Constraints on $Λ(t)$CDM models as holographic and agegraphic dark energy with the observational Hubble parameter data
The newly released observational data (OHD) is used to constrain CDM models as holographic and agegraphic dark energy. By the use of the length scale and time scale as the IR cut-off including Hubble horizon (HH), future event horizon (FEH), age of the universe (AU), and conformal time (CT), we achieve four different CDM models which can describe the present cosmological acceleration respectively. In order to get a comparison between such CDM models and standard CDM model, we use the information criteria (IC), diagnostic, and statefinder diagnostic to measure the deviations. Furthermore, by simulating a larger Hubble parameter data sample in the redshift range of , we get the improved constraints and more sufficient comparison. We show that OHD is not only able to play almost the same role in constraining cosmological parameters as SNe Ia does but also provides the effective measurement of the deviation of the DE models from standard CDM model. In the holographic and agegraphic scenarios, the results indicate that the FEH is more preferable than HH scenario. However, both two time scenarios show better approximations to CDM model than the length scenarios.
a]Zhong-Xu Zhai b,c]Tong-Jie Zhang a]and Wen-Biao Liu \affiliation[a]Department of Physics, Institute of Theoretical Physics, Beijing Normal University, Beijing, 100875, China \affiliation[b]Department of Astronomy, Beijing Normal University, Beijing, 100875, China \affiliation[c]Center for High Energy Physics, Peking University, Beijing, 100871, China \emailAddzzx@mail.bnu.edu.cn \emailAddtjzhang@bnu.edu.cn \emailAddwbliu@bnu.edu.cn
In the past more than ten years, the accelerating expansion of the universe has obtained a lot of attention since the observations of type Ia supernovae [1, 2]. Later on, the discoveries of CMB [3, 4] and BAO [5, 6] observations also got the same results. In order to give a reasonable explanation to this scenario, a great variety of attempts have been done. These theoretical explanations can be classified into two categories. The first is the dark energy which involve the introduction of exotic matter sources. The second is the modified gravity which relates to the changes of the geometry of the spacetime . Among these approaches, CDM, as the standard model, is considered to be the simplest and most natural one which shows great consistence with the observational data. In this model, the cosmological constant is considered to be the dark energy component of the universe. However, this model is also challenged by the fine-tuning problem and the coincidence problem. In order to solve or alleviate these problems, several methods are proposed and discussed [8, 9, 10, 11]. One possible way is to treat no longer a real constant but a time-dependent variable, namely the CDM [12, 13, 14, 15, 16]. In the assumption, the form of is considered to be related to the cosmological length scale or time scale 
where the speed of light is set as , and are the length scale and time scale which can introduce a nonzero and time-varying . Thus, it is natural to connect the CDM model with the holographic and agegraphic dark energy models which are proposed by the holographic principle . In these scenarios, the dark energy takes the form
where is a numerical factor, is the reduced Planck mass, and is the size which represents the IR cut-off or the UV-cutoff. Recently, several kinds of cosmological scales are considered to be the role of IR cut-off. These scenarios are summarized in  including the Hubble horizon, the particle horizon, the future event horizon as the length scale and the age of the universe, the conformal time as the time scale [20, 21, 22, 23, 24]. Therefore, there are five kinds of CDM models so far. The different choices of these scales give various evolutions of and different dynamical behavior of the universe. Thus it is necessary to compare the theoretical analysis with the cosmological observations. This direction has been explored in  which uses the recently compiled “Union2 Compilation” of Type Ia supernovae (SNe Ia) data.
As another kind of the cosmological observation, the Hubble parameter which is directly related to the expansion history of the universe has been tested in several cosmological models [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35]. Different from the distance scale measurements such as SNe Ia, CMB and BAO, there is no integral in calculating . This feature makes the Hubble parameter preserve the information related to the cosmological evolution. Thus investigate the observational data is very rewarding. The previous works showed that the observational Hubble data (OHD) can give a good supplement to the cosmic observations and can also be used as an alternative to the SNe Ia data which is considered as the ”standard candle”[36, 37]. So, whether the OHD can give a compatible constraints with the SNe Ia on the CDM model should be focused on. However, updated and expanded several times, the amount of available data is still scarce compared with SNe Ia luminosity distance data [38, 39, 40]. One way to avoid this embarrassment and alleviate the inaccuracy of lack of data points is to expand the data set of OHD by simulation. In our calculation, we will apply the method proposed in  which shows that as many as 64 independent OHD data points can match the parameter constraining power of SNe Ia by comparing the median figures of merit.
