# Cosmological Constraints on Ghost Dark Energy in the Brans-Dicke Theory by Using MCMC Approach

###### Abstract

By using a Markov Chain Monte Carlo simulation, we investigate
cosmological constraints on the ghost dark energy (GDE) model in the
framework of the Brans-Dicke (BD) theory. A combination of the latest
observational data of the cosmic microwave background radiation data from
seven-year WMAP, the baryon acoustic oscillation data form the
SDSS, the supernovae type Ia data from the Union2 and
the X-ray gas mass fraction data from the Chandra X-ray observations of
the largest relaxed galaxy clusters are used to perform
constraints on GDE in the BD cosmology. In this paper, we consider both flat and non-flat
universes together with interaction between dark matter and
dark energy. The main cosmological parameters are obtained
as: ,
and .
In addition, the Brans-Dicke parameter is estimated as .

Keywords: ghost; dark energy; Brans-Dicke theory;
observational constraints.

## I Introduction

Accelerating expansion of the Universe Riess:1998cb (); Perlmutter:1998np () can be explained either by a missing energy component usually called “dark energy” (DE) with an exotic equation of state, or by modifying the underlying theory of gravity on large scales. The famous examples of the former approach include scalar field models of DE such as quintessence Wetterich:1987fm (); Ratra:1987rm (), K-essence ArmendarizPicon:2000dh (); ArmendarizPicon:2000ah (), tachyon Padmanabhan:2002cp (); Sen:2002in (), phantom Caldwell:1999ew (); Nojiri:2003vn (); Nojiri:2003jn (), ghost condensate ArkaniHamed:2003uy (); Piazza:2004df (), quintom Elizalde:2004mq (); Nojiri:2005sx (); Anisimov:2005ne (), holographic DE Witten:2000zk (), agegraphic DE Cai:2007us (); Wei:2007ty () and so forth. For a comprehensive review on DE models, see Copeland:2006wr (); Li:2011sd (). The latter approach for explanation of the acceleration expansion is based on the modification of the underlying theory of gravity on large scales such as gravity DeFelice:2010aj () and braneworld scenarios Dvali:2000hr (); Carena:2006yr (); Minamitsuji:2008fz (); Sheykhi:2007wn ().

Among various models of DE, the so called ghost dark energy (GDE) has attracted a lot of interests in recent years. The origin of DE in this model comes from Veneziano ghosts in QCD theory Kawarabayashi:1980dp (); Witten:1979vv (); Veneziano:1979ec (); Rosenzweig:1979ay (). Indeed, the contribution of the ghosts field to the vacuum energy in curved space or time-dependent background can be regarded as a possible candidate for DE Ohta:2010in (); Urban:2009vy (). The magnitude of this vacuum energy is of order , where is the Hubble parameter and is the QCD mass scale. With and , gives the right order of magnitude for the observed dark energy density Ohta:2010in (). The advantages of GDE model compared to other DE models is that it is totally embedded in standard model and general relativity, therefore one needs not to introduce any new parameter, new degree of freedom or to modify gravity. The dynamical behavior of GDE model in flat universe have been studied Cai:2010uf (). The study was also generalized to the universe with spacial curvature Sheykhi:2011xz (). The instability of the GDE model against perturbations was studied in Ebrahimi:2011js (). In Sheykhi:2011fe (); Sheykhi:2011nb () the correspondence between GDE and scalar field models of DE were established. In the presence of bulk viscosity and varying gravitational constant, the GDE model was investigated in Sheykhi:2012zzb (). Other features of the GDE model have been investigated in Refs. RozasFernandez:2011je (); Karami:2012yf (); KhodamMohammadi:2012ww (); Malekjani:2012wc (); Feng:2012gr (). The cosmological constraints on this model have been considered by some authors Cai:2010uf (); Feng:2012gr (); Cai:2012fq ().

Scalar tensor theories have been reconsidered extensively, recently. One important example of the scalar tensor theories is the BD theory of gravity which was introduced by Brans and Dicke in 1961 to incorporate the Mach’s principle in the Einstein’s theory of gravity Brans:1961sx (). This theory also passes the observational tests in the solar system domain Bertotti:2003rm (). In addition, BD theory can be tested by the cosmological observations such as the cosmic microwave background (CMB) and large scale structure (LSS) Chen:1999qh (); Acquaviva:2007mm (); Tsujikawa:2008uc (); Wu:2009zb (); Wu:2013 (). Since the GDE model have a dynamic behavior, it is more reasonable to consider this model in a dynamical framework such as BD theory. It was shown that some features of GDE in BD cosmology differ from Einstein’s gravity Ebrahimi:2011ne (). For example, while the original DE is instable in all range of the parameter spaces in standard cosmology Ebrahimi:2011js (), it leads to a stable phase in BD theory Saaidi:2012vs (). In the framework of BD cosmology, the ghost model of DE has been studied Ebrahimi:2011ne (). It is also of great interest to see whether the GDE model in the framework of the BD theory is compatible with observational data or not.

