Revisiting the Cosmological Constraints on the Interacting Dark Energy Models

In this work, we consider the cosmological constraints on the interacting dark energy models. We generalize the models considered previously by Guo {\it et al.}, Costa and Alcaniz, and try to discuss two general types of models: type I models are characterized by $\rho_{_X}/\rho_m=f(a)$ and $f(a)$ can be any function of scale factor $a$, whereas type II models are characterized by $\rho_m=\rho_{m0}\,a^{-3+\epsilon(a)}$ and $\epsilon(a)$ can be any function of $a$. We obtain the cosmological constraints on the type I and II models with power-law, CPL-like, logarithmic $f(a)$ and $\epsilon(a)$ by using the latest observational data.


I. INTRODUCTION
The dark energy has been one of the most active fields in modern cosmology since the discovery of the accelerated expansion of our universe (see e.g. [1] for reviews). Among the conundrums in the dark energy cosmology, the so-called cosmological coincidence problem is the most familiar one. This problem is asking why are we living in an epoch in which the densities of dark energy and matter are comparable? Since their densities scale differently with the expansion of our universe, there should be some fine-tunings. To alleviate the cosmological coincidence problem, it is natural to consider the possible interaction between dark energy and dark matter in the literature (see e.g. [2-9, 31, 32]). In fact, since the nature of both dark energy and dark matter are still unknown, there is no physical argument to exclude the possible interaction between them. On the contrary, some observational evidences of this interaction have been found recently. For example, in a series of papers by Bertolami et al. [10], they shown that the Abell Cluster A586 exhibits evidence of the interaction between dark energy and dark matter, and they argued that this interaction might imply a violation of the equivalence principle. On the other hand, in [11], Abdalla et al. found the signature of interaction between dark energy and dark matter by using optical, X-ray and weak lensing data from 33 relaxed galaxy clusters. Therefore, it is reasonable to consider the interaction between dark energy and dark matter in cosmology.
We consider a flat Friedmann-Robertson-Walker (FRW) universe. In the literature, it is usual to assume that dark energy and dark matter interact through a coupling term Q, according tȯ where ρ m and ρ X are densities of dark matter and dark energy (we assume that the baryon component can be ignored); w X is the equation-of-state parameter (EoS) of dark energy and it is assumed to be a constant; a dot denotes the derivative with respect to cosmic time t; H ≡ȧ/a is the Hubble parameter; a = (1 + z) −1 is the scale factor (we have set a 0 = 1; the subscript "0" indicates the present value of corresponding quantity; z is the redshift). Notice that Eqs. (1) and (2) preserve the total energy conservation equationρ where ρ tot = ρ X +ρ m is the total energy; w eff is the total (effective) EoS. Since there is no natural guidance from fundamental physics on the coupling term Q, one can only discuss it to a phenomenal level. The most familiar coupling terms extensively considered in the literature are Q = ακρ mφ , Q = 3βHρ tot , and Q = 3ηHρ m . The first one arises from, for instance, string theory or scalar-tensor theory (including Brans-Dicke theory) [4][5][6]. The other two are phenomenally proposed to alleviate the coincidence problem in the other dark energy models [7][8][9]. In the usual approach, one should priorly write down the coupling term Q, and then obtain the evolutions of ρ m and ρ X from Eqs. (1) and (2), respectively. In fact, this is the common way to study the interacting dark energy models in the literature. However, there is also an alternative way in the literature [12][13][14][15][16]. One can reverse the logic mentioned above. Due to the interaction Q, the evolutions of ρ m and ρ X should deviate from the ones without interaction, i.e., ρ m ∝ a −3 and ρ X ∝ a −3(1+w X ) , respectively. If the deviated evolutions of ρ m and/or ρ X are given, one can find the corresponding interaction Q from Eqs. (1) and (2). Naively, the simplest example has been considered by Wang and Meng [12], namely where ǫ is a constant which measures the deviation from the normal ρ m ∝ a −3 . Substituting into Eq. (1), it is easy to find the corresponding