Review Article | Open Access
Unified Dark Matter Scalar Field Models
We analyze and review cosmological models in which the dynamics of a single scalar field accounts for a unified description of the Dark Matter and Dark Energy sectors, dubbed Unified Dark Matter (UDM) models. In this framework, we consider the general Lagrangian of -essence, which allows to find solutions around which the scalar field describes the desired mixture of Dark Matter and Dark Energy. We also discuss static and spherically symmetric solutions of Einstein's equations for a scalar field with noncanonical kinetic term, in connection with galactic halo rotation curves.
In the last few decades a standard cosmological “Big Bang” model has emerged, based on Einstein's theory of gravity, General Relativity. Indeed, observations tell us that—by and large—the Universe looks the same in all directions, and it is assumed to be homogeneous on the basis of the “Cosmological Principle”, that is, a cosmological version of the Copernican principle. The request for the Universe to be homogeneous and isotropic translates, in the language of space-time, in a Robertson-Walker metric. Assuming the latter, Einstein equations simplify, becoming the Friedmann equations, and in general the solutions of these equations are called Friedmann-Lemaitre-Robertson-Walker (FLRW) models. The cosmological inhomogeneities we observe on the largest scales as tiny anisotropies of the Cosmic Microwave Background (CMB) are then well explained by small relativistic perturbations of these FLRW “background” models, while on smaller scales the inhomogeneities are larger and call for nonlinear dynamics, but relativistic effects are negligible and Newtonian dynamics is sufficient to explain the formation of the structures we see, that is, galaxies, groups, and clusters forming the observed “cosmic web”. In this context, last decade's observations of large-scale structure, search for Ia supernovae (SNIa) [1–4], and measurements of the CMB anisotropies [5, 6] suggest that two dark components govern the dynamics of the Universe. They are the dark matter (DM), thought to be the main responsible for structure formation, and an additional dark energy (DE) component that is supposed to drive the measured cosmic acceleration [7, 8]. However, the DM particles have not yet been detected in the lab, although there are hints for their existence from cosmic rays experiments [9–11], and there is no theoretical justification for the tiny cosmological constant  (or more general DE component [7, 8]) implied by observations (see also ). Therefore, over the last decade, the search for extended theories of gravity has flourished as a possible alternative to DE [7, 8]. At the same time, in the context of General Relativity, it is very interesting to study the possibility of an interaction between Dark Matter and Dark Energy without violating current observational constraints [7, 8, 14–19] (see also ). This possibility could alleviate the so-called “coincidence problem”, namely, why are the energy densities of the two dark components of the same order of magnitude today. Another more radical explanation of the observed cosmic acceleration and structure formation is to assume the existence of a single dark component: Unified Dark Matter (UDM) models; see, for example, [21–50] (see also [51–55] on how to unify DM, DE, and inflation,  on unification of DM and DE in the framework of supersymmetry, [57–60] on unification of DM and DE from the solution of the strong CP-problem, [61, 62] on unification of DM and DE in connection with chaotic scalar field solutions in Friedmann-Robertson-Walker cosmologies, [63–65] on how to unify dark energy and dark matter through a complex scalar field, and [66–68] on a study of a scalar field, “Cosmos Dark Matter”, that induces a time-dependent cosmological constant).
In comparison with the standard DM + DE models (e.g., even the simplest model, with DM and a cosmological constant), these models have the advantage that we can describe the dynamics of the Universe with a single scalar field which triggers both the accelerated expansion at late times and the LSS formation at earlier times. Specifically, for these models, we can use Lagrangians with a noncanonical kinetic term, namely, a term which is an arbitrary function of the square of the time derivative of the scalar field, in the homogeneous and isotropic background.
Originally this method was proposed to have inflation driven by kinetic energy, called -inflation [69, 70], to explain early Universe's inflation at high energies. Then this scenario was applied to DE [71–73]. In particular, the analysis was extended to a more general Lagrangian [74, 75] and this scenario was called -essence (see also [71, 73, 76–86]).
For UDM models, several adiabatic or, equivalently, purely kinetic models have been investigated in the literature, for example, the generalised Chaplygin gas [22–24] (see also [25, 27–29, 87–94]), the Scherrer  and generalised Scherrer solutions , the single dark perfect fluid with “affine” 2-parameter barotropic equation of state (see [37, 39] and the corresponding scalar field models ), and the homogeneous scalar field deduced from the galactic halo space-time [34, 95]. In general, in order for UDM models to have a background evolution that fits observations and a very small speed of sound, a severe fine-tuning of their parameters is necessary (see, e.g., [25, 27–31, 39, 96]). Finally, one could also easily reinterpret UDM models based on a scalar field Lagrangian in terms of generally nonadiabatic fluids [97, 98] (see also [33, 38]). 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 [70, 99, 100]). In  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 -essence scalar field can cluster (see also [41, 46, 48–50]).
