# Consequences of dark matter-dark energy interaction on cosmological parameters derived from SNIa data

###### Abstract

Models where the dark matter component of the universe interacts with the dark energy field have been proposed as a solution to the cosmic coincidence problem, since in the attractor regime both dark energy and dark matter scale in the same way. In these models the mass of the cold dark matter particles is a function of the dark energy field responsible for the present acceleration of the universe, and different scenarios can be parameterized by how the mass of the cold dark matter particles evolves with time. In this article we study the impact of a constant coupling between dark energy and dark matter on the determination of a redshift dependent dark energy equation of state and on the dark matter density today from SNIa data. We derive an analytical expression for the luminosity distance in this case. In particular, we show that the presence of such a coupling increases the tension between the cosmic microwave background data from the analysis of the shift parameter in models with constant and SNIa data for realistic values of the present dark matter density fraction. Thus, an independent measurement of the present dark matter density can place constraints on models with interacting dark energy.

###### pacs:

98.80.Cq^{†}

^{†}preprint: IFT-P.035/2006

## I Introduction

Supernovae of type Ia (SNIa) chronicle the recent expansion history of the universe. The accumulated data encrypts information about the composition of the universe and the physical properties of its main components, in particular of the dark energy (DE) Reviewdarkenergy that drives the accelerated expansion today.

Data from SNIa SNIaRiess ; SNIaLegacy , the cosmic microwave background (CMB) wmap and large scale structure lss converged to a concordance CDM model concordance , with a nearly flat universe where a cosmological constant is the current dominant energy component, accounting for approximately 74 of the critical density, the remaining being non-relativistic non-baryonic dark matter (DM, 22) and baryonic matter (4).

This simple “vanilla” CDM model is still compatible with data but is not satisfactory mainly because it requires a large amount of fine tuning in order to make the cosmological constant energy density dominant at recent epochs.

DE can also be modelled by a scalar field, the so-called quintessence models, either slowly rolling towards the minimum of the potential or already trapped in this minimum Reviewdarkenergy ; quintessence ; plquintessence ; expquintessence . In this case, the equation of state of DE may vary with cosmological time.

In spite of the success of the concordance model, one should keep in mind that there exists some tension between SNIa and CMB data when the DE equation of state is allowed to be different from a cosmological constant. Best fit models for one set of data alone is usually ruled out by the other set at a large confidence limit tension . SNIa data typically favors large values of non-relativistic dark matter abundance and a phantom-like DE equation of state . Of course these conclusions are valid only in standard models of DE and DM, that is, models where DE and DM are decoupled. This tension has been ameliorated with the new data from the Supernova Legacy Survey(SNLS) SNIaLegacy , as shown in tension2 .

An intriguing possibility is that DM particles could interact with the DE field, resulting in a time-dependent mass for the DM particles and a modification in its equation of state. In this scenario, sometimes called VAMPs (VAriable-Mass Particles) carrolvamps , the mass of the DM particles evolves according to some function of the dark energy field such as, for example, a linear function of the field carrolvamps ; quirosvamps ; peeblesvamps with a inverse power law dark energy potential or an exponential function exp1amendola ; exp2amendola ; exp3amendola ; pietronivamps ; riottovamps with an exponential dark energy potential. Some of these models have a tracker solution, that is, there is a stable attractor regime where the effective equation of state of DE mimics the effective equation of state of DM exp1amendola ; riottovamps .

The tracker behavior is interesting because once the attractor is reached, the ratio between DM energy density and DE energy density remains constant afterwards. This behavior could solve the “cosmic coincidence problem”, that is, why are the DE and DM energy densities similar today. However, even in these cases a large amount of fine-tuning is required for the energy density scale of the scalar potential franca .

In this paper we investigate the effect of a DE-DM coupling in deriving bounds on the DE equation of state and on from SNIa data. In particular, we show that analyzing the SNIa data with a large positive coupling results in a larger value of , thus potentially increasing the tension with CMB data. This is confirmed by an analysis of the CMB shift parameter shift for interacting models with constant equation of state.

