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Brazilian Journal of Physics

Print version ISSN 0103-9733

Braz. J. Phys. vol.40 no.4 São Paulo Dec. 2010 

The cold dark matter model with cosmological constant and the flatness constraint



A.C.B. AntunesI,*; L.J. AntunesII,

IInstituto de Física, Universidade Federal do Rio de Janeiro C.P. 68528, Ilha do Fundão, 21945-970 Rio de Janeiro, RJ, Brazil
IIInstituto de Engenharia Nuclear - CNEN C.P. 68550, Ilha do Fundão, 21945-970 Rio de Janeiro, RJ, Brazil




The Hubble parameter, a function of the cosmological redshift, is derived from the Friedmann-Robertson-Walker 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 type-Ia 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 type-Ia supernovae.

Keywords: Cold dark matter model, Hubble parameter.




From Einstein's equations for the gravitational field in the Robertson-Walker 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 ρm by an equation of state,

with w = 0 for non-relativistic matter [1-4].

Using the vacuum energy density

and introducing the Hubble parameter

the Friedmann equation reads :

The scale factor R and the matter density ρm are related to their present day values R0 and ρ0m 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 ρc = ( 3H02 / 8 πG ) is the critical density and H0 is the Hubble constant.

Now the Hubble parameter can be written explicitly as

Introducing the relative densities Ω0m = ρ0m/ρc and ΩΛ = ρΛ/ρc, 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 left-hand side can be written in terms of x = R0 / R and H(R) = /R. Using


we obtain

Performing the calculation of the right-hand side with

and equating to the above expression for / R containing (3p / c2ρc ) 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 (t0) can be obtained from the Hubble parameter


With the above expression for H2(x) we have

1.1 Determination of the parameters by fitting H(x) to type-Ia supernovae data

Let 0= 65 km·s-1 ·Mpc-1 be a nominal value of the Hubble constant; defining the function


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 type-Ia supernovae. These parameters and their sum, y0= a1+ a2+ a3, give the physical parameters

To fit the Hubble parameter to the data from redshift (z) and luminosity distances (D) measurements of type-Ia supernovae, some changes of variables are in order. The published data set is[5]

with N = 230

The Hubble parameter is




The data set to be used in the fitting is

{ x j , yj , σj ; j = 1,..,n }

which is obtained from the first set using

Details of the fitting are presented in the 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



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 Ωom + ΩΛ = 1. If this condition is imposed on the Hubble parameter (see equation (12)), we have


The polynomial form, equation (26), becomes

where y(x) = ( H(x)/0)2, a1 = y0ΩΛ , a3 = y0Ω0m and y0 = ( H0/ 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 a1 = 0.87 and a3 = 0.32. The deceleration parameter is

and the point of null acceleration is

corresponding to the redshift ze = 0.8 . The age of the Universe is

The sum of the weighted square deviations for this model, as defined in the appendix, is



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 Robertson-Walker metric. The square of the Hubble parameter is an incomplete third-degree polynomial function in the variable x = 1+z. This polynomial is least-squares fitted to data from the measurements of the redshifts and luminosity distances of the type-Ia supernovae, and the three non-null coefficients of the polynomial and the uncertainties and covariances are then computed. The physical parameters are obtained from the three non-null coefficients, showing that these supernovae data are sufficient to determine H0, Ω0m and ΩΛ, 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 type-Ia supernovae data. Both models are consistent with the evidences of an accelerating expansion of the Universe [7-11]. However, in any way there are clear disagreements between models and data. Some observable measurements are model-dependent. 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.


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[2] Rindler,W. , Relativity, Special, General and Cosmological, (Oxford University Press, 2001).         [ Links ]

[3] D'Inverno, R., Introducing Einstein's Relativity, (Clarendon Press, Oxford, 1998).         [ Links ]

[4] Grupen, C. Astroparticle Physics, (Springer, 2005).         [ Links ]

[5] Tonry, J.L., et al., Astro-ph/0305008, Vol. 1 (2003).         [ Links ]

[6] Particle Data Group, Review of Particle Physics, Journal of Physics G, Vol. 33 (July 2006).         [ Links ]

[7] Riess, A.G. et al., Astron. J. 116, 1009 (1988).         [ Links ]

[8] Perlmutter, S. et al., Astrophy. J. 517(1999), 565 .         [ Links ]

[9] Tonry, J.L., et al., Astrophys. J. 594 (2003), 1.         [ Links ]

[10] Riess, A.G. et al., Astrophys. J. 607 (2004), 665.         [ Links ]

[11] Spergell, D.N. Astrophys. J. Suppl. ser. 170(2007) , 377.         [ Links ]



(Received on 29 January, 2009)



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Appendix A

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 least-squares method is used to determine the parameters a1, a2 and a3 which minimize the function

The minimum condition

gives a matricial equation


Inverting the matricial equation above, the parameters are given by

The uncertainties and covariances are respectively


The results of the fitting are

The sum of the weighted square deviations (χ2) defined above is

χ2 = 989.4.

The physical parameters are given by


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/aj) so that σK is given by

The numerical results are

The uncertainty in t0 is