Models of coupled dark matter to dark energy
We present three distinct types of models of dark energy in the form of a scalar field which is explicitly coupled to dark matter. Our construction draws from the pull-back formalism for fluids and generalises the fluid action to involve couplings to the scalar field. We investigate the cosmology of each class of model both at the background and linearly perturbed level. We choose a potential for the scalar field and a specific coupling function for each class of models and we compute the Cosmic Microwave Background and matter power spectra.
During the last decade, observational cosmology has entered an era of unprecedented precision. Measurements of the cosmic microwave background (CMB) (Komatsu et al. (2011),Ade et al. (2013)), the Hubble constant () Riess et al. (2009), the luminosity and distance at high redshift with supernovae Ia Kowalski et al. (2008), and Baryon Acoustic Oscillations (BAO) surveys Lampeitl et al. (2009), suggest that our Universe is currently undergoing a phase of accelerated expansion. The standard cosmological model, consisting of dark energy in the form of a cosmological constant () together with cold dark matter (CDM) fits all the available datasets extremely well, but it also suffers from two fundamental problems, namely the fine-tuning and coincidence problems.
However, there is plenty of margin left for alternative explanations for the nature of the dark sector, and many different approaches have been adopted (see Copeland et al. (2006) for a review). In particular, the problems associated with the cosmological constant have led to a plethora of alternative theories, for example quintessence (Wetterich (1988); Ratra and Peebles (1988); Wetterich (1995)), in which the DE component comes in the form of a scalar field evolving in time. An alternative approach is the modified gravity scenario, which supports a completely different point of view, suggesting that our understanding of gravity through General Relativity might not be applicable on the largest cosmological scales, i.e. that dark energy is due to a modification of gravity (see Clifton et al. (2012) for a review).
Given that the precise nature of the two dark sectors is at present unknown, it may be that dark matter and dark energy have non-zero couplings to each other. This possibility, which can be used to alleviate the coincidence problem, has been investigated in a number of cases in the past. Traditionally dark energy has been modelled as a scalar field . If this field is uncoupled then in order to be a viable model of dark energy (the primary property of which is to provide for cosmic acceleration) this field must today have negative pressure. This is equivalent to saying that it’s energy density is potential dominated. If however, this field is coupled to dark matter, then this may not be necessarily true. Indeed the coupling to dark matter may also induce an effective negative pressure so that cosmic acceleration occurs even if the uncoupled field does not by itself have this property. The Lagrangian of such a coupled system can be written as
where is the mass of the matter fields . If we define the coupling current as
where is the matter energy density, then the Bianchi identities can be written as
If is coupled to CDM only, we also have
so that the total stress-energy momentum tensor of the dark components is conserved. Models of this type have been thoroughly investigated. Amendola Amendola (2000) studied such a coupled quintessence (CQ) model assuming an exponential potential for and a coupling of to the matter sector of the form , and investigated its cosmological consequences. He showed that the system could approach scaling solutions with with an associated accelerated expansion and he constrained the coupling constant () using CMB data.
As the form of the coupling is chosen phenomenologically, there are a rich selection of papers investigating various interacting dark energy models and their cosmological implications, for example the effects of coupling on the Cosmic Microwave Background (CMB) and matter power spectra, on Supernovae, the growth of structure, non-linear perturbations and N-body simulations (see, e.g. Billyard and Coley (2000); Zimdahl and Pavon (2001); Farrar and Peebles (2004); Matarrese et al. (2003); Amendola (2004); Maccio et al. (2004); Amendola et al. (2004, 2007); Guo et al. (2007); Koivisto (2005); Lee et al. (2006); Wang et al. (2007); Mainini and Bonometto (2007); Pettorino and Baccigalupi (2008); Xia (2009); Chimento et al. (2003); Olivares et al. (2005); Sadjadi and Alimohammadi (2006); Brookfield et al. (2008); Boehmer et al. (2008); Caldera-Cabral et al. (2009); He and Wang (2008); Quartin et al. (2008); Valiviita et al. (2010); Pereira and Jesus (2009); Bean et al. (2008a); Gavela et al. (2009); Tarrant et al. (2012); Baldi et al. (2010); Marulli et al. (2012); Salvatelli and Marchini (2013)). A different class of coupled dark energy models are those for which the coupling is to neutrinos rather than CDM (Fardon et al. (2004); Amendola et al. (2008); Brookfield et al. (2006); Mota et al. (2008); Takahashi and Tanimoto (2006a)), more precisely by making the neutrino mass dependent. Several studies have also addressed the appearance of instabilities in coupled models (e.g. Valiviita et al. (2008); Jackson et al. (2009); Afshordi et al. (2005); Bean et al. (2008b); Clemson et al. (2012); Takahashi and Tanimoto (2006b)).