Except that, it is also useful to employ the information criteria to assess these CDM models since their origin of holographic and agegraphic are similar with each other. In this paper, we use the AIC, BIC and AIC as model selection criteria to distinguish different models from the the statistical view. Furthermore, we also adopt the and statefinder diagnostics to evaluate the dark energy models and measure their derivation from the standard CDM model. The corresponding analysis and discussion are presented in section 5
Our paper is organized as follows. In section 2, we present the basic formulas of CDM models with an interaction term between dark energy and dark matter. In section 3, the observational Hubble data is described. The constraints on the CDM models as holographic and agegraphic dark energy are presented in section 4. In section 5, we compare and measure the deviation of such CDM models from CDM model. The simulated Hubble parameter data sample and its corresponding constraints are shown in section 6. Our discussion and conclusion are summarized in section 7.
2 CDM model in the flat FRW universe
In the frame of Einstein gravity, considering the flat FRW metric
with standing for the scale factor, we can get the Friedmann equation
where is Hubble parameter, and are the energy densities of the matter and vacuum (dark energy) respectively. By the use of the dimensionless density parameters and , this equation can be rewritten as
In our scenario, we suppose that the energy can exchange between matter and vacuum energy through the interaction term , thus the local energy-momentum conservation law can yield
where ”.” means the derivative with respect to the cosmic time and the sign in front of will decide the direction of the energy flux. and are the equation of state (EOS) for the matter and vacuum energy respectively. With the definitions of the effective EOS for both the energy term
eq.(6) can be rewritten as
Substituting this equation to eq.(4), we get the evolution of Hubble parameter where the expansion rate can be expressed as
where the subscript ”0” denotes the present value of a quantity.
3 The observational Hubble parameter data
The measurement of Hubble parameter is increasingly becoming important in cosmological constraints. As a direct measurement related to the expansion history of the universe, it can be derived from the derivation of redshift with respect to the cosmic time 
From the observation of galaxy ages, Jimenez et al. demonstrated the feasibility of the method by applying it to a sample . Simon et al. further derived a set of OHD including 8 data points with a redshift range up to from the relative ages of passively evolving galaxies . Additionally, Stern et al. obtained an expanded data set and constrained the cosmological parameters . In our paper, we will focus on these data listed above [34, 41]and constrain the CDM as holographic and agegraphic dark energy models.
By the use of statistics, one can get the best-fitting values of the parameters and its corresponding confident regions
where stands for the parameter vector of each dark energy models and is the prior which is a 3% precision measurement suggested in .
In the calculations, we choose the standard CDM model as the fiducial model which gives the expression of as
The constraints on this fiducial model by OHD are , and . And the relation is plotted in figure 1 It can be seen that the best fit values are consistent with the 7-year WMAP , the BAO  and SNe Ia . In the next section, we will constrain the holographic and agegraphic dark energy models and compare their results with the CDM model.
4 Constraints on the holographic and agegraphic dark energy models
4.1 Hubble horizon as the IR cut-off
The Hubble horizon is defined as
It provides an estimate of the distance that light can travel while the universe expands appreciably. When the Hubble horizon is chosen as the IR cut-off of the holographic dark energy, we can get
where is a constant instead of the speed of light and satisfying 
Thus one can get
Consequently, the expansion rate of this scenario is
So, the choice of Hubble horizon as the IR cut-off (HH) presents a one-parameter cosmological model.
4.2 Particle horizon as the IR cut-off
The particle horizon is defined as
which is the distance that light have traveled since the beginning of the universe. The calculation in  give the effective EOS of vacuum energy
where describe the deviation from the de Sitter universe. It can be seen that the choice of particle horizon can not lead to the present cosmic acceleration because . So there is no need to constrain its undetermined parameters.
