# A Critique of Holographic Dark Energy

AA Holographic Dark Energy Catch-22

Eoin ´O Colg´ain a,b & M. M. Sheikh-Jabbari c a Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea b Department of Physics, Sogang University, Seoul 121-742, Korea c School of Physics, Institute for Research in Fundamental Sciences (IPM),P.O.Box 19395-5531, Tehran, Iran

Abstract

Current cosmological data restricts Holographic Dark Energy (HDE) to a regime of param-eter space with a turning point in the Hubble parameter. In particular, Cosmic MicrowaveBackground (CMB) and baryon acoustic oscillation (BAO) data together favour a turningpoint in the observational regime, thereby alleviating Hubble tension. Nevertheless, includ-ing Pantheon supernovae (SNE) pushes H back to a value consistent with Planck-ΛCDM.Noting that SNE are weighted to lower redshifts relative to BAO, this amounts to an evolu-tion in H with redshift within the HDE model. Since H is an integration constant in anyFLRW cosmology, this suggests that HDE may be at odds with the cosmological paradigm. a r X i v : . [ g r- q c ] M a r Introduction

A turning point in the Hubble diagram H ( z ) - concretely a redshift z ∗ where H (cid:48) ( z ∗ ) = 0 - is anexotic feature, which is precluded by the Null Energy Condition (NEC) within an Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) cosmology in an Einstein gravity theory. Regardless, cosmological modelsexist in this class and Holographic Dark Energy (HDE) [1, 2] can be regarded as a prominent minimalmodel (see also [3–12] for generalisations). HDE is motivated as a solution to the cosmological constantproblem, since dark energy densities with the conventional scaling with the Planck mass ρ de ∼ M can be shown [13] to be inconsistent with the holographic principle [14]. Relative to ﬂat ΛCDM, HDEboasts an additional constant parameter c , which current data strongly constrains to the regime c < . (cid:46) c < z ∗ <

0) [15–21]while for c (cid:46) .

5, it becomes observational ( z ∗ > These exotic features are both a blessing and a curse. On one hand, one may theoretically sidestepthe cosmological constant problem, but on the other, it is more diﬃcult to make sense of eﬀective ﬁeldtheories that violate the NEC (however, see [24]). That being said, HDE may push this to the future,thus postponing it as a pressing concern. Furthermore, a turning point may help observationally withHubble tension [26] - currently one of the biggest puzzles in cosmology - provided the turning point hashappened in the recent past z ∗ ≥

0. This argument originally appeared in [27], before it was realisedin the HDE model [28]. In the latter the presence of a turning point is far from explicit, but thisis not an isolated example in the literature, e. g. [29]. Since the turning point in H ( z ) is at smallpositive redshift [30], this overlaps with late-time transition models [31], which as argued in [32] (seealso [33, 34]) cannot resolve Hubble tension. Here, we are not interested in Hubble tension or falsifyingturning points, but merely exploring the implications of turning points within the HDE model.HDE has proven itself to be adept at mimicking ﬂat ΛCDM to date. If this trend continues and futuredata conﬁrms ﬂat ΛCDM, then HDE will be pushed into a corner and penalised for having an additionalparameter. This is a possible outcome (see [20, 21] for existing constraints). It should be stressed thatdespite an additional parameter, HDE struggles to ﬁt cosmological data as well as ﬂat ΛCDM, so theprognosis is not good. However, here we take a diﬀerent tack and pick up a thread explored in [35].There it was noted that the Hubble constant H is conceptually diﬀerent than other parameters in acosmological model; it is an integration constant within the FLRW framework. This latter fact simplyfollows from the Friedmann equations and is generic to all models once the cosmological principle, i. e.isotropy & homogeneity at cosmological distances, is assumed. In [35] it was argued that “running in H ” is a sign of the breakdown of the FLRW paradigm. Here, we recycle this observation for HDE.Our argument is largely physical and revolves around the inevitability of a turning point in HDE whenconfronted with current cosmological data. In particular, the results that guide our insight and support Within the CPL model [22, 23], one has an observational turning point provided w < − (1 − Ω m ) − . NEC violation does not necessarily signal instability in the theory. For example Casimir energy violates NEC.Moreover, “quantum null energy condition” (QNEC) has been recently proposed [25] which is a more relaxed conditionthan NEC and is expected to hold even if there are NEC violations due to quantum eﬀects. So far, to our knowledge noviolation of QNEC has been reported. Violation of QNEC, as discussed in the literature, is related to nonunitarity inlocal quantum ﬁeld theories. Other examples may exist and it would be nice to document them. “Running in H ” as described in [35], alludes to diﬀerent values one may/will obtain for H within a given model ofcosmology once datasets at diﬀerent redshifts are considered. We should also stress that discussions of [35], and henceanalysis here, are only motivated by the Hubble tension and not intended to tackle the tension. H in late-time modiﬁcations of ﬂat ΛCDM to a value consistentwith Planck. HDE, being a dark energy model, is no exception and we conﬁrm later that ﬁts toCMB+BAO+SNE data results in a Planck value for H . Interestingly, removing SNE allows a highervalue of the Hubble constant, H = 71 . ± .