interaction Q = ǫHρ m [9,12,13,17]. Alternatively, one can consider another type of interacting dark energy model which is characterized by [14,15] where ξ is a constant which measures the severity of the coincidence problem. From Eqs. (1), (2) and (5), one can find that the corresponding interaction is given by [15] Q = −Hρ m Ω X (ξ + 3w X ) = −Hρ X Ω m (ξ + 3w X ) , where Ω i ≡ 8πGρ i /(3H 2 ) for i = m and X, which are the fractional energy densities of dark matter and dark energy, respectively. In fact, Guo et al. [15] considered the cosmological constraints on the interacting dark energy model characterized by Eq. (5) with the 71 SNLS Type Ia supernovae (SNIa) dataset, the shift parameter R from the Wilkinson Microwave Anisotropy Probe 3-year (WMAP3) data, and the distance parameter A of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies. On the other hand, the interacting dark energy model characterized by Eq. (4) has been extended in [16]. It is more realistic that ǫ is a function of time. Costa and Alcaniz [16] considered the interacting dark energy model characterized by in which ǫ(a) was chosen to be where ǫ 0 and ǫ 1 are constants. They obtained the constraints on this model by using the 307 Union SNIa dataset, the CMB constraint Ω m0 h 2 = 0.109 ± 0.006 from the Wilkinson Microwave Anisotropy Probe 5-year (WMAP5) data, and the distance ratio from z BAO = 0.35 to z LS = 1089 measured by SDSS, namely R BAO/LS = 0.0979 ± 0.0036. In the present work, we generalize the interacting dark energy models considered in [15,16], and we call them type I and II models, respectively. The type I models are characterized by where f (a) can be any function of a, beyond the special case in Eq. (5). The type II models are characterized by Eq. (7) but ǫ(a) can be any function of a, beyond the special case in Eq. (8). In the present work, we consider the constraints on these models by using the latest cosmological observations, namely, the 397 Constitution SNIa dataset [18], the shift parameter R from the newly released Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) data [19], and the distance parameter A of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies [20,21]. In the next section, we briefly introduce these observational data. In Sec. III and Sec. IV, we discuss the type I and II models, and consider their cosmological constraints, respectively. A brief summary is given in Sec. V.

II. OBSERVATIONAL DATA
In the present work, we will consider the latest cosmological observations, namely, the 397 Constitution SNIa dataset [18], the shift parameter R from the newly released Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) data [19], and the distance parameter A of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies [20,21].
The data points of the 397 Constitution SNIa compiled in [18] are given in terms of the distance modulus µ obs (z i ). On the other hand, the theoretical distance modulus is defined as where µ 0 ≡ 42.38 − 5 log 10 h and h is the Hubble constant H 0 in units of 100 km/s/Mpc, whereas in which E ≡ H/H 0 , and p denotes the model parameters. Correspondingly, the χ 2 from the 397 Constitution SNIa is given by where σ is the corresponding 1σ error. The parameter µ 0 is a nuisance parameter but it is independent of the data points. One can perform an uniform marginalization over µ 0 . However, there is an alternative way. Following [22,23], the minimization with respect to µ 0 can be made by expanding the χ 2 µ of Eq. (12) with respect to µ 0 as .
Eq. (13) has a minimum for µ 0 =B/C at Since χ 2 µ, min =χ 2 µ, min obviously, we can instead minimizeχ 2 µ which is independent of µ 0 . There are some other observational data, such as the observations of cosmic microwave background (CMB) anisotropy [19] and large-scale structure (LSS) [20]. However, using the full data of CMB and LSS to perform a global fitting consumes a large amount of computation time and power. As an alternative, one can instead use the shift parameter R from the CMB, and the distance parameter A of the measurement of the baryon acoustic oscillation (BAO) peak in the distribution of SDSS luminous red galaxies. In the literature, the shift parameter R and the distance parameter A have been used extensively. It is argued that they are model-independent [24], while R and A contain the main information of the observations of CMB and BAO, respectively.
