A&A 507, 4752 (2009)
Could dark matter interactions be an alternative to dark energy?
S. Basilakos^{1}  M. Plionis^{2}
1  Research Center for Astronomy, Academy of Athens, 11527 Athens,
Greece
2  Institute of Astronomy & Astrophysics, National Observatory
of Athens, Thessio 11810, Athens, Greece &
Instituto Nacional de Astrofísica, Óptica y Electrónica, 72000 Puebla,
Mexico
Received 6 June 2009 / Accepted 3 August 2009
Abstract
We study the global dynamics of the universe within the framework of
the interacting dark matter (IDM) scenario.
Assuming that the dark matter obeys the collisional Boltzmann
equation, we can derive analytical solutions of the global density
evolution, that can accommodate an accelerated
expansion, equivalent to either the quintessence
or the standard
models, with the present time located after the inflection point.
This is possible if there is a
disequilibrium between the DM particle creation and
annihilation processes
with the former process dominating, which creates an effective source
term with negative pressure. Comparing the predicted Hubble expansion
of one of the IDM models (the simplest) with observational
data, we find that the effective
annihilation term is quite small, as suggested by various experiments.
Key words: cosmology: theory  methods: analytical
1 Introduction
The analysis of high quality cosmological data (e.g. supernovae type Ia, CMB, galaxy clustering) have suggested that we live in a flat, accelerating universe, that contains cold dark matter to explain clustering and an extra component with negative pressure, the vacuum energy (or more generally the dark energy), to explain the observed accelerated cosmic expansion (Spergel et al. 2007; Davis et al. 2007; Kowalski et al. 2008; Komatsu et al. 2009, and references therein). Because of the absence of a physically wellmotivated fundamental theory, there have been many theoretical speculations about the nature of the exotic dark energy (DE) including a cosmological constant, or either scalar or vector fields (see Weinberg 1989; Wetterich 1995; Caldwell et al. 1998; Brax & Martin 1999; Peebles & Ratra 2003; Perivolaropoulos 2003; Brookfield et al. 2006; Boehmer & Harko 2007, and references therein).Most papers in this type of study are based on the assumption that DE evolves independently of the dark matter (DM). The unknown nature of both DM and DE implies that we cannot preclude the possibility to find interactions in the dark sector. This is very important because interactions between the DM and quintessence could provide possible solutions to the cosmological coincidence problem (Grande et al. 2009). Several papers have been published in this area (e.g., Amendola et al. 2003; Cai & Wang 2005; Binder & Kremer 2006; Campo et al. 2006; Wang et al. 2006; Das et al. 2006; Olivares et al. 2008; He & Wang 2008, and references therein) proposing that the DE and DM could be coupled, assuming also that there is only one type of noninteracting DM.
However, there are other possibilities. It is plausible, for example, that the dark matter is selfinteracting (IDM) (Spergel & Steinhardt 2000). This possibility was proposed in order to solve discrepancies between theoretical predictions and astrophysical observations, including less cuspy halo profiles, predicted by the IDM model, allowing for the observed gammaray and microwave emission from the center of our galaxy (Flores & Primack 1994; Moore et al. 1999; Hooper et al. 2007; Regis & Ullio 2008, and references therein) and the discrepancy between the predicted optical depth, , inferred from the GunnPeterson test in the spectra of highz QSOs and the WMAPbased value (e.g., Mapelli et al. 2006; Belikov & Hooper 2009; Cirelli et al. 2009, and references therein). It has also been shown that some dark matter interactions could provide an accelerated expansion phase of the universe (Zimdahl et al. 2001; Balakin et al. 2003; Lima et al. 2008). In addition, DM could potentially contain more than one particle species, for example a mixture of cold, warm, or hot dark matter (Farrar & Peebles 2004; Gubser & Peebles 2004), with or without intercomponent interactions.
In this work, we are not concerned with the viability of the different possibilities, nor with the properties of interacting DM models. The single aim of this work is to investigate whether there are repercussions of DM selfinteractions on the global dynamics of the universe and specifically whether these models can yield an accelerated phase of the cosmic expansion, without the need for dark energy. We note that we do not ``design'' the fluid interactions to produce the desired accelerated cosmic evolution, as in some previous works (e.g., Balakin et al. 2003), but investigate the circumstances under which the analytical solution space of the collisional Boltzmann equation, in the expanding universe, allows for a late accelerated phase of the universe.
