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

Authors: Ujjal Debnath
Journal: Advances in High Energy Physics
 (2014)

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## Abstract

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 redshiftz. From observed Hubble data (OHD) set (12 points), we have obtained the bounds of the arbitrary parameters (A,B) of MCG by minimizing the χ2 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 μz against redshift z for our theoretical MCG model in Einstein-Aether gravity has been tested for the best fit values of the parameters and the observed SNe Ia union2 data sample.


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## Body

## 1. 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) [1–5] radiation. The standard explanation invokes an unknown “dark energy” component which has the property of 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 (BAO) [7], WMAP data analysis [8, 9], and so forth. 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 candidates 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, and Chaplygin gas model [12–16]. 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 and Cai [21], respectively. The theoretical models have been tally with the observations with different data sets say TORNY, Gold sample data sets [3, 22–24]. 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 compared to 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 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 toΛCDM 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, 29] 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 [30]. Other modified gravity includes f(T) gravity, f(G) gravity, Gauss-Bonnet gravity, Horava-Lifshitz gravity, and Brans-Dicke gravity [31–35].In the present work, we concentrate on the generalized Einstein-Aether theories as proposed by Zlosnik et al. [36, 37], which is a generalization of the Einstein-Aether theory developed by Jacobson and Mattingly [38, 39]. These years a lot of work has been done in generalized Einstein-aether theories [40–46]. 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 the constraints have been found from observational data [47, 48]. Since modified gravity theory may be treated as alternative to dark energy, so Meng and Du [47, 48] have not taken by hand any types of dark energy in Einstein-Aether gravity and shown that the gravity may generate 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 and the modified Chaplygin gas (MCG) type dark energy. The basic concepts of Einstein-Aether gravity theory are presented in Section 2. The modified Friedmann equations and their solutions are given in Section 3. The observational data analysis tools in observed Hubble data (OHD), OHD + BAO, OHD + BAO + CMB, and OHD + BAO + CMB + SNe for χ2 minimum test will be studied in Section 4 and investigate the bounds of unknown parameters (A,B) of MCG dark energy by fixing other parameters. The best fit values of the parameters are obtained by 66%, 90%, and 99% confidence levels. The distance modulus μ(z) against redshift z for our theoretical model of the MCG in Einstein-Aether gravity models for the best fit values of the parameters and the observed SNe Ia union2 data sample. Finally we present the conclusions of the work in Section 5.


## 2. 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 theAether and the theory coupling the metric and unit timelike vector is called theEinstein-Aether theory [38]. 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 [36, 47]
(1)S=∫‍d4x-gR16πG+LEA+L,
where LEA is the vector field Lagrangian density while L denotes the Lagrangian density for all other matter fields. The Lagrangian density for the vector part consists of terms quadratic in the field [36, 47]:
(2)LEA=M216πGFK+116πGλAaAa+1,(3)K=M-2Kcdab∇aAc∇bAd,(4)Kcdab=c1gabgcd+c2δcaδdb+c3δdaδcb,
where ci 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, Aa is a contravariant vector, gab is metric tensor, and F(K) ia an arbitrary function of K. From (1), we get the field equations
(5)Gab=TabEA+8πGTab,(6)∇aF′Jba=2λAb,
where
(7)F′=dFdK,Jba=2Kbcad∇dAc.
Here Tab is the energy momentum tensor for all matter fluids and TabEA is the energy momentum tensor for the vector field and they are, respectively, given as follows: [47]
(8)Tab=ρ+puaub+pgab,
where ρ and p are, respectively, the energy density and pressure of all matter fluids and ua=(1,0,0,0) is the fluid 4-velocity vector and
(9)TabEA=12∇dJadAb-JdAba-JabAdF′-YabF′+12gabM2F+λAaAb
with
(10)Yab=-c1∇dAa∇dAb-∇aAd∇bAd,
where the subscript (ab) means symmetric with respect to the indices involved and Aa=(1,0,0,0) is nonvanishing time-like unit vector satisfying AaAa=-1.


