Observational Constraints of Modified Chaplygin Gas in Loop Quantum Cosmology
Abstract
We have considered the FRW universe in loop quantum cosmology (LQC) model filled with the dark matter (perfect fluid with negligible pressure) and the modified Chaplygin gas (MCG) type dark energy. We present the Hubble parameter in terms of the observable parameters , and with the redshift and the other parameters like , , and . From Stern data set (12 points), we have obtained the bounds of the arbitrary parameters by minimizing the test. The bestfit values of the parameters are obtained by 66%, 90% and 99% confidence levels. Next due to joint analysis with BAO and CMB observations, we have also obtained the bounds of the parameters () by fixing some other parameters and . From the best fit of distance modulus for our theoretical MCG model in LQC, we concluded that our model is in agreement with the union2 sample data.
I Introduction
The combinations of different observations astrophysical data
continuously testing the theoretical models and the bounds of the
parameters. Different observations of the SNeIa
Perlmutter et al. (1998, 1999); Riess et al. (1998, 2004), large scale redshift
surveys Bachall et al. (1999); Tedmark et al. (2004), the measurements of the cosmic
microwave background (CMB) Miller et al. (1999); Bennet et al. (2000) and WMAP
Briddle et al. (2003); Spergel et al. (2003) indicate that our universe is presently
expanding with acceleration. Standard big bang Cosmology with
perfect fluid fails to accommodate the observational fact. In
Einstein’s gravity, the cosmological constant (which has
the equation of state ) is a suitable candidate
which derive the acceleration, but till now there is no proof of
the origin of . Now assume that there is some unknown
matter which is responsible for this accelerating scenario which
has the property that the positive energy density and sufficient
negative pressure, know as dark energy Padmanabhan (2003); Sahni et al. (2000). The
scalar field or quintessence Peebles et al. (1988) is one of the most
favored candidate of dark energy which produce sufficient negative
pressure to drive acceleration in which the potential dominates
over the kinetic term. In the present cosmic concordance
CDM model the Universe is formed of 26% matter
(baryonic + dark matter) and 74% of a smooth vacuum energy
component. The thermal CMB component contributes only about
0.01%, however, its angular power spectrum of temperature
anisotropies encode important information about the structure
formation process and other cosmic observables.
If we assume a flat universe and further assume that the only
energy densities present are those corresponding to the
nonrelativistic dustlike matter and dark energy, then we need to
know of the dustlike matter and to a very
high accuracy in order to get a handle on or
of the dark energy Choudhury et al. (2007); Padmanabhan et al. (2003). This can be a fairly
strong degeneracy for determining from observations.
TONRY data set with the 230 data points Tonry et al. (2003) alongwith the
23 points from Barris et al Barris et al. (2004) are valid for .
Another data set consists of all the 156 points in the “gold”
sample of Riess et al Riess et al. (2004), which includes the latest
points observed by HST and this covers the redshift range . In Einstein’s gravity and in the flat model of the FRW
universe, one finds , which are
currently favoured strongly by CMBR data (for recent WMAP results,
see Spergel et al. (2003)). In a simple analysis for the most recent
RIESS data set gives a bestfit value of to be
. This matches with the value
obtained by Riess et al
Riess et al. (1998). In comparison, the bestfit for flat
models was found to be Choudhury et al. (2007). The flat
concordance CDM model remains an excellent fit to the
Union2 data with the bestfit constant equation of state parameter
(stat)(stat+sys
together) for a flat universe, or
(stat)(stat+sys
together) with curvature Amanullah et al. (2010). Chaplygin gas is the
more effective candidate of dark energy with equation of state
Kamenshchik et al. (2001) with . It has been generalized
to the form Gorini et al. (2003) and thereafter
modified to the form Debnath et al. (2004). The
MCG best fits with the 3 year WMAP and the SDSS data with the
choice of parameters and Lu et al. (2008)
which are improved constraints than the previous
ones Jun et al. (2005).
