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Interacting Dark Energy -- constraints and degeneracies

In standard cosmologies, dark energy interacts only gravitationally with dark matter. There could be a non-gravitational interaction in the dark sector, leading to changes in the effective DE equation of state, in the redshift dependence of the DM density and in structure formation. We use CMB, BAO and SNIa data to constrain a model where the energy transfer in the dark sector is proportional to the DE density. There are two subclasses, defined by the vanishing of momentum transfer either in the DM or the DE frame. We conduct a Markov-Chain Monte-Carlo analysis to obtain best-fit parameters. The background evolution allows large interaction strengths, and the constraints from CMB anisotropies are weak. The growth of DM density perturbations is much more sensitive to the interaction, and can deviate strongly from the standard case. However, the deviations are degenerate with galaxy bias and thus more difficult to constrain. Interestingly, the ISW signature is suppressed since the non-standard background evolution can compensate for high growth rates. We also discuss the partial degeneracy between interacting DE and modified gravity, and how this can be broken.


I. INTRODUCTION
The late-time acceleration of the expansion of the Universe demands explanation and observational verification. Currently observational tests of the standard ΛCDM cosmological model are not precise enough to adequately rule out the wide variety of alternative dark energy (DE) models that have been proposed to explain the data. Instead it is necessary to obtain constraints on the free parameters of such models and find ways to distinguish between them using observations. One way to test ΛCDM is to describe DE as an effective fluid and promote its equation of state w to a free parameter. This is known as wCDM and allows deviations from the standard value of w = −1; w is still only measured to about 5%-10% accuracy and has a best-fit value of w < −1 for some datasets [1].
Alternatively, one may go beyond wCDM by introducing a new parameter to quantify interactions within the dark sector. It is natural to expect some new physics in the dark sector given the richness of interactions between species in the standard model of particle physics [2]. Indeed taking dark sector interactions to be zero would be an additional assumption of the model. Models of dark sector interactions (see e.g. ) have little concrete guidance from particle physics, but by studying possible interactions and confronting them with observations we can shed light on questions such as which models may lead to unphysical behaviour and which are in best agreement with observations. Further motivation for interacting dark energy (IDE) models includes: (1) they may alleviate the coincidence problem of explaining why the domination of DE roughly coincides with the formation of large-scale structure; (2) IDE affects structure formation and therefore provides a new way to modify the predictions of the standard non-interacting model.
Here we investigate a particular version of the interaction model used in [77][78][79][80][81][82][83][84][85]. The general form of this interaction in the background is Γρ, where Γ is constant. This has also been used to describe particle decays in other contexts [86][87][88]. Interactions of the form αHρ (with α constant) (see e.g. [89] and references therein) appear similar, but they mean that the interaction at any event is affected by the global expansion rate H, as opposed to the locally determined interaction Γρ.
We describe the IDE model in the background universe in Sec. II and in the perturbed universe in Sec. III. In Sec. IV we investigate the effects of the interaction on the CMB and matter power spectra. We find the best-fit models using CMB, baryon acoustic oscillation (BAO) and supernova (SNIa) data, and we discuss the behaviour of some typical models and the implications for the growth of large-scale structure in the Universe. Our conclusions are in Sec. V.

II. IDE IN THE BACKGROUND
An IDE model is characterized in the background by the energy transfer rateQ x = −Q c : where primes denote derivatives with respect to conformal time τ , H = d ln a/dτ and w = p x /ρ x . We can define an effective DE equation of state parameter to be that of a non-interacting DE with the sameρ x (a), i.e.
We can see from Eq (3) that w eff can be dynamical even if w is a constant. Interestingly w eff can be less than −1, or cross −1 during its evolution ifQ x > 0, even though w itself is always greater than −1. A simple model forQ x is a linear function in the dark sector energy densities. IDE withQ x ∝ρ c has been studied in the greatest detail [77][78][79][80][81][82][83][84]. However, for constant w the model suffers from an instability [79]. This instability arises because the model of DE as a fluid with constant w is non-adiabatic. The instability can be cured by allowing w to vary in time [83].
Here we study the version withQ x ∝ρ x , where Γ is a constant transfer rate. The strength of the interaction is measured by |Γ|/H 0 . Γ > 0 corresponds to energy transfer from DM→DE. This appears somewhat unnatural, since the energy transfer is proportional toρ x . For Γ < 0, the interaction can be seen in the background as a decay of DE into DM, which is a more natural model. The solution of (2) is [80] which shows that Γ > 0 leads to exponential growth of DE. By (1), it follows thatρ c eventually becomes negative. The model breaks down if this happens before the current time, which is possible for large Γ/H 0 . Observational constraints require Γ/H 0 < ∼ 1, so that typically the DM density only becomes negative in the future. In this case, we can treat the model as a viable approximation, for the past history of the Universe, to some more complicated interaction that avoids the blow-up of DE in the future. The DE→DM decay model, with Γ < 0, does not have this problem: both energy densities remain positive at all times when evolving forward from physical initial conditions [80]. Furthermore, the Γ < 0 case includes the possibility of beginning with no DM present and having it created entirely from the decay of DE.
We use a phenomenological fluid model for DE, in which we treat w and the soundspeed c s as arbitrary parameters. This is a commonly used model for noninteracting DE, where the model is known as wCDM. We impose the condition w ≥ −1 to avoid 'phantom' instabilities that can arise in scalar field models of DE [90,91]. The limiting case w = −1 is admitted by the background equations, but the perturbation equations have singularities (see below). Therefore we assume For completeness, we consider also the w ≤ −1 case in Appendix A. In the background, the Γ < 0 case appears to be better motivated. However, the analysis of perturbations (see below) shows that these models suffer from an instability when w > −1. The Γ > 0 models avoid this instability.

