Unphysical properties in a class of interacting dark energy models

Models with non-gravitational interactions between the dark matter and dark energy components are an alternative to the standard cosmological scenario. These models are characterized by an interaction term, and a frequently used parameterization is $Q = 3\xi H \rho_{x}$, where $H$ is the Hubble parameter and $\rho_{x}$ is the dark energy density. Although current observations support such a model for negative values of the interaction parameter $\xi$, we show here that this interval of values of $\xi$ leads the model to predict a violation of the Weak Energy Condition (WEC) for the dark matter density, regardless of the value of the equation-of-state parameter of the dark energy component. This violation is accompanied by unphysical instabilities of matter perturbations.

As it is well known, there is no known fundamental principle that prevents a non-minimal coupling between the energy components of the cosmological dark sector. Such a possibility has in fact been explored since the eighties as an alternative to the standard cosmology (see e.g. [1-3]), with its theoretical and observational consequences being of great interest nowadays [4][5][6][7][8][9]. However, in the absence of a natural guidance from fundamental physics on the coupling term, a number of phenomenological models have been proposed and their cosmological consequences investigated in light of the current observational data -we refer the reader to [10] for a recent comparative analysis of different classes of interacting models.
In particular, models in which the coupling or interacting term Q is proportional to the dark energy (DE) density ρ x [11][12][13][14], Q = 3ξHρ x , where H and ξ are the Hubble and interaction parameters respectively, have become popular in the recent years (see e.g. [8] and references therein). However, as will be shown in this short note, this class of models shows unphysical behavior for the interval of values of its parameters currently constrained by observational data [8]. In particular, the model predicts that the pressureless matter density will eventually become negative, violating the Weak Energy Condition (WEC), with further consequences for the evolution of the dark matter and baryon density perturbations.
Let us first consider the balance equations of the model where ρ dm and ρ b are respectively the densities of dark matter (DM) and conserved baryons, for the total pressureless matter. The general solutions of (3)-(4) are In the spatially flat case, C and ρ x0 obey the additional constraint ρ m0 + ρ x0 = 3H 2 0 , where a subscript 0 denotes the value of the corresponding quantity at present time. That is, where Ω x0 = ρ x0 /(3H 2 0 ) is the present DE density parameter. In order to have a positive matter density at early times, the interaction parameter must obey ξ/|ω x | < (1 − Ω x0 ).
For ξ < 0 and ω x < 0, the second term on the right-hand side of equation (5) is negative, and it will eventually dominate ρ m for For example, if ω x = −1, Ω x0 = 0.7 and ξ = −0.5, ρ m becomes negative when the scale factor is a ≈ 1.2. In Fig. 1 we show the scale factor for WEC violation as a function of the interaction parameter ξ. In Fig. 2 we show the behavior of the Hubble function, as well as of the DE and matter densities, as functions of the scale factor for ξ = −0.5, ω x = −1 and Ω x0 = 0.7. The violation of the WEC in this class of interacting models was also pointed out e.g. in [6].
In a model with non-gravitational interaction in the dark sector, the conservation equations for conserved baryons, DM and DE are T µν Combining Eqs. (9) and (10) we obtain We can decompose the energy-momentum transfer Q µ in directions parallel and orthogonal to the fluid 4-velocity u µ , with u µQ µ = 0.
For comoving observers, Q represents the energy transfer between the components, andQ µ represents the momentum transfer.
Assuming that both DE and pressureless matter are adiabatic perfect fluids, perturbing (11) and (12) in the longitudinal gauge, and assuming that there is no momentum transfer in the matter rest frame (i.e. matter follows geodesics), we find the balance and Poisson where a prime means derivative w.r.t. conformal time, H = aH, ρ is the total energy density, δ = δρ/ρ, and θ is the fluid velocity potential.
For ω x = −1, the equations above lead to the Poisson and matter perturbation equations where θ = θ m (the DE velocity remains undefined). In the limit of small scales these equations are reduced to We also have δρ x = −aQθ m /k 2 and δQ = Q θ m /k 2 , which are negligible in that limit. For conserved baryons the balance equations are There is no pressure term in equations (23)-(26). Instabilities are caused by the background interaction term Q = 3ξHρ x due to the violation of the WEC discussed earlier.
Some authors try to fix this issue by taking ω x = −1. In this case, the system of perturbation equations (15)-(19) does not close. Then an additional ansatz for δQ is needed, but it should be consistent to a covariant perturbation of the background ansatz Q = 3ξHρ x .
Another possibility is to consider interacting models with a non-adiabatic DE, with sound speed c 2 s = ω x [4]. In any case, the weak energy condition is still violated by the background in the present case.
In Fig. 3 we plot the evolution of the total matter/baryon density contrast and velocity potential, that shows an instability at a ≈ 1.2, when the matter density becomes negative 0 0.2 0.4 0.6 0.8 1.0 1.2 1 As we have shown, for ω x = −1 the DE component is smooth on sub-horizon scales. This is not true if ω x > −1, when DE clusters and contributes to clustering matter [5]. Therefore, on may argue that the weak energy condition must be satisfied by the total clustering energy, not by pressureless matter alone. In order to verify this possibility, let us decompose the dark energy as where ρ m is pressureless and ρ Λ has EoS parameter −1. The total clustering matter (including the fluctuating DE part) is given by From (5)-(6) we then have It is easy to verify that ρ c is positive defined if, and only if, When this inequality is saturated, the model is equivalent to a ΛCDM model. It is violated, in particular, by the best-fit values obtained from current observations [8].
If (32) is satisfied, the energy flux is from ρ Λ to ρ c . With this interaction term, the subhorizon perturbation equations for the clustering matter are the same as (23)-(24), with δ Λ δ c and δQ c ≈ 0. Under condition (32), there is no instability.
To conclude this analysis, a discussion on the use of the comoving synchronous gauge is in order. In the synchronous gauge the balance equations are given bẏ For the metric potential, we obtain from the Einstein's equations In the case ω x = −1 we can assume, as above, that matter follows geodesics, that is, we can set δQ i = 0, θ = θ m = 0, and (37) is identically satisfied. From (38) we see that δρ x = −δp x = 0 (while θ x remains undefined). Finally, from (40) we have δQ = 0 and our system is reduced toδ h + 2Hḣ = ρ m δ m .
where we have introduced the rate of matter creation Γ = Q/ρ m . It is easy to check that, for ω x = −1, this is not generally possible. Dark energy is perturbed, there is momentum transfer in the matter rest frame, matter does not follow geodesics, which means that it is not comoving with synchronous observers.