Unified Dark Matter models with fast transition
Abstract:
We investigate the general properties of Unified Dark Matter (UDM) fluid models where the pressure and the energy density are linked by a barotropic equation of state (EoS) and the perturbations are adiabatic. The EoS is assumed to admit a future attractor that acts as an effective cosmological constant, while asymptotically in the past the pressure is negligible. UDM models of the dark sector are appealing because they evade the socalled “coincidence problem” and “predict” what can be interpreted as , but in general suffer the effects of a nonnegligible Jeans scale that wreak havoc in the evolution of perturbations, causing a large Integrated SachsWolfe effect and/or changing structure formation at small scales. Typically, observational constraints are violated, unless the parameters of the UDM model are tuned to make it indistinguishable from CDM. Here we show how this problem can be avoided, studying in detail the functional form of the Jeans scale in adiabatic UDM perturbations and introducing a class of models with a fast transition between an early Einstein–de Sitter CDMlike era and a later CDMlike phase. If the transition is fast enough, these models may exhibit satisfactory structure formation and CMB fluctuations. To consider a concrete case, we introduce a toy UDM model and show that it can predict CMB and matter power spectra that are in agreement with observations for a wide range of parameter values.
1 Introduction
In the last three decades the flat CDM model [1, 2] has emerged as the standard “concordance” [3, 4] model of cosmology. It assumes General Relativity (GR) as the correct theory of gravity, with two unknown components dominating the dynamics of the late Universe: i ) a cold collisionless Cold Dark Matter (CDM) describing some weakly interacting particles, responsible for structure formation, ii ) a cosmological constant [5, 6] making up the balance to make the Universe spatially flat and driving the observed cosmic acceleration [7, 8, 9, 10]. The main alternative to the cosmological constant is a more general dynamic component called Dark Energy (DE) [11, 12, 13]. Many independent observations support both the existence of a CDM component and that of a separate DE [10, 14, 15, 16, 17, 18, 19, 20, 21, 22].
Early proposals [1, 2] of the CDM model were adding to CDM in an attempt to conciliate in the simplest possible way the emerging inflationary paradigm, which requires a spatially flat Universe and an almost scaleinvariant spectrum of perturbations, with the observed low density of matter. It should however be recognised that, while some form of CDM is independently expected to exist within any modification of the Standard Model of high energy physics, the really compelling reason to postulate the existence of DE has been the cosmic acceleration measured in the last decade [7, 8, 9, 10, 17, 18, 19, 20, 21]. It is mainly for this reason that it is worth investigating the hypothesis that CDM and DE are the two faces of a single Unified Dark Matter (UDM) component, thereby also avoiding the socalled “coincidence problem” [23].
Other attempts to explain the observed acceleration also exist, most notably by assuming a gravity theory other than GR, or an interaction between DM and DE (see e.g. [11, 24, 25, 26, 27] and [28, 29, 30, 31, 32, 33, 34]). In this paper however we focus on UDM models, where this single matter component provides an explanation for structure formation and cosmic acceleration.
In general, in the CDM model or in most models with DM and DE, the CDM component is free to form structures at all scales, with DE only affecting the general overall expansion [11, 12, 13]. Instead, a general feature of UDM models is the possible appearance of an effective sound speed, which may become significantly different from zero during the Universe evolution, then corresponding in general to the appearance of a Jeans length (i.e. a sound horizon) below which the dark fluid does not cluster (e.g. see [35, 36, 37]). Moreover, the presence of a nonnegligible speed of sound can modify the evolution of the gravitational potential, producing a strong Integrated Sachs Wolfe (ISW) effect [36]. Therefore, in UDM models it is crucial to study the evolution of the effective speed of sound and that of the Jeans length.
