The adiabatic instability on cosmology's dark side

Rachel Bean¹, Éanna É Flanagan² and Mark Trodden³

¹ Department of Astronomy, Cornell University, Ithaca, NY 14853, USA
² Laboratory for Elementary Particle Physics, Cornell University, Ithaca, NY 14853, USA
³ Department of Physics, Syracuse University, Syracuse, NY 13244, USA

E-mail: trodden@physics.syr.edu

Received 20 November 2007
Published 4 March 2008

Contents

• 1. Class of models
• 2. Adiabatic regime
• 3. Adiabatic instability
• 4. Examples of models
• Acknowledgments
• References

In order for our cosmological models to provide an accurate fit to current observational data, it is necessary to postulate two dramatic augmentations of the assumption of baryonic matter interacting gravitationally through Einstein's equations—dark matter and dark energy. A logical possibility is that these dark sectors interact with each other or with the normal matter [1]–[3]. A number of models have been proposed that exploit this possibility to address, for example, the coincidence problem.

Such models face a range of existing constraints arising from both particle physics and gravity. In this paper, we consider perturbations around the cosmological solution and demonstrate the existence of a dynamical instability which we term the adiabatic instability. This instability is characterized by a negative sound speed squared of the effective coupled fluid [4, 5] and was first discovered [6] in a context slightly different to that considered here—the mass varying neutrino model of dark energy. Our aim here is to give a general treatment of the instability, applicable to a wide class of models, to identify the regimes in which the instability occurs, and to delineate the resulting redshift-dependent constraints.

1. Class of models

We begin from the following action

where g_μν is the Einstein frame metric, is a scalar field which acts as dark energy and Ψ_j are the matter fields. Here, we have adopted a signature (–,  + ,  + ,  + ) and defined the reduced Planck mass by m_p²≡(8πG)^–1. The functions α_j() are couplings to the jth matter sector. This general action encapsulates many models studied in the literature [7]. The field equations are:

where we have treated the matter field(s) in the jth sector as a fluid with density and pressure as measured in the frame e^2α_jg_μν, and with 4-velocity u_j μ normalized according to g^μνu_j μu_j ν = –1.

We consider models with a baryonic sector (α_b()) and a composite dark matter sector, with one coupled species with density ρ_c and coupling α_c() = α(), and another uncoupled species with density ρ_co and coupling α_co = 0. Neglecting the gravitational effect of the baryons, using , and defining gives

and ∇_μ∇^μ–V_eff  ' () = 0, where we have defined an effective potential by V_eff() = V() + e^α()ρ_c. The fluid obeys ∇_μ(ρ_cu_c^μ) = 0, and u_c^ν∇_νu_c^μ = –(g^μν + u_c^μu_c^ν)∇_να.

2. Adiabatic regime

The effective potential V_eff() may have a minimum resulting from the competition between the two distinct terms. If the timescale or lengthscale for to adjust to the changing position of the minimum of V_eff is shorter than that over which the background density changes, the field will adiabatically track this minimum [2]. In this case, the cold dark matter (CDM) component together with together act as a single fluid with an effective energy density ρ_eff and effective pressure p_eff:

Here, _m(ρ_c) is the solution of the algebraic equation

for . Eliminating ρ_c between equations (5) and (6) gives the equation of state p_eff = p_eff(ρ_eff).

For cosmological background solutions, we assume that the coupled fluid acts as the source of the cosmic acceleration. In the adiabatic approximation, the effective fluid description is valid for the background cosmology and for linear and nonlinear perturbations. Therefore, the equation of state of perturbations is the same as that of the background cosmology, and the matter and scalar field evolve as one effective fluid, obeying the usual fluid equations of motion with the given effective equation of state.

A necessary condition for the validity of the adiabatic approximation is that the lengthscales or timescales over which the density ρ_c varies are large compared to the inverse of the effective mass

of the scalar field. More precisely, we can show that the condition is [8]

this condition is necessary to justify dropping the terms involving the gradient of from the fluid and Einstein equations. In most situations the logarithmic derivative factor is of the order of unity and can be neglected. In [8], we also derive a non-local sufficient condition for the validity of the approximation, which generalizes conditions in the literature for the chameleon (thin-shell condition) [9, 10] and f(R) modified gravity [12] models. Condition (9) is not very stringent; many dark energy models admit regimes where it is satisfied for the background and for linearized perturbations over a range of scales.

In the adiabatic regime, the inferred dark energy equation of state parameter in the case α_b = 0 is

with α₀≡α(₀) the value today. Thus, w is precisely –1 today, and generically satisfies w < –1 in the past [2, 8].