## 3. Modified Friedmann Equations and Solutions

We consider the Friedmann-Robertson-Walker (FRW) metric of the universe as(11)ds2=-dt2+a2tdr21-kr2+r2dθ2+sin2⁡θdϕ2,
where k(=0,±1) is the curvature scalar and a(t) is the scale factor. From (3) and (4), we get
(12)K=M-2c1gabgcd+c2δcaδdb+c3δdaδcb=3βH2M2,
where β=c1+3c2+c3 is constant. From (5), we get the modified Friedmann equation for Einstein-Aether gravity as follows [36, 47]:
(13)β-F′+F2KH2+H2+ka2=8πG3ρ,(14)βddtHF′+-2H˙+2ka2=8πGρ+p,
where H(=a˙/a) is Hubble parameter. Now we see that if the first expressions of L.H.S. of (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 by
(15)ρ˙+3a˙aρ+p=0.Now, assume that the matter fluid is combination of dark matter and modified Chaplygin gas type dark energy. Soρ=ρm+ρch and p=pm+pch, where ρm and pm are, respectively, the energy density and pressure of dark matter and ρch and pch are, respectively, the energy density and pressure of modified Chaplygin gas. Assume that the dark matter follows the barotropic equation of state pm=wmρm, where wm is a constant. The equation of state of modified Chaplygin gas (MCG) is given by [17]
(16)pch=Aρch-Bρchα,
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 (14), we obtain the conservation equations for dark matter and dark energy in the form:
(17)ρ˙m+3a˙aρm+pm=0,ρ˙ch+3a˙aρch+pch=0.
Using equation of states and the conservation equations (17), we obtain ρm=ρm0(1+z)3(1+wm) and
(18)ρch=B1+A+C1+z3(1+A)(1+α)1/(1+α),
where ρm0 and C are positive constants in which ρm0 represents the present value of the density of dark matter and z=1/a-1 is the cosmological redshift (choosing a0=1). The above expression can be written in the form:
(19)ρch=ρch0As+1-As1+z3(1+A)(1+α)1/(1+α),
where ρch0 is the present value of the MCG density and As=B/((1+A)C+B). So 0≤As≤1.Now sinceF(K) is a free function of K. Some authors have chosen F(K) in the following forms: (i) F(K)=γ(-K)n [36, 43], (ii) F(K)=γ-K+3K/βln⁡(-K) [47, 48]. Here we may choose another form of F(K) for our next calculations in simplified form as F(K)=(2/β)K(1-ϵK), where ϵ is a constant. So solving (13), we obtain the expression of H2 in terms of redshift z in the following:
(20)H2z=M33ϵβ×As+1-As1+z3(1+A)(1+α)1/(1+α)-3k1+z2+8πGρm01+z3(1+wm)+8πGρch0As+1-As1+z3(1+A)(1+α)1/(1+α)1/2.Now defining the dimensionless parametersΩm0=8πGρm0/3H02, Ωch0=8πGρch0/3H02, Ωk0=k/H02, and ΩEA=M/3H0ϵβ, we obtain the form of H(z):
(21)Hz=H0ΩEA×As+1-As1+z31+A1+α1/(1+α)-Ωk01+z2+Ωm01+z3(1+wm)+Ωch0As+1-As1+z31+A1+α1/(1+α)1/4.
Due to the above solution, (13) gives the following relation:
(22)ΩEAΩm0+Ωch0-Ωk0=1.


## 4. Observational Data Analysis Tools

In this section, we will investigate some bounds of the parameters of the modified Chaplygin gas (MCG) in Einstein-Aether gravity by observational data fitting. The parameters are determined by observed Hubble data (OHD), BAO, CMB, and SNe data analysis [47–57]. We will use the χ2 minimization technique (statistical data analysis) from Hubble-redshift data set to get the constraints of the parameters of MCG model in Einstein-Aether gravity.


