Holographic Dark Energy: its Observational Constraints and Theoretical Features
aa r X i v : . [ a s t r o - ph . C O ] M a r Holographic Dark Energy: its ObservationalConstraints and Theoretical Features
Yin-Zhe Ma
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, U.K.
Abstract.
We investigate the observational signatures of the holographic dark energy model in thispaper, including both the original model and a model with an interaction term between the dark en-ergy and dark matter. We first delineate the dynamical behavior of such models, especially whetherthey would have a “Big Rip” for different parameters, then we use several recent observational datato give more reliable and tighter constraints on the models. The results favor the equation of stateof dark energy crossing −
1, and the universe ends in the "Big Rip" phase. By using the Bayesianevidence as a model selection criterion to make the model comparison, we find that the holographicdark energy models are mildly favored by the observations compared with the L CDM model.
Keywords: holographic principle, interacting dark energy, observational constraints
PACS:
INTRODUCTION
The nature of dark matter (DM) and dark energy (DE), which constitute about 95% ofthe total cosmic density, are among the most important problems in modern physics andastronomy [1]. Dark energy may be a problem that has to be solved in the framework offundamental microscopic physics such as string theory [2].Here we consider the dark matter-dark energy interaction in the case of the so-calledholographic dark energy (HDE) model [3, 4, 5, 6, 7]. This is based on the holographicprinciple [8], which was inspired by the Bekenstein entropy bound of black holes [9]. Ifthe quantum zero-point energy density r L is relevant to an ultraviolet cut-off, the totalenergy of the whole system with size L should not exceed the mass of a black hole ofthe same size, thus we have L r L ≤ LM . The largest infrared cut-off L is chosen bysaturating the inequality so that the HDE density is r de = c M L − , (1)where c is a numerical constant, and M pl ≡ / √ p G is the reduced Planck mass. L should be the size of the future event horizon, since only this case would result in a darkenergy component which drives the accelerated expansion of the universe [6], i.e. L = R eh ( t ) = a ( t ) Z ¥ a da ′ H ′ a ′ . (2)If we consider the dark matter and dark energy interaction, the energy conservationequation will be written as ˙ r m + H r m = Q , (3) - - - - - - - z w H z L c = c = c = - z Ρ HD E H z L (cid:144) H M p l c = c = c = FIGURE 1.
EEoS and HDE density in the holographic dark energy model for different value of c . Left:EEoS; Right: HDE density. Here we set W m = . ˙ r de + H ( + w de ) r de = − Q , (4)where r m , r de and w de are matter density, HDE density and HDE equation of staterespectively. Here we consider the physically plausible interaction as Q = a H r de ( a isa parameter describing the strength of interaction), which has been widely used for theinteraction between the matter and massive scalar field [10]. Therefore, by combiningthe above equations and Friedmann equation, we quickly obtain the following equationswhich describe the evolution of the fractional energy density W de , Hubble parameter H ( z ) , and also the HDE effective equation of state (EEoS) (see [7] for details): d W de ( z ) dz + W de + z [( − W de )( + c p W de ) − a W de ] = , (5) dHdz = − H ( z ) + z [ W de ( + a + c p W de ) − ] , (6) w eff ( z ) = w de + a = − − √ W de c . (7)By solving the Eqs. (5), (6), and (7), we can know the dynamics of the HDE model asfollows: For the non-interacting HDE model, i.e. a =
0, the evolution of HDE densityand its EEoS will be affected by the value of parameter c as shown in Fig. 1. If c = − c >
1, the EEoS is always greater than −
1, and the HDE isjust like the quintessence dark energy. On the contrary, if c <
1, the EEoS will cross − a but fix c = a <
0, the dark matter "decays" into IHDE(Eqs. (3) and (4)), which will make the HDE density increase rapidly at the later era ofcosmic evolution. Thus, effectively the IHDE will resemble the phantom dark energywith EEoS less than − a >
0, the IHDE density would decay more rapidly,making the IHDE behave like a quintessence field. - - - - - - - - - z w H z L Α= Α= Α= Α=-
Α=- = - - z Ρ I HD E H z L (cid:144) H M p l Α= Α= Α= Α=-
Α=- = FIGURE 2.
EEoS and IHDE density in the interacting holographic dark energy model for differentvalue of a . Left: EEoS; Right: HDE density. Here we set W m = . However, we want to ask the inverse questions: what values of parameter c and a arepreferred by the current observational data? In addition, which of these dynamic darkenergy models are favored by current observational data compared with the concordance L CDM model?