2 Collisional Boltzmann equation in an expanding universe
It is well established that the global dynamics of a homogeneous,
isotropic, and flat universe is given by the Friedmann
equation
(1) 
where is the total energydensity of the cosmic fluid, containing (in the matterdominated epoch) dark matter, baryons, and any type of exotic energy. Differentiating Eq. (1), we derive the second Friedmann equation, given by:
As we mentioned in the introduction, the dark matter is usually considered to contain only one type of particle that is stable and neutral. In this work, we investigate, using the Boltzmann formulation, the cosmological potential of a scenario in which the dominant ``cosmic'' fluid does not contain dark energy, is not perfect, and at the same time is not in equilibrium^{}. Although our approach is phenomenological, we briefly review a variety of physically motivated dark matter selfinteraction models that have appeared in the literature.
The time evolution of the total density of the cosmic fluid is
described by the collisional Boltzmann equation
where is the Hubble function, is the rate of creation of DM particle pairs, and 0) is given by:
(4) 
where is the crosssection for annihilation, u is the mean particle velocity, and M_{x} is the mass of the DM particle.
We note that, in the context of a spatially flat FLRW
cosmology, for an effective pressure term of:
the collisional Boltzmann equation reduces to the usual fluid equation: . Inserting Eqs. (3) and (5) into Eq. (2), we obtain
Obviously, a negative pressure (whatever its cause) can effectively act as a repulsive force possibly providing a cosmic acceleration.
We investigate the effects of DM selfinteractions on the global dynamics of the universe and under which circumstances they can produce a negative pressure and thus provide an alternative to conventional dark energy. It is well known that negative pressure implies tension rather than compression, which is an impossibility for ideal gases but not for some physical systems that depart from thermodynamic equilibrium (Landau & Lifshitz 1985).
The particle annihilation regime was described by Weinberg (2008), using the collisional Boltzmann formulation, in which the physical properties of the DM interactions are related to massive particles (which are still present) that, if they carry a conserved additive or multiplicative quantum number, would imply that some particles must remain after all the antiparticles have been annihilated (Weinberg calls them Lparticles). The Lparticles may annihilate to form other particles, which during the period of annihilation they can be assumed to be in thermal and chemical equilibrium (see Weinberg 2008). This DM selfinteracting model can affect the global dynamics of the universe (see our Case 2 below).
The corresponding effects on the global dynamics of the particle creation regime, which provides an effective negative pressure, has also been investigated by a number of authors (e.g., Prigogine et al. 1989; Lima et al. 2008, and references therein).
In the framework of a Boltzmann formalism, a negative pressure could in general be the outcome of dark matter selfinteractions, as suggested in Zimdahl et al. (2001) and Balakin et al. (2003), if an ``antifrictional'' force is selfconsistently exerted on the particles of the cosmic fluid. This possible alternative to dark energy has the caveat of its unknown exact nature, which is also however the case for all dark energy models. Other sources of negative pressure have been proposed, including gravitational matter ``creation'' processes (Zeldovich 1970), modeled by nonequilibrium thermodynamics (Prigogine et al. 1989) or even the Cfield of Hoyle & Narlikar (1966). The effects of the former proposal (gravitational matter creation) on the global dynamics of the universe have been investigated, based on the assumption that the particles created are noninteracting (Lima et al. 2008). The merit of all these alternative models is that they unify the dark sector (dark energy and dark matter), since a single dark component (the dark matter) needs to be introduced into the cosmic fluid.
In a unified manner we present, the outcome for the global dynamics of the universe of different type of dark matter selfinteractions, using the Boltzmann formulation in the matterdominated era.
3 The Cosmic density evolution for different DM interactions
We proceed to analytically solve Eq. (3). We change variables from t to and thus Eq. (3) can be writtenwhere
Within this framework, based on Eqs. (5), (7) and (8), we can distinguish four possible DM selfinteracting cases:
Case 1: P=0: If the DM is collisionless or the collisional annihilation and pair creation processes are in equilibrium (i.e., ), the corresponding solution of the above differential equation is (where is the scale factor of the universe), and thus we obtain, as we should, the dynamics of the Einstein deSitter model, with H(t)=2/3t.
Case 2:
:
If we assume that in the matterdominated era the particle creation
term is negligible,
[
], (the case discussed in
Weinberg 2008),
then the corresponding pressure becomes positive. It is clear that
Eq. (7)
becomes a Bernoulli equation, the general solution of which provides
the evolution of the
global energydensity, which is that corresponding to the
IDM ansanz:
Prior to the present epoch ( ), we find that , while at late enough times ( ) the above integral converges, which implies that the corresponding global density tends to evolve again as the usual dark matter (see Weinberg 2008), with
(10) 
where t_{0} is the present age of the universe. The latter analysis, relevant to the usual weakly interacting massive particle case  Weinberg (2008), leads to the conclusion that the annihilation term has no effect resembling that of dark energy, but does affect the evolution of the selfinteracting DM component, with the integral in the denominator rapidly converging to a constant (which depends on the annihilation crosssection).