## Ii A phenomenological model

Variable-mass particles generically arise in models where the quintessence field is coupled to the non-baryonic dark matter field (coupling to baryonic matter is severely restricted couplinglimits ). Such a coupling represents a particularly simple and relatively general form of modified gravity: they appear in fact in scalar-tensor models (in the Einstein frame) and in simple versions of higher-order gravity theories in which the action is a function of the Ricci scalar. From a lagrangian point of view, these couplings could be of the form or for a fermionic or bosonic dark matter represented by and respectively, where the functions and of the quintessence field can in principle be arbitrary.

Instead of postulating a definite model by choosing two functions defining a DE self-coupling potential and DE-DM coupling das , one could alternatively follow an approach that is more model-independent and closer to observations by introducing a parameterization for the DE equation of state and for the coupling function , where is the scale factor of the universe.

The dynamics of the quintessence field, governed by its potential, induces a time dependence in the mass of dark matter particles. Therefore one would have that we will parameterize in terms of a function of the scale factor in the following way:

(1) |

where is the particle mass today. In other words, in addition to the usual parametrizations of the DE variable equation of state we introduce a parametrization for the DM variable mass. Just as for the equation of state , this parametrization allows to study the observational data in a systematic fashion in search of new physical phenomena. The physical interpretation of the coupling is therefore straightforward: it represents the rate of change of the DM particle mass, . In this paper we will focus on the simplest case, a constant coupling .

This variable mass results in the following equation for the evolution of the DM energy density :

(2) |

where is the Hubble parameter. Conservation of the total stress-energy tensor them implies that the dark energy density should obey

(3) |

Recently, Majerotto et al. majerotto (see also AGP ) considered the case of constant and assumed a tracking behaviour of the DM and DE densities () in their analysis of SNIa data. In this class of models the modifications of the abundance of mass-varying DM particles were studied in abundance and consequences for the evolution of the universe were analyzed in CaiWang .

However, in quintessence models the equation of state is generally time-dependent. Therefore in this work we will study the impact of a constant DM-DE coupling on the determination of a redshift dependent DE equation of state and on the best-fit value of the dark matter abundance today from SNIa data.

In the case of a constant interaction, eq. (2) can be easily solved:

(4) |

where is the non-baryonic DM energy density today. Substituting this solution in eq. (3) we obtain a differential equation in the scale factor :

(5) |

Before proceeding to an evolving equation of state, it is instructive to study the particular case of a constant , where the solution to eq. (5) is given by:

(6) |

where is the DE energy density today. The first term of the solution is the usual evolution of DE without the coupling to DM. From this solution it is easy to see that one must require a positive value of the coupling in order to have a consistent positive value of for earlier epochs of the universe (it is to be noted however that negative values of could be allowed if the dark energy is in fact a manifestation of modified gravity, see e.g. fr ). This feature remains in the case of varying and we will consider only positive values of throughout the paper.

Furthermore, if one has a tracking of DE and DM densities at earlier epochs:

(7) |

resulting in

(8) |

Therefore, requiring in the past fixes in this simple case of constant . It is also interesting that one can analytically compute the scale factor where the transition from DM to DE occurs in this simple case:

(9) |

For instance, for and we find (corresponding to ).

For the remaining of this work, we will study an equation of state with the commonly used parameterization:

(10) |

in order to compare with other results in the literature. However, it should be stressed that SNIa data is not currently very sensitive to time variations in . In some cases, as in the comparison with the CMB data, we will use a constant equation of state by fixing .

We obtain a closed form solution for eq. (5) as a function of the redshift :

(11) |

where

(12) |

is the usual evolution of non-interacting (NI) DE density for this parameterization of the equation of state and the correction is given by:

(13) |

where is the generalized incomplete gamma function:

(14) |

Notice that the correction vanishes in the case of no interaction, , as it should be. We have also found solutions for different parameterizations of the dark energy equation of state but will not use them in this paper.

We use the Hubble-free dimensionless luminosity distance defined in terms of the usual luminosity distance as

(15) |

The Hubble-free luminosity distance in our model for the parameterization used is given by (we are assuming a flat universe throughout the paper):

(16) | |||||

where and are the dark matter and the baryonic density fractions today, respectively. In the next section we will compare (or for a given value of the Hubble parameter) obtained from our model to observations in order to study the consequences of the coupling between DE and DM.

