Reconciling Hubble Constant Discrepancy from Holographic Dark Energy
RReconciling Hubble Constant Discrepancy from Holographic Dark Energy
Wei-Ming Dai , , ∗ Yin-Zhe Ma , , , † and Hong-Jian He , , ‡ School of Chemistry and Physics, University of KwaZulu-Natal,Westville Campus, Private Bag X54001, Durban, 4000, South Africa NAOC-UKZN Computational Astrophysics Centre (NUCAC),University of KwaZulu-Natal, Durban, 4000, South Africa Purple Mountain Observatory, CAS, No.10 Yuanhua Road, Qixia District, Nanjing 210034, China Tsung-Dao Lee Institute & School of Physics and Astronomy,Shanghai Key Laboratory for Particle Physics and Cosmology,Shanghai Jiao Tong University, Shanghai 200240, China Institute of Modern Physics & Physics Department, Tsinghua University, Beijing 100084, China Center for High Energy Physics, Peking University, Beijing 100871, China
Holographic dark energy (HDE) describes the vacuum energy in a cosmic IR region whose totalenergy saturates the limit of avoiding the collapse into a black hole. HDE predicts that the darkenergy equation of the state transiting from greater than the − −
1, acceler-ating the Universe slower at the early stage and faster at the late stage. We propose the HDE asa new physical resolution to the Hubble constant discrepancy between the cosmic microwave back-ground (CMB) and local measurements. With
Planck
CMB and galaxy baryon acoustic oscillation(BAO) data, we fit the HDE prediction of the Hubble constant as H = 71 . ± .
78 km s − Mpc − ,consistent with local H measurements by LMC Cepheid Standards (R19) at 1 . σ level. Combining Planck +BAO+R19, we find the HDE parameter c = 0 . ± .
02 and H = 73 . ± .
14 km s − Mpc − ,which fits cosmological data at all redshifts. Future CMB and large-scale structure surveys will fur-ther test the holographic scenario.Phys. Rev. D 102 (2020) 121302 (Rapid Communication), [arXiv:2003.03602 [astro-ph.CO]].
1. Introduction
The cosmological observations derived from the“Early” and “Late” Universe tend to prefer different val-ues of the Hubble constant ( H ), leading to a discrep-ancy between the two types of measurements. The mea-surements of the cosmic microwave background (CMB)from Planck satellite [1] and Atacama Cosmology Tele-scope (ACT) [2] measured the Hubble constant to be H = 67 . ± . − Mpc − ( Planck ), and H =67 . ± . − Mpc − (ACT). In contrast, several lo-cal measurements give consistent higher values. SH0ESmeasurement of Cepheids data (R19) obtained from Hub-ble Space Telescope (HST) gives the improved local de-termination H = 74 . ± .
42 km s − Mpc − [3]. Re-placing the Cepheids with the oxygen-rich Miras dis-covered in NGC4258, Ref. [4] measured the Hubble con-stant H = 73 . ± . − Mpc − . Using the geomet-ric distance to the megamaser-hosting galaxies CGCG074-064 and NGC 4258, Ref. [5] gives H = 73 . ± . − Mpc − . In a complementary probe of usinggravitationally lensed quasars with measured time de-lays in a flat ΛCDM cosmology, the H0LiCOW teamfound H = 73 . +1 . − . km s − Mpc − [6] and more recently82 . +8 . − . [7]. A combination of different local measure-ments yields H = 73 . ± . − Mpc − , which is5 . σ discrepant from the aforementioned Planck
CMBresult [8].Various theories have been proposed to resolve thisdiscrepancy, mainly from two prospects [9]: (i) modify- ing the early-universe physics to shrink down the soundhorizon at the drag epoch r drags [10], such as includingneutrino self-interactions to delay its free-streaming [11,12][13]. (ii) modifying dark energy (DE) evolution, byconsidering the dark sector interactions [14] or early darkenergy (EDE) component [15]. But these new models areeither phenomenologically contrived, or hard to falsifybecause they always recover ΛCDM behaviour if someparameters are tuned. For example, the interacting darksector model assumes a somewhat arbitrary form of in-teraction, with a free coupling. If the constraints areimproved, the interaction coupling is asymptotically ap-proaching zero but can always have a small value, mak-ing it almost impossible to rule out the model. The EDEmodel inserts a short period of fast expansion around re-combination, by demanding “just about” the amount ofdark energy to expand the Universe and exit at the righttime. So the EDE scenario, by its construction, is highlycontrived.In this work, we propose the holographic dark energy(HDE) as a striking new resolution to the H discrepancy.This scenario is built upon the holographic principle ofquantum gravity from ’t Hooft [16] and Susskind [17], andthe Bekenstein-Hawking entropy bound [18, 19] that con-nects an ultraviolet (UV) scale in quantum states to aninfrared (IR) cutoff in the macroscopic scale [20]. Wedemonstrate that with a single HDE parameter c (cid:39) . w > − w < −
1, which can naturally resolve the H discrepancybetween the CMB and local measurements. We also show a r X i v : . [ a s t r o - ph . C O ] D ec that for any value of c , the HDE EoS always evolves as afunction of time and never mimics Einstein’s cosmolog-ical constant Λ, which can be substantiated or falsifiedwith future data.