Case 3: : For the case of a nonperfect DM fluid (i.e., having up to the present time, a disequilibrium between the annihilation and particle pair creation processes), we can have either a positive or a negative effective pressure term. Although the latter situation may or may not appear plausible, even the remote such possibility, i.e., the case in which the DM particle creation term is larger than the annihilation term ( ), is of particular interest because of its effect on the global dynamics of the universe (see for example Zimdahl et al. 2001; Balakin et al. 2003).
It is interesting to note that this case can be viewed as a generalization of the gravitational matter creation model of Prigogine et al. (1989) (see also Lima et al. 2008, and references therein) in which annihilation processes are also included, although the mattercreation component dominates over annihilations. In this scenario, as in any interacting darkmatter model with a leftover residual radiation, a possible contribution from the radiation products to the global dynamics is negligible, as we show in Appendix A.
For
and ,
it is not an easy task
in general to solve analytically Eq. (7), because it
is a nonlinear differential equation (Riccati type).
However, Eq. (7)
could be fully solvable if (and only if) a particular solution is
known. We indeed
find that for some special functional forms of the
interactive term, such as ,
we can derive analytical solutions.
We identified two functional forms for which
we can solve the previous differential equation analytically, only one
of these two is of interest because it provides a
a^{3}
dependence of the scale factor (see Appendix B), which is:
Although, the above functional form was not motivated by physical theory, but rather phenomenologically because it provides analytical solutions to the Boltzmann equation, its exact form can be justified a posteriori within the framework of IDM (see Appendix C).
The general solution of Eq. (7) for the total
energy density, using Eq. (11), is:
where the kernel function has the form
We note that has units of Gyr^{1}, while m, , and are the corresponding constants of the problem. Obviously, Eq. (12) can be rewritten as
where is the density corresponding to the residual matter creation that results from a possible disequilibrium between the particle creation and annihilation processes, while can be viewed as the energy density of the selfinteracting dark matter particles that are dominated by the annihilation processes. This can easily be understood if we define the constant to equal to zero, implying that the creation term is negligible and reducing the current solution (Eq. (14)) to that of Eq. (9). We note that close to the present epoch as well as at late enough times ( ), as also in Case 2, the evolves in a similar way to the usual dark matter (see also Weinberg 2008). Finally, if both and tend to zero, the above cosmological model reduces to the usual EinsteindeSitter model (Case 1).
We note that, since ,
the constant
obeys the restriction
Evaluating now Eq. (12) at the present time (, ), we obtain the presenttime total cosmic density, which is: , with and .
Case 4:
:
In this scenario, we assume that the annihilation term is negligible [
and =0]
and the particle creation
term dominates. This situation is mathematically equivalent to the
gravitational DM particle creation process within the context
of nonequilibrium thermodynamics Prigogine et al. (1989),
the important cosmological consequence of which were studied by Lima
et al. (2008,
and references therein).
Using our nomenclature and ,
Eq. (7)
becomes a first order linear differential equation, a general solution
of which is:
The negative pressure can yield a late accelerated phase of the cosmic expansion (as in Lima et al. 2008), without the need for the required (in ``classical'' cosmological models) dark energy.
4 Case 3: P = /3H
In this section, we investigate the conditions under which Eqs. (12) and (16) could provide accelerating solutions, similar to the usual dark energy case.4.1 Conditions to have an inflection point and galaxy formation
To have an inflection point at , we must have (see Eq. (6)). The latter equality implies that the expression should contain a real root in the interval: . Therefore, with the aid of Eq. (12), (5) and (11), we derive the following condition:from which we obtain that m>2 (where , , and ). Evidently, if we parametrize the constant m according to , we obtain the condition , which implies that the current cosmological model can be viewed as a viable quintessence darkenergy lookalike, as far as the global dynamics is concerned. We remind the reader that the same restriction holds for the conventional dark energy model in which ( ; for more details see Appendix D).
Furthermore, to ensure the growth of spatial density fluctuations, the effective DM should be capable of clustering and providing the formation of galaxies, while the effective dark energy term should be close to being homogeneous. In our case, the effective term that emulates dark energy is homogeneous in the same sense as in the classical quintessence, while the term slightly modifies the pure DM evolution. In any case, the interacting DM term after the inflection point tends to an evolution a^{3}. During the galaxy formation epoch at highz's, we expect (due to the functional form of the DM term) that the slope of the interacting DM term is not far from that of the classical DM evolution (we will explore these issues further in a forthcoming paper).