One of the main issues of these UDM models is whether the single dark fluid is able to cluster and produce the cosmic structures we observe in the Universe today. In fact, a general feature of UDM models is the appearance of an effective sound speed, which may become significantly different from zero during the evolution of the Universe. In general, this corresponds to the appearance of a Jeans length (or sound horizon) below which the dark fluid does not cluster. Thus, the viability of UDM models strictly depends on the value of this effective sound speed [70, 99, 100], which has to be small enough to allow structure formation [27, 31, 32] and to reproduce the observed pattern of the CMB temperature anisotropies [25, 32].
In general, in order for UDM models to have a very small speed of sound and a background evolution that fits the observations, a severe fine tuning of their parameters is necessary. In order to avoid this fine tuning, alternative models with similar goals have been analyzed in the literature. The work in  studied in detail the functional form of the Jeans scale in adiabatic UDM perturbations and introduced a class of models with a fast transition between an early Einstein-de Sitter cold DM-like era and a later CDM-like phase. If the transition is fast enough, these models may exhibit satisfactory structure formation and CMB fluctuations, thus presenting a small Jeans length even in the case of a nonnegligible sound speed. The work in  explored unification of DM and DE in a theory containing a scalar field of non-Lagrangian type, obtained by direct insertion of a kinetic term into the energy-momentum tensor. Finally,  introduced a class of field theories where it comprises two scalar fields, one of which is a Lagrange multiplier enforcing a constraint between the other’s field value and derivative in order to have the sound speed always identically zero on all backgrounds.
This work is organized as follows. In Section 2, considering the general Lagrangian of -essence models, we layout the basic equations. In Section 3 we present an analytical study of the Integrated Sachs-Wolfe (ISW) effect within the framework of UDM. Computing the temperature power spectrum of the Cosmic Microwave Background anisotropies one is able to isolate those contributions that can potentially lead to strong deviations from the usual ISW effect occurring in a CDM Universe. This helps to highlight the crucial role played by the sound speed in the unified dark matter models. Our treatment is completely general in that all the results depend only on the speed of sound of the dark component and thus it can be applied to a variety of unified models, including those which are not described by a scalar field but relies on a single dark fluid; see also . In Section 4 we study and classify UDM models defined by the purely kinetic model. We show that these models have only one late-time attractor with equation of state equal to minus one (cosmological constant), studying all possible solutions near the attractor which describes a unified dark matter fluid; see also . Subsequently, noting that purely kinetic models can be described as adiabatic single fluid, for these Lagrangians it is natural to give a graphical description on pressure—energy density plane (see also ). In Section 5, we present the simplest case of a scalar field with canonical kinetic term which unavoidably leads to an effective sound speed equal to the speed of light. In Section 6, making the stronger assumption that the scalar field Lagrangian is exactly constant along solutions of the equation of motion, we find a general class of -essence models whose classical trajectories directly describe a unified Dark MatterDark Energy (cosmological constant) fluid. In particular we consider more general models that allow for the possibility that the speed of sound is small during Einstein-de Sitter CDM-like era. In Section 7, we investigate the class of UDM models studied in , which designed a reconstruction technique of the Lagrangian, allowing one to find models where the effective speed of sound is small enough, and the -essence scalar field can cluster (see also [41, 46, 48, 50]). In particular, the authors of  require that the Lagrangian of the scalar field is constant along classical trajectories on cosmological scales, in order to obtain a background identical to the background of the CDM model. In Section 8, we develop and generalize the approach studied in . Specifically, we focus on scalar-field Lagrangians with non-canonical kinetic term to obtain UDM models that can mimic a fluid of dark matter and quintessence-like dark energy, with the aim of studying models where the background does not necessarily mimic the CDM background; see also . In Section 9, we investigate the static and spherically symmetric solutions of Einstein's equations for a scalar field with non-canonical kinetic term, assumed to provide both the dark matter and dark energy components of the Universe; see also . We show that there exist suitable scalar field Lagrangians that allow to describe the cosmological background evolution and the static solutions with a single dark fluid. In Section 10, we draw our main conclusions. Finally, in the appendix, for completeness we provide the spherical collapse top-hat solution for UDM models based on purely kinetic scalar eld Lagrangians, which allow us to connect the cosmological solutions to the static configurations.
2. Unified Dark Matter Scalar Field Models
We start recalling the main equations which are useful for the description of most the UDM models within the framework of -essence.
Consider the action where where the symbol denotes covariant differentiation. We adopt units and the signature for the metric (Greek indices run over spacetime dimensions, while Latin indices label spatial coordinates).
The stress-energy tensor of the scalar field has the following form: and its equation of motion reads
If is time like, then describes a perfect fluid , where the pressure is and the energy density is The four-velocity has the following form: Assume a flat, homogeneous Friedmann-Lemaître-Robertson-Walker (FLRW) background metric, that is, where is the scale factor, denotes the unit tensor, and is the conformal time.
Assuming that the energy density of the radiation is negligible at the times of interest, and disregarding also the small baryonic component the background evolution of the Universe is completely characterised by the following equations: where and . (Indeed the density of baryons relative is about 4.5% today and 16.9% prior to Dark Energy domination in the standard cosmological model [5, 6].) The dot denotes differentiation with respect to the cosmic time whereas a prime denotes differentiation wrt the conformal time .