As is well known, the shift parameter R of the CMB is defined by [24,25] R ≡ Ω where the redshift of recombination z * = 1091.3 which has been updated in the Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) data [19]. The shift parameter R relates the angular diameter distance to the last scattering surface, the comoving size of the sound horizon at z * and the angular scale of the first acoustic peak in CMB power spectrum of temperature fluctuations [24,25]. The value of R has been updated to 1.725 ± 0.018 from the WMAP7 data [19]. On the other hand, the distance parameter A of the measurement of the BAO peak in the distribution of SDSS luminous red galaxies [20] is given by where z b = 0.35. In [21], the value of A has been determined to be 0.469 (n s /0.98) −0.35 ± 0.017. Here the scalar spectral index n s is taken to be 0.963, which has been updated from the WMAP7 data [19]. So, the total χ 2 is given by whereχ 2 µ is given in Eq. (14), The best-fit model parameters are determined by minimizing the total χ 2 . As in [26,27], the 68% confidence level is determined by ∆χ 2 ≡ χ 2 − χ 2 min ≤ 1.0, 2.3 and 3.53 for n p = 1, 2 and 3, respectively, where n p is the number of free model parameters. Similarly, the 95% confidence level is determined by ∆χ 2 ≡ χ 2 − χ 2 min ≤ 4.0, 6.17 and 8.02 for n p = 1, 2 and 3, respectively.

A. Equations
As mentioned in Sec. I, the type I models are characterized by Eq. (9), whereas f (a) can be any function of a. From Eq. (9), it is easy to obtain Substituting ρ X = ρ m f (a) into Eq. (2) and usingρ m from Eq. (1), we can find that the corresponding interaction term is given by where a prime denotes the derivative with respect to a. Obviously, if f (a) ∝ a ξ , Eq. (19) reduces to Eq. (6) which has been obtained in [15]. On the other hand, one can recast Eq. (3) as Using Eq. (18), we can integrate Eq. (20) to obtain where const. is an integral constant, which can be determined by requiring the condition ρ tot (a = 1) = ρ tot,0 = 3H 2 0 /(8πG). Once ρ tot is on hand, we can readily find the corresponding E ≡ H/H 0 from Friedmann equation, and then fit it to the observational data. Correspondingly, ρ X = Ω X ρ tot and ρ m = Ω m ρ tot are also available from Eqs. (18) and (21). Finally, it is worth noting that by definition (9), we have which is useful to fix one of parameters in f (a).

B. Cosmological constraints on type I models
In this subsection, we consider the cosmological constraints on type I models by using the observational data given in Sec. II. At first, we consider the power-law case with where ξ is a constant; f 0 can be determined by definition (9) to be the one given in Eq. (22), and hence it is not an independent parameter. In this case, there are three free model parameters, namely, Ω m0 , w X and ξ. Although the cosmological constraints on the model characterized by Eq. (23) has been considered by Guo et al. [15], as mentioned in Sec. I, they have used the earlier observational data. Therefore, it is still worthwhile to consider the cosmological constraints once more in the present work by using the latest observational data mentioned in Sec. II. Substituting Eq. (23) into Eq. (21) and requiring ρ tot (a = 1) = ρ tot,0 , we can determine the integral constant, and finally obtain Substituting into Friedmann equation, we find that By minimizing the corresponding total χ 2 in Eq. (17), we find the best-fit parameters Ω m0 = 0.281, w X = −0.982 and ξ = 2.988, whereas χ 2 min = 465.604. In Fig. 1, we also present the corresponding 68% and 95% confidence level contours in w X − ξ plane, Ω m0 − ξ plane and Ω m0 − w X plane. It is easy to see that these constraints on the model characterized by f (a) = f 0 a ξ are much tighter than the ones obtained by Guo et al. [15], thanks to the latest observational data.  