## Iii Results

We will work with two data sets, the so-called gold data set consisting of 157 SNIa SNIaRiess and the recent SNLS data set of 71 new SNIa with high redshift SNIaLegacy . Since these two data sets were obtained with different detection techniques and use different methods to extract the distance moduli of the supernovae, we will analyze them separately. Other recent analysis can be found in tension2 ; snls1 ; snls2 ; snls3 .

In both cases we used the maximum likelihood method for estimating the cosmological parameters. We find the best-fit values by numerically minimizing the likelihood function and the 68% confidence level contour plots were obtained by direct integration of the likelihood function. We fixed throughout our analysis. Since the impact of on is very weak, the exact value of is not important.

The apparent luminosity of a supernova in terms of its absolute magnitude is usually written as:

(17) |

The absolute magnitude of each SNIa has a spread around a standard value:

(18) |

where is the absolute magnitude of a standard SNIa and represents the corrections arising from fits to the color and light curves of each supernova. Hence one can write the apparent magnitude as:

(19) |

where the so-called nuisance parameter is given by

(20) |

We will use the values of the distance modulus given by

(21) |

in order to derive bounds in the parameters in our model.

For the analysis of the SNIa gold data set, we follow ref. peri ; peri2 ; lazkoz and estimate the likelihood function already marginalized over the nuisance parameter , which includes , for different values of the coupling .

For the analysis of the 71 new SNIa data obtained by SNLS we used the distance moduli values obtained from the best fit values (for ) of the correction parameters ( and ) and , and the corresponding errors reported in SNIaLegacy . The likelihood function is computed as:

(22) |

where is the uncertainty associated with the observational techniques in determining the magnitudes, is associated with the peculiar velocities (and hence negligible for large redshifts) and is due to the intrinsic dispersion of the absolute magnitudes.

In the case where there is no interaction between DE and DM, we find that the best fit values for in a CDM model ( and ) are and for the gold set and SNLS data respectively. The best fit from the SNLS data is in remarkable agreement with the recent analysis of the 3-year data from the Wilkinson Microwave Anisotropy Probe (WMAP3y) wmap alone, which results in , whereas the best fit from the gold data set has a somewhat poorer agreement.

Many parameterizations for the DE equation of state were tested with SNIa data but frequently fixing a particular value of or marginalizing over a flat or gaussian prior around SNIaRiess ; tension ; peri ; peri2 ; lazkoz . However, the agreement between SNIa and CMB gets particularly worse for the gold data set when we allow for and to vary simultaneously without any priors. For instance, fixing , that is, a constant DE equation of state, we find the best fit values are and for the gold set and and for the SNLS data. Allowing for an evolving equation of state does not significantly alter the fits. A possible tension between SNIa and CMB data which was present in the gold data set when one considers models other than the CDM model practically disappeared in the SNLS data tension2 .

We want to investigate in this article the effects of adding a coupling between DE and DM on these fits to SNIa data. The coupling will be characterized by the constant . We analyze first the case of constant (i.e. ) and then generalize to a variable equation-of-state.

### iii.1 Constant equation of state

We show in Figure 1 the 68% confidence level contour plots in the plane for different values of the coupling , keeping (constant equation of state). Notice that the SNLS data results in a better agreement with CMB measurements. One can see the existence of a correlation between and and that turning on the interaction results in a marked tendency towards increasing the best value for , against the CMB results.

However, a direct comparison of our result for with the CMB results is not possible, since the latter has been obtained in the context of uncoupled models. It is therefore important to see whether there are upper limits to which are independent of the cosmological model. An upper limit to which does not depend on the background cosmology can be obtained from the galaxy cluster dynamics. However the current data yield very weak constraints: ref. Feldmanetal gives so that even is not excluded at more than 95% c.l. and actually the strong degeneracy with allows for even higher values.

On the other hand, for a constant equation of state, we can use the CMB shift parameter in order to study the effects of interaction. The shift parameter encapsulates information contained in the detailed CMB power spectrum and, more importantly, its measured values is weakly dependent of assumptions made about dark energy. It has been used recently to put constraints on models with braneworld cosmologies usedshift .