In the background we have that , and therefore the equation of motion (4) for the homogeneous mode becomes An important quantity is the Equation of State (EoS) parameter , which in our case reads We mainly focus on the other relevant physical quantity, the speed of sound, which enters in governing the evolution of the scalar field perturbations. Consider small inhomogeneities of the scalar field, that is, and write the perturbed FLRW metric in the longitudinal gauge as being for . The linearised and Einstein equations are (see [70, 100]) where one defines a “speed of sound” relative to the pressure and energy density fluctuation of the kinetic term  as follows: From the above linearized Einstein's equations one obtains [70, 100] Equation (16) is sufficient to determine the gravitational potential and the perturbation of the scalar field. It is useful to write explicitly the perturbed scalar field as a function of the gravitational potential: Defining two new variableswhere , we can recast (16) in terms of and : where . Starting from (19) we arrive at the following second-order differential equations for : Unfortunately, we do not know the exact solution for a generic Lagrangian. However, we can consider the asymptotic solutions, that is, the long-wavelength and the short-wavelength perturbations, depending whether or , respectively.
There are two regimes of evolution. If and the speed of sound is slowly varying, then the solution of (20) is where is an appropriate integration constant. (This solution is exact if the speed of sound satisfies the equation , which implies , where and are generic constants. A particular case is when , for which the speed of sound is constant.) On these scales, smaller than the Jeans length, the gravitational potential oscillates and decays in time, with observable effects on both the CMB and the matter power spectra .
For large scale perturbations, when , (20) can be rewritten as , with general solution: In this large-scale limit the evolution of the gravitational potential depends only on the background evolution, encoded in ; that is, 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, for example, .
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 [32, 39, 99]). 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 . Therefore, in UDM models it is crucial to study the evolution of the effective speed of sound and that of the Jeans length. In other words, one would conclude that any UDM model should satisfy the condition that for all scales of cosmological interest, in turn giving an evolution for the gravitational potential as in (23): where , is the primordial gravitational potential at large scales, set during inflation, and is the matter transfer function; see, for example, .
Therefore the speed of sound plays a major role in the evolution of the scalar field perturbations and in the growth of the over-densities. If is significantly different from zero, it can alter the evolution of density of linear and nonlinear perturbations . When cs becomes large at late times, this leads to strong deviations from the usual ISW effect of CDM models .
In the next section we will perform an analytical study of the Integrated Sachs-Wolfe (ISW) effect within the framework of Unified Dark Matter models based on a scalar field which aim at a unified description of dark energy and dark matter. Computing the angular power spectrum of the Cosmic Microwave Background temperature anisotropies, we are able to isolate those contributions that can potentially lead to strong deviations from the usual ISW effect occurring in a CDM universe. This helps to highlight the crucial role played by the sound speed in the unified dark matter models.
3. Analytical Approach to the ISW Effect
In this section we focus on the contribution to the large-scale CMB anisotropies which is due to the evolution in time of the gravitational potential from the epoch of last scattering up to now, the so-called late Integrated Sachs-Wolfe (ISW) effect . Through an analytical approach we point out the crucial role of the speed of sound in the unified dark matter models in determining strong deviations from the usual standard ISW occurring in the CDM models. Our treatment is completely general in that all the results depend only on the speed of sound of the dark component and thus it can be applied to a variety of models, including those which are not described by a scalar field, but relies on a single perfect dark fluid. In the case of CDM models the ISW is dictated by the background evolution, which causes the late time decay of the gravitational potential when the cosmological constant starts to dominate . In the case of the unified models there is an important aspect to consider: from the last scattering to the present epoch, the energy density of the Universe is dominated by a single dark fluid, and therefore the gravitational potential evolution is determined by the background and the perturbation evolution of just such a fluid. As a result the general trend is the appearance of a sound speed significantly different from zero at late times corresponding to the appearance of a Jeans length (or a sound horizon) under which the dark fluid does not cluster any more, causing a strong evolution in time of the gravitational potential (which starts to oscillate and decay) and thus a strong ISW effect. Our results show explicitly that the CMB temperature power spectrum for the ISW effect contains some terms depending on the speed of sound which give a high contribution along a wide range of multipoles . As the most straightforward way to avoid these critical terms one can require the sound speed to be always very close to zero. Moreover we find that such strong imprints from the ISW effect come primarily from the evolution of the dark component perturbations, rather than from the background expansion history.