2. Holographic Dark Energy
In the black hole thermodynamics, the Bekensteinentropy bound states that the maximum entropy in abox of volume L grows only as the box’s area [18, 19].Then, ’t Hooft [16] and Susskind [17] observed that dueto the Bekenstein bound the 3+1 dimensional field theo-ries over-count the degrees of freedom (d.o.f), which areproportional to the area of the box surface. They con-jectured the holographic principle: the total d.o.f of anyeffective field theory in a box of size L must be belowthe black hole entropy of the same size, L Λ (cid:54) S BH = πL M , where M Pl = (8 πG ) − / is the reduced Planckmass and Λ denotes the UV cutoff. Subsequently, Cohen,Kaplan and Nelson [20] found that if this condition holds,it implies that there are many states within the box size L with Schwarzschild radii even larger than L . Hencethese states should have collapsed and cannot exist. Toavoid such catastrophe, Ref. [20] tightened up the boundby imposing a constraint on the IR cutoff 1 /L that ex-cludes all states lying within the Schwarzschild radius, L Λ (cid:54) LM . This condition predicts the maximum en-tropy S max = S / and is indeed a tighter bound. In sum,this new holographic condition means that for quantumstates in a box size L to exist without collapsing, theshort-distance UV scale is connected to a long-distanceIR cutoff due to the limit set by the black hole formation.Namely, the maximum total energy set by the UV cutoffΛ in a region of size L should not exceed the mass ofa black hole with the same size, L ρ Λ (cid:46) LM . Hence,the dark energy density is bounded by ρ Λ (cid:46) M L − . Inshort, the HDE construction has made two assumptions.(i). Holographic principle applies to the entire universe,so the box size L should be related to the horizon scalein cosmology; (ii). The dark energy is indeed the quan-tum vacuum energy, so the dark energy density ρ Λ ∼ Λ .Ref. [21] subsequently proposed that, to make the largest L saturate the above new condition, the energy densityof this HDE should be ρ de = 3 c M L − , (1)where c is a constant coefficient. It was also found [21,22] that only if the IR cutoff L is taken as the futureevent horizon of the Universe, L = R eh = a (cid:82) ∞ t d t (cid:48) /a ( t (cid:48) ) ,the dark energy can provide the desired repulsive forceto explain the cosmic acceleration.We combine Eq.(1) with the energy-momentum con- servation and Friedmann equation to obtain:˙ ρ de + 3 H (1 + w de ( z )) ρ de = 0 , (2a)3 M H = (cid:88) j ρ j , (2b)where w de ( z ) is the EoS parameter of the HDE and H is the Hubble parameter. In Eq.(2b), the sum of energydensities includes matter ( ρ m ), radiation ( ρ r ), and darkenergy ( ρ de ). Among these, ρ r = Ω r ρ cr (1 + z ) is fixedby the observed CMB temperature. From Eq.(2), wederive the following differential equations governing thedynamics of background expansion,d ρ de d t = − Hρ de (cid:32) − ρ / √ c M Pl H (cid:33) , (3a)d H d t = 16 M H (cid:88) j ˙ ρ j , (3b)and the EoS for the HDE, w de ( z ) = − − ρ / √ c M Pl H . (4)We numerically solve Eqs. (3a)-(3b) as the backgroundevolution of the Universe, and compute the HDE EoSfrom Eq. (4). We show the w de ( z ) function for the case c = 0 . c asEq.(1), which controls the behaviour of HDE. Given anappropriate c , the HDE can have w de greater than − z (cid:38) − z (cid:46) P = w de ρ ) was weaker at the earlier epoch thanthe present time. Hence, it causes the Universe to havesmaller acceleration earlier on, and faster acceleration atthe later stage, but still keeps the total angular diam-eter distance to the last-scattering surface unchanged.We find that this “ delayed acceleration ” is precisely thedynamical behaviour needed to resolve the H tensionbecause the present-day expansion rate H from the lo-cal measurements is higher than what is measured bythe CMB. But the angular diameter distance to the last-scattering surface is fixed by the high-precision CMBmeasurement. As an analogue, a marathon runner canrun slower at an early stage but accelerate at the laterperiod to keep the total time and distance unchanged.