### 4.1. Analysis with Observed Hubble Data (OHD)

We analyze the MCG model in Einstein-Aether gravity using observed value of Hubble parameter data (OHD) [58, 59] at different redshifts consisting 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 χ2 statistics for OHD is given as follows:
(23)χOHD2=∑‍Hz-Hobsz2σ2z,
where H(z) and Hobs(z) are, respectively, the theoretical and observational values of Hubble parameter at different redshifts and σ(z) is the corresponding error which is given in Table 1. We consider the present value of Hubble parameter H0=72±8 Kms−1 Mpc−1. Here we will determine two parameters of MCG model out of 3 parameters A,B,α by fixing any one parameter from minimizing the above distribution χOHD2. There are other parameters of the model say Ωm0, Ωk0, Ωch0, ΩEA, wm. Fixing the one parameter α of MCG model, the relation between the other parameters (A,B) can be determined by the observational data. Now for OHD analysis, χOHD2 is minimized for best fit values of A=0.238303 and B=0.18176 and the minimum value of χOHD2=7.08613 where we have assumed α=0.1. We also plot the graph for different confidence levels (66%, 90%, 99%) in Figure 1.Table 1
The observed Hubble parameterH(z) and the standard error σ(z) for different values of redshift z.
z
H
(
z
)
σ
(
z
)
0
73
±8
0.1
69
±12
0.17
83
±8
0.27
77
±14
0.4
95
±17.4
0.48
90
±60
0.88
97
±40.4
0.9
117
±23
1.3
168
±17.4
1.43
177
±18.2
1.53
140
±14
1.75
202
±40.4Figure 1
It shows the variations ofA with B for α=0.1 for OHD and OHD + BAO analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


### 4.2. 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 χ2 analysis. The BAO peak parameter may be defined by [47]
(24)A=ΩmΩkEz11/31z1sinh⁡⁡Ωk∫0z1‍dzEz2/3,
where E(z)=H(z)/H0 may be called the normalized Hubble parameter and the redshift z1=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 χ2 function for the BAO measurement can be written as
(25)χBAO2=A-0.46920.0172.Now the total joint data analysis of BAO with OHD for theχ2 function may be defined by
(26)χtotal2=χOHD2+χBAO2.According to OHD + BAO joint analysis the best fit values ofA and B are A=0.238695 and B=0.209932 with χ2 minimum being 7.07842. Finally we draw the contours B versus A for the 66%, 90%, and 99% confidence limits depicted in Figure 2.Figure 2
It shows the variations ofA with B for α=0.1 for OHD and OHD + BAO analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


### 4.3. 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 [60–62]
(27)R=ΩmΩksinh⁡Ωk∫0z2‍dzEz,
where z2 is the value of redshift at the last scattering surface. From 7-year WMAP data [63], the value of the parameter has been obtained as R=1.726±0.018 at the redshift z2=1091.3. Now the χ2 function for the CMB measurement can be written as
(28)χCMB2=R-1.72620.0182.Now when we consider OHD, BAO, and CMB analysis together, the total joint data analysis (OHD + BAO + CMB) for theχ2 function may be defined by
(29)χTOTAL2=χOHD2+χBAO2+χCMB2.Now the best fit values ofA and B with χ2 for joint analysis of BAO and CMB with OHD observational data support the theoretical range of the parameters. The best fit values are A=0.239018 and B=0.240047 with the minimum value of χ2=7.07086. The 66%, 90%, and 99% contours for A and B are plotted in Figure 3.Figure 3
It shows the variations ofA with B for α=0.1 for OHD + BAO + CMB and OHD + BAO + CMB + SNe analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