In this paper, cosmological constraints on GDE in the BD theory (GDEBD) Ebrahimi:2011ne () theory is performed by using the Marko Chain Monte Carlo (MCMC) simulation. The used observational datasets are as follows: cosmic microwave background radiation (CMB) from WMAP7 Komatsu:2010fb (), 557 Union2 dataset of type Ia supernova Amanullah:2010vv (), baryon acoustic oscillation (BAO) from SDSS DR7 Percival:2009xn (), and the cluster X-ray gas mass fraction from the Chandra X-ray observations Allen:2007ue (). To put the constraints, the modified CosmoMC cosmomc () code is used.

The organization of this paper is as follows. In section II, we review the formalism of the GDE in the framework of Brans-Dicke cosmology. In section III the methods which are used in this paper to analyze the data are introduced. Section IV contains the results of the MCMC simulation and we conclude our paper in section V.

## Ii Interacting Ghost Dark Energy in the Brans-Dicke Theory in a Non-Flat Universe

Let us first review the formalism of the interacting GDE in the framework of BD theory in a non-flat universe Ebrahimi:2011ne (). The action of the BD theory in the canonical form may be written Arik:2005ir ()

(1) |

where is the Ricci scalar and is the BD scalar field. Varying the action with respect to the metric and the BD scalar field , yields

(2) | |||

(3) |

where stands for the energy-momentum tensor of the matter fields. The line element of the Friedmann-Robertson-Walker (FRW) universe is

(4) |

where is the scale factor, and is the curvature parameter with corresponding to open, flat, and closed universes, respectively. Nowadays, there are some evidences in favor of closed universe with a small positive curvature () Bennett:2012zja (). Using metric (4), the field equations (2) and (3) reduce to

(5) | |||

(6) | |||

(7) |

where is the Hubble parameter, and are, respectively, the energy density and pressure of DE, and is the pressureless matter density which contains both dark matter (DM) and baryonic matter (BM) densities i.e. where and are the energy densities of dark matter and baryonic matter respectively.

To be more general and because of some observational evidences Bertolami:2007zm (); Tsujikawa:2004dp (), here we propose the case where there is an interaction between GDE and DM. In this case the semi-conservation equations read

(8) | |||

(9) | |||

(10) |

where represents the interaction term between dark matter and dark energy and here we assume that the baryonic matter is conserved separately. We assume with being a constant. Such a choice for interacting term implies the the DE and DM component do not conserve separately while the total density is still conserved through

(11) |

where and .

The ghost energy density is proportional to the Hubble parameter Ohta:2010in ()

(12) |

where is roughly of order and is QCD mass scale. Taking into account the fact that and for the present time, this gives the right order of magnitude for the ghost energy density Ohta:2010in ().

Since the system of equations (5-7) is not closed, we still have another degree of freedom in analyzing the set of equations. As usual we assume the BD scalar field has a power law relation versus the scale factor,

(13) |

An interesting case is when is small whereas is high so that the product results of order unity Banerjee:2007zd (); Sheykhi:2009dz (). In section IV we will consider the condition for constraining the model by observational data. This is interesting because local astronomical experiments set a very high lower bound on Will (); in particular, the Cassini experiment implies that Bertotti:2003rm (); Acquaviva:2007mm (). Now we take the time derivative of relation (13). We arrive at

(14) |

Combining Eqs. (13) and (14) with the first Friedmann equation (5), we get

(15) |

As usual the fractional energy densities are defined as

(16) | |||||

(17) | |||||

(18) | |||||

(19) |

where

(20) |

Using (12) we can rewrite equation (19) as

(21) |

Based on these definitions, equation (15) can be rewritten as

(22) |

where and we have defined

(23) |

Next we take the time derivative of (15), after using (22), we find

(24) |

Combining the above equation with Eqs. (8) and (12), we obtain the EoS parameter as

(25) |

The first and second derivatives of the distance can be combined to obtain the acceleration parameter . It was shown that the zero redshift value of , is independent of space curvature, and can be obtained from the first and second derivatives of the coordinate distance Daly:2007dn (). It was argued that , which indicates whether the universe is accelerating at the current epoch, can be obtained directly from the supernova and radio galaxy data Daly:2007dn (). The acceleration parameter is given by

(26) |

Using (24)the acceleration parameter (26) is obtained as

(27) |

Finally, we obtain the equation of motions of GDE in BD theory. For this purpose, we first take the time derivative of relation (21). We find

(28) |

Substituting from (27) into equation (28) and using relation , we get

(29) |

In the remaining part of this paper we will constrain the GDEBD model by using the most recent observational date in the three different physical models: model I which is the GDEBD model in a flat universe ( and ), model II is the interacting GDEBD model in a flat universe ( and ) and finally model III is the interacting GDEBD model in a non-flat universe ( and ).