The ISW contribution to the CMB power spectrum is given by where is the fractional temperature perturbation due to ISW effect: with and being the present and the last scattering conformal times, respectively, and are the spherical Bessel functions. Let us now evaluate analytically the power spectrum (25). As a first step, following the same procedure of , we notice that, when the acceleration of the Universe begins to be important, the expansion time scale sets a critical wavelength corresponding to . It is easy to see that if we consider the CDM model, then , that is, when . Thus at this critical point we can break the integral (25) in two parts : where As explained in  the ISW integrals (26) take on different forms in these two regimes:where is the change in the potential from the matter-dominated (e.g., at recombination) to the present epoch and is the conformal time when a given k-mode contributes maximally to the angle that this scale subtends on the sky, obtained at the peak of the Bessel function . The first limit in (30a) is obtained by approximating the Bessel function as a constant evaluated at the critical epoch . Since it comes from perturbations of wavelengths longer than the distance, a photon can travel during the time , a kick () to the photons is the main result, and it will correspond to very low multipoles, since is very close to the present epoch . It thus appears similar to a Sachs-Wolfe effect (or also to the early ISW contribution). The second limit in (30b) is achieved by considering the strong oscillations of the Bessel functions in this regime, and thus evaluating the time derivative of the potentials out of the integral at the peak of the Bessel function, leaving the integral  With this procedure, replacing (30a) in (28) and (30b) in (29) we can obtain the ISW contribution to the CMB anisotropies power spectrum (25).
Now we have to calculate, through (22)-(23) and (2.21), the value of for and . As we will see that main differences (and the main difficulties) of the unified dark matter models with respect to the CDM case will appear from the second regime of (30a)-(30b).
3.1. Derivation of for Modes
In the UDM models when , then is always satisfied. This is due to the fact that before the dark fluid starts to behave dominantly as a cosmological constant, for , its sound speed generically is very close to zero in order to guarantee enough structure formation, and moreover the limit involves very large scales (since is very close to the present epoch). For the standard CDM model the condition is clearly satisfied. In this situation we can use the relation (23) and can be expressed as in (24). The integral in (24) may be written as follows: where and is the conformal time at recombination. When , the UDM models behave as dark matter. (In fact the Scherrer  and generalized Scherrer solutions  in the very early Universe, much before the equality epoch, have and . However at these times the dark fluid contribution is subdominant with respect to the radiation energy density and thus there is no substantial effect on the following equations.) In this temporal range the Universe is dominated by a mixture of “matter” and radiation and , where is the value of the scalar factor at matter-radiation equality, and . With these definitions it is easy to see that . Notice that (24) is obtained in the case of adiabatic perturbations. Since we are dealing with UDM models based on a scalar field, there will always be an intrinsic nonadiabatic pressure (or entropic) perturbation. However for the very long wavelengths, , under consideration here such an intrinsic perturbation turns out to be negligible . For adiabatic perturbations  and accounting for the primordial power spectrum, , where is the scalar spectral index, we get from (30a) where we have neglected since it gives a negligible contribution.
A first comment is in order here. There is a vast class of UDM models that are able to reproduce exactly the same background expansion history of the Universe as the CDM model (at least from the recombination epoch on wards). For such cases it is clear that the low contribution (33) to the ISW effect will be the same that is predicted by the CDM model. This is easily explained considering that for such long wavelength perturbations the sound speed in fact plays no role.
3.2. Derivation of for Modes
As we have already mentioned in the previous section, in general a viable UDM must have a sound speed very close to zero for in order to behave as dark matter also at the perturbed level to form the structures we see today, and thus the gravitational potential will start to change in time for . Therefore for the modes , in order to evaluate (30b) into (29) we can impose that which, from the definition of , moves the lower limit of (29) to . Moreover we have that . We can use this property to estimate any observable at the value of . Defining , and , we have , taking , and where . Notice that the expansion (34) is fully justified, since as, already mentioned above, the minimum value of in (29) moves to , making much less than 1. Therefore we can write In this case, during , there will be perturbation modes whose wavelength stays bigger than the Jeans length or smaller than it; that is, we have to consider both possibilities and . In general the sound speed can vary with time, and in particular it might become significantly different from zero at late times. However, just as a first approximation, we exclude the intermediate situation because usually is very close to (see also ).
3.2.1. Perturbation Modes on Scales Bigger than the Jeans Length
We can see that for and for the contribution to the angular power spectrum from the modes under consideration is In other words we find a similar slope as found for the CDM model in [104, 105]. Recalling the results of the previous section, this means that in UDM models the contribution to the ISW effect from those perturbations that are outside the Jeans length is very similar to the one produced in a CDM model. The main difference on these scales will be present if the background evolution is different from the one in the CDM model, but for the models where the background evolution is the same, as those proposed in [30, 33, 36, 106, 107] no difference can be observed.
3.2.2. Perturbation Modes on Scales Smaller than the Jeans Length
When , one must use the solution (22) and through the relation (18a) the gravitational potential is given by In (37) is a constant of integration where and it is obtained under the approximation that for one can use the longwavelength solution (24), since for these epochs the sound speed must be very close to zero. Notice that (37) shows clearly that the gravitational potential is oscillating and decaying in time. Defining , we take the time derivative of the gravitational potential appearing in (30b) by employing the expansion of (35). We thus find that, for , (29) yields the potentially most dangerous term: with . Such a term makes the angular power spectrum to scale as until . This angular scale is obtained by considering the peak of the Bessel functions in correspondence of the cutoff scale , . In fact, for smaller scales, will decrease as . This is due to a natural cut-off in the various integrals which is introduced for those modes that enter the horizon during the radiation dominated epoch, due to the Meszaros effect that the matter fluctuations will suffer until the full matter domination epoch. Such a cut-off will show up in the gravitational potential and in the various integrals of (39) as a factor, where is the wavenumber of the Hubble radius at the equality epoch.