III. IDE IN THE PERTURBED UNIVERSE
The critical difference between the background and perturbed IDE is that there is nonzero momentum transfer in the perturbed universe. As emphasized in [79], a model for energy and momentum transfer does not follow from the background model -and a covariant and gaugeinvariant approach is essential to construct a physically consistent model for energy-momentum transfer.

General IDE
We give a brief summary of the general discussion in [79]. The perturbed Friedmann metric in a general gauge is Each fluid A satisfies an energy-momentum balance equation, where the second condition expresses conservation of the total energy-momentum tensor. For dark sector interactions, the energy-momentum transfer four-vectors satisfy We split Q µ A relative to the total four-velocity u µ , so that where Q A is the energy density transfer rate relative to u µ and F µ A is the momentum density transfer rate relative to u µ . To first order where f A is the (gauge-invariant) momentum transfer potential. We choose each u µ A and the total u µ as the unique four-velocity with zero momentum density, i.e.
Then we have where v A , v are the peculiar velocity potentials. Equations (10) and (14) imply that The evolution equations for δ A and the velocity perturbation are [79]: Here c sA is the physical soundspeed, defined by c 2 sA = (δp A /δρ A ) restframe , and c aA is the adiabatic soundspeed, defined by c 2 aA ≡p ′ A /ρ ′ A . For the adiabatic DM fluid, c 2 sc = c 2 ac = w c = 0. By contrast, the DE fluid is nonadiabatic: c 2 ax = w < 0 and so c ax cannot be the physical soundspeed. The physical soundspeed for the fluid DE model is a phenomenological parameter. It must be real and non-negative to avoid unphysical instabilities. We choose c sx = 1, which is the sounspeed for quintessence (a self-consistent model of DE). Our analysis is insensitive to the value of c sx , as long as c sx is close to one, so that DE does not cluster significantly on sub-Hubble scales. (See [79] for more details.)

DM-baryon bias from IDE
In IDE models, the DE exerts a drag on DM but not on baryons. This leads to a linear DM-baryon bias in the late-time density perturbations, and in general also to a velocity difference [82]. For baryons after decoupling Thus for non-interacting DE models, Thus in standard DE models, there is no DM-baryon velocity difference, and any linear density perturbation difference is determined by initial conditions. For IDE models, the interaction induces a nonconstant difference between δ c and δ b -which is degenerate with the standard galaxy bias. The Euler equation for DM is (20), with c 2 sc = w c = 0. This differs from the standard Euler equation unless k 2 f c =Q(θ − θ c ), which follows only for Q µ c = Q c u µ c , regardless of the form of Q c [82]. In those models that modify the Euler equation for DM, there will also be a velocity bias. Equations (19) and (20) imply Thus there will be a velocity bias, unless Q µ c = Q c u µ c .

Our IDE models
The preceding equations are completely general. A choice must now be made for the energy-momentum transfer in the dark sector. Firstly, the nature of the background energy transfer suggests that we take where δ A ≡ δρ A /ρ A . Thus we are treating Γ as a universal constant. For the momentum transfer, the simplest physical choice is that there is no momentum transfer in the rest frame of either DM or DE [79,82]. This leads to two types of model, with energy-momentum transfer four-vectors parallel to either the DM or the DE fourvelocity: Thus where A = c, x for type Q u c , Q u x . By (17), the momentum transfer relative to the background frame is For both the Q u c and Q u x models, the density perturbation (continuity) equation (19) reduces to The velocity perturbation (Euler) equations are however different. For the Q u c model, (20) gives For the Q u x model: It follows that the Euler equation for DM in the Q u c model has the standard form, whereas it is modified in the Q u x model.

