_{1}

^{*}

Assuming a flat universe expanding under a constant pressure and combining the first and the second Friedmann equations, a new equation, describing the evolution of the scale factor, is derived. The equation is a general kinematic equation. It includes all the ingredients composing the universe. An exact closed form solution for this equation is presented. The solution shows remarkable agreement with available observational data for redshifts from a low of z = 0.0152 to as high as z = 8.68. As such, this solution provides an alternative way of describing the expansion of space without involving the controversial dark energy.

The evolution of the universe has already been investigated through the first Friedmann equation. As discussed by Carrol [

Except for the flatness assumption, the analytical solution is completely general. It includes all the ingredients forming the universe. The ΛCDM-based models only consider specific combinations of limited numbers of ingredients. Also the value of the cosmological density parameter is analytically estimated to be

In the next section we develop the new general equation and derive an analytical estimate of the cosmological density parameter. The new analytical solution together with the two existing analytical solutions is presented in Section 3. Comparison of the new analytical solution with the analytical solution involving matter and lamda is presented in Section 4.1. Comparisons of the new analytical solution with the ΛCDM-based models are presented in Section 4.2. In Section 5, the efficacy of the new analytical solution is shown through comparisons with two ΛCDM-based solutions and through comparisons with three sets of observational data.

The first Friedmann equation, including the curvature, k, and the cosmological constant, Λ, is

Alternatively, including the cosmological term in the total density, the above equation can be represented by

where now the density,

where

The curvature can be represented by

where

Substitutions for

For the present time,

To evaluate Λ from Equation (8), we need to evaluate

To evaluate the present time values of

where

iverse,

Because at all points the expansion is taking place in all directions, to evaluate the rate of increase of space between any two galaxies,

Correcting the velocity in the above relation for the effect of time dilation yields

where c represents the speed of light. The value of

Therefore the corrected expansion velocity at the present time is given by

Also the present time value of the expansion acceleration, according to Equations (13) and (14), is given by

Substitutions for

Solving the above relation for Λ yields

Therefore the cosmological density parameter can be represented by

Now substitution for Λ from Equation (18) back into Equation (7) yields the Friedmann acceleration equation as

In the next section, assuming a flat universe expanding under the constant pressure

where

and

Assuming a flat universe, i.e.,

This value of the energy density parameter is essentially identical to the

Already there exist two analytical solutions for the Friedmann equations. One is for the case of a flat universe containing only matter,

where

Assuming the pressure to be constant implies that in the above equation the term

pressure being equal to the negative of the vacuum density as discussed by Carroll [

It is clear that the above equation satisfies the present time boundary conditions as given in Equations (9), (15) and (16). To non-dimensionalize the above equation, let

where now dot denotes differentiation with respect to

The beauty of the above analytical solution is the fact that it does not involve the fractional components forming the mix of the universe. Using the Mathematica code [

As seen from the above figure, at the present time,

Considering Equation (18), it is clear that in Equation (20), the pressure

In the following subsections the analytical solution given by Equation (29) is compared with the analytical solution for a universe containing only matter and lambda as given by Equation (25). It is also compared with other models based on ΛCDM parameterizations.

In order to compare the analytical solution given by Equation (29) with the one given by Equation (25), we need to first decide on the value of

The time

The ΛCDM-based models characterize the universe with a limited number of energy density parameters as fractions of constituent ingredients. The values of these fractions are estimated through finding the optimum fit to the observationally measured data. The results are presented in terms of distance modulus versus redshift. Here, a comparison of the variation of scale factors versus redshift will be carried out first. The relation between the scale factor and the redshift, z, is defined by

Thus

To express the scale factor given by Equation (29) in terms of redshift, substitution for the scale factor, a, from Equation (29) back into Equation (31) yields the relation between the red shift, z, and the time

Using the above equation, the variation of time versus the redshift is presented in

In

are tabulated for values of

versus the tabulated values of the redshift, z, in

For ΛCDM-based models, considering Equation (30), and rewriting Friedmann Equation (2) in terms of fractions of constituent densities, for a universe containing mass, energy, radiation and curvature, one obtains the

Using the above equation, the values of

In the next subsection the analytical curve and the curves based on ΛCDM parameterization are compared with the observational data.

Parameters | ΛCDM-1 Plank + WP + high L + BAO | ΛCDM-2 WMAP-9 + BAO |
---|---|---|

0.0 | 0.0 | |

0.0 | 0.0 |

In this part the curve of the analytical scale factor given by Equation (29), transformed through Equation (32), is compared with the curves based on ΛCDM, as given by Equation (33), for the two different sets of ingredients presented in

1) A set of 557 SNe data with redshifts from a low of

2) A set of 394 extragalactic distances to 349 galaxies at cosmological redshifts significantly higher than the Union2 Compilation with redshifts from a low of

3) A set of data for a quasar and the three most distant recently confirmed galaxies, as presented in

To compare with the aforementioned observational data, the scale factors have to be presented in terms of distance modulus and redshift. The SNe and the Union2 data are already available in terms of distance modulus and redshift. The data for the galaxies and for the quasar are listed in

where

tions including the instrument corrections and the K-correction. These are well known corrections and they are

considered in various ways [

with the first set of the observational data. Then we check the validity of its value through comparisons with the

second and third sets of observational data as well as with the ΛCDM-based curves. To evaluate

the analytical curve, we only need to have the value for the Hubble constant. The recent estimated values of the

Hubble constant based on observational data are: (the Seven-Year Wilkinson Microwave Anisotropy Probe [

Because this value is remarkably consistent with the observationally estimated values, we will use this value and substitute it together with the values of

Name | Reference | Light Travel Distance, Gly | Redshift, z |
---|---|---|---|

Galaxy, EGSY8p7 | Zitrin, 2015 [ | 13.2 | |

Galaxy, EGS-zs8-1 | Oesch, 2015 [ | 13.044 | 7.730 |

Galaxy, z8GND 5296 | Finkelstein, 2013 [ | 13.02 | 7.51 |

Quasar, ULAS J1120+0641 | Matson, 2011 [ | 12.9 | 7.085 |

As can be seen from

where

Now, to check the validity of the factor

on Equations (29) and (32), and the two ΛCDM curves based on Equation (33), are presented in Figures 6-9. As seen from these figures, in all cases, the analytical curve is remarkably consistent with the observational data as well as with the ΛCDM-based curves. The log-linear plots and the linear plots show how well the curves represent the observational data at the low and high values of the redshifts respectively.

The excellent match of the analytical curve and the ΛCDM curves with the second and the third set of the observational data validates Equation (36) representing the factor

luated value for the Hubble constant as given in Equation (35). It should be noted that, except for the flatness assumption, the analytical solution is completely general. It includes all the ingredients forming the universe. The ΛCDM-based solutions only consider specific combinations of limited numbers of ingredients.

The value of the energy density parameter was analytically estimated to be

tency of the analytical solution with the observational data, as well as with the -based models, provide the necessary confidence in the fidelity of the analytical solution in the representation of reality.

The pressure is cancelled from the Friedmann acceleration equation through the contribution of the cosmological constant. As the result, Equation (28) may be interpreted as a kinematic equation. Its solution, Equation

(29) describes the evolution of the expansion of space. As such, this equation provides an alternative way of describing the expansion of space without involving the controversial dark energy.

Naser Mostaghel, (2016) A New Solution for the Friedmann Equations. International Journal of Astronomy and Astrophysics,06,122-134. doi: 10.4236/ijaa.2016.61010