78 km/s/Mpc [28]. As we will verify in this letter, this isdue to a turning point in the vicinity of z = 0 today. The picture then is intuitive to anyone with a physics background. Let us spell it out. The combinationCMB+BAO+SNE constrains the HDE model to a range of parameter space where it is forced to mimicﬂat ΛCDM. Removing the “guardrails” at low redshift, there is nothing to preclude the turning pointmoving into the observational regime. Moreover, as BAO data improves with DESI [45], instead ofcombining CMB with BAO with low weighted redshifts, one can employ BAO with much higher weightedredshifts. In principle, there is nothing to stop the turning point moving to higher redshift, and if thishappens, it will lead to even higher values of H . This is a potential impending issue for the HDE model,but here our analysis will be restricted by the quality of current data. Concretely, here we will show thatadding and removing Pantheon SNE [46] is enough to lead to ∼ σ displacements in all the cosmologicalparameters in the HDE model, while ΛCDM parameters change only by ∼ . σ . Finally, noting thatthe weighted average redshift for the Pantheon SNE dataset is z ∼ .

28, whereas for the employed BAOit is z ∼ .

36, one has the basis of a statement that within the HDE model H can run with redshift inthe sense discussed in [35]. Recalling that H is an integration constant in the Friedmann equations,this contradicts the inherent assumption in the HDE model that it is an FLRW cosmology. Ultimately,HDE can resolve Hubble tension, but at the high price of seemingly contradicting its own assumptions,which ultimately amounts to a cosmological (model) Catch-22. In HDE the basic idea is that the dark energy density takes the form, ρ de = 3 c M L − , L := a ( t ) (cid:90) ∞ t d t (cid:48) a ( t (cid:48) ) , (2.1)where M pl is the reduced Planck mass and the length scale L is the future event horizon of the Universe.The latter choice may seem unusual, however, if one assumes simply that L is the Hubble radius (particlehorizon), the resulting equation of state (EoS), ω de > − , does not yield an accelerating Universe [47]. In [28] H = 73 . ± .

14 km/s/Mpc and c = 0 . ± .

02 is obtained, but only once a Riess et al. prior [44] is included.Still, this ﬁgure is illustrative. We provide a similar value without a H prior later. Neglecting this redshift running factor, one is still confronted with a current robustness problem for HDE, which canbe traced to the turning point. E d E d z = − Ω de (1 + z ) (cid:18) c (cid:112) Ω de + 12 − r de (cid:19) (2.2)dΩ de d z = − (1 − Ω de )Ω de (1 + z ) (cid:18) c (cid:112) Ω de + 1 + Ω r − Ω de (cid:19) , (2.3)where E ( z ) ≡ H ( z ) /H is the normalised Hubble parameter. Note that the algebraic equation1 = Ω m + Ω de + Ω r , Ω i = ρ i M H , (2.4)is assumed. This last condition tells us that 0 ≤ Ω i ≤ de ( z ) is a monotonically decreasing function of z . To take theanalysis further, recall that the matter and radiation sectors satisfy the equations ˙ ρ r + 3 Hρ r = 0 and˙ ρ r + 4 Hρ r = 0, respectively. This ﬁxes Ω i ( z ) in terms of E ( z ):Ω m ( z ) = Ω m (1 + z ) E ( z ) , Ω r ( z ) = Ω r (1 + z ) E ( z ) . (2.5)Ω m is a constant determined through ﬁts to the data and Ω r is, in analogous fashion to ﬂat ΛCDM, ﬁxedby the temperature of the CMB and N eﬀ , the number of relativistic (neutrino) species. The equationsare solved numerically subject to the conditions that E ( z = 0) = 1 and Ω de ( z = 0) = 1 − Ω m − Ω r .As in the ﬂat ΛCDM cosmological model, radiation is not so relevant at low redshift, and this termcan be safely neglected at smaller values of z . With Ω r removed, it is immediately clear that Ω de stopsincreasing at Ω de = 1 in the future as z → − c = 1. Notethat the RHS of (2.2) and (2.3) vanish in this case, so it corresponds to a ﬁxed-point solution for thesystem. Moreover, when c = 1, the HDE model has the same asymptotic attractor as ﬂat ΛCDM. Thiscan also be seen from the dark energy EoS w de = − − √ Ω de c , (2.6)which clearly approaches w de = − de → c = 1. Further observing that Ω de ≈ . z ∼

