a large-scale stable interacting dark energy model

8/13/2019 A Large-scale Stable Interacting Dark Energy Model

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A large-scale stable interacting dark energy model: Cosmological perturbations and

observational constraints

Yun-He Li1 and Xin Zhang1,2,

1Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China2Center for High Energy Physics, Peking University, Beijing 100080, China

Dark energy might interact with cold dark matter in a direct, non-gravitational way. However, the

usual interacting dark energy models (with constant w) suffer from some catastrophic difficulties.For example, the Q c model leads to an early-time large-scale instability, and the Q demodel gives rise to the future unphysical result for cold dark matter density (in the case of apositive coupling). In order to overcome these fatal flaws, we propose in this paper an interactingdark energy model (with constant w) in which the interaction term is carefully designed to realizethat Q de at the early times and Q c in the future, simutaneously solving the early-timesuper-horizon instability and future unphysical c problems. The concrete form of the interactionterm in this model is Q = 3H dec

de+c, where is the dimensionless coupling constant. We show

that this model is actually equivalent to the decomposed new generalized Chaplygin gas (NGCG)model, with the relation= w. We calculate the cosmological perturbations in this model ina gauge invariant way, and show that the cosmological perturbations are stable during the wholeexpansion history provided that >0. Furthermore, we use the Planck data in conjunction withother astrophysical data to place stringent constraints on this model (with 8 parameters), and wefind that indeed > 0 is supported by the joint constraint at more than 1 level. The excellenttheoretical features and the support from observations all indicate that the decomposed NGCG

model should be payed more attentions and deserves further investigations.

PACS numbers: 95.36.+x, 98.80.Es, 98.80.-k

I. INTRODUCTION

Current universe is dominated by two dark sectors, namely, dark energy (DE) and dark matter (DM), which issupported by the recent astronomical observations [13]. However, we still know little about their natures and canonly indirectly detect them via their gravitational effects. This provides us more space to study such a possibilitythat there exists some direct, non-gravitational interaction between DE and DM. Such a possible interaction can helpsolve or alleviate several theoretical problems of DE, such as the cosmic coincidence problem [ 4], the cosmic doomsday

problem led by phantom[5], and the cosmic age problem caused by old quasar [6], et al. Besides, DE can also exertsa non-gravitational influence on DM by dark sector interaction, inducing new features to structure formation, suchas new large scale bias [7]and violation of weak equivalence principle for DM [8,9]. Thus, it is very meaningful tostudy such an interaction between DE and DM.

The dark sector interaction in the background evolution can be characterized by adding an interaction term Q tothe energy balance equations of DE and DM, i.e.,

de = 3H(1 + w)de+ Qde, (1)c = 3Hc+ Qc, Qde = Qc= Q, (2)

wherede andc are the energy densities of DE and DM (here, specifically, cold dark matter), respectively, H= a/ais the Hubble expansion rate and a dot denotes the derivative with respect to the cosmic time t, a is the scale factorof the Friedmann-Robertson-Walker (FRW) universe, and w = pde/de is the equation of state (EOS) parameter of

DE. Due to the fact that the knowledge about the micro-origin of the dark sector interaction is absent, one has topropose the interacting DE models by writing down the possible forms ofQ by hand. So far, lots of phenomenologicalforms for Q have been put forward [1035]. Among them, the models withQ H and Q (with either theenergy density of DE/DM or the sum of the two) are widely studied.

In recent years, it has been found that the interacting DE models may suffer from a large-scale instability at theearly times if the EOS of DE is taken to be a constant. In Ref. [25], the authors gave a detailed investigation on the

Corresponding authorElectronic address: [email protected]

arXiv:1312.6328

v1

[astro-ph.CO]

22Dec2013
mailto:[email protected]:[email protected]


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perturbation evolutions for the three interacting DE models, Q = c, Q= Hc, and Q = H(c+ de), where andare coupling constants. They found that all of them cannot give stable cosmological perturbations at the earlytimes ifw = const and w >1, while ifw 0. The same stabilitycondition was also pointed out in Ref. [35]for theQ = de model. Thus, it seems that the case withQ proportional

to de and with a positive coupling constant provides us with the most acceptable interacting DE model.1

Indeed,a positive coupling is favored by observations; see, e.g., Refs. [35,3840]. However, this does not mean that there isno problem in this interacting DE model. Actually, a positive coupling in the model with Q proportional to de willlead to a negative value ofc in the future. For example, for theQ = 3H de case, c = c0a

3(1 + r ra33w)for a constantw, wherer de0/[c0( w)] and the subscript 0 denotes the present value of the correspondingquantity, and one can check that c < 0 after a 1.35 if choosing = 0.1, w =0.98, and c0/de0 = 0.36. Thisnon-physical result arises from the fact that a positive coupling results in energy transfer from DM to DE, and theinteraction term Q Hde (orQ de) exacerbates this energy transfer in the DE dominated future. Note that themodels withQ proportional to c do not have this problem.