3.3. Discussion of Some Examples
Most UDM models have several properties in common. It is easy to see that in (32) is negligible because of the low value of . Moreover in the various models usually we have that strong differences with respect to the ISW effect in the CDM case can be produced from perturbations on those scales that are inside the Jeans length as the photons pass through them. For these scales the perturbations of the UDM fluid play the main role. On larger scales instead we find that they play no role and ISW signatures different from the CDM case can come only from the different background expansion histories. We have found that when (see (20)), one must take care of the term in (39). Indeed this term grows faster than the other integrals contained in (39) when increases up to . It is responsible for a strong ISW effect and hence, in the CMB power spectrum , it will cause a decrease in the peak to plateau ratio (once the CMB power spectrum is normalized). In order to avoid this effect, a sufficient (but not necessary) condition is that the models have satisfy the condition for the scales of interest. The maximum constraint is found in correspondence with the scale at which the contribution (39) takes its maximum value, that is, . For example, in the Generalized Chaplygin Gas model (GCG), that is, when and (see Section 4), we deduce that (see [24, 25, 28]). This is also in agreement with the finding of  which performs an analysis on the mass power spectrum and gravitational lensing constraints, thus finding a more stringent constraint.
As far as the generalized Scherrer solution models  are concerned, in these models the pressure of the UDM fluid is given by , where is a suitable constant and (see Section 4). The case corresponds to unified model proposed by Scherrer . In this case we find that imposing the constraint for the scales of interest we get .
If we want now to study in greater detail what happens in the GCG model when , we discover the following things. (i)For , where we are in the “Intermediate case”. Now is very small and the background of the cosmic expansion history of the Universe is very similar to the CDM model. In this situation the pathologies, described before, are completely negligible. (ii)For a very strong ISW effect is produced; one estimates the same order of magnitude for the decrease of the peak to plateau ratio in the anisotropy spectrum (once it is normalized) obtained numerically in  (having assumed that the production of the peaks during the acoustic oscillations at recombination is similar to what happens in a CDM model, since at recombination the effects of the sound speed are negligible).
An important observation arises when considering those UDM models that reproduce the same cosmic expansion history of the Universe as the CDM model. Among these models one can impose the condition which, for example, is predicted by UDM models with a kinetic term of Born-Infeld type [26, 33, 106, 107]. In this case, computing the integral in (39) which gives the main contribution to the ISW effect one can estimate that the corresponding decrease of peak to plateau ratio is about one third with respect to what we have in the GCG when the value of is equal to 1. The special case is called “Chaplygin Gas” (see, e.g., ) and it is characterized by a background equation of state which evolves in a different way to the standard CDM case. From these considerations we deduce that this specific effect stems only in part from the background of the cosmic expansion history of the Universe and that the most relevant contribution to the ISW effect is due to the value of the speed of sound .
Let us now make some comments about a particular class of the generalized Chaplygin gas models where the sound speed can be larger than the speed of light at late times, that is, when (see, e.g., [94, 96, 108]). In particular, in , the author finds that the new constraint . Indeed, for this range of values, the Jeans wavenumber is sufficiently large that the resulting ISW effect is not strong. In this case the Chaplygin gas is characterised by a fast transition . However this particular model is ruled out because the transition from a pure CDM-like early phase to a posttransition CDM-like late epoch is nearly today (). In fact, as discussed in  and in Section 4.4, the fast transition has to take place sufficiently far in the past. 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 the CMB and matter power spectra.
4. Purely Kinetic Lagrangians
In this section we focus mainly on Lagrangians (i.e., the pressure ) that depend only on . (This section is largely based on .) Defining , we have to solve the equation when is time-like. Then, from (10) we get where . We can immediately note that a purely kinetic Lagrangian, through (40), (see, e.g., ), can be described as a perfect fluid whose pressure is uniquely determined by the energy density, since both depend on a single degree of freedom, the kinetic term . In this case corresponds to the usual adiabatic sound speed. Obviously if we consider a barotropic or adiabatic equation of state, can be described through a purely kinetic -essence Lagrangian, if the inverse function of exists. In Section 4.4, we will use the pressure-density plane to analyze the properties that a general barotropic UDM model has to fulfil in order to be viable (see also ).
Now we want to make a general study of the attractor solutions in this case. From (10) (see ) we obtain the following nodes: with being a constant. Both cases correspond to , as one can read from (11).
At this point we can study the general solution of the differential equation (41). For and the solution is  with being a positive constant. This solution has been also derived, although in a different form, in . As , or (or both) must tend to zero, which shows that, depending on the specific form of the function , each particular solution will converge toward one of the nodes above. From (43), for , the value of or (or of both of them) must tend to zero. Then, it is immediate to conclude that is an attractor for and confirms that each of the above solutions will be an attractor depending on the specific form of the function .