3. Two Parametrized Models
To explore the transiting w ( z ) behaviour, we seek twoparametrized models of the dynamical dark energy with 2and 4 more parameters than ΛCDM, which can mimic the z w ( z ) HDECPLTransDE
FIG. 1. Fitting HDE behavior by using TransDE and CPLdark energy models. The blue solid curve shows the EoS ofHDE with c = 0 . Planck best-fitting values [1]. The orange dashedand green dash-dotted curves are the best-fitting w ( z ) of theCPL dark energy and the TransDE, respectively, in the red-shift range z ∈ (0 , ). behaviour of HDE. In general, if a dark energy model hasEoS w de > − w de < − w , w , z t , ∆ z ), w ( x ≡ ln(1+ z )) = w + w (cid:18)
1+ tanh x − x t ∆ x (cid:19) , (5)where x t ≡ ln(1+ z t ) and ∆ x ≡ ∆ z/ (1+ z t ) determinethe redshift z t and width ∆ z of the transition. The ( w + w ) and w control the asymptotic behaviour of EoS atthe infinite past ( z → ∞ ) and infinite future ( z → − w ( a ) = w + w a (1 − a ) , which behaves like the TransDE modelat high- z , but the difference is non-negligible if a rapidtransition of EoS happens at low- z [25].We substitute the EoS of the TransDE and CPLparametrizations into Eq.(2a) and obtain the analyticalsolution for the dark energy density individually, ρ TransDEde = ρ cosh (cid:0) x t ∆ x (cid:1) w x exp (cid:104) (cid:16) w + w (cid:17) x + 32 w ∆ x ln (cid:18) cosh x − x t ∆ x (cid:19)(cid:21) , (6a) ρ CPLde = ρ exp {− w a (1 − a )+(1+ w + w a ) ln a ] } , (6b)where ρ = Ω de ρ is the present-day dark energy den-sity, and ρ = 3 H M is the critical energy densityat present. We fit the HDE scenario with c = 0 . H ( z ) / ( + z ) / ( k m s M p c ) z 354045505560657075 FIG. 2.
Planck +BAO12+R19 constraints on the Hubble pa-rameter for HDE (blue) and ΛCDM cosmology (grey). Thedark (light dark) colored stripes present the 68% (95%) lim-its, and the black solid curve in the center corresponds tothe mean value. The orange dots with 68% error bars arethe (marginalized) measurements. From left to right, the firstpoint is R19, the next three points are BAO DR12 constraints,and the last three points are the eBOSS DR14 QSO, BOSSDR12 Ly α and BOSS DR12 QSOxLy α , which are listed intable 1 of Ref. [27]. The BAO data at z (cid:38) CPL models can mimic the HDE behavior, with the min-imal deviations found by the global optimizer
PyGMO [26].This comparison shows that the HDE model is the mosteconomical model to resolve the H discrepancy.
4. Data Analysis
We combine the R19 data (local measurement), galaxybaryon acoustic oscillation (BAO) data (median red-shifts), and
Planck
CMB data (high redshifts) in ourdata fitting.We use the final full-mission baseline
Planck likelihooddata (the 2018 release), which includes the low- (cid:96) tem-perature likelihood (Commander), low- (cid:96) EE likelihood(SimAll), high- (cid:96) TT, TE and EE likelihood (Plik) [28],and the additional CMB lensing likelihood [29]. In thefollowing, “Planck” denotes the combination of the afore-mentioned
Planck data.The BAO data includes the “consensus” SDSS/DR12data [30], the 6dF [31] data and MGS [32] BAO data. Be-sides, SDSS quasar data and the combination of Lyman- α auto-correlation and Quasar-Lyman- α cross-correlationdata have put BAO constraints at redshifts z > z (cid:38)
2) for comparison.But unlike galaxy BAO measurements, quasar Ly α mea-surements require several additional assumptions of themodeling of metal-line and high-column-density neutralhydrogen and quasar spectra universality, hence are morecomplicated than the galaxy BAO measurements. Be- H (km s Mpc ) w d e ( ) Planck+BAO12+Pantheon+Pantheon( z < 0.2)+Pantheon( z > 0.2) FIG. 3. Joint constraints for the HDE on w de (0) (projectedEoS at z = 0) and H from Planck + BAO12,
Planck +BAO12+Pantheon ( z < . Planck + BAO12 + Pantheon( z > . Planck + BAO12 + Pantheon (full). It showsthat
Planck + BAO12 + partial Pantheon dataset (with either z > . z < .
