Interacting dark energy: Constraints and degeneracies

Timothy Clemson,1 Kazuya Koyama,1 Gong-Bo Zhao,1 Roy Maartens,1,2 and Jussi Valiviita3

1Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, United Kingdom2Department of Physics, University of Western Cape, Cape Town 7535, South Africa

3Institute of Theoretical Astrophysics, University of Oslo, N-0315 Oslo, Norway(Received 13 October 2011; published 9 February 2012)

In standard cosmologies, dark energy (DE) interacts only gravitationally with dark matter (DM). There

could be a nongravitational 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 cosmic

microwave background, baryon acoustic oscillation and supernova 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 cosmic microwave background 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 integrated Sachs-Wolfe signature is suppressed since the nonstandard

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.

DOI: 10.1103/PhysRevD.85.043007 PACS numbers: 98.80.Es, 95.35.+d, 95.36.+x

I. INTRODUCTION

The late-time acceleration of the expansion of theUniverse demands explanation and observational verifica-tion. Currently observational tests of the standard CDMcosmological model are not precise enough to adequatelyrule out the wide variety of alternative dark energy (DE)models that have been proposed to explain the data. Insteadit is necessary to obtain constraints on the free parametersof such models and find ways to distinguish between themusing observations. One way to test CDM is to describeDE as an effective fluid and promote its equation of state wto a free parameter. This is known as wCDM and allowsdeviations from the standard value of w ¼ 1; w is stillonly measured to about 5%–10% accuracy and has a best-fit value of w<1 for some data sets [1].

Alternatively, onemay go beyondwCDM by introducinga new parameter to quantify interactions within the darksector. It is natural to expect some new physics in the darksector given the richness of interactions between species inthe standard model of particle physics [2]. Indeed takingdark sector interactions to be zero would be an additionalassumption of themodel.Models of dark sector interactions(see e.g. [3–76]) have little concrete guidance from particlephysics, but by studying possible interactions and confront-ing them with observations we can shedlight on questions such as which models may lead to un-physical behavior and which are in best agreement withobservations.

Further motivation for interacting dark energy (IDE)models includes: (1) they may alleviate the coincidenceproblem of explaining why the domination of DE roughlycoincides with the formation of large-scale structure;

(2) IDE affects structure formation and therefore providesa new way to modify the predictions of the standard non-interacting model.Here we investigate a particular version of the interac-

tion model used in [77–85]. The general form of thisinteraction in the background is , where is constant.This has also been used to describe particle decays in othercontexts [86–88]. Interactions of the form H (with constant) (see e.g. [89] and references therein) appearsimilar, but they mean that the interaction at any event isaffected by the global expansion rate H, as opposed to thelocally determined interaction .We describe the IDE model in the background universe

in Sec. II and in the perturbed universe in Sec. III. InSec. IV we investigate the effects of the interaction onthe cosmic microwave background (CMB) and matterpower spectra. We find the best-fit models using CMB,baryon acoustic oscillation (BAO) and supernova (SNIa)data, and we discuss the behavior of some typical modelsand the implications for the growth of large-scale structurein the Universe. Our conclusions are in Sec. V.

II. IDE IN THE BACKGROUND

An IDE model is characterized in the background by theenergy transfer rate Qx ¼ Qc:

0c ¼ 3H c þ a Qc; (1)

0x ¼ 3H ð1þ wÞ x þ a Qx; (2)

where primes denote derivatives with respect to conformaltime , H ¼ d lna=d and w ¼ px=x. We can define an

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effective DE equation of state parameter to be that of anoninteracting DE with the same xðaÞ, i.e.

weff ¼ w a Qx

3H x

: (3)

We can see from Eq. (3) that weff can be dynamical even ifw is a constant. Interestingly weff can be less than 1, orcross 1 during its evolution if Qx > 0, even though witself is always greater than 1.

A simple model for Qx is a linear function in the darksector energy densities. IDE with Qx / c has been studiedin the greatest detail [77–84]. However, for constant w themodel suffers from an instability [79]. This instabilityarises because the model of DE as a fluid with constantw is nonadiabatic. The instability can be cured by allowingw to vary in time [83].