In what follows we will provide some examples of stable node solutions of the equation of motion, some of which have been already studied in the literature. The models below are classified on the basis of the stable node to which they asymptotically converge.
4.1. Case 1: Generalized Chaplygin Gas
Plugging the equation of state (44) into the the continuity equation , we can write and as function of . Indeed with . We note that when is small, we have . In other words, this model behaves as DM. Meanwhile, in the late epoch (i.e., ), it behaves as a cosmological constant.
Since , it is necessary for our scopes to consider the case , so that . Note that corresponds to the standard “Chaplygin gas” model. Let us obviously consider and .
Let us conclude this section mentioning two more models that fall into this class of solution. The first was proposed in , in which (with being a suitable constant) satisfying the constraint along the attractor solution . This model, however, is well known to imply a diverging speed of sound. The second was proposed in [36, 37, 39, 112] where there was the single dark perfect fluid with “affine” 2-parameter barotropic equation of state which satisfies the constraint that along the attractor solution . For this model, we have ; that is, the speed of sound is always a constant. The evolution of leads to , where today . When the pressure and the energy density are considered as functions of , we have where is the integration constant derived imposing the value of the fluid energy density at present and is at present time. From the matter power spectrum constraints , it turns out that .
4.2. Case 2: Scherrer Solution
For the solution of case 1 we want to study the function around some . In this case we can approximate as a parabola with : with and being suitable constants. This solution, with and , coincides with the model studied by Scherrer in  (see also [31, 113]).
It is immediate to see that for and the value of goes to zero. Replacing this solution into (43) we obtain while the energy density becomes Now if we impose that today is close to so that then (50) reduces to with and with for . As a consequence, the energy density becomes In order for the density to be positive at late times, we need to impose . In this case the speed of sound (15) turns out to be We notice also that, for , we have for the entire range of validity of this solution. Thus, (54) tells us that our -essence behaves like a fluid with very low sound-speed with a background energy density that can be written as where behaves like a “dark energy” component (.) and behaves like a “dark matter” component (). Note that, from (54), must be different from zero in order for the matter term to be there. (For this particular case the Hubble parameter is a function only of this fluid .)
If the Lagrangian is strictly quadratic in , we can obtain explicit expressions for the pressure and the speed of sound in terms of , namely,
Looking at these equations, we observe that in the early Universe (, i.e., ) the -essence behaves like radiation. Therefore, the -essence in this case behaves like a low sound-speed fluid with an energy density which evolves like the sum of a “dark matter” (DM) component with and a “dark energy” (DE) component with . The only difference with respect to the standard CDM model is that in this -essence model, the dark energy component has . Starting from the observational constraints on and , the value of is determined by the fact that the -essence must begin to behave like dark matter prior to the epoch of matter-radiation equality. Therefore, , where is the scale factor at the epoch of equal matter and radiation, given by (where we have imposed that the value of the scale factor today is ). At the present time, the component of corresponding to dark energy in (54) must be roughly twice the component corresponding to dark matter, so . Substituting into this equation, we get 
In practice, if we assume that has a local minimum that can be expanded as a quadratic form and when (52) is not satisfied (i.e., for ), we cannot say anything about the evolution of and . The stronger bound is obtained by Giannakis and Hu , who considered the small-scale constraint that enough low-mass dark matter halos are produced to reionize the Universe. On the other hand the sound speed can be made arbitrarily small during the epoch of structure formation by decreasing the value of . One should also consider the usual constraint imposed by primordial nucleosynthesis on extra radiation degrees of freedom, which however leads to a weaker constraint. Moreover the Scherrer model differs from CDM in the structure of dark matter halos both because of the fact that it behaves as a nearly pressure-less fluid instead of a set of collisionless particles. Analytically we will discuss this problem when we will study the static configuration of the UDM models; see Section 9 or . Practically, we will see that when , the energy density of the Scherrer model is negative. Thus, and must depend strongly on time. In other words, this model will behave necessarily like a fluid and, consequently, there is the strong possibility that it can lead to shocks in the non-linear regime .
4.3. Case 2: Generalized Scherrer Solution
Starting from the condition that we are near the attractor , we can generalize the definition of , extending the Scherrer model in the following way: with and and being suitable constants.
The density reads If , (43) reduces to (where ) and so becomes with for . We have therefore obtained the important result that this attractor leads exactly to the same terms found in the purely kinetic model of , that is, a cosmological constant and a matter term. One can therefore extend the constraint of  to this case, obtaining . A stronger constraint would clearly also apply to our model by considering the small-scale constraint imposed by the Universe reionization, as in . If we write the general expressions for and , we have For one obtains a result similar to that of , namely, On the contrary, when , we obtain In this case one can impose a bound on so that at early times and/or at high density the -essence evolves like dark matter. In other words, when , unlike the purely kinetic case of , the model is well behaved also at high densities.
In the appendix we study spherical collapse for the generalized Scherrer solution models.