Several adiabatic fluid models and models based on non canonical kinetic Lagrangians have been investigated in the literature. For example, the generalised Chaplygin gas [38, 39, 40] (see also [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51]), the Scherrer [52] and generalised Scherrer solutions [53], the single dark perfect fluid with “affine” 2parameter barotropic equation of state (see [54, 37] and the corresponding scalar field models [55]) and the homogeneous scalar field deduced from the galactic halo spacetime [56, 57]. In general, in order for UDM models to have a background evolution that fits observations and a very small speed of sound, a severe finetuning of their parameters is necessary (see for example [37, 47, 48, 49, 50, 52, 58, 59]). Finally, one could also easily reinterpret UDM models based on a scalar field Lagrangian in terms of generally nonadiabatic fluids [60, 61] (see also [53, 62]). For these models the effective speed of sound, which remains defined in the context of linear perturbation theory, is not the same as the adiabatic speed of sound (see [35], [63] and [64]). In [62] a reconstruction technique is devised for the Lagrangian, which allows to find models where the effective speed of sound is small enough, such that the kessence scalar field can cluster (see also [65]).
In the present paper we investigate the possibility of constructing adiabatic UDM models where the Jeans length is very small, even when the speed of sound is not negligible. In particular, our study is focused on models that admit an effective cosmological constant and that are characterised by a short period during which the effective speed of sound varies significantly from zero. This allows a fast transition between an early matter dominated era, which is indistinguishable from an Einstein–de Sitter model, and a more recent epoch whose dynamics, background and perturbative, are very close to that of a standard CDM model.
To consider a concrete example, we introduce a 3parameter class of toy UDM adiabatic models with fast transition. One of the parameters is the effective cosmological constant or, equivalently, the corresponding density parameter ; the other two are and , respectively regulating how fast the transition is and the redshift of the transition. Studying the Jeans scale in these models we find an approximate analytical relation that sets a constraint on these two parameters, a sufficient condition that and have to satisfy in order for the models to be minimally viable. This relation can be used to fix for any given : in this case, with respect to a flat CDM, in practice our models have one single extra parameter. With the help of this relation we establish our main result: if the fast transition takes place early enough, at a redshift when the effective cosmological constant is still subdominant, then the predicted background evolution, Cosmic Microwave Background (CMB) anisotropy and linear matter power spectrum are in agreement with observations for a broad range of parameter values. In practice, in our toy models the predicted CMB and matter power spectra do not display significant differences from those computed in the CDM model, because the Jeans length remains small at all times, except for negligibly short periods, even if during the fast transition the speed of sound can be large. In other words, this kind of adiabatic UDM models evade the “nogo theorem” of Sandvik et al [50] who, studying the generalised Chaplygin gas UDM models, showed that this broad class must have an almost constant negative pressure at all times in order to satisfy observational constraints, making these models in practice indistinguishable from the CDM model (see also [37]).
The paper is organised as follows: in section 2 we introduce the basic equations describing the background and the perturbative evolution. In section 3 we use the pressuredensity plane to analyse the properties that a general barotropic UDM model has to fulfil in order to be viable. In section 4 we introduce our toy UDM model with fast transition and study its background evolution, comparing it to a CDM. In section 5 we analyse the properties of perturbations in this model, focusing on the the evolution of the effective speed of sound and that of the Jeans length during the transition. Then, using the CAMB code [66], in section 6 we compute the CMB and the matter power spectra. Finally, section 7 is devoted to our conclusions.
2 Background and perturbative equations for a UDM model
We assume a spatially flat FriedmannLemaîtreRobertsonWalker (FLRW) cosmology. The metric then is , where is the cosmic time, is the scale factor and is the Kronecker delta. The total stressenergy tensor is that of a perfect fluid: , where and are, respectively, the energy density and the pressure of the fluid, while is its fourvelocity. Starting from these assumptions, and choosing units such that , Einstein equations imply the Friedmann and Raychaudhuri equations:
(1)  
(2) 
where is the Hubble expansion scalar and the dot denotes derivative with respect to the cosmic time. Assuming that the energy density of the radiation is negligible at the times of interest, and disregarding also the small baryonic component, and represent the energy density and the pressure of the UDM component.