Here we study the version with Qx / x,

Q x ¼ Qc ¼ x; (4)

where is a constant transfer rate. The strength of theinteraction is measured by jj=H0. > 0 corresponds toenergy transfer from DM ! DE. This appears somewhatunnatural, since the energy transfer is proportional to x.For < 0, the interaction can be seen in the background asa decay of DE into dark matter (DM), which is a morenatural model. The solution of (2) is [80]

x ¼ x0a3ð1þwÞ exp½ðt t0Þ; (5)

which shows that > 0 leads to exponential growth of DE.By (1), it follows that c eventually becomes negative. Themodel breaks down if this happens before the current time,which is possible for large =H0. Observational constraintsrequire =H0 & 1, so that typically the DM density onlybecomes negative in the future. In this case, we can treatthe model as a viable approximation, for the past history ofthe Universe, to some more complicated interaction thatavoids the blowup of DE in the future. The DE ! DMdecay model, with < 0, does not have this problem: bothenergy densities remain positive at all times when evolvingforward from physical initial conditions [80]. Furthermore,the < 0 case includes the possibility of beginning withno DM present and having it created entirely from thedecay of DE.

We use a phenomenological fluid model for DE, inwhich we treat w and the soundspeed cs as arbitraryparameters. This is a commonly used model for noninter-acting DE, where the model is known as wCDM. Weimpose the condition w 1 to avoid ‘‘phantom’’ insta-bilities that can arise in scalar field models of DE [90,91].The limiting case w ¼ 1 is admitted by the backgroundequations, but the perturbation equations have singularities(see below). Therefore we assume

w>1; w ¼ const: (6)

For completeness, we consider also the w 1 case inAppendix A. In the background, the < 0 case appears to

be better motivated. However, the analysis of perturbations(see below) shows that these models suffer from an insta-bility when w>1. The > 0 models avoid thisinstability.

III. IDE IN THE PERTURBED UNIVERSE

The critical difference between the background andperturbed IDE is that there is nonzero momentum transferin the perturbed universe. As emphasized in [79], a modelfor energy and momentum transfer does not follow fromthe background model—and a covariant and gauge-invariant approach is essential to construct a physicallyconsistent model for energy-momentum transfer.

A. General IDE

We give a brief summary of the general discussion in[79]. The perturbed Friedmann metric in a general gauge is

ds2 ¼ a2fð1þ 2Þd2 þ 2@iBddxi

þ ½ð1 2c Þij þ 2@i@jEdxidxjg: (7)

Each fluid A satisfies an energy-momentum balanceequation,

rTA ¼ Q

A ; AQA ¼ 0; (8)

where the second condition expresses conservation of thetotal energy-momentum tensor. For dark sector interac-tions, the energy-momentum transfer four-vectors satisfy

Qx ¼ Q

c : (9)

We split QA relative to the total four-velocity u, so that

QA ¼ QAu

þFA ; QA ¼ QA þ QA; uF

A ¼ 0;

(10)

where QA is the energy density transfer rate relative tou and F

A is the momentum density transfer rate rela-

tive to u. To first order

FA ¼ a1ð0; @ifAÞ; (11)

where fA is the (gauge-invariant) momentum transferpotential.We choose each u

A and the total u as the unique four-

velocity with zero momentum density, i.e.

TAu

A ¼ Au

A ; T

u ¼ u; (12)

A ¼ A þ A; AA ¼ þ : (13)

Then we have

uA ¼ a1ð1; @ivAÞ; u ¼ a1ð1; @ivÞ;(14)

ðAA þ ApAÞv ¼ AðA þ pAÞvA; (15)

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where vA, v are the peculiar velocity potentials. Equations(10) and (14) imply that

QA0 ¼ a½ QAð1þÞ þ QA; (16)

QAi ¼ a@i½ QAðvþ BÞ þ fA: (17)

The evolution equations for A and the velocity pertur-bation

A ¼ k2ðvA þ BÞ; (18)

are [79]:

0A þ 3H ðc2sA wAÞA þ ð1þ wAÞAþ 9H 2ð1þ wAÞðc2sA c2aAÞ

Ak2

3ð1þ wAÞc 0 þ ð1þ wAÞk2ðB E0Þ

¼ a QA

A

A þ 3H ðc2sA c2aAÞ

Ak2

þ a

A

QA;

(19)

0A þH ð1 3c2sAÞA c2sAð1þ wAÞ k

2A k2

¼ a

ð1þ wAÞ A

f QA½ ð1þ c2sAÞA k2fAg: (20)