In recent years, loop quantum gravity (LQG) is a outstanding effort to describe the quantum effect of our universe. Nowadays several dark energy models are studied in the frame work of loop quantum cosmology (LQC). Quintessence and phantom dark energy models Wu et al. (2008); Chen et al. (2008) have been studied in the cosmological evolution in LQC. When the Modified Chaplying Gas coupled to dark matter in the universe is described in the frame work LQC by Jamil et al Jamil et al. (2011) who resolved the famous cosmic coincidence problem in modern cosmology. In another study Fu et al. (2008) the authors studied the model with an interacting phantom scalar field with an exponential potential and deduced that the future singularity appearing in the standard FRW cosmology can be avoided by loop quantum effects. Here we assume the FRW universe in LQC model filled with the dark matter and the MCG type dark energy. We present the Hubble parameter in terms of the observable parameters , and with the redshift . From Stern data set (12 points), we obtain the bounds of the arbitrary parameters by minimizing the test. The bestfit values of the parameters are obtained by 66%, 90% and 99% confidence levels. Next due to joint analysis with BAO and CMB observations, we also obtain the bounds and the best fit values of the parameters () by fixing some other parameters and . From the best fit of distance modulus for our theoretical MCG model in LQC, we concluded that our model is in agreement with the union2 sample data.
Ii Basic Equations and Solutions for MCG in Loop Quantum Cosmology
We consider the flat homogeneous and isotropic universe described by FRW metric, so the modified Einstein’s field equations in LQC are given by
(1) 
and
(2) 
where is the Hubble parameter defined as
with is the scale factor. Where
is
called the critical loop quantum density, is the
dimensionless BarberoImmirzi parameter. When the energy density
of the universe becomes of the same order of the critical density
, this modification becomes dominant and the universe
begins to bounce and then oscillate forever. Thus the big bang,
big rip and other future singularities at semi classical regime
can be avoided in LQC. Let us note here it has been suggested that
by the black hole thermodynamics in LQC. In
LQG, this parameter is fixed by the requirement of the validity
of BekensteinHawking entropy for the Schwarzschild black hole.
The physical solutions are allowed only for . For
, it is called bounce. The maximum value of the
Hubble factor is reached for
and the
maximum value of Hubble factor is .
Here and , where is the density of matter (with vanishing pressure) and , are respectively the energy density and pressure contribution of some dark energy. Now we consider the Universe is filled with Modified Chaplygin Gas (MCG) model whose equation of state(EOS) is given by
(3) 
We also consider the dark matter and and the dark energy are separately conserved and the conservation equations of dark matter and dark energy (MCG) are given by
(4) 
and
(5) 
From first conservation equation (4) we have the solution of as
(6) 
From the conservation equation (5) we have the solution of the energy density as
(7) 
where is the integrating constant, is the cosmological redshift (choosing ) and the first constant term can be interpreted as the contribution of dark energy. So the above equation can be written as
(8) 
where is the present value of the dark energy
density.
In the next section, we shall investigate some bounds of the
parameters in loop quantum cosmology by observational data
fitting. The parameters are determined by  (Stern), BAO
and CMB data analysis Wu et al. (2007); Thakur et al. (2009); Paul et al. (2010, 2011); Ghose et al. (2011). We shall
use the minimization technique (statistical data
analysis) from Hubbleredshift data set to get the
constraints of the parameters of MCG model in LQC.
Iii Observational Data Analysis Mechanism
From the solution (8) of MCG and defining the dimensionless density parameters and we have the expression for Hubble parameter in terms of redshift parameter as follows ()
(9) 
From equation (9), we see that the value of depends on so the above equation can be written as
(10) 
where
(11) 
Now contains four unknown parameters and . Now we will fixing two parameters and by observational data set the relation between the other two parameters will obtain and find the bounds of the parameters.

