Abstract
The Hubble parameter, a function of the cosmological redshift, is derived from the FriedmannRobertsonWalker equation. The three physical parameters H0, Ω0m and ΩΛ are determined fitting the Hubble parameter to the data from measurements of redshift and luminosity distances of typeIa supernovae. The best fit is not consistent with the flatness constraint (k = 0). On the other hand, the flatness constraint is imposed on the Hubble parameter and the physical parameters used are the published values of the standard model of cosmology. The result is shown to be inconsistent with the data from typeIa supernovae.
Cold dark matter model; Hubble parameter
The cold dark matter model with cosmological constant and the flatness constraint
A.C.B. Antunes^{I,}^{*} * Electronic address: antunes@if.ufrj.br Electronic address: leila@ien.gov.br ; L.J. Antunes^{II,}^{} * Electronic address: antunes@if.ufrj.br Electronic address: leila@ien.gov.br
^{I}Instituto de Física, Universidade Federal do Rio de Janeiro C.P. 68528, Ilha do Fundão, 21945970 Rio de Janeiro, RJ, Brazil
^{II}Instituto de Engenharia Nuclear  CNEN C.P. 68550, Ilha do Fundão, 21945970 Rio de Janeiro, RJ, Brazil
ABSTRACT
The Hubble parameter, a function of the cosmological redshift, is derived from the FriedmannRobertsonWalker equation. The three physical parameters H_{0}, Ω_{0m} and Ω_{Λ} are determined fitting the Hubble parameter to the data from measurements of redshift and luminosity distances of typeIa supernovae. The best fit is not consistent with the flatness constraint (k = 0). On the other hand, the flatness constraint is imposed on the Hubble parameter and the physical parameters used are the published values of the standard model of cosmology. The result is shown to be inconsistent with the data from typeIa supernovae.
Keywords: Cold dark matter model, Hubble parameter.
1. THE HUBBLE PARAMETER FROM THE FRIEDMANN EQUATION
From Einstein's equations for the gravitational field in the RobertsonWalker metric, one can derive the Friedmann differential equation
and the acceleration equation
where R is the scale factor, k is the curvature index and Λ the cosmological constant. The pressure p is related to the matter density ρ_{m} by an equation of state,
with w = 0 for nonrelativistic matter [14].
Using the vacuum energy density
and introducing the Hubble parameter
the Friedmann equation reads :
The scale factor R and the matter density ρ_{m} are related to their present day values R_{0} and ρ_{0m} by
Defining an adimensional variable, the cosmological frequency redshift,
where z is the redshift, the equation above becomes
For current values, corresponding to x = 1, this equation gives
where ρ_{c} = ( 3H_{0}^{2} / 8 πG ) is the critical density and H_{0} is the Hubble constant.
Now the Hubble parameter can be written explicitly as
Introducing the relative densities Ω_{0m} = ρ_{0m}/ρ_{c} and ΩΛ = ρ_{Λ}/ρ_{c}, the Hubble parameter reads
The function containing the curvature index and the present day scale factor becomes
The acceleration equation
can be rewritten as
The lefthand side can be written in terms of x = R_{0} / R and H(R) = /R. Using
in
we obtain
Performing the calculation of the righthand side with
and equating to the above expression for / R containing (3p / c^{2}ρ_{c} ) we obtain p = 0 . Thus, the acceleration equation is finally reduced to
This result permits to obtain the value of x at the equilibrium point corresponding to = 0:
The dimensionless deceleration parameter
can be calculated at the present day condition (x = 1):
The age of the universe (t_{0}) can be obtained from the Hubble parameter
then
With the above expression for H^{2}(x) we have
1.1 Determination of the parameters by fitting H(x) to typeIa supernovae data
Let
_{0}= 65 km·s^{1} ·Mpc^{1} be a nominal value of the Hubble constant; defining the function
with
and also the parameters
the above equation for the Hubble parameter gives
The three parameters of this polynomial function can be determined by fitting data from measurements of the luminosity distances and the redshift of the typeIa supernovae. These parameters and their sum, y_{0}= a_{1}+ a_{2}+ a_{3}, give the physical parameters
To fit the Hubble parameter to the data from redshift (z) and luminosity distances (D) measurements of typeIa supernovae, some changes of variables are in order. The published data set is[5]
with N = 230
The Hubble parameter is
As
and
then
The data set to be used in the fitting is
{ x _{j} , y_{j} , σ_{j} ; j = 1,..,n }
which is obtained from the first set using
Details of the fitting are presented in the ^{appendix}appendix. The results of the fitting give the following values for the physical parameters:
and the age of the universe
The point of null acceleration is
which corresponds to the redshift
2. THE FLATNESS CONSTRAINT
The cosmic microwave background (CMB) observations suggest that the spacial geometry of the Universe is very close to flat. According to equation (13) the zero curvature corresponds to the condition Ω_{om} + Ω_{Λ} = 1. If this condition is imposed on the Hubble parameter (see equation (12)), we have
with
The polynomial form, equation (26), becomes
where y(x) = ( H(x)/_{0})^{2}, a_{1} = y_{0}Ω_{Λ} , a_{3} = y_{0}Ω_{0m} and y_{0} = ( H_{0}/ _{0} )^{2}.