Here csA is the physical soundspeed, defined by c2sA ¼ðpA=AÞrestframe, and caA is the adiabatic soundspeed,defined by c2aA p0

A= 0A. For the adiabatic DM fluid,

c2sc ¼ c2ac ¼ wc ¼ 0. By contrast, the DE fluid is nonadia-batic: c2ax ¼ w< 0 and so cax cannot be the physicalsoundspeed. The physical soundspeed for the fluid DEmodel is a phenomenological parameter. It must be realand non-negative to avoid unphysical instabilities. Wechoose csx ¼ 1, which is the soundspeed for quintessence(a self-consistent model of DE). Our analysis is insensitiveto the value of csx, as long as csx is close to 1, so that DEdoes not cluster significantly on sub-Hubble scales. (See[79] for more details).

B. DM-baryon bias from IDE

In IDE models, the DE exerts a drag on DM but not onbaryons. This leads to a linear DM-baryon bias in the late-time density perturbations, and in general also to a velocitydifference [82]. For baryons after decoupling

0b þ b 3c 0 þ k2ðB E0Þ ¼ 0;

0b þHb k2 ¼ 0:(21)

Thus for noninteracting DE models,

c b ¼ ðc bÞi aia (22)

We can choose ðb cÞi ¼ 0, so that

c b ¼ 0; c b ¼ ðc bÞi: (23)

Thus in standard DE models, there is no DM-baryonvelocity difference, and any linear density perturbationdifference is determined by initial conditions.For IDE models, the interaction induces a nonconstant

difference between c and b—which is degenerate withthe standard galaxy bias. The Euler equation for DMis (20), with c2sc ¼ wc ¼ 0. This differs from the stan-dard Euler equation unless k2fc ¼ Qð cÞ, whichfollows only for Q

c ¼ Qcuc , regardless of the form of

Qc [82]. In those models that modify the Euler equationfor DM, there will also be a velocity bias. Equations (19)and (20) imply

ðc bÞ0 þ a Qc

c

ðc bÞ þ ðc bÞ

¼ a

c

½Qc þ Qcð bÞ; (24)

ðc bÞ0 þH þ a Qc

c

ðc bÞ

¼ a

c

½k2fc þ Qcð bÞ: (25)

Thus there will be a velocity bias, unless Qc ¼ Qcu

c .

C. Our IDE models

The preceding equations are completely general. Achoice must now be made for the energy-momentum trans-fer in the dark sector. Firstly, the nature of the backgroundenergy transfer suggests that we take

Qx ¼ x ¼ xð1þ xÞ ¼ Qc; (26)

where A A= A. Thus we are treating as a universalconstant. For the momentum transfer, the simplest physicalchoice is that there is no momentum transfer in the restframe of either DM or DE [79,82]. This leads to two typesof model, with energy-momentum transfer four-vectorsparallel to either the DM or the DE four-velocity:

Qx ¼ Qxu

c ¼ Q

c type Qjjuc; (27)

Qx ¼ Qxu

x ¼ Q

c type Qjjux: (28)

Thus

Qx ¼ a x½1þ x þ; @iðvA þ BÞ; (29)

where A ¼ c, x for type Q k uc, Q k ux. By (17), themomentum transfer relative to the background frame is

fx ¼ xðvc vÞ ¼ fc for Qjjuc; (30)

fx ¼ xðvx vÞ ¼ fc for Qjjux: (31)

For both the Q k uc and Q k ux models, the densityperturbation (continuity) equation (19) reduces to

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0c þ c 3c 0 þ k2ðB E0Þ ¼ a

x

c

ðc x Þ;(32)

0x þ 3H ð1 wÞx þ ð1þ wÞx þ 9H 2ð1 w2Þx

k2

3ð1þ wÞc 0 þ ð1þ wÞk2ðB E0Þ¼ a

þ 3H ð1 wÞ x

k2

: (33)

The velocity perturbation (Euler) equations are howeverdifferent. For the Q k uc model, (20) gives

0c þHc k2 ¼ 0; (34)

0x 2Hx k2x

ð1þ wÞ k2 ¼ a

ð1þ wÞ ðc 2xÞ:(35)

For the Q k ux model:

0c þHc k2 ¼ ax

c

ðc xÞ; (36)

0x 2Hx k2x

ð1þ wÞ k2 ¼ axð1þ wÞ : (37)

It follows that the Euler equation for DM in the Qjjucmodel has the standard form, whereas it is modified inthe Qjjux model.