0, we see for c (cid:46) .

83 that one encounters a phantom crossing in the observational regimeat positive redshift z . As we shall see in due course, data places us in a regime with a phantom crossingand hence a de Sitter attractor in the HDE model is disfavoured by data. One interesting feature of the HDE model is the turning point in the Hubble parameter for c < There appears to be a typo in equation (24) of [48]. Flipping the sign in the Ω r term in (2.3) allows Ω de to increaseat higher redshift. Note that Ω de = ( cLH ) − , where L, H are respectively the event and cosmological horizon radii. Therefore, theΩ de = 1 , c = 1 ﬁxed point corresponds to LH = 1. E (0) = 1, Ω de (0) = 0 . c < c decreases, the turning point moves to higher redshift.in the future can inﬂuence the Hubble parameter in the past provided it is close enough to z = 0.Concretely, a future turning point leaves one with a model that largley tracks ﬂat ΛCDM well at lowredshift and it may be diﬃcult to distinguish. However, this conclusion is challenged if data favours c in the regime c (cid:46) . r is negligible, (2.2) implies that a turning point happens once1 c (cid:112) Ω de + 12 ≈ de . (2.7)Since Ω de < c <

1. More precisely, sinceΩ de ≈ . c (cid:46) . . (cid:46) c <

1, thereis always a turning point in the future. We document this feature in Figure 1 for a range of values of c .As can be seen, as c decreases from unity, the turning point crosses over from the future into the past,i.e. within the range of observation. Now, it is worth noting that resolving the Hubble tension withHDE places us in a regime where the turning point is happening today, c = 0 . ± .

02 [28], providedΩ de ≈ . H valuethat is close to Planck-ΛCDM [42]. Removing the SNE removes the guardrails and allows the HDEmodel to exploit its natural turning point to raise H . As is clear from Figure 1, the Hubble parameter ismonotonically decreasing in time, just as in ﬂat ΛCDM, but it starts to increase again after the turningpoint. This physics is responsible for higher H values once the turning point is in the observationalregime. This becomes more pronounced when one employs a local prior on H [44], but even without,one of the ﬁndings of [28] is that CMB+BAO data is enough to raise H . The key point to bear in mindis that HDE resolves the H tension and achieves higher values of H (compared to ΛCDM) preciselyby having a turning point close enough to z = 0 in the presence of the “right” data.It should be noted that SNE appears to have some tolerance to a turning point in H ( z ) at low redshift.To appreciate this, observe that E ( z ) ≡ H ( z ) /H at z ∼ .

07 allows values below unity, E ( z ) < .21 0.24 0.27 0.30 0.33 m H c m

68 72 76 80 H CMB+BAOCMB+BAO+

SNE

Figure 2: The shift in the HDE model parameters following the inclusion of the Pantheon SNE datasetto CMB and BAO. This is to be compared with Figure 3 of [28].within the 1 σ conﬁdence interval, whereas E ( z = 0) = 1 by deﬁnition. In particular, since E ( z ) can beexpanded as E ( z ) = 1 + (1 + q ) z + . . . at low redshift, this tells one that 1 + q < z ∼ . E ( z ) >

1, so a turning point is only consistent with SNE observations within 1 σ below z < . Our goal in this letter is to show that this is a double-edged sword: one can ﬁnd diﬀerent combinationsof reputable data for which the diﬀerence in H determinations within a given HDE model is sizable.Bluntly put, it is diﬃcult to buy into the notion that H is a constant, as required within the FLRWcosmology framework. We follow [28] and employ the same data. We employ Planck CMB data [36], isotropic BAO determi-nations at z = 0 .