The accepted values for the physical parameters of the cold dark matter model with cosmological constant (ΛCDM) subjected to the flatness constraint (k = 0) are [6]:
So, the coefficients of the polynomial in equation (32) are a_{1} = 0.87 and a_{3} = 0.32. The deceleration parameter is
and the point of null acceleration is
corresponding to the redshift z_{e}= 0.8 . The age of the Universe is
The sum of the weighted square deviations for this model, as defined in the ^{appendix}appendix, is
3. CONCLUSION
The cold dark matter model with cosmological constant (ΛCDM) is expressed by the Hubble parameter as a function of the cosmological redshift x = 1+z. This function is derived from the Friedmann equation in the RobertsonWalker metric. The square of the Hubble parameter is an incomplete thirddegree polynomial function in the variable x = 1+z. This polynomial is leastsquares fitted to data from the measurements of the redshifts and luminosity distances of the typeIa supernovae, and the three nonnull coefficients of the polynomial and the uncertainties and covariances are then computed. The physical parameters are obtained from the three nonnull coefficients, showing that these supernovae data are sufficient to determine H_{0}, Ω_{0m} and Ω_{Λ}, the three fundamental parameters of the ΛCDM model. The results of this model are compared with the published results of the ΛCDM model with the flatness constraint (k = 0)[6](see Fig. 1).
In this second model, the ΛCDM (k = 0), the measurements from the cosmic microwave background (CMB) are also taken into account. The results of these models disagree. The ΛCDM model fitted to the Ia supernovae data implies a positive curvature index (k = +1) and a large age for the Universe. This conflicts with the results of the CMB measurements. On the other hand, the ΛCDM (k = 0) model presents large deviations from the typeIa supernovae data. Both models are consistent with the evidences of an accelerating expansion of the Universe [711]. However, in any way there are clear disagreements between models and data. Some observable measurements are modeldependent. These observables are related to the parameters of the model. The values of the parameters must be fixed so that these observables can be computed from other observable measurements.
Thus, it is a contradictory result that the ΛCDM (k = 0) model, which is used in the computation of the luminosity distances of these Ia supernovae, be in disaccord with these same data.
(Received on 29 January, 2009)
In this appendix we collect some formulas used in the fitting of the polynomial function
to the data set
obtained from the published data[5] by the transformations
The leastsquares method is used to determine the parameters a_{1}, a_{2} and a_{3} which minimize the function
The minimum condition
gives a matricial equation
where
Inverting the matricial equation above, the parameters are given by
The uncertainties and covariances are respectively
and
The results of the fitting are
The sum of the weighted square deviations (χ^{2}) defined above is
χ^{2} = 989.4.
The physical parameters are given by
and
The uncertainties in these parameters are respectively
The integral that gives the age of the universe is
In order to obtain the uncertainty in K we must compute the derivatives K'_{j}=(∂K/∂a_{j}) so that σ_{K} is given by
The numerical results are
The uncertainty in t_{0} is
 [1] Weinberg, S., Gravitation and Cosmology Principles and Applications of the General Theory of Relativity, (JohnWiley and Sons, 1972).
 [2] Rindler,W. , Relativity, Special, General and Cosmological, (Oxford University Press, 2001).
 [3] D'Inverno, R., Introducing Einstein's Relativity, (Clarendon Press, Oxford, 1998).
 [4] Grupen, C. Astroparticle Physics, (Springer, 2005).
 [5] Tonry, J.L., et al., Astroph/0305008, Vol. 1 (2003).
 [6] Particle Data Group, Review of Particle Physics, Journal of Physics G, Vol. 33 (July 2006).
 [7] Riess, A.G. et al., Astron. J. 116, 1009 (1988).
 [8] Perlmutter, S. et al., Astrophy. J. 517(1999), 565 .
 [9] Tonry, J.L., et al., Astrophys. J. 594 (2003), 1.
 [10] Riess, A.G. et al., Astrophys. J. 607 (2004), 665.
 [11] Spergell, D.N. Astrophys. J. Suppl. ser. 170(2007) , 377.
appendix
Publication Dates

Publication in this collection
03 Jan 2011 
Date of issue
Dec 2010
History

Received
29 Jan 2009