤ 1. Now we assume that there is no interaction between dark matter and dark energy. So they are separately conserved. From equation (14), we obtain the conservation equations for dark matter and dark energy in the form: Using equation of states and the conservation equations (17), we obtain ρ m = ρ m0 (1 + z) 3(1+wm) and where ρ m0 and C are positive constants in which ρ m0 represents the present value of the density of dark matter and z = 1 a − 1 is the cosmological redshift (choosing a 0 = 1). The above expression can be written in the form: where ρ ch0 is the present value of the MCG density and A s = B (1+A)C+B . So 0 ≤ A s ≤ 1. Now since F (K) is a free function of K. Some authors have chosen F (K) in the following forms: (i) [46,47]. Here we may choose another form of F (K) for our next calculations in simplified form as F (K) = 2 β K(1 − ǫ K), where ǫ is a constant. So solving equation (13), we obtain the expression of H 2 in terms of redshift z in the following: Now defining the dimensionless parameters Ω m0 = 8πGρ m0 Due to the above solution, the equation (13) gives the following relation:

A. Analysis with Observed Hubble Data (OHD)
We analyze the MCG model in Einstein-Aether gravity using observed value of Hubble parameter data (OHD) [57,58] at different redshifts consists of twelve data points. The observed values of Hubble parameter H(z) and the standard error σ(z) for different values of redshift z are listed in Table 1. The χ 2 statistics for OHD is give as follows: the minimum value of χ 2 OHD = 7.08613 where we have assumed α = 0.1. We also plot the graph for different confidence levels (66%, 90%, 99%) in figure 1.

B. Analysis with OHD+BAO
Another constraint is from the Baryonic Acoustic Oscillations (BAO) traced by the Sloan Digital Sky Survey (SDSS). The BAO peak parameter value has been proposed by Eisenstein et al [7]. Here we examine the parameters A and B for MCG gas model from the measurements of the BAO peak for low redshift (with range 0 < z < 0.35) using standard χ 2 analysis. The BAO peak parameter may be defined by [46] where E(z) = H(z)/H 0 may be called the normalized Hubble parameter, the redshift z 1 = 0.35 is the typical redshift of the SDSS. The value of the parameter A for the universe is given by A = 0.469 ± 0.017 using SDSS data [7]. Now the χ 2 function for the BAO measurement can be written as Now the total joint data analysis of BAO with OHD for the χ 2 function may be defined by According to OHD+BAO joint analysis the best fit values of A and B are A = 0.238695 and B = 0.209932 with χ 2 minimum is 7.07842. Finally we draw the contours B vs A for the 66%, 90% and 99% confidence limits depicted in figure 2.

C. Analysis with OHD+BAO+CMB
In addition to OHD and BAO analysis, we use the Cosmic Microwave Background (CMB) shift parameter.
The CMB shift parameter (CMB power spectrum first peak) is defined by [59][60][61] where z 2 is the value of redshift at the last scattering surface. From 7 year WMAP data [62], the value of the parameter has obtained as R = 1.726 ± 0.018 at the redshift z 2 = 1091.3. Now the χ 2 function for the CMB measurement can be written as Now when we consider OHD, BAO and CMB analysis together, the total joint data analysis (OHD+BAO+CMB) for the χ 2 function may be defined by and its redshift z [63,64]. Now, take recent observational data (including SNe Ia) consists of 557 data points and belongs to the Union2 sample [65]. From the type Ia observations, the luminosity distance determines the dark energy density. The luminosity distance d L (z) is defined by and the distance modulus µ(z) for Supernovas is given by The χ 2 function for SNe Ia is given by where µ obs (z) is observational value of distance modulus parameter at different redshifts and σ(z) is the corresponding error. In this work, we take Union2 data set consisting of 557 supernovae data. Now we consider four cosmological tests together, the total joint data analysis (Stern+BAO+CMB+SNe) for the χ 2 function may be defined by