In short, for the interacting DE models with constant w, the knowledge acquired from the above discussions canbe briefly summarized as: Q c leads to a large-scale instability at the early times, and Q de gives rise to anegativec in the future. Therefore, it is fairly natural to design an interacting DE model (with constant w) in whichthe interaction termQ is proportional to de at the early times and proportional to c in the future. We expect thatin this model the cosmological perturbations will always be stable during the whole expansion history of the universeand the negative value ofc will not occur. We shall show that such a reasonable interacting DE model can emergefrom an existing unified dark fluid scenario, namely, the new generalized Chaplygin gas (NGCG) scenario[41].

Let us consider the interaction form,

Q= 3H deRc, (3)

with Rc cde+c . It is clear to see that this form ofQ satisfies the above conditions. The interacting DE modelwith this form ofQ and a constant w can be obtained from the NGCG scenario [41] by setting = w with theNGCG model parameter (for the detailed derivation, see AppendixA). So, this interacting DE model is equivalent to adecomposed NGCG model. Actually, a decomposed generalized Chaplygin gas (GCG) model has been discussed[42].However, we should notice that the decomposed GCG model [ 42] describes DM interacting with the vacuum energy(w= 1), and thus the perturbation of DE is always zero in the DM-comoving frame. In our work, we focus on thew= const case, for which one must seriously treat the DE perturbation, as there may exist the large-scale instabilitymentioned above. We shall show that the model with Q given by Eq. (3) and a constant w (or, the decomposedNGCG model) is a reasonable, large-scale stable interacting DE model.

Using Eqs. (1)(3), we can obtain the background energy densities of DE and DM,

de = de0a3(1+w)

Rc0+ (1 Rc0)a3(w) w

, (4)

c = c0a3

Rc0+ (1 Rc0)a3(w) w

. (5)

From the above equations, one can clearly see that both the energy densities of DE and DM are always positive fromthe past to the future no matter what sign oftakes, since 0< Rc0 < 1. Thus, this interacting DE model overcomesthe flaw that a positive coupling leads to the future non-physical evolution of c in the Q = de or Q = H demodel. Furthermore, we will show that this model can also give stable cosmological perturbations at the early times.

Our paper is organized as follows. In Sec.II, we give the general gauge-dependent perturbation equations for the

present interacting DE model. Following Ref.[39], we will consider the perturbation ofHin Eq. (3) in order to derivethe gauge invariant evolution equations. In Sec. III, we discuss the stability of cosmological perturbations using agauge invariant way. In Sec.IV,we use the Planck data and other observations to constrain the model. We will showthat a positive coupling constant required by the stable perturbations is also favored by the current observations.We will give conclusions in the final section. In our analysis, we only care about the w 1 case to avoid futureinstability of our universe.

1 It was shown that a time-dependent wcan help solve the early-time instability problem[37]. However, a time-dependent wwill introduceat least one more free parameter.


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II. PERTURBATION EQUATIONS

In this section, we give the general gauge-dependent perturbation equations for the considered interacting DEmodel. For simplicity, we only consider a flat universe. Extending the result to a non-flat universe is straightforward.We follow the notation of Ref. [25] and from here on we use the conformal time (defined as d = dt/a) as theindependent variable instead of the cosmic time t. So, the conformal Hubble expansion rate isH = Ha. The flatFRW metric with scalar perturbations can be written in general as:

ds2 =a2 (1 + 2)d2 + 2iB ddxi + (1 2)ij+ 2ijEdxidxj, (6)

where , B, and Eare the gauge-dependent scalar metric perturbation quantities. For the given metric (6), onedoes not need to modify the linear Einstein equations for the dark sector interaction case, but needs to modify theconservation equations for A-fluid,

TA =QA ,A

QA= 0, (7)

whereQA denotes the energy-momentum transfer for A-fluid, and TA is the A-fluid energy-momentum tensor,

TA= (A+pA)uAu

A +pA

+

A, (8)

where

A is the A-fluid anisotropic stress, and we note that A and pA contain the contributions of correspondingperturbations A andpA, respectively. The A-fluid four-velocity is given by

uA= a1

1 , ivA

, uA =a

1 , i[vA+ B]

, (9)

with vA theA-fluid peculiar velocity potential. In our work, we use the A-fluid volume expansion rate A [43],

A= k2(vA+ B), (10)wherek is the comoving wavenumber in the Fourier space.