The energy conservation equation is:
(3) 
where is the equation of state (hereafter EoS) “parameter”. In this paper we investigate a class of UDM models based on a barotropic EoS , i.e. those models for which the pressure is function of the density only (see e.g. [67] and [68] for a discussion of general properties of barotropic fluids as dark components). In this case, if the EoS allows the value , the barotropic fluid admits an effective cosmological constant energy density, i.e. a fixed point of Eq. (3) [67, 69, 54], which we will denote as . Under very reasonable conditions (see the discussion below) this effective cosmological constant is unavoidable for barotropic fluids^{1}^{1}1Since this effective cosmological constant trivially satisfies an energy conservation equation (3) on its own, a fluid admitting an effective is always equivalent to two separate components, namely itself and an “aether” fluid, see [67] and [68]. Obviously, this is more general; for instance, a scalar field with a potential admitting a non vanishing minimum at  say  , is equivalent to a cosmological constant and a scalar field in a potential . [67, 69].
In order to properly describe the dynamics of the fluid we must consider the background EoS as well as the speed of sound which regulates the growth of fluid perturbations on different cosmological scales. In the following we shall confine our study to the simplest hypothesis that the EoS remains of the barotropic form when we allow for perturbations: in this case our models will be adiabatic, and the effective and adiabatic speeds of sound will coincide, see e.g. [70, 71, 72, 35]. Other choices for the perturbed spacetime are possible, see [37] for a recent practical example.
Let us consider small perturbations of the FLRW metric in the longitudinal gauge, using conformal time : , where is the gravitational potential.
Defining
(4) 
and linearising the 00 and 0i components of Einstein equations, for a planewave perturbation one obtains the following second order differential equation [73, 58, 36]:
(5) 
where the prime is the derivative with respect to the conformal time , is the effective speed of sound and
(6) 
with the redshift, . In general, the adiabatic speed of sound is ; for an adiabatic fluid .
Starting from Eq. (5), let us define the squared Jeans wave number [36]:
(7) 
Its reciprocal defines the squared Jeans length: .
There are two regimes of evolution. If and the speed of sound is slowly varying, then the solution of Eq. (5) is
(8) 
where is an appropriate integration constant^{2}^{2}2This solution is exact if the speed of sound satisfies the equation , which implies
For large scale perturbations, when , Eq. (5) can be rewritten as , with general solution
(9) 
In this large scale limit the evolution of the gravitational potential depends only on the background evolution, encoded in , i.e. it is the same for all modes. The first term is the usual decaying mode, which we are going to neglect in the following, while is related to the power spectrum, see e.g. [64].
3 Analysis of barotropic UDM models on the pressuredensity plane
A common way to study the properties of the EoS of DE is to consider the phase space (see e.g. [74, 75, 76, 77]). Here we follow another approach, studying our models in the pressuredensity plane, see Fig. 1. There are several motivations for this choice. First of all, in the barotropic case we are considering the pressure is a function of the density only, so it is natural to give a graphical description on the plane. Second, this plane gives an idea of the cosmological evolution of the dark fluid. Indeed, in an expanding Universe () Eq. (3) implies for a fluid satisfying the null energy condition [78] during its evolution, hence there exists a onetoone correspondence between time and energy density. Finally, in the adiabatic case the effective speed of sound we have introduced in Eq. (5) can be written as , therefore it has an immediate geometric significance on the plane as the slope of the curve describing the EoS .
For a fluid, it is quite natural to assume , which then implies that the function is monotonic, and as such crosses the line at some point .^{3}^{3}3Obviously, we are assuming that during the evolution the EoS allows to become negative, actually violating the strong energy condition [78], i.e. at least for some , otherwise the fluid would never be able to produce an accelerated expansion. From the point of view of the dynamics this is a crucial fact, because it implies the existence of an attracting fixed point () for the conservation equation (3) of our UDM fluid, i.e. plays the role of an unavoidable effective cosmological constant. The Universe necessarily evolves toward an asymptotic deSitter phase, a sort of cosmic nohair theorem (see [79, 80] and refs. therein and [67, 69, 54]).
We now summarise, starting from Eqs. (15) and taking also into account the current observational constraints and theoretical understanding, a list of the fundamental properties that an adiabatic UDM model has to satisfy in order to be viable. We then translate these properties on the plane, see Fig. 1.

We assume the UDM to satisfy the weak energy condition: ; therefore, we are only interested in the positive half plane. In addition, we assume that the null energy condition is satisfied: , i.e. our UDM is a standard (nonphantom) fluid. Finally, we assume that our UDM models admit a , so that an asymptotic is built in.