3. Adiabatic instability

We write the potential V() as a function V(α) of the coupling function α() by eliminating . This gives, from equations (5) and (7),

The square of the adiabatic sound speed, is then given by

In the adiabatic regime the effective sound speed, relating to local perturbations in pressure and density, c_s²(k,a)≡δP(k,a)/δρ(k,a), tends towards the adiabatic sound speed and is always negative, since dV/dα must be negative so that equation (7) admits a solution, and d²V/dα² must be positive so that (8) yields a positive m_eff². From here in, we consider the regime in which this adiabatic limit has been reached and take c_s² = c_a².

Consider now, a perturbation with lengthscale . In order to be in the adiabatic regime we require . The negative sound speed squared will cause an exponential growth of the mode, as long as the growth timescale is short compared to the local gravitational timescale . Combining equations (5), (7) and (8) yields c_s²m_eff² = (α ')²V_{, α} = (α ')²ρ_eff/(V/V_{, α}–1), and therefore the instability will operate in the range of lengthscales given by

Here, the quantity d lnV/dα(α) on the right hand side is expressed as a function of using α = α(), and then as a function of the density using  = _m(ρ_c). In order for this range of scales to be non-empty, the dimensionless coupling m_p|α '| must be large compared to unity, i.e. the scalar mediated interaction between the dark matter particles must be strong compared to gravity (see also [13, 14, 15]).

There are two different ways of describing and understanding the instability, depending on whether one thinks of the scalar-field mediated forces as `gravitational' or `pressure' forces. In the Einstein frame, the instability is independent of gravity, since it is present even when the metric perturbation due to the fluid can be neglected. In the adiabatic regime the acceleration due the scalar field is a gradient of a local function of the density, which can be thought of as a pressure. The net effect of the scalar interaction is to give a contribution to the specific enthalpy h(ρ_c) = ∫dp/ρ_c of any fluid which is independent of the composition of the fluid. If the net sound speed squared of the fluid is negative, then there exists an instability in accord with our usual hydrodynamic intuition.

In the Jordan frame description, however, the instability involves gravity. The effective Newton's constant describing the interaction of dark matter with itself is

where k is a spatial wavevector [8]. At long lengthscales the scalar interaction is suppressed and G_cc≈G. At short lengthscales, the scalar field is effectively massless and G_cc asymptotes to a constant. However, when m_p|α '|  1 there is an intermediate range

over which the effective Newton's constant increases like G_cck². This interaction behaves just like a (negative) pressure in the hydrodynamic equations. This explains why the effect of the scalar interaction can be thought of as either pressure or gravity in the range of scales (15). Note that the range of scales (15) coincides with the range (13) derived above, up to a logarithmic correction factor.

From this second, Jordan-frame point of view, the instability is simply a Jeans instability. In a cosmological background the CDM fractional density perturbation traditionally exhibits power-law growth on subhorizon scales because Hubble damping competes with the exponential (Jeans) instability one might expect on a timescale of . In our case, however, the gravitational self-interaction of the mode is governed by G_cc(k) instead of G, and consequently in the range (15) where G_cc  G the timescale for the Jeans instability is much shorter than the Hubble damping time. Therefore, the Hubble damping is ineffective and the Jeans instability causes approximate exponential growth.

4. Examples of models

For single component dark matter models, one can find coupled models in the adiabatic regime [2, 12]. However, in the strong coupling limit m_p|α '|  1 of interest here, they typically do not yield acceptable background cosmologies. Therefore, we focus on composite dark matter models.

As a first example, we consider a constant coupling function and an exponential potential

where and, C  <  0 and λ are constants. The Friedmann equation in the adiabatic limit is then 3m_p²H² = V  +  e^αρ_co + ρ_c, in which the first two terms on the right hand side act like a fluid that, for |C|  1, approaches a cosmological constant. Thus, the background cosmology is close to ΛCDM for large enough |C|. Since the fraction of coupled dark matter is Ω_co = e^αρ_co/(3m_p²H²), in the asymptotic adiabatic regime, Ω_co = λ(1–Ω_c)/(λ–βC), and Ω_c~0.3 today, Ω_co must be small for large coupling, |C|  1. If the parameters of the model are chosen so that Ω_c~1 today, then the maximum and minimum lengthscales for the instability are and . Taking the Jeans view, it is then possible to show [8] that the instability should operate whenever modes are inside the horizon and in this range.