To complete Eq.(7), one needs a covariant energy-momentum transfer form. However, we cannot obtain it from thefirst principle. In our work, we construct it using the background interaction term (3). First, we follow Refs. [44,45]and splitQA relative to the total four-velocity,

QA0 = a

QA(1 + ) + QA

, (11)

QAi = ai

fA QA

k2

, (12)

wherefA represents the momentum transfer potential and is the total velocity perturbation. Then, the energy andmomentum balance equations for A-fluid from Eq. (7) are given by[25]

A+ 3H(c2sA wA)A+ (1 + wA)A+ 3H

3H(1 + wA)(c2sA wA) + wAA

k2

3(1 + wA) + (1 + wA)k2

B E = aQAA

A+ 3H(c2sA wA)

Ak2

+

a

AQA , (13)

A+ H1 3c2sAA c2sA

(1 + wA)k

2

A+

2

3a2(1 + wA)A k

4

A k2

= aQA

(1 + wA)A

(1 + c2sA)A a

(1 + wA)Ak2fA , (14)

where A = AA

, the prime denotes the derivative with respect to the conformal time , and c2sA is the sound speed

ofA-fluid. For a barotropic fluid,c2sA = c2aA with c

2aA the adiabatic sound speed ofA-fluid defined by c

2aA pA/A.

However, for the DE perturbation, we cannot take c2s,de = c2a,de, since c

2a,de = w < 0 leads to instability in the dark

energy [46]. So, it is necessary to assume that DE is a non-adiabatic fluid and impose c2s,de> 0 by hand. In our work,

as usual, we take c2s,de= 1; this is what is done in the CAMBcode[47].


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Next, we calculate Q for our interacting DE model. From Eq. (3), we have

Q = Q

H

H + Rcde+ (1 Rc)c

. (15)

Note that here we consider the perturbation of the Hubble parameter, which is indispensable for the gauge invariantequations (28) and (30) in the next section. That is to say, without the help of the term HH , one cannot get thegauge invariant equations for a dark sector coupling case, if the interaction term Q is proportional to H. In Ref.[39],

the authors pointed out this problem and tried to solve it by considering the perturbation ofH for the first time.In our work, we follow Ref. [39] and takeK 1H3 H + k

2

3 (B E)

as the perturbation of H. (Note

that their notation of the metric perturbations is different from ours; the corresponding relationships are A = andHL= 13k2E ). Substituting Eq. (3) into Eq. (15), and taking HH = K, we have

aQc = aQde = 3HdeRc [K + Rcde+ (1 Rc)c] . (16)The momentum transfer potentialfAcannot be derived from the background interaction term ( 3), and one needs to

specify it by hand. In our work, we choose it by assuming that the energy transfer is parallel to the DM four-velocity,so that the momentum transfer vanishes in the DM-rest frame, i.e.,

aQc = aQde= 3HdeRcuc . (17)Using Eq. (9), one can get

aQc = aQde = 3HadeRc

1 + + K + Rcde+ (1 Rc)c , i (vc+ B)

. (18)

Comparing Eq. (18) with Eq. (12), one can find

ak2fc = ak2fde= 3HdeRc(c ). (19)Finally, with the help of Eqs. (16) and (19), c = 0 =de, and c2s c = wc = 0 =w

, for our interacting DE model,Eqs. (13) and (14) can be written as

de+ 3H(1 w)de+ (1 + w)

de+ k2(B E)+ 9H2(1 w2) de

k2 3(1 + w)

= 3

HRc K + (1 Rc)(c de) + 3H(1 w)

de

k2

+ , (20)de 2Hde

k2

(1 + w)de k2= 3H

1 + wRc (c 2de) , (21)

c+ c+ k2(B E) 3 = 3H(1 Rc)[K + Rc(de c) + ] , (22)

c+ Hc k2= 0 . (23)