D. Instability

There is an obvious instability in the Euler equations forDE, (35) and (37), as w ! 1. Thus we must exclude thevalue w ¼ 1. This instability is different from that for adynamical DE model with w crossing 1, in which casethe DE perturbation is well-defined, but at least one moredegree of freedom is required, usually leading to its inter-pretation as a sign of modified gravity. Here though, theDE is not dynamical and the DE perturbation is ill-defined at w ¼ 1.

These equations also reveal an instability for w 1 incertain regions of parameter space. The underlying causeof this instability is the choice of c2sx ¼ 1, which meansthat the DE fluid is nonadiabatic, as discussed above. It isqualitatively similar to the instability first discovered forconstantw IDE in [79]. (See also [34,92–96] for the case ofmodels with replaced by H). This is a DE velocityinstability, which then drives an instability in the DE andDM density perturbations.

On large scales, we can drop the x and terms in theDE Euler equations (35) and (37). In (35) we can also setc ¼ 0 by (34). Then we can integrate to find that

x

ð¼0Þx

¼ exp

1þ wðt t0Þ

; (38)

where ð¼0Þx is the DE velocity in the noninteracting case,

and ¼ 2, 1 for Qjjuc, Qjjux. It follows that

ð1þ wÞ> 0 ) instability: (39)

Note that although one can choose a reference frame wherex 0, the instability is still present in the velocity differ-ence, which is gauge invariant. Given our assumption (6),the stable models must have positive , i.e.

w>1 and > 0 ) noinstability; (40)

for bothQjjuc andQjjux. This defines for us the physicallyacceptable models. In Appendix Awe allow for any sign of and 1þ w. In order for the instability to affect signifi-cantly the perturbation evolution by today, the time scale ofgrowth of x in (38) should be shorter than the Hubbletime, i.e., the models with

H0ð1þ wÞ *1 for Qjjuc;2 for Qjjux:

(41)

may not be viable. The results from our full parameter scanconfirm this (see the excluded wedges near to w ¼ 1 inFig. 9).

IV. ANALYSIS

The evolution of wCDM models was computed nu-merically using a modified version of the CAMB

BOLTZMANN code [97], including implementation of the

initial conditions derived in Appendix B. The code wasadapted (a) to allow for the nonstandard background evo-lution caused by the interactions; (b) to evolve the DMvelocity perturbation (ordinarily set to zero); (c) to sup-press perturbations when j1þ wj< 0:01 due to the blowupof terms in (35) and (37) as w ! 1. It is useful to includethe w ¼ 1 limit for comparison with CDM.Insight into the physical implications of the interaction

can be gained by running the modified CAMB code withfixed input parameters, varying only the interaction rate .Figure 1 shows the CMB power spectrum for three valuesof with all other cosmological parameters set to typicalvalues (see Table I for details).Positive describes a transfer of energy from DM to

DE, so with fixed c today, the DM energy density wouldhave been correspondingly greater in the past than withoutinteractions. Hence the amplitude of the CMB power spec-trum is decreased and the position of the peaks shifted,since a larger proportion of DM at early times implies asmaller amount of baryonic matter and therefore a moresignificant effect from photon driving before decoupling.The present-day matter power spectrum for these choicesof shows that a relative increase in the past DM densitynaturally leads to more structure formation and an increasein the amplitude of the matter power spectrum.

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The modified CAMB code was integrated into theCOSMOMC [98] Markov-Chain Monte Carlo (MCMC) code

in order to explore the parameter space (see Appendix Afor 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 b from big-bang nucleosynthesis [102]. Figure 2 shows the 68% and95% likelihood contours in the w- plane for the twodifferent momentum transfer models Qjjuc and Qjjux,where all other parameters have been marginalized over.

The likelihood regions are very similar for the twomodels since they differ only in their perturbations andthe observations predominantly constrain the backgroundevolution. The best-fit values for the Qjjuc and Qjjuxmodels are different due to the integrated Sachs-Wolfe(ISW) effect on the CMB, as shown in Table I. For theQjjux model the best-fit value is a genuine global maxi-mum. For the Qjjuc model however the mean likelihoodfunction of w and is essentially one-tailed, with the trueglobal maximum lying outside of the region we considerphysical (see Appendix A). Indeed the 2 of this point isclose to that of CDM (see Table I). This is becauseQjjucmodels in this region can closely mimic the ISW signatureof CDM.