In order to make further progress at the phenomenological level, it is desirable to construct general models of coupled dark energy. A natural question that arises is whether the models considered so far saturate the possibilities. In other words, what is the most general phenomenological model one can construct? As we will show in this article, the models considered so far are only one small subset in the space of possibilities. The common feature of those models is a coupling that involves the energy density (and sometimes the pressure) of the dark fluid (whether CDM or neutrinos). However, it is also possible to couple to the velocity of the fluid as we shall show below (note that the consequences of a velocity coupling associated with dark matter scattering elastically with dark energy, leading to pure momentum transfer, were investigated in Simpson (2010)).
Our investigation highlights three classes of coupled models. The first two involve both energy and momentum transfer between the two components of the dark sector (albeit with distinctively different coupling mechanisms), while the third is identified as a pure momentum transfer model.
In Section II we give a short overview of the pull-back formalism for a fluid as is used in GR in order to construct an action from which the dynamics of the fluid are derived. We use this formalism to construct three distinct general classes (Types) of coupled dark energy models in Section III, and we derive the field equations of motion. In Section IV we study the background cosmology for the three Types of models. In Section V we derive the perturbed cosmological equations and investigate the observational implications on the CMB and matter power spectra for three specific cases (one for each Type). We conclude in Section VI.
Ii The pull-back formalism for fluids
The coupled models of dark energy investigated in the past introduce the phenomenological coupling by modifying the field equations. However, it may be desirable to introduce the coupling at the level of the action. The reason is that when building theories, using an action principle is in most cases more intuitive. Although we are interested in phenomenological models of fluids for the case of CDM, rather than the actual fundamental field that may be CDM, using an action principle can provide better insight as to how such couplings may emerge. We shall return to this further below.
We need a description of fluids at the level of the action. Fortunately such a description has already been formulated and is called the fluid pull-back formalism. The fluid pull-back (FPB) formalism is a way to construct an action from which the dynamics of the fluid are derived. It was formulated independently by Kijowski, Smólski and Górnicka Kijowski et al. (1990), by Brown Brown (1993) and by Comer and Langlois Comer and Langlois (1993, 1994), building on earlier work by Taub Taub (1954) and Carter Carter (1973). The reader is referred to the review of Andersson and Comer Andersson and Comer (2005) for further study. Before embarking on describing the three coupled models of dark energy, we first give a short overview of the pull-back formalism for a fluid as is used in GR.
One of the assumptions regarding perfect fluids is that collisions do not occur. Or put differently, the time taken to traverse a distance equal to the mean free path of the particles comprising the fluid is much larger than the time for which our description is supposed to hold. In the case of cosmology, the mean free path of such fluids corresponds to times longer than many Hubble times. The absence of collisions means that the trajectories of particles, called the worldlines, do not intersect. Furthermore, if the distribution of particles is continuous (as it should be for the fluid description to hold) then the space of worldlines forms a three-dimensional manifold . We may then introduce coordinates on this manifold which we denote as with . Each point in denotes a unique worldline associated with a distinct particle, meaning that the coordinates may be thought as particle labels. Since the worldlines cannot intersect then there can be one and only one particle associated with each point in .
The space is time-less, i.e. each particle has its own specific label for all times. If this were not the case and it was possible for to change as time flows then it would also be possible for the worldlines to intersect. This means that the fluid description using the space of worldlines is the relativistic analogue to the Lagrangian coordinate system in Newtonian mechanics, i.e. the system of coordinates which follows the particles as they move through space.
ii.1 The decomposition
In order to describe the fluid using the manifold we need to embed it in spacetime . Since is three-dimensional it is in one-to-one correspondence with a three-dimensional spacelike surface at some particular time . However, we have an infinite number of surfaces , one for each .