Let us consider a Taylor expansion of the UDM EoS about the present energy density :
(10) i.e. an “affine” EoS model [37, 54, 55, 67] where is the adiabatic speed of sound at the present time. Clearly, these models would be represented by straight lines in Fig. 1, with the slope. The CDM model, interpreted as UDM, corresponds to the affine model (10) with (see [67] and [54, 37]) and thus it is represented in Fig. 1 by the horizontal line . From the matter power spectrum constraints on affine models [37], it turns out that . Note therefore that, from the UDM perspective, today we necessarily have .
Few comments are in order. From the points above, one could conclude that any adiabatic UDM model, in order to be viable, necessarily has to degenerate into the CDM model, as shown in [50] for the generalised Chaplygin gas and in [37] for the affine adiabatic model^{5}^{5}5From the point of view of the analysis of models in the plane of Fig. 1, the constraints found by Sandvik et al [50] on the generalised Chaplygin gas UDM models and by [37] on the affine UDM models simply amount to say that the curves representing these models are indistinguishable from the horizontal CDM line. (see [82, 83, 84] for an analysis of other models). In other words, one would conclude that any UDM model should satisfy the condition at all times, so that for all scales of cosmological interest, in turn giving an evolution for the gravitational potential as in Eq. (9):
(11) 
where , is the primordial gravitational potential at large scales, set during inflation, and is the matter transfer function, see e.g. [85].
On the other hand, let us write down the explicit form of the Jeans wave number:
(12) 
Clearly, we can obtain a large not only when , but also when changes rapidly, i.e. when the above expression is dominated by the term. When this term is dominating in Eq. (12), we may say that the EoS is characterised by a fast transition.
Thus, viable adiabatic UDM models can be constructed which do not require at all times if the speed of sound goes through a rapid change, a fast transition period during which can remain large, in the sense that for all scales of cosmological interest to which the linear perturbation theory of Eq. (5) applies. From point 3 above, at late times we must have ; on the other hand, at recombination we have and the speed of sound is negligible, implying . Therefore, the transition will mark the passage from a very small (possibly vanishing) almost constant to the asymptotic value or, in other words, from a pure CDMlike early phase to a posttransition CDMlike late epoch. In addition, we may expect the transition to occur at relatively high redshifts, high enough to make the UDM model quite similar to the CDM model at late times. Indeed, again from point 3 above, we infer that the fast transition should take place sufficiently far in the past, in particular during the dark matter epoch, when . Otherwise, we expect that it would be problematic to reproduce the current observations related to the UDM parameter , for instance it would be hard to have a good fit of supernovae and ISW effect data.
In the rest of the paper, in order to quantitatively investigate observational constraints on UDM models with fast transition, we introduce and discuss a toy model. In particular, we will explore which values of the parameters of this toy model fit the observed CMB and matter power spectra.
Finally, let us make a last remark on building phenomenological UDM (or DE) fluid models intended to represent the homogeneous FLRW background and its linear perturbations. A fast transition in a fluid model could be characterised by a large value of , even larger than . As far as the FLRW background evolution is concerned, this fact does not raise any issue: the background is homogeneous, and does not actually represent a speed of sound, as there is nothing that could propagate in this case. For linear perturbations, at scales such that the solution of the Eq. (5) for the gravitational potential is Eq. (11). Therefore, for such scales there is no superluminal propagation. This is because Eq. (5) is the Fourier component of a wave equation with potential , and this potential does not allow propagation for . In building a phenomenological fluid model, we can therefore choose values for the parameters of the model in order to always satisfy the condition for all of cosmological interests to which linear theory applies, hence such a fluid model will be a good causal model for all scales that intends to represent. In other words, we can always build the fluid model in such a way that all scales smaller than the Jeans length correspond to those in the nonlinear regime, i.e. scales beyond the range of applicability of the model. So, for these scales, no conclusions can be derived from the linear theory on the behaviour of perturbations of a UDM model with . To investigate these scales, one needs to be beyond the perturbative regime investigated here, possibly also increasing the sophistication of the fluid model in order to properly take into account the greater complexity of small scale nonlinear physics and to maintain causality.