The CMB data is best fit by a particular ISW signal, andthe twomomentum transfermodels differ somewhat in theirstructure formation histories. DM in the Qjjux model re-ceives a change inmomentum from theDEperturbations, asexpressed by its modified Euler equation (34), leading tomore structure growth relative to the Qjjuc model. Thismeans that DE can be weaker for the Qjjux model in orderto give the same amount of ISW signal as theQjjuc model.In order to assess the relative merits of these models we

have included the change in 2 from aCDM baseline. Tohelp put this quantity into context we have also includedtwo best-fit CDM models with H0 fixed at 69 and70 km=s=Mpc. The mean likelihoods of the samples varylittle in the direction of the degeneracy in the w- plane.For example, the difference in 2 between the Qjjuxbest-fit and the CDM best-fit is less than the differencebetween the two fixed-H0 CDM models (CDM69 andCDM70 in Table I).In Fig. 6, we show the effective DE equation of state and

the a3-scaled energy density for DM for a selection ofmodels in comparison with CDM. Interestingly, we findthat the weff for the best-fit wCDM models with w ¼0:85 and0:95 crosses1 during its evolution, showinga quintomlike behavior [103].

FIG. 1. CMB and total matter power spectra from the modified CAMB code for 3Qjjuc models with different values of but identicalvalues of their remaining parameters (see wCDM A, B, C in Table I).

TABLE I. Cosmological parameters for IDE models (see Appendix A for more general constraints).

Model QA 2 =H0 w H0 bh

2 ch2 ns As rei

CDM best-fit - 0 - 1 69.8 0.0223 0.113 0.960 2:16 109 0.0844

CDM69 - 0.774 - 1 69.0 0.0221 0.114 0.958 2:18 109 0.0855

CDM70 - 0:0200 - 1 70.0 0.0224 0.112 0.962 2:16 109 0.0844

wCDM best-fit - 0:220 - 1:03 70.7 0.0222 0.113 0.960 2:18 109 0.0883

wCDM A Qjjuc - 0 0:98 70.0 0.0226 0.112 0.960 2:10 109 0.0900

wCDM B Qjjuc - 0.2 0:98 70.0 0.0226 0.112 0.960 2:10 109 0.0900

wCDM C Qjjuc - 0.4 0:98 70.0 0.0226 0.112 0.960 2:10 109 0.0900

wCDM 1a Qjjuc 0:00830 0.4 0:95 70.9 0.0222 0.0702 0.961 2:16 109 0.0816

wCDM 1b Qjjuc 0.702 0.7 0:85 70.0 0.0223 0.0311 0.963 2:15 109 0.0832

wCDM 2a Qjjux 0:236 0.4 0:95 71.0 0.0224 0.0701 0.966 2:19 109 0.0870

wCDM 2b Qjjux 0:0420 0.7 0:85 70.2 0.0224 0.0305 0.966 2:15 109 0.0819

0, w 1 best-fit Qjjuc 0:0522 0.366 0:964 71.0 0.0224 0.0748 0.963 2:18 109 0.0849

0, w 1 best-fit Qjjux 0:322 0.798 0:851 70.4 0.0224 0.0194 0.965 2:18 109 0.0870

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We have focused on the stable > 0 models withw>1. These models do have a problem of negativeDM energy densities in the future, but we assume thatthis can be cured by a more realistic model to which ourmodel is a good approximation when c > 0.

Growth of structure

Models wCDM 1a, 1b, 2a, 2b (see Table I) wereselected for further study. COSMOMC was rerun with and w fixed, to obtain the best-fit values of the other non-derived parameters for input back into CAMB, namelybh

2, ch2, H0, ns (scalar spectral index), As (scalar

amplitude) and rei (optical depth of reionization).Figure 3 shows the CMB power spectrum of the best-fit

parameter sets for the chosen values of and w. The onlysignificant difference between the CMB spectra is in theISW feature, although this is not very large becauseCOSMOMC has fit them well to the data from WMAP7.