Instability
There is an obvious instability in the Euler equations for DE, (35) and (37), as w → −1. Thus we must exclude the value w = −1. This instability is different from that for a dynamical DE model with w crossing −1, in which case the DE perturbation is well-defined, but at least one more degree of freedom is required, usually leading to its interpretation as a sign of modified gravity. Here though, the DE is not dynamical and the DE perturbation is illdefined at w = −1.
These equations also reveal an instability for w = −1 in certain regions of parameter space. The underlying cause of this instability is the choice of c 2 sx = 1, which means that the DE fluid is non-adiabatic, as discussed above. It is qualitatively similar to the instability first discovered for constant w IDE in [79]. (See also [34,[92][93][94][95][96] for the case of models with Γ replaced by αH). This is a DE velocity instability, which then drives an instability in the DE and DM density perturbations.
On large scales, we can drop the δ x and φ terms in the DE Euler equations (35) and (37). In (35) we can also set θ c = 0 by (34). Then we can integrate to find that where θ is the DE velocity in the non-interacting case, and α = 2, 1 for Q u c , Q u x . It follows that Note that although one can choose a reference frame where θ x ≡ 0, the instability is still present in the velocity difference, which is gauge invariant. Given our assumption (6), the stable models must have positive Γ, i.e.
w > −1 and Γ > 0 ⇒ no instability, for both Q u c and Q u x . This defines for us the physically acceptable models. In Appendix A we allow for any sign of Γ and 1 + w. In order for the instability to affect significantly the perturbation evolution by today, the time scale of growth of θ x in (38) should be shorter than the Hubble time, i.e., the models with may not be viable. The results from our full parameter scan confirm this (see the excluded wedges near to w = −1 in Fig. 9).

IV. ANALYSIS
The evolution of ΓwCDM models was computed numerically using a modified version of the CAMB Boltzmann code [97], including implementation of the initial conditions derived in Appendix B. The code was adapted (a) to allow for the non-standard background evolution caused by the interactions; (b) to evolve the DM velocity perturbation (ordinarily set to zero); (c) to suppress perturbations when |1 + w| < 0.01 due to the blow up of terms in (35) and (37) as w → −1. It is useful to include the w = −1 limit for comparison with ΛCDM.
Insight into the physical implications of the interaction can be gained by running the modified CAMB code with fixed input parameters, varying only the interaction rate Γ. Figure 1 shows the CMB power spectrum for three values of Γ with all other cosmological parameters set to typical values (see Table I for details).
Positive Γ describes a transfer of energy from DM to DE, so with fixed Ω c today, the DM energy density would have been correspondingly greater in the past than without interactions. Hence the amplitude of the CMB power spectrum is decreased and the position of the peaks shifted, since a larger proportion of DM at early times implies a smaller amount of baryonic matter and therefore a more significant effect from photon driving before decoupling. The present-day matter power spectrum for these choices of Γ shows that a relative increase in the past DM density naturally leads to more structure formation and an increase in the amplitude of the matter power spectrum.
The modified CAMB code was integrated into the Cos-moMC [98] Markov-Chain Monte-Carlo (MCMC) code in order to explore the parameter space (see Appendix A for more details). The data used in the MCMC analysis were: CMB (WMAP7 [1]), BAO [99], HST [100], and SNIa (SDSS [101]) data, as well as a prior on Ω b from bigbang nucleosynthesis [102]. Figure 2 shows the 68% and 95% likelihood contours in the w − Γ plane for the two different momentum transfer models Q u c and Q u x , where all other parameters have been marginalized over.
The likelihood regions are very similar for the two models since they differ only in their perturbations    Table I).  Table I).
FIG. 3: CMB and total matter power spectra from the modified CAMB code for the WMAP7 wCDM best-fit values and the ΓwCDM 1a,1b,2a,2b models chosen from the 95% confidence range for further analysis (see Table I). The best-fit values of standard cosmological parameters were found using CosmoMC. Models 1a,1b have Γ = 0.4H0 and Q uc while 2a,2b have Γ = 0.7H0 and Q ux .  Table I. For the Q u x model the best-fit value is a genuine global maximum. For the Q u c model however the mean likelihood function of w and Γ is essentially one-tailed, with the true global maximum lying outside of the region we consider physical (see Appendix A). Indeed the χ 2 of this point is close to that of ΛCDM (see Table I). This is because Q u c models in this region can closely mimic the ISW signature of ΛCDM.
The CMB data is best fit by a particular ISW signal, and the two momentum transfer models differ somewhat in their structure formation histories. DM in the Q u x model receives a change in momentum from the DE perturbations, as expressed by its modified Euler equation (34), leading to more structure growth relative to the Q u c model. This means that DE can be weaker for the Q u x model in order to give the same amount of ISW signal as the Q u c model.
In order to assess the relative merits of these models we have included the change in χ 2 from a ΛCDM baseline. To help put this quantity into context we have also included two best-fit ΛCDM models with H 0 fixed at 69 and 70 km/s/Mpc. The mean likelihoods of the samples vary little in the direction of the degeneracy in the w − Γ plane. For example, the difference in ∆χ 2 between the Q u x best-fit and the ΛCDM best-fit is less than the difference between the two fixed-H 0 ΛCDM models (ΛCDM69 and ΛCDM70 in Table I).
In Fig. 6, we show the effective DE equation of state and the a 3 -scaled energy density for DM for a selection of models in comparison with ΛCDM. Interestingly, we find that the w eff for the best-fit ΓwCDM models with w = −0.85 and −0.95 crosses −1 during its evolution, showing a quintom-like behaviour [103].
We have focused on the stable Γ > 0 models with w > −1. These models do have a problem of negative DM energy densities in the future, but we assume that this can be cured by a more realistic model to which our model is a good approximation when ρ c > 0.