Consider a surface at time and a timelike vector field which can be chosen to be normal to . By using the integral curves of parameterized by , we can then foliate by associating a surface normal to for each . What remains is to associate each to . This is done with the help of a map , where the index is to take into account the fact that the two spaces involved are three-dimensional. If (with ) are coordinates on , then under the map we have that . In other words, we have an infinite number of maps labelled by , which map the points of each to . For each the mapping is one-to-one. Although the points in are fixed, motion is perceived through the embedding of in . In other words, the maps map the same point at coordinates to a point in and by stacking together all mappings for all in a continuous way we have the appearance of motion of the particles in spacetime. This is shown in Fig. 1. From the perspective, the maps form a set of three scalar fields.
Now even at fixed the choice of is not unique but depends on the choice of a timelike vector field to which is normal to. There is one particular choice for which is the -velocity of the fluid particles, , and without loss of generality we shall make this choice hence forth.
The fluid velocity obeys and we can use it to define a metric on the hypersurface as
The tensor is also a projector, i.e. it obeys . Furthermore we have that and . Note that indices are raised and lowered using the metric .
Any tensor can be projected onto parts parallel to and parts orthogonal to it via . For instance, consider the completely anti-symmetric tensor defined by where is the metric determinant and is the Levi-Civita anti-symmetric tensor density so that in all coordinate systems. We find and repeating the procedure on the 2nd term (and so on) we finally get
The last relation can be inverted to give
The tensor is completely antisymmetric and is related to the -dimensional Levi-Civita tensor density as where is the -metric determinant and in all coordinate systems.
Nothing discussed so far permits us to uniquely construct the tensors , and . Their construction involves knowledge about the fluid and as we shall see further below, the fundamental variable from which these tensors are derived is the dual fluid number density tensor on , . It is therefore not allowed to use the relations (7) and (8) to obtain the variations and .
ii.2 Pullbacks and pushforwards for metrics, volume forms and connections
So far we have considered tensors on the spacetime manifold . However, we also have tensors defined on the fluid manifold . For instance, the fluid manifold may also have a metric , or a completely anti-symmetric tensor playing the role of a volume form, where is the determinant of .
Tensors on are related to tensors on and by extension along the integral curves of to the whole of . This is achieved through the maps via the pull-back and push-forward operations (hence the name for this formalism). More specifically we can pull-back covariant vector fields (forms) from to and push-forward contravariant vector fields from to with the help of
This is remiscent of coordinate transformations, in fact if and had the same dimensionality and the map was invertible then it reduces exactly to a coordinate transformation. With the help of (9) we pull-back a form from on to a form on as
and push-forward a vector field from to a vector field on as
We can apply the formalism above to and and relate them to and as
respectively. Clearly, a geometry on induces a geometry on .
Let’s now make things more interesting. We consider connections. Given we define the connection , s.t. , and associated Christoffel symbol . Likewise on we define a connection with associated Christoffel symbol . The pull-back maps (9) can then be used to relate the action of the two connections on forms on each corresponding space. In particular for scalars we have
and on a form
The Christoffel symbols are also related. From the above equation a straightforward calculation gives
The above relation is a generalization of the usual transformation law of Christoffel symbols under coordinate transformations, only here it is valid even if the dimensionality of the two spaces involved is different.
ii.3 The relativistic fluid
ii.3.1 Kinematical setup
We can now proceed to fluids. Consider the fluid space whose points denote particles for all time (worldlines). The density of points of the fluid space should then be related to the particle number density in some sense. Indeed, we can define a tensor called the dual number density, which measures the number of particles within a region. The total number of particles is then given by
The corresponding tensor in spacetime is given by the pull-back of
from which, in turn, we can get the particle number density as
The number of particles can be obtained directly from spacetime as .