106 by the 6dF survey [52], SDSS-MGS survey at z = 0 .

15 [53] and anisotropic BAOby BOSS-DR12 from z = 0 . , .

51 and z = 0 .

61 [51]. We also use Pantheon SNE [46]. As pointedout in the introduction, the data at redshift lower than z ∼ . z ∼ .

28 and for the BAO mentioned above it is z ∼ . The main result of our letter can be found in Table 1, where we have quoted the best-ﬁt values and theerrors obtained from marginalised constraints for both the HDE model and ﬂat ΛCDM to the data. The We thank Adam Riess for discussion on this point. .29 0.30 0.31 0.32 0.33 m H m CMB+ BA OCMB+

BAO +SNE

Figure 3: The shift in the ΛCDM model parameters following the inclusion of the Pantheon SNE datasetto CMB and BAO.corresponding plots can be found in Figures 2 and 3. Since, the combination BAO+SNE anchors theHubble parameter at low redshift [41–43], we quote values with and without the Pantheon SNE data.Note that we have not employed a local prior on H and all our data is cosmological in nature. We ranMarkov Chain Monte Carlo (MCMC) for the HDE model, but for ﬂat ΛCDM we have used the existingPlanck MCMC chains [36]. The results of our MCMC analysis, while qualitatively similar to [28], diﬀerthrough the inclusion of dark energy perturbations.It is a simple back of the envelope calculation to determine the diﬀerence in H with and withoutSNE. We ﬁnd a discrepancy of ∼ . σ . This can be contrasted with the analogous number for ﬂatΛCDM, namely 0 . σ , which simply underscores the fact that ﬂat ΛCDM has an aﬃnity to the data.One can conﬁrm from Table 1 that this (cid:38) σ displacement is not conﬁned to H and is also evidentlythere in both Ω m and c . This displacement serves as sharp contrast to the ﬂat ΛCDM model. Sincethe combination CMB+BAO+SNE is consistent within ﬂat ΛCDM, it is robust to the addition of SNEdata: the addition, as it is intuitively expected, slightly shrinks the conﬁdence ellipses while preservinga signiﬁcant overlap within 1 σ . This is made explicit in Figure 3 as well as in Table 1.Model Data H (km/s/Mpc) Ω m c HDE CMB+BAO 72 . +1 . − . . ± .

014 0 . ± . . ± .

80 0 . ± .

008 0 . ± . . ± .

42 0 . ± .

006 -CMB+BAO+SNE 67 . ± .

40 0 . ± .

005 -Table 1: Best-ﬁt values of the cosmological parameters at 68%C.L.The HDE model is, however, not robust to the addition of SNE: as depicted in Figure 2 and is6een in Table 1, there is a clear jump once SNE is added to CMB+BAO. It should be noted that thequoted sigma discrepancies should be treated with caution since there are overlapping datasets, namelyCMB+BAO and the actual discrepancy is bounded above by 2 . σ . Nevertheless, we believe a likefor like comparison between ﬂat ΛCDM and HDE using the same methodology is meaningful. It isinstructive to also record the χ values, which we do in Table 2. Clearly, despite having an additionalparameter, HDE ﬁts the data worse than ﬂat ΛCDM. That point aside, note that the jump in χ whenSNE are added is consistent with the introduction of ∼ m = 0 . , c = 0 .

621 (withSNE), the turning point is located at z ∗ ≈ − .

1, while for Ω m = 0 . , c = 0 .

507 (without SNE), theturning is at z ∗ ≈ .