Growth of structure
Models ΓwCDM 1a,1b,2a,2b (see Table I) were selected for further study. CosmoMC was rerun with Γ and w fixed, to obtain the best-fit values of the other nonderived parameters for input back into CAMB, namely Ω b h 2 , Ω c h 2 , H 0 , n s (scalar spectral index), A s (scalar amplitude) and τ rei (optical depth of reionization). Figure 3 shows the CMB power spectrum of the bestfit parameter sets for the chosen values of Γ and w. The only significant difference between the CMB spectra is in the ISW feature, although this is not very large because CosmoMC has fit them well to the data from WMAP7. By contrast, there are dramatic differences between the total matter power spectra at z = 0 for these models (see also Fig. 4). We chose not to fit the matter power spectrum to observational data -because the modification to the growth of matter perturbations δ m due to the interactions is degenerate with the galaxy-DM bias b in observations of galaxy number density fluctuations: δ g = bδ m . This degeneracy is governed by equations (24) and (25). Figure 3 does not include any bias.
Note that Ω c can be very small in models with large Γ, since it can be compensated for by a higher w in order to obtain a sensible H 0 . This explains the correlation in the Γ − w plane shown in Fig. 2, so the late-time effect of the DM may be proportionately even greater than one might think at first glance.
The combination of similar ISW signatures and large differences in the growth of structure is unusual -in a ΛCDM cosmology for example, different growth rates lead to correspondingly dissimilar ISW signatures. The mechanisms behind this are clearest from the growth of DM perturbations in the Newtonian limit: on sub-Hubble scales at late-times, in the Newtonian gauge (B = 0 = E). The evolution of synchronous gauge density perturbations in CAMB matches that of perturbations in the Newtonian gauge. The ISW effect comes from gravitational potentials determined by the Poisson equation, and the left panel of Fig. 4 shows that there are indeed large differences between the models in their growth rates at late times. The reason the ISW effects can remain small for these models is that the non-standard background evolution (see Fig. 6) can counteract the growth of δ c in (43) and lead to relatively stable gravitational potentials. The right panel of Fig. 4 shows that the relevant combination, a 2ρ c δ c , can remain comparable for models with very different structure formation histories such as those considered here. Note how well the ΓwCDM 1a model mimics the ΛCDM behaviour of a 2ρ c δ c , effectively leading to the same χ 2 (see Table I).
This important feature of IDE models has implications for any cosmological test which assumes a standard evolution of the DM energy density during matter domination, such as those for detecting deviations from GR. It may also be useful for distinguishing between IDE and modified gravity models [104], which have standard background evolutions.
Using (1), (2), (32), (33), (34), (43) and the Friedmann equation, H 2 = 8πGa 2 ρ tot /3, a velocity independent equation of motion for δ c can be derived for the Q u c model: Thus the DM perturbations experience effectively different values of H and G due to the interactions: The Q u x model by contrast has a non-standard Euler equation (36), and there remains a term proportional to θ x which can not in general be neglected: Nevertheless, for stable models θ x remains small enough to be negligible and we can define the deviations from standard growth due to the interactions via These equations show that the differences in momentum transfer lead to a greater modification to the growth via H eff for the Q u x model and via G eff for the Q u c model, as can be seen in Fig. 5. It is clear that DM perturbations in the models with large couplings are already beginning to grow exponentially at the present day (compare [48,94,96,105]). In models with Q µ x = Γρ c u µ c , as studied in [79,83,84], there is no interaction source term in the synchronous gauge version of (32) and so the DM perturbations are stable.