4 A toy model with fast transition
In the present section we introduce a toy model based on a hyperbolic tangent EoS, which is conveniently parametrised as
(13) 
and is depicted in Fig. 2 for a typical shape. In the EoS (13) is the typical energy scale at the transition, is related to the rapidity of the transition, is the effective cosmological constant, i.e. . This model reduces to a CDM, which in the UDM language of the previous section is described by , in two limits:
and .
The main properties of the EoS (13) are the following:

The asymptotic behaviour of the pressure for is . From the considerations of the previous section, we expect , which corresponds to a minimum value of the redshift . For instance, we have if we want to have . In Figs. 3 we plot the evolution of as a function of redshift, for (left panel) and (right panel), for three different choices of . The solid line represents the CDM model, while the horizontal lines respectively represent: a pure CDM model for ; the boundary between the decelerated and the accelerated expansion phases of the Universe for . Clearly, from both panels, models with larger ratio have a background evolution more similar to that of the CDM model at all times. On the other hand, a smaller ratio implies a faster transition between the CDMlike phase and the CDM phase. In addition, we have the confirmation that the transition has to take place sufficiently far in the past, i.e. , in order for the late time evolution of to be in any case close to that of the CDM model.
Figure 3: Evolution of the UDM parameter in the hyperbolic tangent model, for (left panel) and (right panel). For reference we also plot: the line representing a flat pure CDM model (an EdS Universe); the line representing the boundary between the decelerated and the accelerated phases; the solid curve representing the evolution of for a typical CDM model with . In each panel, the three dashed, dashdotted and dotted lines respectively correspond to . Clearly, the dotted lines correspond to UDM models with a slow transition, almost indistinguishable from a CDM at all times, while the dashed lines well represent UDM models with a very fast transition from a pure CDM to a typical CDM behaviour. The higher , the earlier the UDM transits to that of a CDM. Figure 4: Illustrative plots of the speed of sound and as functions of the energy density for the hyperbolic tangent model. The parameters values are and . The energy density and the pressure are normalised to . 
The speed of sound is the following:
(14) illustrated in the left panel of Fig. 4. It attains its maximum value
(15) in . For our analysis, there are two main cases to consider, assuming :

. In this case, , so that the model is close to a CDM at all times.

. In this case, we have two subcases: bi) , for which or bii) , for which . The latter subcase may in principle imply superluminal perturbations; fortunately, as we shall see, acausal effects can be avoided if the transition is sufficiently fast.


As we explained in the previous section, in order to have a fast transition we must have in Eq. (12) for the Jeans wave number. This quantity is depicted in the right panel of Fig. 4. For the EoS (13) the derivative of is
(16) which attains its extrema at . The maximum corresponds to the minus sign.
Clearly, the derivative of is important only in the case b) of the previous point. In this case the maximum is:
(17) For subcase bii), , we always have , while in subcase bi) there is also the possibility that be small.
Let us now consider the case when the transition takes place at the lower limit , corresponding to . In this case, from Eq. (17), the maximum is . Therefore, in order to have a fast transition, we must have . Then, it is inevitable from point bii) that, if we want the fast transition to take place just before the accelerated phase of the expansion of the Universe, we must have . In this case, shortly after the transition the pressure rapidly approaches the asymptotic value .
5 Analysis of the Jeans wave number during the transition
The Jeans length is a crucial quantity in determining the viability of a UDM model, because of its effect on perturbations, which is then revealed in observables such as the CMB and matter power spectra. We now focus on the Jeans wave number for the toy UDM model introduced in the previous section and investigate its behaviour as a function of the speed of sound, in particular around , in the middle of the transition where the speed of sound is at its peak.
Starting from the classification we presented in point 2 of the previous section, we are interested in the case b), namely , because in this regime a fast transition in the EoS takes place. The majority of the adiabatic UDM models considered so far in the literature belongs to the case a) of point 2 of section 4. For the toy model Eq. (13) as well, implies that the pressure tends to at all times, i.e. to a CDM, as shown in Fig. 3.