04. This backs up our earlier claim that the turning point is in the observationalregime just considering CMB+BAO data alone. See [30] for an earlier exposition of an observationalturning point in a holographic dark energy model with a view to resolving Hubble tension [27] (seealso [30]). Model Data χ HDE CMB+BAO 2818 . . . . χ values for diﬀerent models with diﬀerent data. Our goal here is not to rule out HDE observationally in the traditional sense using a χ comparison,although as is clear from Table 2, HDE performs worse than ﬂat ΛCDM, so the outlook for the model isnot good. This will require better quality data from upcoming experiments. Nevertheless, as we do here,one can in tandem comment on the theoretical assumptions going into the HDE model and whetherthey are borne out in observations. This serves as an important consistency check. Based on vanillacosmological data, namely CMB, BAO and SNE, we ﬁnd evidence for ∼ σ running in cosmologicalparameters, especially in H . Such a feature is not evident in ﬂat ΛCDM, so HDE is in conﬂict withsome component of the data and robustness appears to be a problem. As argued in the introduction,this lack of robustness follows from the turning point in H ( z ), which has not received due attention inobservational cosmology studies of the model. Remarkably, it fails to feature in the review [2].Within these cosmological parameters, we single out H as being special on the grounds that it isan integration constant in the Friedmann equations, and thus common to all FLRW cosmologies. Ourresults, based on current data, are in noticeable tension with the idea that H is a constant. Toappreciate this, observe that for c < H ( z ) within HDE is unavoidable. The overallcombination CMB+BAO+SNE imposes strong enough constraints that HDE is forced to mimic ﬂat We thank Yin-Zhe Ma for discussion on this technical point. We are conscious that this tension with FLRW may not be a problem in the end. It is imperative to test thecosmological principle, e. g. [56]. See [57] for a resolution of the Hubble tension where FLRW is relaxed. H that is discrepant at the ∼ σ level with the value ﬁxed by CMB+BAO+SNE.While admittedly the ∼ σ discrepancy is an overestimation, because there is overlapping data, thesame overestimation logic applies to the ∼ . σ discrepancy seen in the ﬂat ΛCDM model. Clearly, thelike for like comparison has meaning. As can be argued, the eﬀective SNE redshift is lower ( z ∼ . z ∼ . H withthe redshift of the data, as explained in [35]. We expect similar conclusions to hold for generalisationsof the holographic dark energy paradigm. Upcoming DESI releases will provide better quality BAOto much higher redshifts, which may permit the turning point in H ( z ) to venture deeper into the past,resulting in even higher H inferences. This can be investigated through comprehensive jackknifes ofthe forthcoming data. This “running H ”, if substantiated, may provide a means to rule out the HDEmodel, and potentially related models, without resorting to χ comparison.Note that our conclusions can be squared with other results in the literature, in particular Figure 3of Dai et al. [28]. Tellingly, the grey contours CMB+BAO+SNE are consistent with Planck. Removingthe Pantheon SNE for z < .

2, in addition to removing them completely, leads to the blue and greencontours, respectively, and the resultant higher values of H . This is consistent with a turning point atpositive redshift. The only contour that is mysterious is the red contour. But here again, there is aninteresting explanation. It is well documented that Pantheon prefers a lower value of Ω m (eﬀectivelythe deceleration parameter) below z ∼ . H , Ω m ) as ﬂat ΛCDM. This means that as Ω m goes down (and it goes down in Pantheon for z (cid:46) . H driven by CMB+BAO data. Consideration of a H prior [44] only makes this trend, which is driven bythe turning point, more pronounced. So, yes, HDE can alleviate Hubble tension, but at the relativelyhigh price of violating the underlying FLRW assumption that H is a constant. This is in addition toviolating the NEC. This is essentially a clash between theory, or model assumptions, and observation.Finally, let us make one last pertinent comment. Observe that we have arrived at our conclusions byusing purely cosmological data. We have not invoked a H prior [44] - largely on the grounds that ithas not been substantiated that Hubble tension is cosmological in origin - so here one is making a faircomparison to ﬂat ΛCDM. In short, our analysis here is independent of Hubble tension. Acknowledgements

We thank Stephen Appleby, Qing-Guo Huang, Yin-Zhe Ma, Adam Riess and Xin Zhang for discussionand/or comments on earlier drafts. E ´OC is funded by the National Research Foundation of Korea(NRF-2020R1A2C1102899). MMShJ acknowledges SarAmadan grant No. ISEF/M/99131.We are also indebted to Tao Yang and Lu Yin for discussions, running MCMC analysis, analysingthe MCMC chains and producing plots. Every eﬀort was made to keep our junior collaborators in theproject without compromising the integrity of the scientiﬁc narrative. We are conscious that the HDEmodel may be a sensitive subject for scientists who have built careers on it, and the outcome may have The model presented in [27] has one parameter less than HDE and it is enough to add or remove Lyman- α BAO [54,55]determinations of the Hubble parameter (instead of the SNE) to ﬁnd displacements in H . Simply put, if Lyman- α BAOholds up, then the HDE model presented in [27] can be falsiﬁed.