### 4.4. Redshift-Magnitude Observations from Supernovae Type Ia: Analysis with OHD + BAO + CMB + SNe Ia

The observations of type Ia supernovae (SNe Ia) provide an excellent tool for probing the expansion history of the universe. The main evidence for the existence of dark energy is provided by the supernova type Ia experiments [1–3]. The type Ia observations directly measure the distance modulus of supernovae and its redshift z [64, 65]. Now, take recent observational data (including SNe Ia) which consists of 557 data points and belongs to the Union2 sample [66]. From the type Ia observations, the luminosity distance determines the dark energy density. The luminosity distance dL(z) is defined by
(30)dLz=1+zΩksinh⁡⁡Ωk∫0z‍dz′Ez′
and the distance modulus μ(z) for supernovas is given by
(31)μz=5log⁡10dLz/H01MPc+25.Theχ2 function for SNe Ia is given by
(32)χSNe2=∑‍μz-μobsz2σ2z,
where μobs(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 points. Now we consider four cosmological tests together, the total joint data analysis (Stern + BAO + CMB + SNe) for the χ2 function may be defined by
(33)χTOTAL2=χOHD2+χBAO2+χCMB2+χSNe2.From the joint analysis, we found the minimum value ofχ2 which is 7.06716. The best fit values of the parameters are A=0.239158 and B=0.255814. The confidence contours are drawn in Figure 4. The best fit value of distance modulus μ(z) for our theoretical model and the supernova type Ia union2 sample are drawn in Figure 5 for our best fit values of A and B. From the curves, we see that the theoretical MCG model in Einstein-Aether gravity is in agreement with the union2 sample data.Figure 4
It shows the variations ofA with B for α=0.1 for OHD + BAO + CMB and OHD + BAO + CMB + SNe analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.Figure 5
It shows the variation of distance modulusμ(z) versus redshift z for our model (solid line) and the Union2 sample (dotted points).


## 4.1. Analysis with Observed Hubble Data (OHD)

We analyze the MCG model in Einstein-Aether gravity using observed value of Hubble parameter data (OHD) [58, 59] at different redshifts consisting 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 χ2 statistics for OHD is given as follows:
(23)χOHD2=∑‍Hz-Hobsz2σ2z,
where H(z) and Hobs(z) are, respectively, the theoretical and observational values of Hubble parameter at different redshifts and σ(z) is the corresponding error which is given in Table 1. We consider the present value of Hubble parameter H0=72±8 Kms−1 Mpc−1. Here we will determine two parameters of MCG model out of 3 parameters A,B,α by fixing any one parameter from minimizing the above distribution χOHD2. There are other parameters of the model say Ωm0, Ωk0, Ωch0, ΩEA, wm. Fixing the one parameter α of MCG model, the relation between the other parameters (A,B) can be determined by the observational data. Now for OHD analysis, χOHD2 is minimized for best fit values of A=0.238303 and B=0.18176 and the minimum value of χOHD2=7.08613 where we have assumed α=0.1. We also plot the graph for different confidence levels (66%, 90%, 99%) in Figure 1.Table 1
The observed Hubble parameterH(z) and the standard error σ(z) for different values of redshift z.
z
H
(
z
)
σ
(
z
)
0
73
±8
0.1
69
±12
0.17
83
±8
0.27
77
±14
0.4
95
±17.4
0.48
90
±60
0.88
97
±40.4
0.9
117
±23
1.3
168
±17.4
1.43
177
±18.2
1.53
140
±14
1.75
202
±40.4Figure 1
It shows the variations ofA with B for α=0.1 for OHD and OHD + BAO analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


## 4.2. 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 χ2 analysis. The BAO peak parameter may be defined by [47]
(24)A=ΩmΩkEz11/31z1sinh⁡⁡Ωk∫0z1‍dzEz2/3,
where E(z)=H(z)/H0 may be called the normalized Hubble parameter and the redshift z1=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 χ2 function for the BAO measurement can be written as
(25)χBAO2=A-0.46920.0172.Now the total joint data analysis of BAO with OHD for theχ2 function may be defined by
(26)χtotal2=χOHD2+χBAO2.According to OHD + BAO joint analysis the best fit values ofA and B are A=0.238695 and B=0.209932 with χ2 minimum being 7.07842. Finally we draw the contours B versus A for the 66%, 90%, and 99% confidence limits depicted in Figure 2.Figure 2
It shows the variations ofA with B for α=0.1 for OHD and OHD + BAO analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