In the case of a fast transition, from Eq. (12) for the Jeans wave number, it is interesting to compare the term with the remaining ones contained in the squared brackets, namely:
(18) 
In Figs. 57 we plot , and the Jeans wave number as functions of . For the calculation of we use , with and the critical energy density , where is the Hubble constant. We choose in order to consider a transition sufficiently back in the Dark Matter epoch (see point 1 of section 4) and vary the ratio , with , in order to show examples of faster transitions.
From the plots in Figs. 57 it is clear that the smaller is, the larger is the difference between and .
Moreover, from Fig. 5, is negative for , then for it increases becoming positive and intersecting the curve a first time for . For smaller values of the energy density, decreases again to zero, again intersecting the curve. In Figs. 67, the same behaviour of the curves takes place and since the difference between the two curves is much larger, we have chosen a logarithmic scale. Therefore, the negative part of has been omitted.
The intersection points between the curves and represent the moments at which the Jeans wave number vanishes, as it can be seen from the right panels of Figs. 57. In general, around these points the corresponding Jeans length becomes very large, possibly causing all sort of problems to perturbations, with effects on CMB and structure formation in the UDM model. On the other hand, for sufficiently small the transition is fast enough that i) in general the Jeans wave number becomes larger and ii) it becomes vanishingly small for extremely short times, so that the the effects caused by its vanishing are sufficiently negligible, as we are going to show in the next section for the CMB and matter power spectra. As illustrated in Figs. 57, by choosing progressively smaller values of we can obtain progressively larger Jeans wave numbers, while the curve starts to show a plateau shape around the transition.
Clearly, we are interested in the value of during the transition, because before and after that the negligible speed of sound implies a vanishing Jeans length, or a very large . In essence, for a fast enough transition the “average” value of around the transition is approximated by its value on the plateau  say  and this is, on average, the minimum value of , i.e. the maximum Jeans length for the given values of the parameters and . Thus, we now want to establish a relation between and for any given . This, fixing a which allows for an acceptable matter power spectrum which fits observational data, will help us to find the needed to have the transition at .
The relative maximum of the Jeans wave number between the two zeros of the curve corresponds approximatively to where assumes its maximum value, i.e. in , as we have shown in point 3 of section 4. Thus, let us define with the value of for : as required, it is of the same order of the plateau value (see for example Fig. 7) of the Jeans wave number during the fast transition.
Evaluating the analytical expression (12) of at under the assumption we obtain the following approximate expression:
(19) 
Defining and making sure that we have for , we can then extract from (19) the required relation between and :
(20) 
In the left panel plots of Fig. 8, we compare the analytical approximation (20) with the numerical calculations from Eq. (12), for and for Mpc, as functions of . The agreement between our analytical approximation and the numerical calculation is clearly very good. As we can see from the figure, if we require larger values of then must be smaller, i.e. a faster transition is needed. On the other hand, if the transition takes place farther in the past, i.e. for increasing values of , this constraint is less stringent.
Having established a good approximation for , we now want to determine for which values of and this quantity is well representative of around the transition, i.e. when we have a plateau as in Fig. 7. In particular, this can be estimated from the difference of the values of and at . The larger is the difference, the faster is the transition and the higher is the plateau effect. We therefore define the efficiency parameter which, under the assumption , can be analytically approximated from Eq. (12):
(21) 
From this, we obtain a new relation between and :
(22) 
In the right panel of Fig. 8, we compare the analytical approximation (22) with the numerical calculations, for , with very good agreement. Notice that the larger is the efficiency , the smaller must be, i.e. a faster transition is required.^{6}^{6}6Assuming in Eq. (22) implies . This limit in can be seen in the right panel of Fig. 8 and in Fig. 10, where in this limit .
It is important to stress that a large efficiency is relevant in order for to be a more representative “minimum on average” value of during the transition, i.e. it is not a necessary condition in order to have a fast transition or a model in good agreement with observation. This can be understood observing the multiplicative term in front of the expression (12) of the Jeans wave number . Indeed, for increasing values of , this term amplifies the difference during the transition, giving a larger . So, one can obtain models in good agreement with observation even if the efficiency is low.