Table 1: The Hubble parameter and the standard error for different values of redshift .
iii.1 Analysis with Stern () Data Set
Using observed value of Hubble parameter at different redshifts (twelve data points) listed in observed Hubble data by Stern et al. (2010) we analyze the model. The Hubble parameter and the standard error for different values of redshift are given in Table 1. For this purpose we first form the statistics as a sum of standard normal distribution as follows:
(12) 
where and are theoretical and observational values of Hubble parameter at different redshifts respectively and is the corresponding error for the particular observation given in table 1. Here, is a nuisance parameter and can be safely marginalized. We consider the present value of Hubble parameter = 72 8 Kms Mpc and a fixed prior distribution. Here we shall determine the parameters and from minimizing the above distribution . Fixing the two parameters , the relation between the other parameters can be determined by the observational data. The probability distribution function in terms of the parameters and can be written as
(13) 
where is the prior distribution function for . We now plot the graph for different confidence levels. In early stage the Chaplygin Gas follow the equation of state where . So, as per our theoretical model the two parameters should satisfy the two inequalities and . Now our best fit analysis with Stern observational data support the theoretical range of the parameters. The 66% (solid, blue), 90% (dashed, red) and 99% (dashed, black) contours are plotted in figures 1, 2 and 4 for and . The best fit values of and are tabulated in Table 2.

Table 2:  (Stern): The best fit values of , and the minimum values of for different values of .
iii.2 Joint Analysis with Stern BAO Data Sets
The method of joint analysis, the Baryon Acoustic Oscillation (BAO) peak parameter value has been proposed by Eisenstein et al. (2005) and we shall use their approach. Sloan Digital Sky Survey (SDSS) survey is one of the first redshift survey by which the BAO signal has been directly detected at a scale 100 MPc. The said analysis is actually the combination of angular diameter distance and Hubble parameter at that redshift. This analysis is independent of the measurement of and not containing any particular dark energy. Here we examine the parameters and for Chaplygin gas model from the measurements of the BAO peak for low redshift (with range ) using standard analysis. The error is corresponding to the standard deviation, where we consider Gaussian distribution. Lowredshift distance measurements is a lightly dependent on different cosmological parameters, the equation of state of dark energy and have the ability to measure the Hubble constant directly. The BAO peak parameter may be defined by
(14) 
Here is the normalized Hubble parameter, the redshift is the typical redshift of the SDSS sample and the integration term is the dimensionless comoving distance to the to the redshift The value of the parameter for the flat model of the universe is given by using SDSS data Eisenstein et al. (2005) from luminous red galaxies survey. Now the function for the BAO measurement can be written as
(15) 
Now the total joint data analysis (Stern+BAO) for the function may be defined by
(16) 
According to our analysis the joint scheme gives the best fit
values of and in Table 3. Finally we draw the contours
vs for the 66% (solid, blue), 90% (dashed, red) and 99%
(dashed, black) confidence limits depicted in figures for and .

Table 3:  (Stern) + BAO : The best fit values of , and the minimum values of for different values of .
iii.3 Joint Analysis with Stern Bao CMB Data Sets
One interesting geometrical probe of dark energy can be determined by the angular scale of the first acoustic peak through angular scale of the sound horizon at the surface of last scattering which is encoded in the CMB power spectrum Cosmic Microwave Background (CMB) shift parameter is defined by Bond et al. (1997); Efstathiou et al. (1999); Nessaeris et al. (2007). It is not sensitive with respect to perturbations but are suitable to constrain model parameter. The CMB power spectrum first peak is the shift parameter which is given by
(17) 
where is the value of redshift at the last scattering surface. From WMAP7 data of the work of Komatsu et al Komatsu et al. (2011) the value of the parameter has obtained as at the redshift . Now the function for the CMB measurement can be written as
(18) 
Now when we consider three cosmological tests together, the total joint data analysis (Stern+BAO+CMB) for the function may be defined by
(19) 
Now the best fit values of and for joint analysis of BAO and CMB with Stern observational data support the theoretical range of the parameters given in Table 4. The 66% (solid, blue), 90% (dashed, red) and 99% (dashed, black) contours are plotted in figures 79 for and .

