Observational constraints on the interacting Ricci dark energy model
Physics Department, College of Science and Technology, Nihon University, 1-8-14, Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
We consider an extension of the holographic Ricci dark energy model by introducing an interaction between dark energy and matter. In this model, the dark energy density is given by , where is the Ricci scalar curvature, is the reduced Planck mass, and is a dimensionless parameter. The interaction rate is given by , where is the Hubble expansion rate, and is a dimensionless parameter. We investigate current observational constraints on this model by applying the type Ia supernovae, the baryon acoustic oscillation and the CMB anisotropy data. It is shown that a nonvanishing interaction rate is favored by the observations. The best fit values are and for the present dark energy density parameter .
Recent observations of type Ia supernovae (SNIa) have revealed that the present Universe is undergoing accelerated expansion . This indicates that the energy density of the Universe at present epoch is dominated by dark energy with equation of state , where is the energy density of dark energy, and is its pressure. Combining with observations of CMB anisotropy  and the baryon acoustic oscillation (BAO) , the density parameter is determined as , where is the critical density today and is the present Hubble parameter.
The simplest way to explain dark energy is to introduce a cosmological constant . However, this model suffers from two problems . The first one is the fine-tuning problem: the observed cosmological constant is extremely small compared to the fundamental Planck scale , requiring an incredible fine-tuning. The second one is the cosmic coincidence problem: why the cosmological constant and matter have comparable energy density today even though their time evolution is so different. A variety of models have been proposed to solve these problems, such as quintessence , phantom , quintom  and holographic dark energy [9, 10, 11, 12].
In particular, the holographic dark energy (HDE) models have been discussed extensively in recent years. These models are motivated by the holographic principle of quantum gravity . From the condition that a system with size would not form a black hole, it is required that the total vacuum energy should not exceed the mass of the black hole of the same size. Therefore, the dark energy density must satisfy , where is the reduced Planck mass. Saturating this inequality, is defined by 
where is a dimensionless parameter. In Ref. , the size , which is regarded as an IR cut-off, was chosen to be the inverse Hubble expansion rate to naturally explain the observed vacuum energy density . However, it was shown that this choice can not explain the accelerated expansion of the Universe at present . This problem was solved in Ref.  by choosing to be the future event horizon . Extensive studies on this model have been done, and the HDE model with the future event horizon as the IR cut-off is found to be consistent with current observational data . Various aspects on the HDE model have been discussed in Ref. . There have also been further developments on the HDE model by introducing an interaction between dark energy and matter [18, 19]. However, it was pointed out that this model with has a conceptual problem that the future event horizon, which determines , depends on the future evolution of the Universe, hence violates causality .
Subsequently, inspired by the HDE model, the holographic Ricci dark energy (RDE) model  was proposed in which is proportional to the Ricci scalar curvature . It was shown that this model does not only avoid the causality problem and is phenomenologically viable, but also naturally solves the coincidence problem. Also, it was found that the causal connection scale consistent with cosmological observations is given by which is proportional to in a flat universe . This may provide us with a physical motivation for the RDE model. Cosmological constraints on this model were studied in Ref. . Similar models have also been studied in Ref. .
In this paper, we consider the RDE model with an interaction between dark energy and matter, and call it the interacting Ricci dark energy (IRDE) model. We investigate the observational constraints on this model obtained from SNIa, CMB and BAO data. We organize this paper as follows. In section 2, we describe the IRDE model, and obtain analytic expressions for cosmic time evolution. In section 3, we discuss the observational constraints on this model. We summarize our results in section 4.
2 The interacting Ricci Dark Energy model
We consider the spatially homogeneous and isotropic Universe described by the Friedmann-Robertson-Walker metric
where for closed, flat and open geometries. The time evolution of the scale factor is described by the Friedmann equation
where and represent energy density of dark energy, matter, radiation and curvature, respectively, and is the Hubble parameter.