III. LARGE-SCALE STABILITY AND INITIAL CONDITIONS FOR PERTURBATIONS

As mentioned in Sec. I, many interacting DE models suffer from the early-time large-scale instabilities. Suchinstabilities arise from the fact that the non-adiabatic mode soon dominates and leads to rapid growth of curvatureperturbation at the early times, even if the adiabatic initial conditions are utilized [25, 37]. Thus, analyzing such

instabilities is closely related to analyzing the initial conditions for cosmological perturbations. In Ref. [ 48], the authorspresented a systematic approach to obtaining the initial conditions of cosmological perturbations in a non-interactingdark sector case using a gauge invariant way. In that approach, the solutions to the perturbation equations of eachcomponent are reduced to those of a first order differential matrix equation,

dU

d ln x= A(x)U(x), (24)

wherex = k , and

UT =

c, Vc, , V, b, , V, , de, Vde

. (25)


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Here, the subscripts , b and represent photons, baryons and neutrinos, respectively. A, VA and A are gaugeinvariant variables for matters devised by Bardeen [49]:

A= A+ H1 A

A , VA= k

1A+ k(B E) , A= A. (26)

Note that VA and A in Eq. (25) are the corresponding rescaled quantities, namely, VA = VA/x and A = A/x2,

respectively. Besides the gauge invariant variables of matters, the metric gauge invariant variables and are also

used, constructed by[49]

= + H(B E) , = + H (B E) + (B E) . (27)In Ref. [37], the authors generalized the analyzing approach in Ref. [48] to the dark sector coupling case. In this

part, we apply it to our analysis of the perturbation stability and initial conditions for our interacting DE model.First, we rewrite the perturbation equations for each component in terms of the gauge invariant variables. Since wecare about the solutions in the early radiation dominated epoch, we can take H =1. Then, Eqs. (20)(23) become

dcd ln x

= x2Vc 3(1 Rc)

Rc(de c) + x2

3V

, (28)

dVcd ln x

= 2Vc+ , (29)

dded ln x

= 3(w 1)

de+ 3(1 + w)

+

+ (1 + w)

3 x23(w 1)

Vde

+3Rc

(1 Rc)(c de) + 3(1 w)

Vde+ +

+

x2

3V

, (30)

dVded ln x

= de1 + w

+Vde+ 3+ 4 + 3RcVc 2Vde

1 + w , (31)

with given by

=

A=c,b,,,de A

A+ 3 (1 + wA)VA

A=c,b,,,de 3 (1 + wA)A+

23

x2 . (32)

Here, we have used the Einstein equation = and defined A A/critfor A-fluid withcrit the criticaldensity of our universe. For other components, they satisfy the same differential equations as those of the uncoupledcase, given in Ref. [48].

Next, we give the coefficient matrix A(x) in Eq.(24). At the early times, x 1, A(x) can be reduced to a constantmatrix A0, as long as no divergence occurs when x 0. We can also take a = H0

r0 and A A/r, since the

early universe is dominated by radiation; thus we have

b =br

=b0r0

a= b0

r0

H0k

x= 1 x , c =c0R

/(w)c0r0

H0k

x= 2 x ,

de=de0R

/(w)c0

r0

r0H0

k

13(w)x13(w) =3 x

13(w),

=/r = R, = 1

b

c

de

. (33)

Here, for DE and DM, we have used Eqs. (4) and (5), and neglected the (1 Rc0)a3(w) term. Note that there isno divergence term in Eq. (33) whenx 0 under the assumptionsw < 1/3 and small coupling constantrequired


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For the Re(d)> 0 case, according to the discussion of Ref. [37], the DE perturbation will dominate at the earlytimes and drag other perturbations onto non-adiabatic blow-up even if they are adiabatic at the initial times. Thus,Re(d)> 0 corresponds to the instable case. From Eq. (36), we find that the parameter interval that can give stable

cosmological perturbations (Re(d)< 0) is >0 under the assumptionw > 1, which is the same as that in modelsQ= 3H de andQ = de. As an example for the stable case, we plot the evolutions of gauge invariant matter andmetric perturbations in Fig. 1,for k = 0.01Mpc1, k = 0.1Mpc1 and k = 1.0Mpc1. Here we choosew =0.98, = 0.1, and fix other cosmological parameters at the best-fit values from Planck. We can clearly see that all theperturbation evolutions are normal and stable. Besides, we can also see from Fig. 1 that due to the existence of DE,

the late-time evolutions of the metric perturbations and for k = 0.01Mpc1 suddenly change at log10 a 0.4,

which is the source of the late-time integrated Sachs-Wolfe effect [50] on the large scale. Figure1 also presents anexotic feature that the perturbation of DE oscillates when baryons and photons are tightly coupled. This oscillationfeature for DE arises from the Kterm in Eq. (20), since Kcontains the total velocity which oscillates when baryonsand photons are tightly coupled. However, as pointed out in Ref. [39], theK term does not significantly affect theobservational constraint results.