References [1] M. Li, “A Model of holographic dark energy,” Phys. Lett. B (2004), 1 [arXiv:hep-th/0403127[hep-th]].[2] S. Wang, Y. Wang and M. Li, “Holographic Dark Energy,” Phys. Rept. (2017), 1-57[arXiv:1612.00345 [astro-ph.CO]].[3] Q. G. Huang and M. Li, “The Holographic dark energy in a non-ﬂat universe,” JCAP (2004),013 [arXiv:astro-ph/0404229 [astro-ph]].[4] D. Pavon and W. Zimdahl, “Holographic dark energy and cosmic coincidence,” Phys. Lett. B (2005), 206-210 [arXiv:gr-qc/0505020 [gr-qc]].[5] B. Wang, Y. g. Gong and E. Abdalla, “Transition of the dark energy equation of state in an inter-acting holographic dark energy model,” Phys. Lett. B (2005), 141-146 [arXiv:hep-th/0506069[hep-th]].[6] R. G. Cai, “A Dark Energy Model Characterized by the Age of the Universe,” Phys. Lett. B (2007), 228-231 [arXiv:0707.4049 [hep-th]].[7] C. Gao, F. Wu, X. Chen and Y. G. Shen, “A Holographic Dark Energy Model from Ricci ScalarCurvature,” Phys. Rev. D (2009), 043511 [arXiv:0712.1394 [astro-ph]].[8] L. P. Chimento, M. I. Forte and M. G. Richarte, “Self-interacting holographic dark energy,” Mod.Phys. Lett. A (2013), 1250235 [arXiv:1106.0781 [astro-ph.CO]].[9] L. P. Chimento and M. G. Richarte, “Interacting dark matter and modiﬁed holographic Riccidark energy induce a relaxed Chaplygin gas,” Phys. Rev. D (2011), 123507 [arXiv:1107.4816[astro-ph.CO]].[10] L. P. Chimento, M. G. Richarte and I. E. S´anchez Garc´ıa, “Interacting dark sector with variablevacuum energy,” Phys. Rev. D (2013), 087301 [arXiv:1310.5335 [gr-qc]].[11] M. Tavayef, A. Sheykhi, K. Bamba and H. Moradpour, “Tsallis Holographic Dark Energy,” Phys.Lett. B (2018), 195-200 [arXiv:1804.02983 [gr-qc]].[12] E. N. Saridakis, “Barrow holographic dark energy,” Phys. Rev. D (2020), 123525[arXiv:2005.04115 [gr-qc]].[13] A. G. Cohen, D. B. Kaplan and A. E. Nelson, “Eﬀective ﬁeld theory, black holes, and the cosmo-logical constant,” Phys. Rev. Lett. (1999), 4971-4974 [arXiv:hep-th/9803132 [hep-th]].[14] G. ’t Hooft, “Dimensional reduction in quantum gravity,” Conf. Proc. C (1993), 284-296[arXiv:gr-qc/9310026 [gr-qc]]. 915] Q. G. Huang and Y. G. Gong, “Supernova constraints on a holographic dark energy model,” JCAP (2004), 006 [arXiv:astro-ph/0403590 [astro-ph]].[16] X. Zhang and F. Q. Wu, “Constraints on holographic dark energy from Type Ia supernova obser-vations,” Phys. Rev. D (2005), 043524 [arXiv:astro-ph/0506310 [astro-ph]].[17] Z. Chang, F. Q. Wu and X. Zhang, “Constraints on holographic dark energy from x-ray gas massfraction of galaxy clusters,” Phys. Lett. B (2006), 14-18 [arXiv:astro-ph/0509531 [astro-ph]].[18] X. Zhang and F. Q. Wu, “Constraints on Holographic Dark Energy from Latest Supernovae, GalaxyClustering, and Cosmic Microwave Background Anisotropy Observations,” Phys. Rev. D (2007),023502 [arXiv:astro-ph/0701405 [astro-ph]].