4.3 Future event horizon as the IR cut-off
The future event horizon is defined as
which is the distance that light will be able to travel in the future. By the use of the same method as the Hubble horizon, we can get
where is taken account to fill the deviation from the de Sitter universe. Several steps after, one can get the evolution of with respect to the redshift
Furthermore, the effective EOS for both matter and vacuum energy can also be achieved
Substituting these to eq.(9), we can get the evolution of the expansion rate .
Apparently, the choice of future event horizon (FEH) derives a two-parameters cosmological model which can lead to the accelerating phase. When performing the statistics for the OHD to constrain the parameters , we get and at the 68.7% confidence level. This best results show a universe with fewer matter comparing with the observations. However, at the confident level, the value of is consistent with the observations. The probability contours on the plane is plotted in figure 2 (the right panel) with .
4.4 Age of the universe as the IR cut-off
The age of the universe is defined as
Under this scenario, we obtains
where is a constant to fill the derivation from the de Sitter universe . After several steps of calculation, we can get the evolution of satisfying
and two EOSs for matter and vacuum energy
The same as the previous section, the choice of age of the universe as the IR cut-off (AU) also gives a two-parameter cosmological model. Using the OHD and statistics, we get the best-fit results of the model parameters: and at the 68.3% confidence level. It should be noticed that the OHD can not give a finite best-fit value and upper limit of while the constraint just shows a lower limit of . In figure 3, we plot the confident regions on the plane and we set as the cut-off.
4.5 Conformal time as the IR cut-off
The conformal time is defined as
which is the total comoving distance light could travel . In this case, we can get
By the use of the same method as presented in the previous section, one can obtain the differential relation of with respect to redshift and the effective EOSs for both matter and vacuum energy
The corresponding expansion rate of this model can be obtained by substituting this equation into eq.(9).
Similar to the previous models, the choice of conformal time as the IR cut-off (CT) also derives a two-parameter cosmological model. The constraint of OHD gives the best-fit results and which is almost the same as the scenario of age of the universe. The absences of the best-fit value and upper limit of also exist in this model. The only difference is a little bit larger than the one obtained in the age of the universe scenario which is . figure 3 shows the confident region on the plane while setting as the cut-off.
5 Model Identification
5.1 Information criteria
It is widely known that statistics is effective to find the best-fit values of parameters in a model. However it can not be used to decide which model is the best one among various models sufficiently. In our work, we have chosen the standard CDM model as the fiducial model, so it is necessary to assess the holographic and agegraphic dark energy models with the fiducial model. The information criteria (IC) is an effective way to assess different models. In the following, we will use the BIC and AIC as model criteria. In this criteria frame, models which give lower values of BIC and AIC will be preferred[45, 46, 47, 48, 49].
The BIC is given by
where is the maximum likelihood and can be calculated as , is the number of the model parameters and is the number of the data points. The difference in is denoted as where stands for different models. The model that can give a smaller will be considered closer to the fiducial one.
The AIC is defined as
where and have the same meanings as in BIC. Similarly, the one which gives the smallest value of AIC will be considered as the best model. The difference of AIC from the fiducial one is which describes the deviation from the fiducial model. Additionally, for the small data sample, there is a corrected AIC version which is defined as
This formula is useful when . In our calculation, there is only 13 data points in OHD, so is suitable.
Our results of information criteria are listed in Table. I. We can see that all three information criteria results show that the standard CDM model is the best one and this is consistent with the assumption that setting CDM as the fiducial model at the beginning of our calculation. Although the OHD constraint on FEH scenario give the smallest value of , the information criteria (both BIC and AIC) show a punishment because of its complexity. Except that, the large value of information criteria in HH shows a great deviation from the fiducial model. From the previous constraints, we see that the best-fit results obtained in AU and CT scenarios are very similar. Furthermore, their values of information criteria both located around 2. This situation shows a positive evidence that the AU and FT scenarios are close to the fiducial model.