The shift parameter is defined by:

(23) |

where is the redshift of matter-radiation decoupling and . We will use the value , obtained from WMAP3y data shift . Clearly for such calculations a radiation component with standard conservation equation was appropriately included.

In Figs. 2 and 3 we compare the resulting bounds obtained from and from SNIa data. Figure 2 clearly shows the tension between CMB and the gold data set referred to earlier. It is clear that the introduction of the coupling makes the CMB and the gold data set more incompatible with each other. The coupling favors smaller values of from CMB at the same time favoring larger values of the same quantity from SNIa data. The same qualitative behaviour is seen for the SNLS data, although since there is already a good agreement with , the situation is less drastic in this case. It is interesting to notice that assuming a constant and requiring seems to rule out a coupling from the bounds arising from the shift parameter.

### iii.2 Variable equation of state

Allowing does not change the qualitative features of Figure 1. We show in Figure 4 the confidence level contour plots in the plane obtained with a marginalization over and as expected the allowed region gets somewhat broader, with the best fit of shifting to larger values.

In Figure 5 we show the marginalized likelihood function over and in order to obtain the confidence level estimation for . The main effect of the coupling is to increase the estimation of . It is interesting to observe that for the SNLS dataset the WMAP3y value for is rejected at more than 95 c.l. already for : however, as we anticipated, such a direct comparison is not correct.

These results are summarized in Table I, where we show the best fit values of and with their corresponding errors obtained by marginalizing over and (in the case of ) and and (in the case of ) for both sets of SNIa data. Although the best fit values for are quite large when is included, the distribution is non-zero even for small values of .

Gold | SNLS | Gold | SNLS | |
---|---|---|---|---|

0.0 | ||||

0.2 | ||||

0.6 |

It is also interesting to study the consequences of the DM-DE coupling in the determination of the parameters characterizing the DE equation of state. In Figure 6 we plot the confidence level contour plots in the plane obtained by marginalizing the likelihood function over using a gaussian prior , as obtained from the WMAP3y data. The effect of the coupling is very small when the prior on the DM density is taken into account.

## Iv Conclusions

There is a vast amount of work studying the possibility of having an interaction between the dark energy and the dark matter components of our universe. In this paper we analyzed a simple model for the interaction between these two fluids, in which the mass of the dark matter particles increases at a constant rate. We have shown that introducing a coupling between the dark energy component of the universe with dark matter particles has the effect of increasing the best fit values of the DM density today obtained from current SNIa data.

We performed a comparison with CMB data for the case of constant equation of state, using the shift parameter. We found that the introduction of the coupling results in a poorer compatibility between CMB and SNIa data. Our results showed that assuming a constant and requiring , a coupling seems to be ruled out. This must be checked by model-independent measurements of from large scale structure and also with a more careful analysis of the CMB data including more observables in addition to the shift parameter. We also found that introducing the coupling does not change significantly the determination of the DE equation of state when a prior on is adopted.

We worked with a simple parameterization of the dark energy equation of state and for the DM-DE coupling but we believe that our results are fairly general for the type of interaction that we introduced. It would be interesting to extend our analysis to more general parameterizations for both the interaction and the equation of state available in the literature. However, it should be stressed that SNIa data is not currently very sensitive to a time-varying .

## Acknowledgments

This work was supported by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP), grant 01/11392-0, and by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq).