By contrast, there are dramatic differences between thetotal matter power spectra at z ¼ 0 for these models (seealso Fig. 4). We chose not to fit the matter power spectrumto observational data—because the modification to thegrowth of matter perturbations m due to the interactionsis degenerate with the galaxy-DM bias b in observations of

galaxy number density fluctuations: g ¼ bm. This de-

generacy is governed by Eqs. (24) and (25). Figure 3 doesnot include any bias.Note that c can be very small in models with large ,

since it can be compensated for by a higher w in order toobtain a sensible H0. This explains the correlation in the-w plane shown in Fig. 2, so the late-time effect of theDM may be proportionately even greater than one mightthink at first glance.The combination of similar ISW signatures and large

differences in the growth of structure is unusual—in aCDM cosmology for example, different growth rateslead to correspondingly dissimilar ISW signatures. Themechanisms behind this are clearest from the growth ofDM perturbations in the Newtonian limit: on sub-Hubblescales at late-times,

c ¼ c ; x ¼ 0 ¼ c 0 ¼ 0; (42)

in the Newtonian gauge (B ¼ 0 ¼ E). The evolution ofsynchronous gauge density perturbations in CAMB matchesthat of perturbations in the Newtonian gauge. The ISWeffect comes from gravitational potentials determined bythe Poisson equation,

w

Γ

−1 −0.9 −0.8 −0.70

0.2

0.4

0.6

0.8

1

Γ

Γ

w

Γ

−1 −0.9 −0.8 −0.70

0.2

0.4

0.6

0.8

1

Γ

Γ

FIG. 2. Smoothed 68% and 95% contours of the marginalized probability distribution for IDE model with Qjjuc (left) andQjjux(right) in the range of stability, w>1 and 0. Crosses identify models chosen to be analyzed in more detail (see Table I).

FIG. 3. CMB and total matter power spectra from the modified CAMB code for the WMAP7 wCDM best-fit values and thewCDM 1a, 1b, 2a, 2b models chosen from the 95% confidence range for further analysis (see Table I). The best-fit values ofstandard cosmological parameters were found using COSMOMC. Models 1a, 1b have ¼ 0:4H0 and Qjjuc while 2a, 2b have ¼ 0:7H0 and Qjjux.

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k2 ¼ 4Ga2ðcc þ bbÞ; (43)

and the left panel of Fig. 4 shows that there are indeed largedifferences between the models in their growth rates at latetimes. The reason the ISW effects can remain small forthese models is that the nonstandard background evolution(see Fig. 6) can counteract the growth of c in (43) and leadto relatively stable gravitational potentials. The right panelof Fig. 4 shows that the relevant combination, a2 cc, canremain comparable for models with very different structureformation histories such as those considered here. Notehow well the wCDM 1a model mimics the CDM be-havior of a2 cc, effectively leading to the same 2 (seeTable I).

This important feature of IDE models has implicationsfor any cosmological test which assumes a standard evo-lution of the DM energy density during matter domination,such as those for detecting deviations from GR. It may alsobe useful for distinguishing between IDE and modifiedgravity models [104], which have standard backgroundevolutions.

Using (1), (2), (32)–(34), and (43) and the Friedmannequation, H 2 ¼ 8Ga2tot=3, a velocity independentequation of motion for c can be derived for the Qjjucmodel:

00c þH

1 a

Hx

c

0c

¼ 4Ga2fbb þ cc

1þ 2tot

3c

a

Hx

c

2 3w

þ a

H

1þ x

c

: (44)

Thus the DM perturbations experience effectively differentvalues of H and G due to the interactions:

H eff

H¼ 1 a

Hx

c

; (45)

Geff

G¼1þ2tot

3c

a

Hx

c

23wþ a

H

1þx

c

: (46)

FIG. 4. Left: Normalized growth rates for CDM and the same best-fit models as in Fig. 3. Right: The same models but showing anormalized combination of a2 cc which is important for the ISW effect.

FIG. 5. Deviations from CDM of the effective Hubble parameter (left) and effective Newton constant for c (right), for the samebest-fit models as in Fig. 3.