FIG. 1: The evolutions of gauge invariant matter perturbations and metric perturbations fork = 0.01Mpc1, k = 0.1Mpc1

and k = 1.0Mpc1. Here, we choose w = 0.98 and = 0.1, and fix other cosmological parameters at the best-fit values fromPlanck.

IV. OBSERVATIONAL CONSTRAINTS

In this section, we constrain our interacting dark energy model using current observational data. What we most careabout here is whether a positive required by stable cosmological perturbations is consistent with the observations.We modify the CAMB code[47]for our interacting dark energy model. In the synchronous gauge (= B = 0, = ,


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and k2E= h2 3), Eqs. (20)(23) become:

de = 3H(1 w)

de+ 3H(1 + w) dek2 (1 + w)de (1 + w) h

2

+ 3HRc

(1 Rc)(c de) + 3H +

h

6H + 3H(1 w)dek2

, (40)

de= 2

Hde+

k2de

(1 + w)

3H

1 + w

Rc(2de

c), (41)

c = c h

2 3H(1 Rc)

Rc(de c) + 3H +

h

6H

, (42)

c = Hc. (43)We use the adiabatic initial conditions obtained in the last section to solve the cosmological perturbation equations,and setc = 0 at the initial times so that DM is always at rest in the synchronous gauge.

We use the public Markov-Chain Monte-Carlo (MCMC) package CosmoMC [51]to explore the space of the cosmo-logical parameters. The free parameter vector is

b0h2, c0h

2, H0, , w, , ns, ln(1010As)

. (44)

Here,h is the Hubble constantH0 in units of 100 km s1 Mpc1,is the optical depth to reionization, and ln(1010As)

and ns are the amplitude and the spectral index of the primordial scalar perturbation power spectrum for the pivotscale k0 = 0.05Mpc1. The priors of all the free parameters used in running MCMC are listed in Table I.Note that

we directly use H0 as a free parameter in place of the commonly used parameter MC defined as the approximationto the ratio of the comoving sound horizon at z = z (with z the redshift when the optical depth equals unity).CosmoMCusing MC instead ofH0 is due to that MC is much better constrained than H0. However, the value ofzused to derive MC comes from a fitting formula in Ref. [52], which assumes a standard non-interacting backgroundevolution. In our work, we fix the effective number of neutrinos Neff= 3.046 and the total mass of standard neutrinosm= 0.06 eV, adopted as the same as Ref. [53].

For the observations, we use the following data sets:

The Cosmic Microwave Background (CMB) observations including the high-lTT likelihood atl = 502500 andthe low-l TT likelihood at l < 50 from Planck and low-l TE, EE, BB likelihood (polarization measurements)from 9-year WMAP. All the data can be downloaded from Planck Collaboration[54].

The type Ia supernova (SN) observations of 580 data from Union2.1 sample (without considering the systematicerrors)[55]. The baryon acoustic oscillation (BAO) data at z = 0.106 from the 6dF Galaxy Survey [56], z = 0.35 from the

SDSS DR7 measurement [57]and z = 0.57 from BOSS DR9 measurement[58].

The Hubble constant measurement, H0 = 73.8 2.4 k m s1 Mpc1, from the HST[59].Our fit results are summarized in Table I and Fig. 2. The best fit of the coupling constant is 0.1385 and its

68% limits are 0.178+0.0810.097, which are greater than 0 at more than 1 confidence level. This result is consistent withthat obtained in a latest fit work[40] using Planck data to constrain the Q= 3H de model. From Fig.2, we findthat there exists a strong anti-correlation between the coupling constant and the physical cold dark matter densityc0h2, which results in a low value of m0 and a high value of de0 as shown in Table I, since a positive couplingconstantis favored by observations. These results can be easily understood. For our interacting dark energy model,a positive coupling constant leads to the energy transfer from DM to DE, and so the stronger coupling is, the lower

energy density of matter becomes.

