# Constraints on dark energy from the observed density fluctuations spectrum and supernova data

###### Abstract

One of the greatest challenges in cosmology today is to determine the nature of dark energy, the source of the observed present acceleration of the universe. High precision experiments are being developed to reduce the uncertainties in the observations. Recently, we showed that the agreement to an accuracy of of measurements of the present density fluctuations , derived from galaxy distribution (GD) data and cosmic microwave background (CMB) anisotropies in the CDM model, puts very strong limits on the possible decay of the vacuum energy into cold dark matter. Using this agreement, combined with the evidence that the matter density and that the universe is approximately flat, we show that the vacuum metamorphosis model (VMM) and the popular brane-world model (BWM), both used to explain dark energy, can be discarded. When we relax the requirement, we find that an agreement within can be obtained only with for the VMM and for the BWM, both of which are not consistent with observations. The agreement of the CMB and GD data and previous constraints from SNIa data exclude, or put strong limits on, other dark energy models, which have been suggested, that can be described by the parametrized equation of state (EOS) , where and are constants, is the cosmological scale factor and is the pressure (energy density) of the dark energy. We find that the supergravity (SUGRA) model with and can be discarded. In general, we find best values with . For redshifts , where the supernova data is sensitive, for this parametrized EOS.

###### pacs:

98.80.-k, 95.35.+d, 98.70.Vc, 04.62.+v## I Introduction

Many dark energy models have been suggested to explain the recent acceleration of the universe, first indicated by SNIa observations Riess ; Perl . The nature of dark energy is one of the major problems in cosmology. Theories in which gravity is modified as well as parametrizations of the dark energy equation of state (EOS) , where is the pressure (energy density) of the dark energy, have been suggested wag86 ; lin88 ; fpoc . Based on observations, various constraints have been put on the EOS for a variety of models (e.g., fhlt ; huttur ; hutstark ; somehu ; linsl ).

We begin by analyzing two interesting models that have been suggested for modifying gravity, which allow for a description of dark energy in terms of an effective EOS. The first is the five-dimensional brane-world model (BWM) of Defayet et al. bwm , where gravity is modified by adding a five-dimensional Einstein-Hilbert action that dominates at distances which are greater than the crossover distance , where is the Planck mass and is a five-dimensional Planck mass. The second model is the vacuum metamorphosis model (VMM) of Parker and Raval vmm , which assumes the existence of a quantized non-interacting scalar field coupled to the Ricci scalar curvature. In the VMM, the quantum vacuum undergoes a phase transition at a redshift : from a zero value at to a non-zero value for .

Linder lindastro analyzed the linear growth of a density perturbation and the gravitational potential for both of these models, comparing them with the simplest linear parametrization of the dark energy EOS as a function of : , where and are constants and is the cosmological scale factor.

Measurements of the present density fluctuations , derived from the cosmic microwave background (CMB) anisotropies in the CDM model, have been compared with those derived from the 2dF Galaxy Redshift Survey (2dFGRS) 2df ; 2df1 . It was found that their difference is no more than 2df . We recently showed that an agreement within (i.e., ) of the two sets of puts strong limits on a possible decay of the vacuum energy into CDM vdecay . Here we make a similar analysis to show that the BWM and the VMM as dark energy candidates can be discarded, in the face of the present evidence that , the matter density and that the universe is flat, indicated by recent CMB data.

We use the agreement to within of the present observed between the CMB and galaxy distribution (GD) data and the constraints from the Gold SNIa data comp to restrict the parameters of the EOS, . In particular, we analyze a model suggested by supergravity (SUGRA) sugra , studied by Linder lindastro and recently by Solevi et al. solevi ).

In II, we discuss the effect of dark energy on the linear growth of in the models: 1) the vacuum energy decay into CDM; 2) the BWM and VMM; 3) dark energy models parametrized by ; and 4) the SUGRA model. Our conclusions are presented in section III.

## Ii Dark energy and the growth of density fluctuations

The nature of dark energy is still unknown and there are many alternative models to explain it. One possibility is that instead of a constant vacuum energy, described by a cosmological constant, we have a vacuum energy which is decaying. Other possibilities are models that modify gravity and phenomenological models that parametrize the dark energy EOS in the form , setting values for and . In this section, all of the above models will be analyzed. For these models, the Friedmann equation can be written in a general form in terms of an effective EOS lindmnras . Modeling the dark energy as an ideal fluid in a flat universe, we can write the Friedmann equation as

(1) |

or

(2) |

where is the present value for the Hubble parameter, is the present normalized matter density and depends on the phenomenological model lindmnras . The EOS for the dark energy can be written as

(3) |

The linear growth of a density fluctuation, , depends on the EOS. We define the growth factor , where , the cosmological scale factor, and is normalized to unity at , the recombination epoch. In terms of , we have

(4) |

where is defined as

The linear mass power spectrum is proportional to and we define the deviation from the standard CDM model by

(5) |

where is the density fluctuation in the standard CDM model. Since derived from GD data differs from derived from the CMB anisotropies by no more than 10 per cent, the maximum value of is .