These expectations are confirmed (figure 1) by a numerical analysis of a two component coupled model. We use λ = 2, C = –20, H₀ = 70 km s^–1 Mpc^–1, baryon fractional energy density, Ω_b = 0.05, uncoupled CDM component, Ω_c = 0.2, coupled component, Ω_co = 0.05, and potential fractional energy density, Ω_V = 0.7. We fix initial conditions of /m_p = 10^–10 and at a = 10^–10 (initial conditions at least within /m_p = 10^–30–1 give the same evolution because of a scalar dynamical attractor), and assume that the CDM components have the same initial fractional density perturbations δ_c = δ_co, fixed by the usual adiabatic initial conditions. As shown in the bottom panel of figure 1, the background evolution is consistent with a ΛCDM like scenario, with w_eff = –0.69 today, approaching w_eff~–0.89 asymptotically. In the top panel, we see that once the scalar field has entered the adiabatic regime, giving rise to accelerative expansion, the density perturbations undergo significantly increased growth, in stark contrast to the ΛCDM scenario in which accelerative expansion is typically associated with late-time suppression of growth.

Figure 1. Bottom panel: the two component coupled dark energy (CDE) model, with λ = 2 and coupling C = -20 with H0 = 70 km s–1 Mpc–1, Ωb = 0.05, Ωc = 0.2, Ωco = 0.05, and ΩV = 0.70. At late times the scalar field finds the adiabatic minimum with asymptotic equation of state, and sound speed  = -1/(1 + γ) = -0.89, able to reproduce a viable background evolution consistent with supernovae, cosmic microwave background (CMB) angular diameter distance and big bang nucleosynthesis (BBN) expansion history constraints. The figure shows the evolution of the effective equation of state, weff = Ptot/ρtot = (2/3)(d lnt/d lna)- 1 (black full line), adiabatic speed of sound, , (blue long dashed line) and effective speed of sound for cs2 = δPtot/δρtot at k = 0.01 Mpc–1 (red dot-dashed line). The effective equation of state for a comparable ΛCDM model with Ωc = 0.25, Ωb = 0.05 and ΩΛ = 0.7 is also shown (black dashed line). Top panel: the growth of the fractional over-density δ = δρ/ρ for k = 0.01 Mpc–1 for the coupled CDM component, δco, (red long dashed line) and uncoupled component, δc, (black full line) in comparison to the growth for the ΛCDM model (black dashed line). At late times the adiabatic behavior triggers a dramatic increase in the rate of growth of both uncoupled and coupled components, leading to structure predictions inconsistent with observations.

In summary, these models provide a class of theories for which the background cosmology is compatible with observations, but which are ruled out by the adiabatic instability of the perturbations.

Another interesting class is the chameleon models [9, 10] for which the adiabatic regime has been previously demonstrated in static solutions for macroscopic bodies like the Earth, and also in cosmological models [11]. One well-studied example of these has inverse power law potentials, together with the constant coupling function in (16), for which the effective potential is then

where M is a mass scale and, n > 0 and λ are constants. The existence of a local minimum, and hence an adiabatic regime, in (17) requires C  <  0. We shall restrict attention to the regime ρ_c  ρ_crit≡nλM⁴(–βCM/m_p)ⁿ. The sound speed squared is

which is always negative as expected.

The range of spatial scales over which the instability operates for a given density ρ_c  ρ_crit is non-empty for β|C|  1, and is given by

If behaves as dark energy, we require ρ_crit~H₀²m_p². Then for ρ_c~ρ_crit, the maximum lengthscale is of order H₀^–1, and the minimum is ~(H₀β|C|)^–1. Thus, a large set of cosmological models are in the unstable regime at ρ_c~ρ_crit (if β|C|  1), ruling them out in this regime.

In this paper, we have demonstrated the existence and broad applicability of the adiabatic instability—operating in models in which there exists a nontrivial coupling between dark matter and dark energy. We have presented general expressions for the conditions under which the adiabatic instability is relevant, and, when so, the lengthscales over which it operates. This work provides a new way to constrain interactions in the dark sector, and heavily restricts the class of models consistent with cosmic acceleration.

In a companion paper [8], we derive in detail the results presented in this letter, and apply the results to a wide class of coupled models including couplings to both CDM and neutrinos.

Acknowledgments

We thank Ole Bjaelde, Anthony Brookfield, Steen Hannestad, Carsten Van der Bruck, Ira Wasserman and Christoph Wetterich for useful discussions. RB's work is supported by National Science Foundation grants AST-0607018 and PHY-0555216, EF's by NSF grants PHY-0457200 and PHY-0555216, and MT's by NSF grant PHY-0354990 and by Research Corporation.

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