4.2 Relation to the Standard Cosmology
As an example, we show that for m=0 (or ), the global dynamics, provided by Eq. (12), is equivalent to that of the traditional cosmology. To this end, we use and the basic kernel (Eq. (13)) becomeswhere t_{0} is the present age of the universe. In addition, the integral in Eq. (12) (see also Eq. (15)) now takes the form and . We note that at the present time we have G(1)=0. Therefore, using the above formula, the global density evolution (Eq. (12)) can be written
As expected, at early enough times ( ) the overall density scales according to , while close to the present epoch the density evolves according to
which is approximately equivalent to the corresponding evolution in the cosmology in which the term resembles the constantvacuum term ( ) and the term resembles the density of matter (). We note that the effective pressure term (Eq. (5)), for , becomes , which implies that: . Therefore, this case relates to the traditional cosmology, since corresponds to (see Eq. (20)). We now investigate in detail the dynamics of the m=0 model.
Figure 1: Left panel: the solution space provided by fitting our model to the earlytype galaxy Hubble relation of Simon et al. (2005). Right panel: the corresponding solution space. 

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From Eq. (19),
using the usual unitless like
parametrization, we derive after some algebra that
where and , which in the usual cosmology relates to and , respectively.
We can now attempt to compare
the Hubble function of Eq. (21) to that
corresponding to the usual
model.
To this end, we use a
minimization between the different
models (our IDM Eq. (21) or the
traditional CDM model)
and the Hubble relation derived directly from earlytype galaxies at
high redshifts (Simon et al. 2005). For
the case of our IDM model, we
simultaneously fit the two free
parameters of the model, i.e.,
and for a flat
background (
)
with
H_{0}=72 km s^{1} Mpc^{1}
and Gyr
which is roughly the age of the universe of
the corresponding
cosmology.
This procedure yields the bestfit model parameters
and (with
a stringent upper limit 3,
but unconstrained towards lower values)
where
(see left panel of Fig. 1).
Using Eq. (4) we can now relate the range of values of to the mass
of the DM particle, from which we obtain that
(22) 
(see also right panel of Fig. 1) and since is unbound at small values, it is consistent with currently accepted lower bounds of (e.g., Cirelli et al. 2009, and references therein). The corresponding Hubble relation (Fig. 2), provided by the bestfit model free parameters, is indistinguishable from that of the traditional CDM model, because of the very small value of . For completeness, we also show, as the dashed line, the IDM solution provided by eV ( ), which is the stringent lower bound found by our analysis. In this case, the predicted Hubble expansion deviates significantly from the traditional model at small values indicating that it would probably create significant alterations to the standard BBN (e.g. Iocco et al. 2009, and references therein).
Figure 2: Comparison of the Hubble function provided by the traditional CDM model, which coincides with our m=0 model (for the bestfit model of the two free parameters  see text). The dashed line corresponds to our m=0 IDM model for the highest bound, provided by our fitting procedure (10^{3}). The dotdashed line corresponds to our IDM model ( Case 4) for the bestfit model parameters ( and ). Finally, the points correspond to the observational data (Simon et al. 2005). 

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Although the present analysis does not provide any important constraints on M_{x}(within our model), we plan on the future to use a large amount of cosmologically relevant data to attempt to place stronger M_{x} constraints, also in the general case (see Eq. (12)).
5 Case 4: P = /3H
We now prove that for
(negligible annihilation),
the global dynamics resembles that of the traditional quintessence
cosmology (
). Using again the
phenomenologically selected form of ,
provided by Eq. (11),
we obtain
.
It is then straightforward
to obtain the density evolution from Eq. (16), as:
where . The conditions in which the current model acts as a quintessence cosmology, are given by , , and , which implies that to have an inflection point, the following should be satisfied: or m>2 (see Appendix D). We note, that the Hubble flow is now given by
(24) 
where and . Finally, by minimizing the corresponding , we find that the bestfit model values are and ( ) with . The corresponding Hubble flow curve is shown in Fig. 2 as the dotdashed line. We note that this solution is mathematically equivalent to that of the gravitational matter creation model of Lima et al. (2008).