5.2 The diagnostic
In order to get a more transparent comparison between the fiducial model and CDM models, we use the diagnostic which is given by 
Apparently, for CDM. Thus this diagnostic is demonstrated to be useful to distinguish CDM from other DE models. Except that, relies only on the knowledge of Hubble parameter. This makes the errors in the reconstruction of are bound to be small. In figure 4, we plot the evolutions of for each model. We can see that the HH and FEH scenarios behaves different from CDM model apparently. However, the two agegraphic dark energy models show great consistence with CDM where remains constant. It is natural because the best-fit value of in the two agegraphic dark energy models takes the limit . According to eq.(27) and eq.(31), these two model both have and . This limit makes them return to the standard CDM model and get to a situation that the interaction term can be neglected.
5.3 Statefinder diagnostic
The statefinder diagnostic was first introduced in  which is capable to discriminate various dark energy models. This parameter pair is given as
where is the deceleration parameter. Obviously, this statefinder pair is constructed directly from the spacetime metric, i.e the second and third derivatives of the scale factor . Since the trajectories in the plane belong to different cosmological models show qualitatively different behaviors, the statefinder diagnostic can be seen as a powerful tool to distinguish different cosmological models. So far, many works have been done by means of the statefinder diagnostic analysis, such as the standard CDM model, the quintessence with and without interacting models, the holographic dark energy models, the phantom model, and the tachyon model [55, 56, 57, 58, 59, 60, 61, 62].
Firstly, from the definition of the deceleration factor and eq.(4), one can get
Because the parameter in eq.(36) can be rewritten as
substituting the relation
to eq.(38), we can obtain
In the flat FRW universe, the standard CDM model corresponds to a fixed point
This value is independent on the parameter of CDM model and cosmic time (or equivalently redshift ). The statefinder diagnostic applied to the above CDM models as holographic and agegraphic dark energy derives several formulas of , these results are summarized in Table. II.
In figure 5 and 6, we plot the evolutionary trajectories of statefinder for the CDM as holographic and agegraphic dark energy models in the plane. The same as the standard CDM model, the statefinder pair in the HH model is also a fixed point but is dependent on the parameter . By the use of the best fit results from OHD constraints, we obtain in HH scenario which shows a great derivation from CDM model as the previous analyze. The arrows in the curve show that the universe full of vacuum energy in the HH scenario can give the same statefinder pair as CDM model .
Different from the CDM model and HH scenario, the statefinder pairs of the three two-parameter models FEH, AU and CT are all parameter-dependent and relying on the redshift . Applying the constraining results of OHD, we present several trajectories of statefinder for these models with different values. From the right panel of figure 5, we get that there is a quintessence-like to phantom-like transition of the dark energy with the cosmic time increasing. Naturally, at the moment of transition, the statefinder returns to CDM model. From the numerical solution of in , it can be seen that approaches to 0 as the redshift increases. This result derives that the statefinder of FEH will reach to a limit at the very beginning of the universe.
The evolutionary trajectories of statefinder for the two age cosmological constant models AU and CT are quite different (see figure 6). This diagnostic distinguishes AU and CT helpfully. However, we should note that the values of used here are just located in the confident level . When applying , the statefinder pairs of AU and CT in the plane will approach to the fixed point (0,1) as the standard CDM model which are still indistinguishable. Certainly, with the increasing of , the range of the statefinder curve decreases and finally degenerates to a point. Except that, no matter what finite value of is chosen, the will reach to the CDM situation at the very beginning of the universe.
6 Generation of simulated datasets and constraints
So far, we have used the OHD to constrain the CDM as holographic and agegraphic dark energy models. However, the shortcoming of OHD is also apparent because the amount of data points is too scarce to be comparable with other cosmic observations such as SNe Ia. According to , by comparing the median figures of merit, 64 independent measurements of will give a constraining results which can match the parameter constraining power of SNe Ia. By applying the same method, we choose the standard CDM model as the fiducial one to generate 64 data points as a simulation sample (In the rest we will use the abbreviation ”SHD” for the simulated data) to constrain the CDM as holographic and agegraphic dark energy models. The simulated data sample is presented in figure 7.