Table 4:  (Stern) + BAO + CMB : The best fit values of , and the minimum values of for different values of .
iii.4 RedshiftMagnitude Observations from Supernovae Type Ia
The Supernova Type Ia experiments provided the main evidence for
the existence of dark energy. Since 1995, two teams of High
Supernova Search and the Supernova Cosmology Project have
discovered several type Ia supernovas at the high redshifts
Perlmutter et al. (1998, 1999); Riess et al. (1998, 2004). The observations
directly measure the distance modulus of a Supernovae and its
redshift Riess et al. (2007); Kowalaski et al. (2008). Now, take recent
observational data, including SNe Ia which consists of 557 data
points and belongs to the Union2 sample Amanullah et al. (2010).
From the observations, the luminosity distance determines the dark energy density and is defined by
(20) 
and the distance modulus (distance between absolute and apparent luminosity of a distance object) for Supernovas is given by
(21) 
The best fit of distance modulus as a function of redshift for our theoretical model and the Supernova Type Ia Union2 sample are drawn in figure 10 for our best fit values of , , and . From the curves, we see that the theoretical MCG model in LQC is in agreement with the union2 sample data.
Iv Discussions
Modified Chaplygin gas (MCG) is one of the candidate of unified
dark matterdark energy model. We have considered the FRW universe
in loop quantum cosmology (LQC) model filled with the dark matter
(perfect fluid with negligible pressure) and the modified
Chaplygin gas (MCG) type dark energy. We present the Hubble
parameter in terms of the observable parameters ,
and with the redshift and the other
parameters like , , and . We have chosen the
observed values of , and
= 72 Kms Mpc. From Stern data set (12
points), we have obtained the bounds of the arbitrary parameters
by minimizing the 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 () by fixing some
other parameters and . The bestfit
values and bounds of the parameters are obtained by 66%, 90% and
99% confidence levels are shown in figures 19 for Stern,
Stern+BAO and Stern+BAO+CMB analysis. The distance modulus
against redshift has been drawn in figure 10 for our
theoretical model of the MCG in LQC for the best fit values of the
parameters and the observed SNe Ia Union2 data sample. So our
predicted theoretical MCG model in LQC permitted the observational
data sets. The observations do in fact severely constrain the
nature of allowed composition of matterenergy by constraining the
range of the values of the parameters for a physically viable MCG
in LQC model. We have checked that when is large, the
best fit values of the parameters and other results of LQC model
in MCG coincide with the results of the ref. [29] in Einstein’s
gravity. When is small, the best fit values of the
parameters and the bounds of parameters spaces in different
confidence levels in LQC distinguished from Einstein’s gravity for
MCG dark energy model. Also, in particular, if we consider the
generalized Chaplygin gas (), the best fit value of critical
BarberoImmirzi parameter is 0.2486, where we have
assumed the values of other parameters and
for our convenience. In summary, the conclusion of this discussion
suggests that even though the effect that quantum aspect of
gravity have on the CMB are small, cosmological observation can
put upper bounds on the magnitude of the correction coming from
quantum gravity that may be closer to the theoretical expectation
than what one would expect.
Acknowledgments
The authors are thankful to IUCAA, Pune, India for warm
hospitality where part of the work was carried out.
References
 Perlmutter et al. (1998) Perlmutter, S. J. et al, 1998, Nature 391, 51.
 Perlmutter et al. (1999) Perlmutter, S. J. et al, 1999, Astrophys. J. 517, 565.
 Riess et al. (1998) Riess, A. G. et al., 1998, Astron. J. 116, 1009.
 Riess et al. (2004) Riess, A. G. et al., 2004, Astrophys. J. 607, 665.
 Bachall et al. (1999) Bachall, N. A. et al, 1999, Science 284, 1481.
 Tedmark et al. (2004) Tedmark, M. et al, 2004, Phys. Rev. D 69, 103501.
 Miller et al. (1999) Miller, D. et al, 1999, Astrophys. J. 524, L1.
 Bennet et al. (2000) Bennet, C. et al, 2000, Phys. Rev. Lett. 85, 2236.