4.4. Studying Purely Kinetic Models in the Pressure-Density Plane
In this subsection we report some results of . Noting that purely kinetic models can be described as adiabatic single fluid , for these Lagrangians it is natural to give a graphical description on the plane; see Figure 1. Indeed, this plane gives an idea of the cosmological evolution of the dark fluid. Indeed, in an expanding Universe () (41) implies for a fluid satisfying the null energy condition during its evolution; hence there exists a one-to-one correspondence between (increasing) time and (decreasing) energy density. Finally, in the adiabatic case the effective speed of sound we have introduced in (15) can be written as ; therefore it has an immediate geometric meaning 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 it reaches the line at some point . (Obviously, we are assuming that during the evolution the EoS allows to become negative, actually violating the strong energy condition, that is, 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 (41) of our UDM fluid; that is, plays the role of an unavoidable effective cosmological constant. The Universe necessarily evolves toward an asymptotic de-Sitter phase, a sort of cosmic no-hair theorem (see [114, 115] and references therein and [37, 112, 116]).
We now summarise, starting from (9) and (41) 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 Figure 1.(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: ; that is, our UDM is a standard (nonphantom) fluid. Finally, we assume that our UDM models admit a cosmological constant solution Lambda at late time, so that an asymptotic equation of state is built in.(2)We demand a dust-like behaviour back in the past, at high energies, that is, a negligible pressure for . (Note that we could have , and yet, if , the Universe would still be in a matter-like era.) In particular, for an adiabatic fluid we require that at recombination ; see [36, 37, 39, 117].(3)Let us consider a Taylor expansion of the UDM EoS about the present energy density : that is, an “affine” EoS model [36, 37, 39, 112] where is the adiabatic speed of sound at the present time. Clearly, these models would be represented by straight lines in Figure 1, with being the slope. The CDM model, interpreted as UDM, corresponds to the affine model (66) with (see [37, 39, 112]) and thus it is represented in Figure 1 by the horizontal line . From the matter power spectrum constraints on affine models , 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  for the generalised Chaplygin gas and in  for the affine adiabatic model. (From the point of view of the analysis of models in the plane of Figure 1, the constraints found by Sandvik et al.  on the generalised Chaplygin gas UDM models and by  on the affine UDM models simply amount to say that the curves representing these models are indistinguishable from the horizontal CDM line.) 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 (23).
On the other hand, let us write down the explicit form of the Jeans wave-number: Clearly, we can obtain a large not only when , but also when changes rapidly, that is, when the above expression is dominated by the term. When this term is dominating in (67), we may say that the EoS is characterised by a fast transition.
In  the authors investigate observational constraints on UDM models with fast transition, introducing and discussing a toy model. In particular, they explore which values of the parameters of such a toy model fit the observed CMB and matter power spectra.
5. UDM Scalar Field with Canonical Kinetic Term
Starting from the barotropic equation of state we can describe the system either through a purely kinetic -essence Lagrangian, as we already explained in the last section, or through a Lagrangian with canonical kinetic term, as in quintessence-like models (see ).
In the second case we have to solve the two differential equations: where is time-like. In particular, if we assume that our model describes a unified dark matter/dark energy fluid, we can proceed as follows: starting from and we get up to an additive constant which can be dropped without any loss of generality. Inverting (69), that is, writing we are able to get . Now we require that the fluid has constant pressure , that is, that the Lagrangian of the scalar field is constant along the classical trajectory corresponding to perfect fluid behavior. In other words one arrives at an exact solution with potential (see also [106, 107]). For large values of , (equivalently, for large values of , ) and our scalar field behaves just like a pressureless dark matter fluid. Indeed, this asymptotic form, in the presence of an extra radiation component, allows to recover one of the stable nodes obtained in  for quintessence fields with exponential potentials, where the scalar field mimics a pressureless fluid. Under the latter hypothesis we immediately obtain which can be inverted to give the scalar field potential of (70) as . One then obtains which can be immediately integrated, to give for , with such that . Replacing this solution in the expression for the energy density one can easily solve the Friedmann equation for the scale-factor as a function of cosmic time: which coincides with the standard expression for a flat, matter plus Lambda model , with , and being the cosmological constant and matter density parameters, respectively.
Using standard criteria (e.g., ) it is immediate to verify that the above trajectory corresponds to a stable node even in the presence of an extra-fluid (e.g., radiation) with equation of state , where and are the fluid pressure and energy density, respectively. Along the above attractor trajectory our scalar field behaves precisely like a mixture of pressureless matter and cosmological constant. Using the expressions for the energy density and the pressure we immediately find, for the matter energy density, The peculiarity of this model is that the matter component appears as a simple consequence of having assumed the constancy of the Lagrangian.
A closely related solution was found by Salopek and Stewart , using the Hamiltonian formalism.
To conclude this section, let us stress that, like any scalar field with canonical kinetic term [73, 121], our UDM model predicts , as it is clear from (15), which inhibits the growth of matter inhomogeneities. In summary, we have obtained a “quartessence” model which behaves exactly like a mixture of dark matter and dark energy along the attractor solution, whose matter sector, however, is unable to cluster on subhorizon scales (at least as long as linear perturbations are considered).