The energy density of dark energy in the IRDE model is defined as 
where is a dimensionless parameter. Note that is proportional to the Ricci scalar curvature
Moreover, it is assumed that there is an interaction between dark energy and matter. The energy densities and obey the following equations 
The interaction rate is given by 333 If the interaction rate is given by , it turns out that the matter density becomes negative for . This problem does not occur for . However, the case is disfavored by observational constraints. 
where is a dimensionless parameter. The energy density of radiation is given by , where is the present value of radiation density. We adopt a convention that for the present age of the Universe Gyr. According to eq. (7) with in eq. (8), the interaction can be relevant if and are comparable, whether or not the Universe is in the radiation-dominated epoch. In literatures, various types of interactions between dark energy and matter have been considered, including = , , , , and so forth. At present, no definite mechanisms to determine the interaction are established. Thus, we simply assume eq. (8) as a phenomenological model.
where . The solution to eq. (9) is obtained as
, and . Note that can be imaginary for sufficiently large and . This implies that there is a parameter region where has oscillatory behavior. However, this region is not phenomenologically viable. The constants and are the present value of and , respectively. The constants , and are given by
Likewise, the matter density is
The equation of state of dark energy can be found by substituting eq. (15) into the following expression:
In eqs. (15) and (16), the term proportional to is dominant in the past 1, while the term proportional to is dominant in the future 1. As an illustration, let us consider the case 0.45 and 0.15 which corresponds to 0.25 and . In the past 1, the ratio of eq. (16) to eq. (15) is 3.4, while 0.045 in the future 1.
Notice that both and include contributions with non-standard time evolution. This typically leads to a constant ratio of to , and it may help to solve the coincidence problem. As pointed out in Ref. , the coincidence problem is ameliorated in the original RDE model without interaction where and were comparable with each other in the past universe. In this case, starts to increase at low redshift, and the ratio rapidly grows in the future, since in the absence of interaction. On the other hand, in the IRDE model, the behavior in the past is similar to that in the original RDE model, but the ratio is constant even in the future.
3 Observational constraints
In this section, we study the cosmological constraints on the IRDE model obtained from SNIa, CMB and BAO observations. In what follows, we focus only on the spatially flat Universe ().
The SNIa observations measure the distance modulus of a supernova and its redshift . The distance modulus is defined by
where is the luminosity distance given by
Here, and are the redshift and the distance modulus of the -th supernova, respectively. The corresponding error is denoted by .
The CMB shift parameter is one of the least model dependent parameters extracted from the CMB data. Since this parameter involves the large redshift behavior (), it gives a complementary bound to the SNIa data (). The shift parameter is defined as
where is the redshift at recombination, and is the matter fraction at present. We use the value obtained from the WMAP5 data . The CMB constraints are given by minimizing
where and .
Observations of large-scale galaxy clustering provide the signatures of the baryon acoustic oscillation (BAO). We use the measurement of the BAO peak in the distribution of luminous red galaxies (LRGs) observed in SDSS . It gives
where the parameter is given by
and . The for BAO is
In Fig.1, we plot the probability contours for SNIa (red), CMB (blue), BAO (green) observations in the ()-plane in the case without interaction (). The , and contours are drawn with solid, dashed and dotted lines, respectively. The joined constraints using are shown as shaded contours. The best fit values with error are and with . The joint region is outside the regions of CMB and BAO.
The same contour plots as Fig.1 in the presence of interaction with are shown in Fig.2. Compared to Fig.1, the CMB contours are shifted to the right. The best fit values are shifted as and . It is remarkable to note that, unlike the case , the joined region is included in all the regions of SNIa, CMB and BAO.
Dependence on the interaction parameter is presented in the ()-plane for in Fig.3. The best fit values are and with , and . It is found that the CMB constraint strongly depends on . As a result, the joined region appears in the range with sizable . This implies that existence of interaction between dark energy and matter is favored by cosmological constraints in the IRDE model.
We have considered the IRDE model where the interaction rate is given by eq. (8). We have derived the analytic expressions for the Hubble parameter (10) and the energy density of dark energy (15) and matter (16). Both and include contributions with non-standard time evolution. We have also investigated current observational constraints on this model from SNIa, CMB and BAO observations. In particular, the CMB constraint is strongly affected by the interaction. We have shown that a nonvanishing interaction rate is favored by the observations, giving a reduction of . The best fit values are , and .
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