2) gives a higher value of H , but with the fulldataset the value becomes much lower. This is because thePantheon dataset has large correlation between high- z andlow- z samples (cf. text). sides, the HDE and ΛCDM are nearly indistinguishablefor the Hubble parameter in the corresponding redshiftrange. For these reasons, we do not include the Ly α BAO in the parameter constraints, but use 6dF, MGSand SDSS/DR12 data as “BAO12”. (Ref. [1] gives simi-lar strategy and reason)Table I enumerates the effective redshift for each mea-surement, ranging from 0 .
106 to 0 .
61. The parameter r d is the sound horizon at drag epoch and D M denotes thecomoving angular diameter distance. D V is determinedby the angular diameter distance D A and the Hubble pa-rameter H ( z ) via D V = [ cD z (1+ z ) /H ( z )] / . The 6dFand MGS data give the measurement of r d /D V at red-shift z eff = 0 .
106 and the measurement of D V /r d at red-shift z eff = 0 .
15, respectively. BOSS DR12 data include D M r fid , d /r d and Hr d /r fid , d at redshifts z eff = { . , . . } , where r fid , d = 147 .
78 Mpc is a fiducial sound hori-zon. Since DR12 data are correlated between differentredshifts, we include all their full covariance matrix inour
CosmoMC likelihood package.R19 is the measurement of H from Large MagellanicCloud Cepheid Standards by Riess et al. [3], which gives H = 74 . ± .
42 km s − Mpc − , deviating from Planck measurement at 4 . σ level. Pantheon is a new set of light-curve supernovae (SNe), with 1048 samples spanning theredshift range 0 . < z < . H because it is degenerate withthe absolute magnitude M . The constraint comes indi-rectly from the joint datasets with Planck , because theSN samples can put constraints on Ω m (fractional mat-ter density) and w de that have covariance with H . In c H ( k m s M p c ) PlanckPlanck+BAO12Planck+BAO12+R19
FIG. 4. Marginalized constraints on the Hubble constant H versus the HDE parameter c , at 68% C.L. (contours with darkcolors) and 95% C.L. (contours with light colors). The com-binations of three datasets are shown in the legend. Fig. 3, we plot the joint constraint of w de (0) (projecteddark energy EoS at z = 0 ) and H from Planck +BAO12,
Planck + BAO12 + Pantheon ( z < . Planck +BAO12+Pantheon ( z > . Planck +BAO12+ Pantheon (full dataset). It shows that the combined
Planck +BAO12 with either subset of the Pantheon datagives consistent results of higher H value, but the fullPantheon samples shift to a lower value. We find thatthis inconsistency between the full Pantheon samples andeach subset of samples is due to the large correlation be-tween high- z and low- z samples of Pantheon. In Ref. [37],it shows that Planck +BAO and
Planck +Pantheon giveinconsistent results at more than 95% C.L., suggestingthat there are uncounted systematics in either Pantheonor R19 data. Since there are several local measure-ments that support R19 results (e.g., TRGB and lensing),we will not adopt
Planck +BAO+Pantheon as a baselinedataset, instead we use
Planck + BAO12 and
Planck +BAO12 + R19 as two baseline datasets.
5. Results and Discussions
We modify the Boltzmann camb code [38] to embedthe HDE and TransDE models into the background ex-pansion of the Universe, and use public code
CosmoMC (version of July 2019) to explore the parameter spacewith Markov chains Monte Carlo technique [39]. We as-sume spatially flat cosmology with six base cosmologi-cal parameters (Ω c h , Ω b h , θ ∗ , n s , A s , τ ) to vary, andset only one of the three generations of neutrinos havingmass 0 .
06 eV (under normal neutrino mass ordering). Wethen add the dark energy sector into the analysis whichhas one, two and four dark energy parameters for theHDE, CPL and TransDE models, respectively.Figure 4 presents the marginalized 2D contour of theHDE parameter c versus H . The Planck -only con-
TABLE I. BAO measurements. D V , D M D H and Hubbleparameter H are computed at the effective redshifts z eff . Dataset z eff Measurement Constraint6dF 0 . r d /D v . ± . . D v /r d . ± . . D M r fid , d /r d (1512 . ± .