V. CONCLUSIONS

There exists an important possibility that dark energy interacts with cold dark matter in some direct, non-gravitational way. In this paper, we focus on the interacting dark energy models with constant w (andw > 1); thisclass of models may also be called interacting wCDM model. For the widely studied forms of interaction, Q c (orQ Hc) and Q de (or Q Hde), there are some fatal flaws in the model. For instance, Q c leads to alarge-scale instability at the early times, andQ de (with a positive coupling) gives rise to an unphysical result for


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TABLE I: The fit results for the free parameters and some derived parameters. We give their best-fit values as well as themarginalized 68% confidence limits. We also present the prior ranges of the free parameters used in running MCMC.

Parameter Prior Best fit 68% limits

b0h2 [0.005, 0.1] 0.02208 0.02208+0.000250.00025

c0h2 [0.001, 0.99] 0.0987 0.0934+0.01100.0109

H0 [20, 100] 70.0 69.7+1.21.2

[0.01, 0.8] 0.082 0.089+0.0120.014w [1, 0.3] 0.9908 0.9657+0.00710.0342 [0, 1.0] 0.1385 0.178+0.0810.097ns [0.9, 1.1] 0.9630 0.9616

+0.00620.0062

ln(1010As) [2.7, 4.0] 3.074 3.087+0.0250.025

de0 0.7521 0.7603+0.02860.0290

m0 0.2479 0.2397+0.02900.0286

zre 10.42 10.00+1.091.09

Age/Gyr 13.744 13.755+0.0380.038100 1.04184 1.04151

+0.000580.00058

the evolution of cold dark matter density, i.e., negative c in the future. In order to overcome these flaws, we propose

in this paper an interacting wCDM model withQ = 3H decde+c , and show that this model is a reasonable, large-scale

stable interacting dark energy model.By carefully designing the form of Q, this model gets excellent features: At early times, Q de, and so the

early-time large-scale instability can be avoided; in the future, Qc, and thus the problem of negative c can beeliminated.

We have calculated the cosmological perturbations in this model. We also considered the perturbation of the HubbleparameterH in the calculation in order to get the gauge invariant equations for the dark matter and dark energyperturbations. We find that the cosmological perturbations in this interacting wCDM model (withw > 1) are stableduring the whole expansion history provided that >0. We have also used the CMB temperature data from Planckand CMB polarization data from 9-yr WMAP, in conjunction with the SN data, BAO data, and H0 measurement, toplace stringent constraints on this model (with 8 parameters). We find support for >0 from the joint constraint:0.081<


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0 . 9 6 0 . 9 2 0 . 8 8 0 . 8 4

w

6

8

0

2

4

5

6

7

8

5

0

5

5

0

5

0 . 0 2 1 6 0 . 0 2 2 2 0 . 0 2 2 8

b 0

h

2

6

2

8

4

0 . 0 6 0 . 0 8 0 . 1 0 0 . 1 2

c 0

h

2

0 . 9 4 0 . 9 5 0 . 9 6 0 . 9 7 0 . 9 8

n

s

6 7 . 5 7 0 . 0 7 2 . 5

H

0

0 . 1 5 0 . 3 0 0 . 4 5

FIG. 2: The 1D marginalized distributions and 2D marginalized 68% and 95% contours, for the parameters in our interactingdark energy model.

where A(a) =wAa3(1+w)(1+), with a dimensionless parameter and A a positive constant. The NGCG isdesigned as a unification scheme for DE and DM; however, on the other hand, it can also be viewed as an interactingwCDM model, provided that it is decomposed into the two components, DE (with constant w) and CDM,

Ch = de+ c. (A2)

The continuity equations for DE and DM are given by Eqs. (1) and (2). Since DM is pressureless, the pressure of theNGCG is provided only by DE, i.e., pCh = pde. Therefore, from Eqs. (A1)and (A2), we have

A= de(de+ c)a3(1+w)(1+). (A3)


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SinceA is a constant, we have A= 0. Furthermore, using A= 0 and Eqs. (1), (2)and (A3), we obtain the interactionterm,

Q= 3wHdeRc. (A4)Comparing Eq. (A4) with Eq. (3), we find the relation =w. So, the interactingwCDM model with such aninteraction term is actually equivalent to the decomposed NGCG model.

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a large-scale stable interacting dark energy model

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