We apply the above description to the following models used to explain the recent acceleration of the universe, suggested in the literature: a phenomenological vacuum energy decay model; two models in which gravity is modified, BWM and VMM; and the SUGRA model.

### ii.1 Vacuum energy decay into CDM

In a previous paper, we put limits on the rate of a possible decay of the vacuum energy (i.e., the decay of the cosmological constant) into CDM from the observed agreement, to within , of the derived from the CMB and the galaxy survey data (vdecay, ).

Let us consider the vacuum energy decay model described, in a flat universe, by a power law dependence

(6) |

where . From conservation of energy and Eq.(6), we have

(7) |

where . Eq.(7) modifies the Friedmann equation [Eq.(2)] by a factor

(8) |

Comparing the vacuum energy decay model with the CDM model, the deviation is shown in Table 1, assuming , the observed value. It is to be noted that occurs only for .

0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | |

0.06 | 0.12 | 0.18 | 0.25 | 0.33 | 0.41 | 0.5 | 0.59 |

### ii.2 Brane-world and vacuum metamorphosis models

In the BWM bwm , gravity is modified by adding a five-dimensional Einstein-Hilbert action that dominates at distances which are larger than the crossover length that defines an effective energy density for a flat universe. The factor in Eq. (2) then becomes

(9) |

In the VMM vmm , the vacuum contributions are due to a quantized massive scalar field, which is coupled to gravity. For , the in Eq. (2) is

(10) |

where and . Both the BWM and the VMM can be described by the EOS,

(11) |

with and , respectively lindastro .

The growth of the density fluctuation as a function of for the BWM, the VMM, and the CDM model is shown in Fig. 3. We note that is greater than the maximum allowed value, , for the BWM and the VMM.

0.26 | 0.27 | 1.4 | 0.29 | 11 | 1.4 |

0.28 | 0.27 | 1.4 | 0.24 | 11 | 1.4 |

0.3 | 0.26 | 1.4 | 0.20 | 11 | 1.3 |

0.32 | 0.25 | 1.5 | 0.16 | 11 | 1.2 |

0.34 | 0.25 | 1.5 | 0.13 | 11 | 1.2 |

0.36 | 0.24 | 1.6 | 0.095 | 10 | 1.1 |

0.72 | 0.11 | 3.6 | 0.24 | 8 | 0.5 |

It can be seen that an agreement within between the VMM and the CDM is possible only for the matter density . For the BWM, an agreement with the CDM within is possible only if the matter density . Both of these values for are greater than the observational estimate .

### ii.3 Dark energy models described by a parametrized EOS

We now discuss the simplest parametrization of the EOS, that has been widely used for dark energy models, since it is well-behaved at high redshifts (unlike which diverges at high ). This parametrization [Eq.(11)], was introduced by Linder lindprl . The best fit parameters and that are consistent with the Gold SNIa dataset were found to be in the intervals and comp . Assuming and , we further restrain the best fit values of and . These values, whose ranges are and , are shown in Table 3.

1.53 | |
---|---|

1.63 | |

1.73 | |

1.83 | |

1.93 | -1.82-0.01 |

2.03 | -1.89-0.01 |

2.9 | -1.86-0.02 |

### ii.4 Supergravity model

The SUGRA model sugra is an attractive model to possibly explain the acceleration of the universe. This model can be described by the EOS of II-C with and lindmnras . This equation of state is in agreement with observations for the low redshift SNIa dataset sugra and GD data solevi .

Fig. 4 shows that the growth of is smaller for the SUGRA model than for the CDM model. The for the SUGRA model is for , which is appreciably greater than the maximum allowed value .

## Iii Conclusions

We calculate the value of numerically for well-known dark energy models from the growth equation for . From observations, the maximum value of is . A -decay into CDM model, described by a power law dependence (studied in our previous paper vdecay ), was first considered. It was found that the factor increases as the exponent increases. The maximum possible value of was found to be ().

The BWM and VMM were then analyzed. We showed that these models as dark energy candidates can be discarded, assuming that , and that the universe is flat.

We combined the constraints from the Gold SNIa data comp and the condition that to restrict the values of the parameters of the linear EOS for dark energy, . It was found that the best fit values of and are with . For , where the supernova data is sensitive, for this parametrized EOS .

Finally, we also analyzed the SUGRA model for the above parametrized EOS with and . was found to be very large: for , which is appreciably greater than the maximum value .

Acknowledgments. R.O. thanks the Brazilian agencies FAPESP (grant 00/06770-2) and CNPq (grant 300414/82-0) for partial support. A.M.P thanks FAPESP for financial support (grants 03/04516-0 and 00/06770-2).

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