Next, we consider a new case with which can be regarded as a linear expansion of f (a) with respect to a, similar to the familiar Chevallier-Polarski-Linder (CPL) parameterization for EoS w(a) = w 0 + w a (1 − a) [28]. Again, f 0 can be determined by definition (9) to be the one given in Eq. (22), and hence it is not an independent parameter. Thus, there are three free model parameters, namely, Ω m0 , w X and ξ. Substituting Eq. (26) into Eq. (21) and requiring ρ tot (a = 1) = ρ tot,0 , we can determine the integral constant, and finally obtain Substituting into Friedmann equation, we find that Noting Eq. (9) and imposing the condition ρ X ≥ 0, we have ξ ≥ 1 − Ω −1 m0 from Eqs. (26) and (22). Under this condition, by minimizing the corresponding total χ 2 in Eq. (17), we find the best-fit parameters Ω m0 = 0.218, w X = −0.483 and ξ = −3.584, whereas χ 2 min = 563.77. In Fig. 2, we also present the corresponding 68% and 95% confidence level contours in w X − ξ plane, Ω m0 − ξ plane and Ω m0 − w X plane. It is easy to see that the 68% and 95% confidence level contours are very close. On the other hand, χ 2 min = 563.77 is fairly larger than the degree of freedom dof ∼ 400. So, we give up the condition ξ ≥ 1 − Ω −1 m0 in the case of f (a) = f 0 + ξ(1 − a). This means that ρ X might be negative in the early universe. In fact, Guo et al. [15] also explicitly include this possibility. Since such a negative energy appears in phantom models [29] and modified gravity models [30], it is reasonable to consider this possibility. Without the condition ξ ≥ 1 − Ω −1 m0 , by minimizing the corresponding total χ 2 in Eq. (17), we find the best-fit parameters Ω m0 = 0.288, w X = −0.868 and ξ = −2.957, whereas χ 2 min = 467.718. In Fig. 3, we also present the corresponding 68% and 95% confidence level contours in w X − ξ plane, Ω m0 − ξ plane and Ω m0 − w X plane. Obviously, these results are significantly better than the ones with the condition ξ ≥ 1 − Ω −1 m0 , whereas the corresponding χ 2 min = 467.718 is also better. Finally, one might consider the logarithmic case with which can be regarded as a linear expansion of f (a) with respect to the so-called e-folding time N = ln a in the literature. This case seems attractive since in Eq. (21) the integration is with respect to ln a. However, in this case, f (a) (and hence ρ X ) diverges when a → 0 in the early universe. So, we do not consider the logarithmic case in type I model.

A. Equations
In this section, we turn to type II models, which are characterized by Eq. (7) whereas ǫ(a) can be any function of a. Substituting Eq. (7) into Eq. (1), we can easily find that the corresponding interaction term is given by It is worth noting that in type II models, there is no condition like Eq. (22) in type I models to reduce the number of free model parameters. If there are at least two parameters in ǫ(a), adding Ω m0 and w X , we should have four free model parameters or even more. In this case, the constraints will be very loose, and the calculations will be very involved. Instead, we follow Costa and Alcaniz [16] to consider the case with a fixed w X = −1, namely, the role of dark energy is played by a decaying Λ [12,16]. Therefore, Eq. (2) becomesρ Substituting Eq. (7) into Eq. (31), we have We can integrate Eq. (32) to obtain Substituting Eqs. (33) and (7) into Friedmann equation, we find that where

B. Cosmological constraints on type II models
In this subsection, we consider the cosmological constraints on type II models by using the observational data given in Sec. II. At first, we consider the power-law case with where ǫ 0 and ǫ 1 are constants. In this case, there are three free model parameters, namely, Ω m0 , ǫ 0 and ǫ 1 .