We further define a number density current as the dual of as
so that . Thus is timelike, and normalizing it defines the fluid velocity as
The push-forward of on the matter space vanishes, i.e. . The (20) relation can be inverted so that
Eq. (21) uniquely defines the fluid velocity in terms of the fluid number density and dual number density . Having obtained via (21) we can then use (5) to get the three metric (and then push-forward to get on the matter space). We can also obtain and pull-back it to spacetime to get . Alternatively the tensor can also be obtained via (7) once is defined by (21).
ii.3.2 The adiabatic fluid action and variations
We are now ready to discuss the dynamics of fluids as they are derived from an action. The dynamical variables are the spacetime metric and the fluid coordinates on which from the spacetime perspective are the maps which may be considered as three scalar fields.
As we have seen in the last subsection, the quantity of interest which depends on is the number density (or from the spacetime perspective). Therefore, the action for General Relativity coupled to an adiabatic fluid is taken to be
where is an arbitrary function. We shall further see below that the equation of state and its speed of sound is determined entirely by the form of . Specifically, for pressureless matter is proportional to the number density: .
The field equations are obtained via a variational principle. For brevity let us define
a definition which will be useful when we discuss cosmological perturbation theory. Furthermore, we will find it useful to work with which is related to by .
Since we need the variation of the number density . To find that we need the variation of which from (18) is given by
The above expression may look non-covariant as it contains partial derivatives. But as we now show, it is. Since and using (16) we find
where is the Lie derivative along . In fact the last relation is very useful and we shall use it hence forth. To find is now straightforward. We use to get
Although, is not needed to derive the field equations in the case of GR, we derive it here for completeness as it will be needed when we couple the fluid to a scalar field. We use (21) to get
ii.3.3 The fluid equations
Using (28) it is straightforward to find the fluid equations. The Einstein equations are obtained as usual by varying with the metric, and the stress-energy tensor is given by
where is the density and the pressure. From the variation of the fluid action and using (28) we can match and to the fluid function as
Since we see that the speed of sound of the fluid is given by
where and , it is clear that the fluid we have been considering so far is adiabatic. We shall use this notation for the rest of the paper so that for a function we have and and so on.
Varying with respect to the field equations for the fluid are obtained as
and momentum transfer equation
Interestingly, the fluid coordinates and number density do not appear in the above equations and are no longer needed.
Iii Generic models for coupled fluids
We now construct a model where the adiabatic fluid described in the previous section is explicitly coupled to a scalar field which will play the role of dark energy. The most general action formed from the fluid variable through the pull-back , as well as the scalar field and its derivative
We need the most general form for . Since we must have a tensor with -indices to contract with and since must reduce to the GR case if is absent, then the only dependence of on must come through and therefore from (21) also . Therefore we can write
Next we need to form invariants of the tensors , , and . Clearly we have as before the invariant which once again plays the role of the fluid number density. But we have also other invariants, namely
which can be used to construct a kinetic term for and
which plays the role of a direct coupling of the fluid velocity to the gradient of the scalar field. Since is completely antisymmetric any contraction of it with or vanishes. There may be combinations involving nd derivatives of the metric, or which for miraculous reasons give 2nd order field equations as in the case of Horndeski theory, but we ignore this possibility for now. Therefore our final functional form is
It is clear that ordinary GR with a quintessence field and a fluid is described by while k-essence is described by .
The above entity we have constructed is still fairly general. In particular we cannot split it into a scalar field part and a fluid part which are then coupled, without further assumptions. We shall proceed to do that shortly.
iii.2 Field equations for the generic fluid
To get the field equations we proceed as in the case of ordinary relativistic fluid. We have where
Varying with we find the scalar field equations as
Varying with we get the Einstein equations to be once again where the total energy-momentum tensor is
Finally varying with we find the evolution equation for the number density as
Contracting with we find the conservation law
which when used in (46) gives the purely spatial equation
As we have already stressed, we cannot in general separate out a field and a fluid: we are dealing with a single entity. Problems such as this have been discussed in Kunz (2009). We may proceed, however, to do that under further assumptions. In particular we will consider three types of models. Type-1 models are those for which there is no dependence and furthermore the function can be separated as . In both Type-2 and Type-3 models the function has dependence and the difference between the two is how this dependence appears. For Type-2, can be separated as while for Type-3 the separation is as . We now proceed to reduce the general equations above for each of the three types of models and in the process eliminate the variable and instead describe the fluid evolution in terms of an energy density and pressure .
iii.2.1 Type-1 coupled dark matter to a dark energy scalar field
Type-1 models, are classified via
where we have set . This later condition of separability of is not needed in order to simplify the field equations, but is needed in order to be able to solve them uniquely without resorting to the variable . These types of models describe a -essence scalar field coupled to matter. By further choosing , we can also describe coupled quintessence models.