Using the SHD to constrain CDM model, we get and (figure 8). This results is consistent with the one under OHD and the interval is tighter. Thus it is natural to expect that with more data points, the constraints on the holographic and agegraphic dark energy models will be more compact. The constraining results are summarized in Table.III
In figure 9 and 10, we plot the confident regions for each models. Combined with Table.III, we see that the constraint in HH scenario is more compact, however the results of large matter density and big value show a great deviation from CDM model and observations. For other three scenarios, the more data points do not provide a much more tighter constraint of the , and interval. However, the confident regions of these three models are tighter and narrower. This phenomena is consistent with . It implies: when defining the FoM (Figure of Merit) as the area enclosed by the contour of possibility distribution at some confident level, the FoM will increase with enlarging the data sample. Additionally, similar to the OHD constraint, the FEH scenario also gives the smallest . Except that, the problem of lacking best-fit value and upper limit of still exists in the two time-scale models. This situation implies that we need some other constraint conditions from the physical view. Compared with the constraints using ”Union 2 Compilation” of type Ia SNe, we see that all the 3 two-parameter models give consistent trends of confident regions. And the best-fit values and confident interval of the undetermined parameters are also almost identical with the SNe Ia ones.
7 Discussion and conclusion
We have explored the possibility of Hubble parameter as the cosmological constraint on the CDM as the holographic and agegraphic dark energy models. As pointed out in [36, 37], the OHD play almost the same role as that of SNe Ia for the joint constraints on the CDM model. As a promotion of standard CDM, the CDM may provide a wider range of adaptation of cosmological observations. Recently, through the introduction of a length scale or time scale to give a form has been investigated which is equivalent to the holographic and agegraphic dark energy models. In order to track the deviation of these scenarios from the standard CDM model, we use OHD to constrain such CDM models and measure the deviations. We calculate the models introduced by the Hubble horizon (HH), the future event horizon (FEH), the age of universe (AU) and the conformal time (CT) since these scales can provide the present cosmic acceleration.
In order to avoid the dilemma that the amount of OHD is so scarce, we simulate a data sample of (SHD) in the redshift range of by the use of CDM as the fiducial model. From the constraint results, we find that the enlarging of the data sample increases the parameter constraints apparently for all four CDM models. Although the , and confident intervals of the three two-parameters models do not become narrower evidently, the areas of the confident regions in the parameter space shrink. This is expectable due to the bigger data sample.
However, in the constraints of two time scale scenarios, there are lacks of the best-fit value and upper limits of parameter . This is similar to the SNe Ia constraints . According to eq.(26) and (30), this phenomena reveals since is insensitive to large value of . Except that, when , the equation of state of these two models will return to the standard CDM model which is a constant and favors a non-interaction term. However, the lack of constraint of will derive to be a non-physical value. So we need some other constraint conditions from the physical view in the future.
In order to measure the deviation of these CDM models from the standard CDM model. We apply the and statefinder diagnostics and the information criteria. These three methods can distinguish the dark energy models from the standard CDM model on various aspects and focuses. The calculation with the use of OHD shows quite consistent results of these methods. Choosing HH as the IR cut-off will derive a quite different holographic dark energy model from the standard CDM model and its relative derivation is apparent. As a another choice of length scale, the FEH scenario shows better consistence with CDM but still occupies prominent deviation. On the other hand, the two time models AU and CT scenarios both possess very evident consistence from CDM and can provide effective approximations.
From the above analyze and constraints, we find that our results from observational data and simulated data are believable. The OHD can not only provide sufficient constraints on the CDM models as holographic and agegraphic dark energy as the SNe Ia supply, but also can measure the deviation of the cosmological models from standard CDM model. Therefore, with more future high-, high-accuracy determinations, OHD will give more and more important contributions in future cosmological researches.
This research is supported by the National Natural Science Foundation of China (Grant Nos. 10773002, 10875012), Scientific Research Foundation of Beijing Normal University under Grant No. 105116, the National Science Foundation of China (Grants No. 10473002, No. 11033005), the Fundamental Research Funds for the Central Universities, the Bairen program from the Chinese Academy of Sciences and the National Basic Research Program of China grant No. 2010CB833000.
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