Although the cosmological constraints on the model characterized by Eq. (36) has been considered by Costa and Alcaniz [16], as mentioned in Sec. I, they have used the earlier observational data. Therefore, it is still worthwhile to consider the cosmological constraints once more in the present work by using the latest observational data mentioned in Sec. II. Substituting Eq. (36) into Eqs. (34) and (35), we can then fit this model to the observational data. By minimizing the corresponding total χ 2 in Eq. (17), we find the best-fit parameters Ω m0 = 0.282, ǫ 0 = −0.129 and ǫ 1 = 1.263, whereas χ 2 min = 465.635. In Fig. 4, we also present the corresponding 68% and 95% confidence level contours in ǫ 0 − ǫ 1 plane, Ω m0 − ǫ 0 plane and Ω m0 − ǫ 1 plane. It is easy to see from the ǫ 0 − ǫ 1 plane that if ǫ 0 is close to zero, ǫ 1 cannot be too negative. On the other hand, the constraint on the parameter ǫ 1 is still very loose.

V. SUMMARY AND DISCUSSIONS
In this work, we considered the cosmological constraints on the interacting dark energy models. We generalized the models considered previously by Guo et al. [15], Costa and Alcaniz [16], and we have discussed two general types of models: type I models are characterized by ρ X /ρ m = f (a) and f (a) can be any function of scale factor a, whereas type II models are characterized by ρ m = ρ m0 a −3+ǫ(a) and ǫ(a) can be any function of a. We obtained the cosmological constraints on the type I and II models with power-law, CPL-like, logarithmic f (a) and ǫ(a) by using the latest observational data.
Some remarks are in order. Firstly, here we briefly justify the interaction forms considered in the present work. We take type I models as examples. For the power-law case with f (a) = f 0 a ξ in Eq. (23), noting that in the case without interaction ρ X ∝ a −3(1+w X ) and ρ m ∝ a −3 , from definition Eq. (9), it is reasonable to parameterize f (a) = ρ X /ρ m ∝ a ξ , where ξ measures the severity of the coincidence problem [14,15]. For the CPL case with f (a) = f 0 + ξ(1 − a) in Eq. (26) and the logarithmic case with f (a) = f 0 + ξ ln a in Eq. (29), noting that the Taylor series expansion of any function F (x) is given by . . , the CPL and logarithmic cases can be regarded as the Taylor series expansion of f with respect to the scale factor a and the e-folding time N = ln a up to first order (linear expansion), similar to the well-known EoS parameterizations w(a) = w 0 + w a (1 − a) and w(z) = w 0 + w 1 z.
Secondly, we would like to briefly consider the comparison of these models. For convenience, we also consider the well-known ΛCDM model in addition. Fitting ΛCDM model to the observational data considered in the present work, it is easy to find the corresponding best-fit parameter Ω m0 = 0.278, whereas χ 2 min = 466.317. A conventional criterion for model comparison in the literature is χ 2 min /dof , in which the degree of freedom dof = N − k, whereas N and k are the number of data points and the number of free model parameters, respectively. We present the χ 2 min /dof for all the 7 models in Table I. On the other hand, there are other criterions for model comparison in the literature, such as Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC). The BIC is defined by [33,35] BIC = −2 ln L max + k ln N , where L max is the maximum likelihood. In the Gaussian cases, χ 2 min = −2 ln L max . So, the difference in BIC between two models is given by ∆BIC = ∆χ 2 min + ∆k ln N . The AIC is defined by [34,35] AIC = −2 ln L max + 2k .
The difference in AIC between two models is given by ∆AIC = ∆χ 2 min + 2∆k. In Table I, we also present the ∆BIC and ∆AIC of all the 7 models considered in this work. Notice that ΛCDM has been chosen to be the fiducial model when we calculate ∆BIC and ∆AIC. From Table I, it is easy to see that the rank of models is coincident in all the 3 criterions (χ 2 min /dof , BIC and AIC). The ΛCDM model is the best one, whereas ICPLw model is the worst one. This result is consistent with the one obtained in e.g. [35]. However, it is well known that ΛCDM model is plagued with the cosmological constant problem and the coincidence problem (see e.g. [1]). On the other hand, as mentioned in the beginning of Sec. I, there are some observational evidences for the interaction between dark energy and dark matter, and the coincidence problem can be alleviated in the interacting dark energy models. Therefore, it is still worthwhile to study the interacting dark energy models.