Considering the plots in Figs. 57, where for we obtain respectively , in order to have a fast transition and a pronounced plateau, we infer that is needed. In addition, this requirement was also confirmed by the study of the matter power spectrum, see the next section.
Substituting Eq. (20) in Eq. (21), we can now obtain an approximate expression of as a function of the redshift of the transition , which allows us to estimate the range of (and thus ) for which the efficiency is above a certain threshold, for a given . In Fig. 9 we show the plot vs for Mpc (from top to bottom). For increasing values of , the range of in which becomes larger, as expected.
We can now use the relation (20) to understand how large the speed of sound can be during the transition. To this purpose, we substitute Eq. (20) in the maximum value of , at . We plot this in Fig. 10 as a function of for Mpc (from top to bottom). On the same figure we also plot the curve of the maximum of the speed of sound fixing the value of the efficiency at (solid red line). Then, the vertical dashed line corresponds to the value of for which in Eq. (22), giving in Eq. (15). In order to have , we must consider the area above the solid red line. We can see that, for increasing values of , the value of required to have a fixed decreases. For example, for Mpc, in order to have , the transition has to take place at , while if we have .
6 The CMB and matter Power spectra: toy model predictions
In order to compare the predictions of our toy UDM model with observational data, we have used a properly modified version of CAMB^{7}^{7}7http://camb.info/ [66] for the computation of the CMB and the matter power spectra. In particular, we have modified the original definition of the density contrast for the case of adiabatic UDM models. Indeed, we have to define the UDM density contrast as [37], where here is the “aether” part of the UDM fluid [67, 68]. In this case, starting from the perturbation theory we outlined in section 2, we can infer the link between the density contrast and the gravitational potential via the Poisson equation in the following way:
(23) 
for scales smaller than the cosmological horizon and , where is the recombination redshift ().
We compare the theoretical predictions of our toy model with the WMAP 5year data [18, 19, 20] and the luminous red galaxies power spectrum measured by the SDSS collaboration [15]. The CMB data used in our plots are available on the LAMBDA^{8}^{8}8http://lambda.gsfc.nasa.gov website, while those regarding the matter power spectrum are implemented in a modified version of the CosmoMC software^{9}^{9}9http://cosmologist.info/cosmomc/.
We consider as reference the flat CDM model described by the bestfit parameters found by combining WMAP5 data with measurements of Type Ia supernovae and Baryon Acoustic Oscillations [18, 20], with values provided on the LAMBDA website (68% CL uncertainties): , , , km s Mpc, and . For our toy model, we keep the same amount of baryons but choose a vanishing CDM content.
In Figs. 1114 we plot the theoretical predictions of our model, those of the reference CDM and the observed CMB and matter power spectra data. Each of Figs. 1114 respectively correspond to , i.e. to . Guided by the analysis in section 5, for each transition density we have chosen values of which clearly show the progressive enhancement of the agreement between the predicted matter power spectrum and the observed one. Moreover, in the matter power spectrum plots, for each choice we draw a vertical dashed line representing the corresponding value of .
We can see from Figs. 11 and 12 that the CMB anisotropies predicted by the reference CDM and by our toy model are indistinguishable for a large range of . However, while at the higher transition redshift of Fig. 11 the matter power spectrum also allows the same broad range of values, at the smaller of Fig. 12 we start to see the need for a faster transition, i.e. a smaller , to have an acceptable power spectrum. As expected from the analysis of section 5, the effect becomes more pronounced as the transition occurs at the smaller and smaller redshifts of Figs. 13 and 14. In the case of Fig. 13 the first acoustic peak of CMB is higher with respect to the observational data. This effect can be explained by looking at the matter power spectrum. Indeed, the latter moves away from the reference CDM before the equivalence wavenumber Mpc. In other words, the gravitational potential starts to oscillate and to decay for , therefore affecting those modes entering the horizon before the matterradiation equivalence epoch.