METHODOLOGY
We utilize several data sets to constrain the parameters of the HDE and IHDE model,including the 182 high-quality type Ia supernovae [12], the baryon acoustic oscillationmeasurement from the Sloan Digital Sky Survey [13], 42 latest X-ray gas mass frac-tion data from Chandra observations [14], 27 GRB samples generated with E peak − E g correlation [15], and the CMB shift parameter from the WMAP 3 year results[16]. To break the degeneracy and explore the power and differences of the con-straints for these data sets, we use them in several combinations to perform our fitting:SN sel + BAO , SN sel + BAO + f gas , and SN sel + BAO + f gas + GRB + CMB.For comparing different models, one must choose a statistical variable. The c min isthe simplest one and is widely used. However, for models with different numbers ofparameters, the comparison using c may not be fair, as one would expect that modelswith more parameters tend to have lower c . Instead, we use the Bayesian evidence (BE)as a model selection criterion. The Bayesian evidence of a model M takes the formBE = Z L ( d | q , M ) p ( q | M ) d q , (8)where L ( d | q , M ) is the likelihood function given the model M and parameters q , and p ( q | M ) is the priors of the parameters. The BE may be the best model selection criterion,as it is the average of likelihood of a model over its prior of the parameter space andautomatically includes the penalties of the number of parameters and data, so it is moredirect, reasonable and unambiguous than the c min model selection [17]. The logarithmof BE can be used as a guide for model comparison, and we choose the L CDM as thereference model: D ln BE = ln BE model − ln BE L CDM . The strength of the evidence forthe model is considered according to the numerical value of BE: D ln BE < < D ln BE < . . < D ln BE < D ln BE > RESULTS
In Table 1, we give the best fits and the 1 s CL of the HDE model parameters, as well asthe value of ln D BE for the three data set combinations. The best fit of c varies slightlyacross the different data sets, it is 0.761 for the SN+BAO data set, but decreases slightlywhen the f gas , GRB and CMB data are included. However, for all data sets, we have c < . s , indicating that the EEoS of HDE will cross − TABLE 1.
The fitting result for the HDE model.SN + BAO SN + BAO + f gas SN + BAO + f gas + GRB + CMB W m . + . − . . + . − . . + . − . c . + . − . . + . − . . + . − . D ln BE 0 . ± .
12 0 . ± .
18 0 . ± . HDE model fits about equally well (ln BE=0.09) as the L CDM when we only use theSNIa and BAO data. With f gas , GRB and CMB data added, it fits mildly better thanthe L CDM, but with the data presently available the difference is not significant (ln BE=0 . ∼ . TABLE 2.
The Fitting results the IHDE model.SN + BAO SN + BAO + f gas SN + BAO + f gas + GRB + CMB W m . + . − . . + . − . . + . − . c 0 . + . − . . + . − . . + . − . a − . + . − . . + . − . − . + . − . D ln BE 0 . ± .
12 0 . ± .
18 0 . ± . and 1 s CL, the fractional energy density of dark matter is still around 0.27, and theparameter c is always less than 1. The difference from the HDE model is that IHDEhas the interacting parameter a here and its value is either positive or negative, but - - - - - - z w e ff H z L SN sel + BAO + f gas + GRB + CMBSN sel + BAO + f gas SN sel + BAO - - z Ρ I HD E H z L (cid:144) H M p l SN sel + BAO + f gas + GRB + CMBSN sel + BAO + f gas SN sel + BAO
FIGURE 3.
EEoS and energy density evolution for the best fit IHDE models.
IGURE 4.
The contours of c vs. a in the IHDE model. always very small and around zero. If you look closer at it, zero is always coveredin the 1 s CL, indicating that there is little evidence for the interaction. Furthermore,the Bayesian Evidence is always positive but less than 1, if you compare them withthe previous Bayesian Evidence of non-interacting case, you can see that the number isslightly greater than the previous case, which means that the IHDE model is a little morefavored than the HDE model but the evidence is still not strong.Using the best fitting values for the three data sets, we also plot the effective equationof state and dark energy density in Fig. 3. We can see that all three data sets suggest theequation of state crosses − c and a is plotted in Fig. 4. In Fig. 4, the contours correspondingto the data sets SN+BAO+ f gas +GRB+CMB are much tighter than the other data sets.This is because we use the data set GRB and CMB, which has very large redshiftdistribution so they break the degeneracy of the parameter c and a . We also mark the a − c values for which w e f f = − c ∼ .
6, so c <
1. The value of a varies more, but all consistent withbeing 0 within 1 . s , which suggests the evidence for the interaction is very weak. In anycase, for all three data set combinations, the best fit value resides in the phantom-likeregion, although a large area of quintessence-like region is also allowed. CONCLUSION
In this paper we first introduced the holographic dark energy model with the interact-ing term Q = a H r de , so the non-interacting case could be viewed as the special casewith a =
0. We illustrated the dynamical behavior of these models by choosing somerepresentative values of the parameters c and a . The condition for the model to havea “Big Rip” is determined. Second, we utilize several data sets from the recent obser-vations to constrain the models. The best-fits for the three data sets are given in Table for the HDE model, and Table 2 for the IHDE model. For both the HDE and IHDEmodels, the data favors “phantom” behavior slightly, i.e. the dark energy initially has w e f f > −
1, but eventually crossing the phantom dividing line, and the model ends witha “Big Rip”. However, quintessence-like behavior is also allowed with the present data.Next, we utilize the Bayesian evidence (BE) as a model selection criterion to comparethe holographic models with the L CDM model for the three data sets. Both the HDEand the IHDE model are mildly favored by the current observational data set, althoughthe evidence is weak.In brief, we conclude that according to the observational data, the holographic darkenergy model, especially the interacting holographic dark energy model is mildly fa-vored by the current data, and for the best fit model the equation of state for both theHDE and IHDE crosses −
1, for which the Universe ends up in a "Big Rip".
ACKNOWLEDGMENTS
I acknowledge the support from the Cambridge Overseas Trust and a studentship fromTrinity College, Cambridge, as well as the National Science Foundation in China. I alsothank Xuelei Chen, Yan Gong and many other colleagues for the helpful discussions,and David Cline for inviting me to give this talk.
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