The fact that we can separate the function in this way allows us to separate the general energy-momentum tensor (45) into an energy-momentum tensor for as
and an energy-momentum tensor for the fluid given by (30) with and identified as
so that . The scalar field energy density and pressure are read-off from (50) as
We now proceed to simplify the field equations for this type of coupled models. The scalar field equation (44) becomes
while the fluid equation (48) gives
The evolution of the fluid density is found from the conservation equation (47) as
iii.2.2 Type-2 coupled dark matter to a dark energy scalar field
Type-2 models are classified via
In these type of models, the energy-momentum tensor of the scalar field is given by (50) as in the Type-1 case while the energy-momentum of the fluid is again given by (30). However, unlike the Type-1 case, the energy density of the fluid is now identified with
while the pressure is still given by (52). In the case that the scalar is coupled to CDM (), the pressure equation (52) can be solved to give . Since we are only interested in the case we shall use this functional form for in this type of coupled model. Furthermore, for reasons that will become clear, rather than using we introduce a new coupling function so that
Once again, since we have these types of models describe a -essence scalar field coupled to matter and by further choosing , we can also reduce it to coupled quintessence models.
Let us now proceed to the field equations. The scalar field equation (44) for these type of models simplifies to
The last field equation is the momentum transfer equation for the fluid (48) which reduces for this type of models to
The equations (62), (63) and (64) still need some processing. In particular (62) should be solved in terms of a term and should not contain terms and similarly for the other two equations. Starting from (63) we can rearrange it so that it be comes
from which we find
Let us now proceed to the scalar equation (62). We need
and also . Using equation (66) the latter gives
iii.2.3 Type-3 coupled dark matter to a dark energy scalar field
Our third and final class of models is defined by
This also allows us to separate the energy-momentum tensor for from the energy-momentum tensor of the fluid so that
giving the field energy density as
Proceeding to the field equations, the scalar field equation (44) simplifies to
Finally, the coupling current is found to be
Having the required field equations at hand we now proceed to apply them to cosmology. For simplicity we shall only consider the case of coupled quintessence, although in a follow up paper we will discuss more general cases.
iv.1 Cosmological equations
To study cosmology and in particular the observational effects of the coupled models on the Cosmic Microwave Background and Large Scale Structure, we consider linear perturbations about the FRW spacetime. As is common, we choose the synchronous gauge so that the metric is
where is the conformal time, is the covariant derivative associated with , i.e. and is the traceless derivative operator . The unit-timelike vector field is perturbed as
A dot is derivative with respect to conformal time: . We will use a ”bar” over a variable to denote it’s FRW reduction, i.e. is the FRW background energy density for the fluid.
The Einstein equations for all types of coupling are as usual given by the Friedman equation
at the background level, where . At the perturbed level we have
where and with the label running over all fluids including the scalar field. We now exhibit the field equations for the (coupled) quintessence field and for CDM for the three types of cases. Let us note that if no label is placed on either or , then they are meant to refer to the CDM fluid.
Coupled quintessence in the Type-1 case is described by the function . We also consider only the case for which the fluid is CDM so that . The scalar field energy density and pressure for this type are given by
for the background FRW.
The scalar field equation (44) in the case of quintessence simplifies to
while the evolution of the CDM density is found from the conservation equation (47) as
which has a formal solution
Note how this no longer falls off as conventional matter because of the coupling to . The fluid equation (48) is identically satisfied.
The perturbed scalar field energy density and pressure are read-off from (50) with as
while the momentum divergence for the scalar field is
The scalar perturbation evolves according to
iv.1.2 Type 2
We consider a coupled quintessence function of the form . The scalar field energy density and pressure are as for the Type-1 case given by (83) and (84) respectively. The scalar field equation (70) becomes
which (assuming , then allows us to find as
Thus the fluid equation (65) gives