Finally, in Fig. 14 the first CMB spectrum peak is lower than the observed one for any value of . Note in the left panel of Fig. 3 that, for , the background evolution is sensibly different from the reference CDM. Indeed, in this case our model behaves like a pure CDM Einstein–de Sitter for a much longer time. One possibility to avoid this discrepancy is to slightly increase . Therefore, again using and , in Fig. 15 we have chosen , with the first value again corresponding to the above mentioned reference CDM and the other two corresponding to the bestfit and its upper uncertainty obtained by WMAP5 using CMB data only (see [19] and the LAMBDA website). The agreement between the CMB prediction and the observational data is again good for , with a good matter power spectrum.
7 Conclusions
The last decade of observations of large scale structure [14, 15, 16, 17, 21, 22], the search for Ia supernovae (SNIa) [7, 8, 9, 10] and the measurements of the CMB anisotropies [3, 18, 19, 20] are very well explained by assuming that two dark components govern the dynamics of the Universe. They are DM, thought to be the main responsible for structure formation, and an additional DE component that is supposed to drive the measured cosmic acceleration [11, 12, 13].
At the same time, in the context of General Relativity, it is very interesting to study possible alternatives. A popular one is that of an interaction between DM and DE, without violating current observational constraints [11, 28, 29, 30, 31, 32, 33]. This possibility could alleviate the so called “coincidence problem” [23], namely, why are the energy densities of the two dark components of the same order of magnitude today. Another attractive, albeit radical, explanation of the observed cosmic acceleration and structure formation is to assume the existence of a single dark component: UDM models [38, 39, 50, 48, 49, 47, 52, 58, 36, 53, 57, 55, 62, 54, 37, 65] where, by definition, there is no coincidence problem.
In the present paper we have investigated the general properties of UDM fluid models where the pressure and the energy density are linked by a barotropic equation of state (EoS) and the perturbations are adiabatic. Using the pressuredensity plane, we have analysed the properties that a general barotropic UDM model has to fulfil in order to be viable. We have assumed that the EoS of UDM models admits a future attractor which acts as an effective cosmological constant, while asymptotically in the past the pressure is negligible, studying the possibility of constructing adiabatic UDM models where the Jeans length is very small, even when the speed of sound is not negligible. In particular, we have focused on models that admit an effective cosmological constant and that are characterised by a short period during which the effective speed of sound varies significantly from zero. This allows a fast transition between an early epoch that is indistinguishable from a standard matter dominated era, i.e. an Einstein de Sitter model, and a more recent epoch whose dynamics, background and perturbative, are very close to that of a standard CDM model.
In the second part of the paper, in order to quantitatively investigate observational constraints on UDM models with fast transition, we have introduced and discussed a toy model based on a hyperbolic tangent EoS [see Eq. (13)]. We have shown that if the transition takes place early enough, at a redshift when the effective cosmological constant is still subdominant, being also fast enough, then these models can avoid the oscillating and decaying time evolution of the gravitational potential that in many UDM models causes CMB and matter fluctuations incompatible with observations. Consequently, the background evolution, the CMB anisotropy and the linear matter power spectrum predicted by our model do not display significant differences from those computed in a reference CDM [18, 20], because the Jeans length , where is the Jeans wave number [see Eq. (12)], remains small at all times, except for negligibly short periods, even if during the fast transition the speed of sound can be large. In this way, our toy models (and more in general UDM models with a similar fast transition) can evade the “nogo theorem” of Sandvik et al [50], as we discussed in the introduction.
Specifically, we have analysed the properties of perturbations in our toy model, focusing on the the evolution of the effective speed of sound and that of the Jeans length during the transition. In this way, we have been able to set theoretical constraints on the parameters of the model, predicting sufficient conditions for the model to be viable. Finally, guided by these predictions and using the CAMB code [66], we have computed the CMB and the matter power spectra showing that our toy model, for a wide range of parameters values, fits observation.
The full likelihood analysis for this model and its parameters would be an interesting extension of the study carried out here, which we will address in a future work.
Acknowledgments.
OFP and DB wish to thank the ICG Portsmouth for the hospitality during the development of this project. The authors also thank N. Bartolo, B. Bassett, R. Crittenden, R. Maartens, S. Matarrese, S. Mollerach, M. Sasaki and M. Viel for discussions and suggestions. DB research has been partly supported by ASI contract I/016/07/0 “COFIS”.References
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