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The Qjjux model by contrast has a nonstandard Eulerequation (36), and there remains a term proportional to xwhich can not in general be neglected:

00c þH

1 2

a

Hx

c

0c

¼ 4Ga2fbb þ cc

1þ 2tot

3c

a

Hx

c

2 3w

þ a

H

þ a

x

c

x: (47)

Nevertheless, for stable models x remains small enough tobe negligible and we can define the deviations from stan-dard growth due to the interactions via

H eff

H¼ 1 2

a

Hx

c

; (48)

Geff

G¼ 1þ 2tot

3c

a

Hx

c

2 3wþ a

H

: (49)

These equations show that the differences in momentumtransfer lead to a greater modification to the growth viaH eff for theQjjux model and viaGeff for theQjjuc model,as can be seen in Fig. 5. It is clear that DM perturbations inthe models with large couplings are already beginningto grow exponentially at the present day (compare[48,94,96,105]). In models with Q

x ¼ cu

c , as studied

in [79,83,84], there is no interaction source term in thesynchronous gauge version of (32) and so the DM pertur-bations are stable.

V. CONCLUSIONS

We have studied a model of dark sector interactions withan energy transfer proportional to the DE energy density,and with momentum transfer vanishing either in the DM orthe DE rest frame. We performed an MCMC analysis andfound the best-fit parameters using a data compilation that

predominantly constrains the background evolution. Wefound model constraints to which CDM is a good fit,although parameter degeneracies do allow for significantinteraction rates at the present day and even admit the twoextreme cases of zeroDMat early times and zeroDM today.We analyzed the growth of structure in this model and

found that the effects of large growth rates on the ISWsignature in the CMB can be suppressed by the nonstan-dard background evolution. We also showed that interac-tions can greatly enhance growth in these models viaeffective Hubble and Newton constants, in varying degreesdepending on the momentum transfer.There appears to be some tension between the back-

ground evolution and structure formation. The CMB, SNeand BAO data slightly favor interactions, while the growthrate of DM perturbations likely rules out large interactionrates. There is a degeneracy with galaxy bias, which de-serves further investigation. This would allow the use offull range of large-scale structure data and would signifi-cantly improve the constraints on the IDE models consid-ered here.Interacting models are known to be degenerate with

modified gravity models [71,74,82,105–107]. It is impor-tant to break this degeneracy, in order to strengthen cos-mological tests of GR—currently devised tests do notincorporate the possibility of a dark sector interaction.The key distinguishing features of IDE and modified grav-ity (MG) occur in: (1) the late-time anisotropic stress, i.e. c ; (2) the evolution of the background DM density,cð1þ zÞ3; (3) the DM-baryon bias:

MG IDE

c 0 ¼ 0cð1þ zÞ3 ¼ const constb c ¼ const constb c ¼ 0 Can be nonzero

These features are the basis for breaking the degeneracy.For example, any difference between the metric potentials

FIG. 6. Comparison with CDM of the effective DE equation of state (left) and the DM density (right), for the same best-fit modelsas in Fig. 3.

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can be tested via peculiar velocities (a probe of ), weaklensing and ISW (both sensitive to þ c ).

ACKNOWLEDGMENTS

T. C. is funded by a UK Science & Technology FacilitiesCouncil (STFC) PhD studentship. K.K., G. Z., R.M. aresupported by the STFC under Grant No. ST/H002774/1.R.M. is supported by a South African SKA ResearchChair, and by a NRF (South Africa)/Royal Society (UK)exchange grant. K. K. is supported by a European ResearchCouncil Starting Grant and the Leverhulme trust. J. V. issupported by the Research Council of Norway. Numericalcomputations were done on the Sciama High PerformanceCompute (HPC) cluster which is supported by the ICG,SEPNet and the University of Portsmouth.

APPENDIX A: MCMC ANALYSIS

Using COSMOMC we explore the full parameter space ofthe wCDM model, as illustrated in Fig. 7. The code was

modified to vary the two new parameters and w andfurther coding was necessary to ensure that models withnegative DM energy densities were rejected from theMCMC analysis. In addition, the scaling solution for the

background used to find the redshift of the two BAO datapoints was replaced by coding to take into account thenonstandard background evolution of the models.COSMOMC varies a parameter ¼ 100 times the ratio of

the sound horizon to the angular diameter distance, in placeof H0, because it is more efficient. However the derivationof also assumes a standard background evolution. Wetherefore chose not to use , but to constrain H0 directlyinstead. Note that the HST prior onH0 assumes a particularmodel of CDM for evolving Hðz ¼ 0:04Þ up to thepresent day and so has slight model dependence, whichwe neglect.There is a plane of degeneracy in the wc

parameter space which allows for an entire range of pos-sibilities from zero DM at early times to zero DM at thepresent day—see Figs. 7 and 8.