## 4.3. 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 [60–62]
(27)R=ΩmΩksinh⁡Ωk∫0z2‍dzEz,
where z2 is the value of redshift at the last scattering surface. From 7-year WMAP data [63], the value of the parameter has been obtained as R=1.726±0.018 at the redshift z2=1091.3. Now the χ2 function for the CMB measurement can be written as
(28)χCMB2=R-1.72620.0182.Now when we consider OHD, BAO, and CMB analysis together, the total joint data analysis (OHD + BAO + CMB) for theχ2 function may be defined by
(29)χTOTAL2=χOHD2+χBAO2+χCMB2.Now the best fit values ofA and B with χ2 for joint analysis of BAO and CMB with OHD observational data support the theoretical range of the parameters. The best fit values are A=0.239018 and B=0.240047 with the minimum value of χ2=7.07086. The 66%, 90%, and 99% contours for A and B are plotted in Figure 3.Figure 3
It shows the variations ofA with B for α=0.1 for OHD + BAO + CMB and OHD + BAO + CMB + SNe analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.


## 4.4. Redshift-Magnitude Observations from Supernovae Type Ia: Analysis with OHD + BAO + CMB + SNe Ia

The observations of type Ia supernovae (SNe Ia) provide an excellent tool for probing the expansion history of the universe. The main evidence for the existence of dark energy is provided by the supernova type Ia experiments [1–3]. The type Ia observations directly measure the distance modulus of supernovae and its redshift z [64, 65]. Now, take recent observational data (including SNe Ia) which consists of 557 data points and belongs to the Union2 sample [66]. From the type Ia observations, the luminosity distance determines the dark energy density. The luminosity distance dL(z) is defined by
(30)dLz=1+zΩksinh⁡⁡Ωk∫0z‍dz′Ez′
and the distance modulus μ(z) for supernovas is given by
(31)μz=5log⁡10dLz/H01MPc+25.Theχ2 function for SNe Ia is given by
(32)χSNe2=∑‍μz-μobsz2σ2z,
where μobs(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 points. Now we consider four cosmological tests together, the total joint data analysis (Stern + BAO + CMB + SNe) for the χ2 function may be defined by
(33)χTOTAL2=χOHD2+χBAO2+χCMB2+χSNe2.From the joint analysis, we found the minimum value ofχ2 which is 7.06716. The best fit values of the parameters are A=0.239158 and B=0.255814. The confidence contours are drawn in Figure 4. The best fit value of distance modulus μ(z) for our theoretical model and the supernova type Ia union2 sample are drawn in Figure 5 for our best fit values of A and B. From the curves, we see that the theoretical MCG model in Einstein-Aether gravity is in agreement with the union2 sample data.Figure 4
It shows the variations ofA with B for α=0.1 for OHD + BAO + CMB and OHD + BAO + CMB + SNe analysis, respectively, for different confidence levels say 66% (solid, blue), 90% (dashed, red), and 99% (dashed, black) contours.Figure 5
It shows the variation of distance modulusμ(z) versus redshift z for our model (solid line) and the Union2 sample (dotted points).


## 5. Discussions and Concluding Remarks

We have assumed FRW model of the universe in Einstein-Aether gravity filled with dark matter and modified Chaplygin gas (MCG) type dark energy. Dark matter has the equation of state parameterwm, which is small. We assumed the dark matter and dark energy separately conserved and hence we found the solutions in this gravity. Since F(K) is a free function of K, so we have chosen quadratic form of F(K) for simplicity of the calculation. Defining dimensionless parameters, 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 χ2 test where we have chosen α=0.1. The minimum values of the parameters are A=0.238303 and B=0.18176 for OHD analysis. 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). The best fit values of the parameters (i) for OHD + BAO are A=0.238695 and B=0.209932 and (ii) for OHD + BAO + CMB are A=0.239018 and B=0.240047. 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 the parameters (A,B). The best fit values of the parameters for OHD + BAO + CMB + SNe are A=0.239158 and B=0.255814. 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 in Figures 1–4. The distance modulus μ(z) against redshift z for our theoretical MCG model in Einstein-Aether gravity has been tested for the best fit values of the parameters and the observed SNe Ia union2 data sample and drawn in Figure 5. The observations do in fact severely constrain the nature of allowed composition of matter energy by constraining the range of the values of the parameters for a physically viable MCG in Einstein-Aether gravity model.



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*Source: 653630-2014-12-30.xml*