99) MpcDR12 Hr d /r fid , d (81 . ± .
37) km s − Mpc − . D M r fid , d /r d (1975 . ± .
10) Mpc Hr d /r fid , d (90 . ± .
33) km s − Mpc − . D M r fid , d /r d (2306 . ± .
08) Mpc Hr d /r fid , d (98 . ± .
50) km s − Mpc − TABLE II. Model comparison. ∆AIC is the difference of AICfrom the ΛCDM model with the same dataset.
Data Set Model Best-fitting H χ ∆AIC[km s − Mpc − ] Planck +BAO12+R19 HDE 73 .
47 2791 . − . .
40 2789 . − . .
60 2787 . − . .
23 2799 .
25 0 straint on H is relatively weak, but including the BAO12and BAO12+R19 data tightens up the bounds. In Fig. 2,we plot the evolution of Hubble parameter H ( z ) as afunction of the redshift within range z ∈ [0 ,
20] for theHDE (blue) and ΛCDM cosmology (grey) under 1 σ and2 σ variations. We see that the “delayed acceleration”effect of HDE can match the BAO data and local R19data better than ΛCDM model. This fact is also re-flected by the χ and ∆AIC values listed in Table II.The Akaike information criterion (AIC) is a metric toquantify the “goodness-of-fit” by compensating the ad-ditional parameter(s) in the model. Comparing the HDE(1 extra parameter), TransDE (4 extra parameters) andCPL (2 extra parameters) with the benchmark ΛCDMcosmology, the HDE model fits the data better than theTransDE and ΛCDM models, while it also predicts the H value consistent with both the local R19 and stronglensing measurements. Although the CPL model per-forms better in terms of the AIC, it does not recover thelocal H value (Table II).Figure 5 shows the projected distribution of H in the HDE, TransDE, CPL, and ΛCDM cosmolo-gies for both Planck +BAO12 (dashed curves) and
Planck +BAO12+R19 (solid curves) datasets. One cansee that with R19 data (solid), all three dark energymodels prefer higher values of H which are consistentwith the vertical grey bands (R19 data). But with-out R19 data (dashed), the TransDE and CPL dark en-ergy models shift its projected H value to lower val-ues and become less consistent with the R19 value.Hence, it is the R19 data that dictates the TransDEand CPL models to have higher values of H , without which the recovery does not exist. However, even with-out the R19 data, the HDE predicts a projected valueof H = 71 . ± .
78 km s − Mpc − , which is fully con-sistent with the R19 data within the 1 . σ range. Thisresolution is due to its inherent behaviour of w ( z ). With Planck +BAO12+R19 results (solid curves) in Fig. 5, wederive Hubble constant H = 73 . ± .
14 km s − Mpc − and the input parameter c = 0 . ± .
02. Hence, thecombined constraints for HDE give the closest value of H to the R19 and strong lensing measurements. Thisfully resolves the H tension between the CMB and localmeasurements.Finally, we emphasize that, in contrast to the CPLand TransDE parametrizations and other phenomeno-logical approaches, the HDE is physically well motivatedfrom the holographic principle [16]-[19] of quantum grav-ity that connects the total energy of vacuum state tothe cosmic horizon scale (as the infrared cutoff) [20][21].It naturally provides dynamical dark energy with onlyone free parameter that keeps the total angular diam-eter distance to LSS unchanged, while it increases thelocal expansion. This behaviour resolves the H discrep-ancy between the CMB and local H measurements inan exquisite and economical way. More importantly, theHDE model is generically different from the ΛCDM dueto its transiting equation of the state. Thus future mea-surements will improve the constraints and possibly sub-stantiate or falsify the HDE from the benchmark ΛCDMUniverse. This is evident in Fig. 2 that the major dif-ference between the HDE and ΛCDM lies at the redshiftrange 0 < z < Acknowledgements
Y.Z.M. acknowledges the support of NRF-120385,NRF-120378, and NSFC-11828301. H.J.H. wassupported by NSF of China (No. 11675086 andNo. 11835005), CAS Center for Excellence in Parti-cle Physics (CCEPP), National Key R & D Program ofChina (No. 2017YFA0402204), by the Key Laboratory forParticle Physics, Astrophysics and Cosmology (MOE),and by the Office of Science and Technology, ShanghaiMunicipal Government (No. 16DZ2260200). ∗ [email protected] † [email protected] ‡ [email protected][1] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ash-
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