 Briddle et al. (2003) Briddle, S. et al, 2003, Science 299, 1532.
 Spergel et al. (2003) Spergel, D. N. et al, 2003, Astrophys. J. Suppl. 148, 175.
 Padmanabhan (2003) Padmanabhan, T., 2003, Phys. Rept. 380, 235.
 Sahni et al. (2000) Sahni, V. and Starobinsky, A. A., 2000, Int. J. Mod. Phys. D 9, 373.
 Peebles et al. (1988) Peebles, P. J. E. and Ratra, B., 1988, Astrophys. J. Lett., 325, L17.
 Choudhury et al. (2007) Choudhury, T. R. and Padmanabhan, T., 2007, Astron. Astrophys. 429, 807.
 Padmanabhan et al. (2003) Padmanabhan, T. and Choudhury, T. R., 2003, MNRAS 344, 823.
 Tonry et al. (2003) Tonry, J. L. et al., 2003, ApJ, 594, 1.
 Barris et al. (2004) Barris, B. J. et al., 2004, ApJ, 602, 571.
 Amanullah et al. (2010) Amanullah, R. et al., 2010, Astrophys. J. 716, 712.
 Kamenshchik et al. (2001) Kamenshchik, A. et al., 2001, Phys. Lett. B 511, 265 (2001).
 Gorini et al. (2003) Gorini, V., Kamenshchik, A. and Moschella, U., 2003, Phys. Rev. D 67, 063509.
 Debnath et al. (2004) Debnath, U., Banerjee, A. and Chakraborty, S., 2004, Class. Quantum Grav. 21, 5609.
 Lu et al. (2008) Lu, J. et al, 2008, Phys. Lett. B 662, 87.
 Jun et al. (2005) DaoJun, L. and XinZhou, L., 2005, Chin. Phys. Lett., 22, 1600.
 Wu et al. (2008) Wu, P. and Zhang, S. N., 2008, JCAP 06, 007.
 Chen et al. (2008) Chen, S., Wang, B. and Jing, J., 2008, Phys. Rev. D 78, 123503.
 Jamil et al. (2011) Jamil, M. and Debnath, U., 2011, Astrophys Space Sci. 333, 3.
 Fu et al. (2008) Fu, X., Yu, H. and Wu, P., 2008, Phys. Rev. D 78, 063001.
 Wu et al. (2007) Wu, P. and Yu, H., 2007, Phys. Lett. B 644, 16.
 Thakur et al. (2009) Thakur, P., Ghose, S. and Paul, B. C., 2009, Mon. Not. R. Astron. Soc. 397, 1935.
 Paul et al. (2011) Paul, B. C., Ghose, S. and Thakur, P., arXiv:1101.1360v1 [astroph.CO].
 Paul et al. (2010) Paul, B. C., Thakur, P. and Ghose, S., arXiv:1004.4256v1 [astroph.CO].
 Ghose et al. (2011) Ghose, S., Thakur, P. and Paul, B. C., arXiv:1105.3303v1 [astroph.CO].
 Stern et al. (2010) Stern, D. et al, 2010, JCAP 1002, 008.
 Eisenstein et al. (2005) Eisenstein, D. J. et al, 2005, Astrophys. J. 633, 560.
 Bond et al. (1997) Bond, J. R. et al, 1997, Mon. Not. Roy. Astron. Soc. 291, L33.
 Efstathiou et al. (1999) Efstathiou, G. and Bond, J. R., 1999, Mon. Not. R. Astro. Soc. 304, 75.
 Nessaeris et al. (2007) Nessaeris, S. and Perivolaropoulos, L., 2007, JCAP 0701, 018.
 Komatsu et al. (2011) Komatsu, E. et al, 2011, Astrophys. J. Suppl. 192, 18.
 Riess et al. (2007) Riess, A. G. et al., 2007, Astrophys. J. 659, 98.
 Kowalaski et al. (2008) Kowalaski et al, 2008, Astrophys. J. 686, 749.