−3 −2 −1−6

−5

−4

−3

−2

−1

0

1

2

−3 −2 −1−6

−5

−4

−3

−2

−1

0

1

2ΓΓ

FIG. 7. 68% and 95% contours of the marginalized probability distribution for the Qjjuc model (left) and Qjjux model (right). Thedashed lines cross at the position of CDM, the crosses indicate the best-fits in each case and the circles indicate the median samples.Note that some areas appear only due to smoothing of the distributions (see Fig. 9).

Ω

−3 −2 −10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ω

−3 −2 −10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

FIG. 8. 68% and 95% contours of the marginalized probability distribution in them w plane for theQjjuc model (left) andQjjuxmodel (right).

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The cosmological parameters of the median and best-fitmodels from COSMOMC for w ¼ 1 and when the entireparameter space is considered are shown in Table II. ForQjjuc models, the ISW creates a preference in the meanlikelihood function for < 0, as was found previously forthe Q

c ¼ cu

c models [84].

For Qjjux models, there is a preference for > 0.Despite this the median samples have < 0 and w<1for both the Qjjuc and Qjjux models. The background datatherefore shows a slight preference for values of < 0 andw<1. Both the best-fits and the median samples how-ever are relatively close toCDM, given the wide range ofinteraction strengths allowed. The w ¼ 1 results areincluded here to show the proximity of to 0 for thesemodels, in line with CDM.

The singularity in the perturbations at w ¼ 1 leads usto impose j1þ wj< 0:01, allowing us to explore the entireparameter space. The effect of the w 1 instability (38)is illustrated in Fig. 9. The wedged gaps in the distributionof accepted MCMC chain steps are given by the boundariesof the instability region, defined by (41).

APPENDIX B: INITIAL CONDITIONS

In synchronous gauge, ¼ B ¼ 0 and ordinarily theresidual gauge freedom is eliminated by setting c ¼ 0.For the Qjjux model the interaction term in the DM Euler

equation (36) does not in general allow for c ¼ 0.However, since ’ H 0 H in (32)–(37), the interac-tions can be neglected at early times. Using 3c 0 þ k2E0 ¼h=2, where h is the synchronous gauge variable [108],the evolution equations used to find the initial conditionsfor the dark sector are

20c þ h0 ¼ 0; 0c ¼ 0; (B1)

0x þ 3H ð1 wÞx þ ð1þ wÞxþ 9H 2ð1 w2Þx

k2þ ð1þ wÞ h

0

2¼ 0; (B2)

0x 2Hx k2x

ð1þ wÞ ¼ 0: (B3)

The dominant growing mode solution for h found in [108]leads to the standard adiabatic initial conditions for DM,

ci ¼ 12h ¼ 1

2CðkÞ2; ci ¼ 0: (B4)

For DE, we find the leading order solutions, in agreementwith [109],

xi ¼ Cð1þ wÞk2212w 14

; xi ¼ Ck43

12w 14: (B5)

TABLE II. Cosmological parameters of the median and best-fit samples from COSMOMC for w ¼ 1 and when the entire parameterspace is considered.

Model QA 2 =H0 w H0 bh

2 ch2 ns As rei

w ¼ 1 CDM best-fit Qjjuc 0:146 0.154 1 70.8 0.0222 0.0974 0.959 2:16 109 0.0824

w ¼ 1 CDM best-fit Qjjux 0:0522 0.0916 1 70.0 0.0222 0.105 0.959 2:18 109 0.0852

all , all w best-fit Qjjuc 0:294 0:806 1:23 70.5 0.0222 0.180 0.956 2:17 109 0.0822

all , all w best-fit Qjjux 0:0879 0.302 0:951 70.1 0.0222 0.0823 0.959 2:18 109 0.0879

all , all w median Qjjuc - 1:01 1:29 70.7 0.0221 0.194 0.956 2:18 109 0.0841

all , all w median Qjjux - 0:578 1:19 70.7 0.0222 0.164 0.958 2:18 109 0.0840

FIG. 9. Distributions of accepted steps in the MCMC chains for the models with Qjjuc (left) and Qjjux (right).

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