## Iii Data fitting method

In this section we discuss the data fitting method in the Markov Chain Monte Carlo (MCMC) simulation to estimate the parameters of the model in section II using cosmological data.

To get the best fit values of the relevant parameters, the maximum likelihood method is used. The total likelihood function is defined as the product of the separate likelihood functions of uncorrelated observational data with

(30) |

where SNIa stands for type Ia supernovae, CMB for cosmic microwave background radiation, BAO for baryon acoustic oscillation and gas stands for X-ray gas mass fraction data. Best fit values of parameters are obtained by minimizing . In this paper we use the cosmic microwave background radiation data from seven-year WMAP Komatsu:2010fb (), type Ia supernovae data from 557 Union2 Amanullah:2010vv (), baryon acoustic oscillation data from SDSS DR7 Percival:2009xn (), and the cluster X-ray gas mass fraction data from the Chandra X-ray observations Allen:2007ue (). In the rest of this section we discuss each in detail.

To obtain , we use seven-year WMAP data Komatsu:2010fb () with the CMB data point . The shift parameter R, which parametrize the changes in the amplitude of the acoustic peaks is given by Bond:1997wr ()

(31) |

where is the redshift of recombination (see (36)), c is the speed of light in vacuum, is the present value of the matter density parameter (a ”0“ subscript shows the present value of the related quantity), and . In addition, the acoustic scale , which characterizes the changes of the peaks of CMB via the angular diameter distance out to recombination is defined as Bond:1997wr ()

(32) |

The comoving distance is defined

(33) |

and the comoving sound horizon distance at recombination is given by

(34) |

in terms of the sound speed , defined by

(35) |

The seven-year WMAP observations gives and Komatsu:2010fb ().

The redshift of recombination is obtained by using the fitting function proposed by Hu and Sugiyama Hu:1995en ()

(36) |

where

(37) |

Then one can define as , with Komatsu:2010fb ()

(38a) | ||||

(38b) |

where is the inverse covariant matrix.

To obtain , the SNIa Union2 data Amanullah:2010vv () is used which includes 577 type Ia supernovae. The expansion history of the universe can be given by a specific cosmological model. To test this model, we can use the observational data for some predictable cosmological parameter such as luminosity distance . Assume that the Hubble parameter is used to describe the Universe, where parameters are predicted by a theoretical cosmological model. For such a theoretical model we can predict the theoretical ’Hubble-constant free’ luminosity distance as

(39) | |||||

where , is the redshift parameter, and

Then one can write the theoretical modulus distance

(40) |

where . On the other hand, the observational modulus distance of SNIa, , at redshift is given by

(41) |

where and are apparent and absolute magnitudes of SNIa respectively. Then the parameters of the theoretical model, s, can be determined by a likelihood analysis by defining in (30) as

where the nuisance parameter, , can be marginalized over as

(43) |

Here we should mention an important point about using supernovae data to constrain the Brans-Dicke theories which have a varying gravitational coupling constant. Variations of gravitational coupling constant and apparent magnitude of supernovae are correlated as follows. The luminosity of a supernova is powered by Nickel-35 mass which is proportional to the Chandrasekhar mass

(44) |

Moreover, the luminosity distance is the integral over the inverse Hubble parameter, which is proportional to . Therefore, the apparent magnitude

(45) |

varies with a change in the gravitational coupling constant as

(46) |

On the other hand, in the Brans-Dicke theory we have

(47) |

where is the Brans-Dicke scalar field. In the slow roll approximation we can write

(48) |

where is the decelaration parameter. The average value of between today and (the redshift when the SN measurement are probing the dark energy) is of order unity, and by using , we can write

(49) |

By using Eqs. (46) and (49) we obtain

(50) |

In Union2 data set, the redshift interval is between and , i.e. , with the systematic error of order in the measurement of apparent magnitude. Therefore, the systematic error can induce a bias roughly of order on parameter , which is three order of magnitudes larger than the statistical errors, as we will discuss in the next section. Therefore, in order to constrain with a higher precision, we combine the supernovae data with other cosmological data sets as follows. For more detailed discussion on possible evolution of the gravitational constant from cosmological type Ia supernovae see Gaztanaga:2001fh ().