6 Conclusions
We have investigated the evolution of the global density of the universe in the framework of an interacting DM scenario by solving analytically the collisional Boltzmann equation in an expanding universe. A disequilibrium between the DM particle creation and annihilation processes, regardless of its cause and in which the particle creation term dominates, can create an effective source term with negative pressure, which acting like dark energy, provides an accelerated expansion phase of the universe. There are also solutions for which the present time is located after the inflection point. Finally, comparing the observed Hubble function of a few highredshift elliptical galaxies with that predicted by our simplest IDM model (m=0), we find that the effective annihilation term is quite small. In a forthcoming paper, we propose to use a multitude of cosmologically relevant observations to jointly fit the predicted, by our generic IDM model, Hubble relation and thus possibly provide more stringent constraints on the free parameters of the models. We also plan to derive the perturbation growth factor to study structure formation within the IDM model. AcknowledgementsWe thank P.J.E. Peebles for critically reading our paper and for useful comments. Also, we would like to thank the anonymous referee for his/her useful comments and suggestions.
Appendix A: The effect of the decay products
Here we attempt to investigate in the matterdominated era, whether the possible radiation products related to dark matter interactions can affect the global dynamics. A general coupling can be viewed by the continuity equations of interacting dark matter and residual radiation ,where Q is the rate of energy density transfer. If Q<0, then the IDM fluid transfers to residual radiation. As an example, we can use a generic model with , where . Thus, Eq. (26) has an exact solution
(27) 
where t_{0} is the present age of the universe. This shows that the contribution of the residual radiation to the global dynamics was negligible in the past, since there is not only the usual a^{4} dependence of the background radiation but also a further exponential drop, and thus . We therefore conclude that we can approximate the total energydensity with that of the interacting darkmatter density ( ). Note, that can be viewed as the mean lifetime of the residual radiation particles.
Appendix B: Solutions of the Riccati equation
With the aid of differential equation theory we present solutions that are relevant to our Eq. (7). In general, a Riccati differential equation is given byy'=f(x)y^{2}+g(x)y+R(x)  (28) 
and it is fully solvable only when a particular solution is known. Below, we present two cases in which analytical solutions are possible:
 Case 1: for the case where
(29)
the particular solution is x^{m} and thus the corresponding general solution can be written as(30)
where(31)
and are the integration constants. Using now Eq. (8), we obtain .  Case 2: for the case where
(32)
the particular solution is h(x) (in our case we have ). The general solution now becomes(33)
where(34)
In this framework, using Eq. (8) we finally obtain .
Appendix C: Justification of the functional form of
We assume that we have a nonperfect cosmic fluid in a disequilibrium phase with energy density then from the collisional Boltzmann equation, we have that(35) 
Furthermore, we assume that for a convenient period of time, the cosmic fluid, in an expanding Universe, is slowly diluted according to (). From a mathematical point of view, the latter assumption simply means that a solution of the form is a particular solution of the Boltzmann equation. Therefore, we have finally that:
Appendix D: Correspondence between our model and conventional dark energy models
We remind the reader that for homogeneous and isotropic flat cosmologies ( ), controlled by nonrelativistic DM and a DE with a constant equation of state parameter (w), the density evolution of the cosmic fluid can be written aswhere and are the presentday DM and DE densities, respectively.
The necessary criteria for cosmic acceleration and an inflection point in our past ( t_{i}<t_{0}), are (a) P<0 and (b) , which leads to the conditions
 Dark Energy models: , P_{m}=0 with w<1/3.
 IDM models: and m>2 (or ).
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Footnotes
 ... equilibrium^{}
 Initially, the total energy density is . We consider that the selfinteracting dark matter does not interact significantly with the background radiation, and thus in the matterdominated epoch, radiation is irrelevant to the global dynamics (because of the wellknown dependence: ). Therefore, taking the above considerations into account and assuming that there are no residual radiation products of the DM interactions (otherwise see Appendix A), we conclude that in the matterdominated era the total cosmic darkmatter density reduces to that of the IDM density ( ), which obeys the collisional Boltzmann equation (see Eq. (3)).
All Figures
Figure 1: Left panel: the solution space provided by fitting our model to the earlytype galaxy Hubble relation of Simon et al. (2005). Right panel: the corresponding solution space. 

Open with DEXTER  
In the text 
Figure 2: Comparison of the Hubble function provided by the traditional CDM model, which coincides with our m=0 model (for the bestfit model of the two free parameters  see text). The dashed line corresponds to our m=0 IDM model for the highest bound, provided by our fitting procedure (10^{3}). The dotdashed line corresponds to our IDM model ( Case 4) for the bestfit model parameters ( and ). Finally, the points correspond to the observational data (Simon et al. 2005). 

Open with DEXTER  
In the text 
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