6. UDM Scalar Field with Noncanonical Kinetic Term
We can summarize our findings so far by stating that purely kinetic -essence cannot produce a model which exactly describes a unified fluid of dark matter and cosmological constant, while scalar field models with canonical kinetic term, while containing such an exact description, unavoidably lead to , in conflict with cosmological structure formation. In order to find an exact UDM model with acceptable speed of sound we consider more general scalar field Lagrangians (see ).
6.1. Lagrangians of the Type
Let us consider Lagrangians with non-canonical kinetic term and a potential term, in the form The energy density then reads while the speed of sound keeps the form of (15). The equation of motion for the homogeneous mode reads
One immediately finds
One can rewrite the equation of motion (78) in the form It is easy to see that this equation admits 2 nodes, namely: (1)(2).
In all cases, for , the potential should tend to a constant, while the kinetic term can be written around the attractor in the form where is a suitable mass-scale and a constant. The trivial case obviously reduces to the one of Section 4.
Following the same procedure adopted in the previous section we impose the constraint , which yields the general solution .
This allows to define as a solution of the differential equation:
As found in the case of -essence, the most interesting behavior corresponds to the limit of large and in (81), for which we obtain leading to , and . The Lagrangian of this model is similar to that analyzed in .
6.2. Lagrangians of the Type
Let us now consider Lagrangians with a non-canonical kinetic term of the form (see ).
Imposing the constraint , one obtains , which, inserted in the equation of motion yields the general solution:
The latter equation, together with (82), defines our general prescription to get UDM models describing both DM and cosmological constant-like DE.
As an example of the general law in (84) let us consider an explicit solution. Assuming that the kinetic term is of Born-Infeld type, as in [26, 106, 107, 122], with being a suitable mass-scale, which implies , we get where and is the scale-factor at a generic time . In order to obtain an expression for , we impose that the Universe is dominated by our UDM fluid, that is, . This gives which, replaced in our initial ansatz , allows to obtain the following expression (see also [106, 107]):
If one expands around , and , one gets the approximate Lagrangian Note that our Lagrangian depends only on the combination , so that one is free to reabsorb a change of the mass-scale in the definition of the filed variable. Without any loss of generality we can then set , so that the kinetic term takes the canonical form in the limit . We can then rewrite our Lagrangian as
This model implies that for values of and , the scalar field mimics a dark matter fluid. In this regime the effective speed of sound is , as desired.
To understand whether our scalar field model gives rise to a cosmologically viable UDM solution, we need to check if in a Universe filled with a scalar field with Lagrangian (90), plus a background fluid of, for example, radiation, the system displays the desired solution where the scalar field mimics both the DM and DE components. Notice that the model does not contain any free parameter to specify the present content of the Universe. This implies that the relative amounts of DM and DE that characterize the present Universe are fully determined by the value of . In other words, to reproduce the present Universe, one has to tune the value of in the early Universe. However, a numerical analysis shows that once the initial value of is fixed, there is still a large basin of attraction in terms of the initial value of , which can take any value such that .
The results of a numerical integration of our system including scalar field and radiation are shown in Figures 2–4. Figure 2 shows the density parameter, as a function of redshift, having chosen the initial value of so that today the scalar field reproduces the observed values and . Notice that the time evolution of the scalar field energy density is practically indistinguishable from that of a standard DM plus Lambda (CDM) model with the same relative abundances today. Figure 3 shows the evolution equation of state parameter ; once again the behavior of our model is almost identical to that of a standard CDM model for . Notice that, since , the effective speed of sound of our model is close to zero, as long as matter dominates, as required. In Figure 4 we finally show the redshift evolution of the scalar field variables and : one can easily check that the evolution of both quantities is accurately described by the analytical solutions above, (86) and (87), respectively (the latter being obviously valid only after the epoch of matter-radiation equality).
However in this model, as discussed in , the non-negligible value of the sound speed today gives a strong contribution to the ISW effect and produces an incorrect ratio between the first peak and the plateau of the CMB anisotropy power-spectrum .
7. How the Scalar Field of Unified Dark Matter Models Can Cluster
The authors of  proposed a technique for constructing UDM models where the scalar field can have a sound speed small enough to allow for structure formation and to avoid a strong integrated Sachs-Wolfe effect in the CMB anisotropies which typically plague UDM models (see also [41, 46]). (This section is largely based on .) In particular, they studied a class of UDM models where, at all cosmic times, the sound speed is small enough that cosmic structure can form. To do so, a possible approach is to consider a scalar field Lagrangian of the form Therefore, by introducing the two potentials and , we want to decouple the equation of state parameter , and the sound speed . This condition does not occur when we consider either Lagrangians with purely kinetic terms or Lagrangians like or (see, e.g.,  and the previous Sections 6.1 and 6.2). In the following subsections we will describe how to construct UDM models based on (92), following the analysis of .
7.1. How to Construct UDM Models
Let us consider the scalar field Lagrangian of (92). The energy density , the equation of state and the speed of sound are