The baryon acoustic oscillation data from the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) Percival:2009xn () is used here for constraining model parameters. The data constrain parameter , where is the comoving sound horizon distance (see (34)) at the drag epoch (where baryons were released from photons) and is given by Eisenstein:2005su ()

(51) |

The drag redshift is given by the fitting formula Eisenstein:1997ik ()

(52) |

where

(53) |

Then we can obtain by , where

(54) |

and its covariance matrix is given by Percival:2009xn ()

(55) |

The ratio of X-ray gas mass to the total mass of a cluster is defined as the X-ray gas mass fraction Allen:2007ue (). The model fitted to the CDM model is Allen:2007ue ()

(56) |

The elements in equation (60) are defined as follows: and are the proper angular diameter distance in CDM and the alternative theoretical model respectively, where

(57) |

The angular correction factor

(58) |

is caused by changes in angle for the alternative theoretical model compared to , where Allen:2007ue () is the slope of the data within the radius ( is the radius of the gas core in units).

The bias factor in equation (60) contains information about the uncertainties in the cluster depletion factor and the parameter accounts for departures from the hydrostatic equilibrium. The function denotes the uncertainties of the baryonic mass fraction in stars with a Gaussian prior for , with Allen:2007ue (). The factor describes the combined effects of the residual uncertainties, such as the instrumental calibration. A Gaussian prior for the ’calibration’ factor is considered as Allen:2007ue ().

At the end of this section we should assert that the data points parameters of the CMB and BAO data sets which we use in this paper are the best fit values for CDM and the error estimates are also based on the CDM model. Therefore they are not completely accurate in this application. However they are the only parameters which we have to constrain our model.

## Iv Results

Finally we apply a Markov chain Monte Carlo simulation on the parameters of the GDEBD model by using the publicly available CosmoMC code Lewis:2002ah (). The parameter vectors are , , for the flat non-interacting (model I), flat interacting (model II) and non-flat interacting (model III) models respectively. The basic cosmological parameters are taken in the following priors: , and . In addition, as we mentioned in section II, for the model fitting, we consider condition. The results are presented in table 1.

From table 1 one can see that the main cosmological parameters , , and in all three models are compatible with the results of the CDM model Bennett:2012zja (). In addition, in the presence of interaction between DM and DE, the parameter decreases and so the Brans-Dicke parameter increases. The best fit value of the parameter in all three models is compatible with the results of other cosmological constraining works (however see the discussion following Eq. (43)). For example in Chen:1999qh () the authors by using the CMB temperature and polarization anisotropy data, found . Wu and Chen in Wu:2009zb (); Wu:2013 () by using the five-year WMAP and SDSS data obtained . For other cosmological constraints on the BD theory see Tsujikawa:2008uc (); Nagata:2002tm (); Nagata:2003qn (); Acquaviva:2004ti (); Schimd:2004nq (); Acquaviva:2007mm (). This estimated value is also compatible with the results of the solar system tests of the scalar-tensor theories such as the Cassini experiment where it has been obtained Bertotti:2003rm (); Acquaviva:2007mm (). The positive best fit value of parameter describe a conversion of dark matter to dark energy although both in flat and non-flat universes, in 1- CL, an inverse conversion is possible as well. The interacting DE and DM models have been constrained by observational data by many authors with different parametrization of the interacting parameter Chimento:2003iea (); Zimdahl:2002zb (); Valiviita:2009nu (); Pettorino:2012ts (); Salvatelli:2013wra (); He:2010im (); Clemson:2011an (). He et al. in He:2010im () have parametrized the interaction parameter as in the present paper although they have chosen the prior on parameter as . They obtained the best fit value of parameter as .

Parameter | Non-interacting | Interacting | Interacting non-flat |
---|---|---|---|

… | … | ||

… | |||

## V Conclusion

In this paper we considered the cosmological constraints on the parameters of the GDE in the framework of BD theory by using a Markov Chain Monte Carlo simulation. We used the SNIa+ CMB+ BAO+X-ray gas mass fraction data for the model fitting. The best fit values of the cosmological parameters in this model are compatible with the results of the CDM model. In addition, we obtained the best fit values of parameters and where is the BD parameter and is the interacting parameter. The best fit values of these parameters are also compatible with the results of previous constraining works. However as we mentioned in section III, due to large systematic error in by using the supernovae data only, to constrain this parameter with a higher precision, we should combine the supernovae data with other cosmological data sets as the CMB and BAO data sets. In addition we should assert one more time that the data points parameters of the CMB and BAO data which we have used in this paper are the best fit values for CDM and the error estimates are also based on the CDM model. Therefore they are not completely accurate in this application. The numerical results can be improved in the future works by using more recent data such as nine-yaer WMAP Bennett:2012zja () or the Planck Ade:2013zuv () projects.

## Acknowledgements

We would like to thank the anonymous referee for constructive comments. Hamzeh Alavirad would also like to thank Qader Dorosti for helpful discussions. The work of A. Sheykhi has been supported financially by Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran.

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