## References

- (1) For reviews on dark energy, see e.g. P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559 (2003) [astro-ph/0207347]; T. Padmanabhan, Phys. Rept. 380, 235 (2003) [hep-th/0212290]; E. J. Copeland, M. Sami and S. Tsujikawa [hep-th/0603057].
- (2) A. G. Riess et al., Astrophys. J. 607, 665 (2004) [astro-ph/0402512].
- (3) P. Astier et al., Astron. Astrophys. 447, 31 (2006) [astro-ph/0510447].
- (4) D. N. Spergel et al., Astrophys. J. Supp. 148, 175 (2003) [astro-ph/0603449].
- (5) See e.g. K. Abazajian et al., Astron. J. 126, 2081 (2003) [astro-ph/0305492].
- (6) S. L. Bridle, O. Lahav, J. P. Ostriker and P. J. Steinhardt, Science 299, 1532 (2003) [astro-ph/0303180]; M. Tegmark et al., Phys. Rev. D69, 103501 (2004) [astro-ph/0310723].
- (7) R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80, 1582 (1998) [astro-ph/9708069].
- (8) B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406 (1988); P. J. E. Peebles and B. Ratra, Astrophys. J. Lett. 325, L17 (1988).
- (9) C. Wetterich, Nucl. Phys. B302, 668 (1988).
- (10) H. K. Jassal, J. S. Bagla and T. Padmanabhan, Phys. Rev. D 72, 103503 (2005) [astro-ph/0506748].
- (11) H. K. Jassal, J. S. Bagla and T. Padmanabhan, astro-ph/0601389.
- (12) G. W. Anderson, S. M. Carrol, astro-ph/9711288.
- (13) J. A. Casas, J. García-Bellido, M. Quirós, Class. Quantum Grav. 9, 1371 (1992) [hep-ph/9204213].
- (14) G. R. Farrar, P. J. E. Peebles, Astrophys. J. 604, 1 (2004) [astro-ph/0307316].
- (15) L. Amendola, Phys. Rev. D 62, 043511 (2000) [astro-ph/9908023] .
- (16) L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 64, 043509 (2001) [astro-ph/0011243].
- (17) L. Amendola, Mon. Not. Roy. Astron. Soc. 342, 221 (2003) [astro-ph/0209494].
- (18) M. Pietroni, Phys. Rev. D 67, 103523 (2003) [hep-ph/0203085].
- (19) D. Comelli, M. Pietroni and A. Riotto, Phys. Lett. B 571, 115 (2003) [hep-ph/0302080].
- (20) U. França and R. Rosenfeld, Phys. Rev. D 69, 063517 (2004) [astro-ph/0308149].
- (21) Y. Wang and P. Mukherjee, Astrophys. J. 650, 1 (2006) [astro-ph/0604051].
- (22) S. Baessler et al., Phys. Rev. Lett. 83, 3385 (1999) [astro-ph/0308149].
- (23) S. Das, P. S. Corasaniti and J. Khoury, Phys. Rev. D 73, 083509 (2006) [astro-ph/0510628]; G. Huey and B. D. Wandelt, Phys. Rev. D 74, 023519 (2006) [astro-ph/0407196].
- (24) E. Majerotto, D. Sapone and L. Amendola, astro-ph/0410543.
- (25) L. Amendola, M. Gasperini and F. Piazza, JCAP 0409, 14 (2004) [astro-ph/0407573].
- (26) R. Rosenfeld, Phys. Lett. B624, 158 (2005) [astro-ph/0504121].
- (27) R.-G. Cai and A. Wang, JCAP 0503, 002 (2005) [hep-th/0411025].
- (28) L. Amendola, D. Polarski, S. Tsujikawa, astro-ph/0603703.
- (29) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D72, 123519 (2005) [astro-ph/0511040].
- (30) V. Barger, E. Guarnaccia and D. Marfatia, Phys.Lett. B635, 61 (2006) [hep-ph/0512320].
- (31) G. B. Calvo and A. L. Maroto, astro-ph/0604409.
- (32) S. Nesseris and L. Perivolaropoulos, Phys. Rev. D70, 043531-1 (2004) [astro-ph/0401556].
- (33) L. Perivolaropoulos, Phys. Rev. D71, 063503 (2005) [astro-ph/0412308].
- (34) R. Lazkoz, S. Nesseris and L. Perivolaropoulos, JCAP 0511, 010 (2005) [astro-ph/0503230].
- (35) H. Feldman et al., Astrophys. J. 596, L131 (2003) [astro-ph/0305078].
- (36) R. Maartens and E. Majerotto, Phys. Rev. D74, 023004 (2006) [astro-ph/0603353]; R. Lazkoz, R. Maartens and E. Majerotto, Phys. Rev. D74, 083510 (2006) [astro-ph/0605701].