V. CONCLUSIONS
We have studied a model of dark sector interactions with an energy transfer proportional to the DE energy density, and with momentum transfer vanishing either in the DM or the DE rest frame. We performed an MCMC analysis and found the best-fit parameters using a data compilation that predominantly constrains the background evolution. We found model constraints to which ΛCDM is a good fit, although parameter degeneracies do allow for significant interaction rates at the present day and even admit the two extreme cases of zero DM at early times and zero DM today.
We analyzed the growth of structure in this model and found that the effects of large growth rates on the ISW signature in the CMB can be suppressed by the nonstandard background evolution. We also showed that interactions can greatly enhance growth in these models via effective Hubble and Newton constants, in varying degrees depending on the momentum transfer.
There appears to be some tension between the background evolution and structure formation. The CMB, SNe and BAO data slightly favour interactions, while the growth rate of DM perturbations likely rules out large interaction rates. There is a degeneracy with galaxy bias, which deserves further investigation. This would allow the use of full range of large-scale structure data and would significantly improve the constraints on the IDE models considered here.
Interacting models are known to be degenerate with modified gravity models [71,74,82,[105][106][107]. It is important to break this degeneracy, in order to strengthen cosmological tests of GR -currently devised tests do not incorporate the possibility of a dark sector interaction. The key distinguishing features of IDE and modified gravity (MG) occur in: (1) the late-time anisotropic stress, i.e. φ − ψ; (2) the evolution of the background DM density, ρ c (1 + z) −3 ; (3) the DM-baryon bias: These features are the basis for breaking the degeneracy. For example, any difference between the metric potentials can be tested via peculiar velocities (a probe of φ), weak lensing and ISW (both sensitive to φ + ψ). Using CosmoMC we explore the full parameter space of the ΓwCDM model, as illustrated in Fig. 7. The code was modified to vary the two new parameters Γ and w and further coding was necessary to ensure that models with negative DM energy densities were rejected from the MCMC analysis. In addition, the scaling solution for the background used to find the redshift of the two BAO data points was replaced by coding to take into account the non-standard background evolution of the models. Cos-moMC varies a parameter θ = 100 times the ratio of the sound horizon to the angular diameter distance, in place of H 0 , because it is more efficient. However the derivation of θ also assumes a standard background evolution. We therefore chose not to use θ, but to constrain H 0 directly instead. Note that the HST prior on H 0 assumes a particular model of ΛCDM for evolving H(z = 0.04) up to the present day and so has slight model dependence, which we neglect.

Acknowledgments
There is a plane of degeneracy in the Γ−w−Ω c parameter space which allows for an entire range of possibilities from zero DM at early times to zero DM at the present day -see Figs. 7 and 8.
The cosmological parameters of the median and bestfit models from CosmoMC for w = −1 and when the entire parameter space is considered are shown in Table II. For Q u c models, the ISW creates a preference in the mean likelihood function for Γ < 0, as was found previously for the Q µ c = Γρ c u µ c models [84]. For Q u x models, there is a preference for Γ > 0. Despite this the median samples have Γ < 0 and w < −1 for both the Q u c and Q u x models. The background data therefore shows a slight preference for values of Γ < 0 and w < −1. Both the best-fits and the median samples however are relatively close to ΛCDM, given the wide range of interaction strengths allowed. The w = −1 results are included here to show the proximity of Γ to 0 for these models, in line with ΛCDM.
The singularity in the perturbations at w = −1 leads us to impose |1 + w| < 0.01, allowing us to explore the entire parameter space. The effect of the w = −1 instability (38) is illustrated in Fig. 9. The wedged gaps in the distribution of accepted MCMC chain steps are given by the boundaries of the instability region, defined by (41).

Appendix B: Initial Conditions
In synchronous gauge, φ = B = 0 and ordinarily the residual gauge freedom is eliminated by setting θ c = 0. For the Q u x model the interaction term in the DM Euler equation (36) does not in general allow for θ c = 0. However, since Γ ≃ H 0 ≪ H in (32)- (37), the interactions can be neglected at early times. Using 3ψ ′ + k 2 E ′ = −h/2, where h is the synchronous gauge variable [108], the evolution equations used to find the initial conditions for the dark sector are, The dominant growing mode solution for h found in [108] leads to the standard adiabatic initial conditions for DM, For DE, we find the leading order solutions, in agreement with [109], . (B5)