[19] L. Xu, “Constraints to Holographic Dark Energy Model via Type Ia Supernovae, Baryon AcousticOscillation and WMAP,” Phys. Rev. D (2012), 123505 [arXiv:1205.2130 [astro-ph.CO]].[20] M. Li, X. D. Li, Y. Z. Ma, X. Zhang and Z. Zhang, “Planck Constraints on Holographic DarkEnergy,” JCAP (2013), 021 [arXiv:1305.5302 [astro-ph.CO]].[21] M. M. Zhao, D. Z. He, J. F. Zhang and X. Zhang, “Search for sterile neutrinos in holographicdark energy cosmology: Reconciling Planck observation with the local measurement of the Hubbleconstant,” Phys. Rev. D (2017) no.4, 043520 [arXiv:1703.08456 [astro-ph.CO]].[22] M. Chevallier and D. Polarski, “Accelerating universes with scaling dark matter,” Int. J. Mod.Phys. D , 213 (2001) [gr-qc/0009008].[23] E. V. Linder, “Exploring the expansion history of the universe,” Phys. Rev. Lett. , 091301 (2003)[astro-ph/0208512].[24] V. A. Rubakov, “The Null Energy Condition and its violation,” Usp. Fiz. Nauk (2014) no.2,137-152 [arXiv:1401.4024 [hep-th]].[25] R. Bousso, Z. Fisher, S. Leichenauer and A. C. Wall, “Quantum focusing conjecture,” Phys. Rev. D (2016) no.6, 064044 [arXiv:1506.02669 [hep-th]]; R. Bousso, Z. Fisher, J. Koeller, S. Leichenauerand A. C. Wall, “Proof of the Quantum Null Energy Condition,” Phys. Rev. D (2016) no.2,024017 [arXiv:1509.02542 [hep-th]].[26] L. Verde, T. Treu and A. G. Riess, “Tensions between the Early and the Late Universe,” NatureAstron. , 891 [arXiv:1907.10625 [astro-ph.CO]].[27] Maurice H. P. M. van Putten, “Evidence for Galaxy Dynamics Tracing Background Cosmology Be-low the de Sitter Scale of Acceleration”, 2017 The Astrophysical Journal, 848 28 [arXiv:1709.05944[hep-th]][28] W. M. Dai, Y. Z. Ma and H. J. He, “Reconciling Hubble Constant Discrepancy from HolographicDark Energy,” Phys. Rev. D (2020), 121302 [arXiv:2003.03602 [astro-ph.CO]].[29] E. Di Valentino, A. Mukherjee and A. A. Sen, “Dark Energy with Phantom Crossing and the H tension,” [arXiv:2005.12587 [astro-ph.CO]]. 1030] E. ´O Colg´ain, M. H. P. M. van Putten and H. Yavartanoo, “de Sitter Swampland, H tension &observation,” Phys. Lett. B (2019), 126-129 [arXiv:1807.07451 [hep-th]].[31] G. Benevento, W. Hu and M. Raveri, “Can Late Dark Energy Transitions Raise the Hubble con-stant?,” Phys. Rev. D (2020) no.10, 103517 [arXiv:2002.11707 [astro-ph.CO]].[32] D. Camarena and V. Marra, “On the use of the local prior on the absolute magnitude of Type Iasupernovae in cosmological inference,” [arXiv:2101.08641 [astro-ph.CO]].[33] G. Alestas, L. Kazantzidis and L. Perivolaropoulos, “A w − M phantom transition at z t < . G eﬀ at z t (cid:39) .

01 as a solution of theHubble and growth tensions,” [arXiv:2102.06012 [astro-ph.CO]].[35] C. Krishnan, E. ´O Colg´ain, M. M. Sheikh-Jabbari and T. Yang, “Running Hubble Tension and aH0 Diagnostic,” [arXiv:2011.02858 [astro-ph.CO]].[36] N. Aghanim et al. [Planck Collaboration], “Planck 2018 results. VI. Cosmological parameters,”arXiv:1807.06209 [astro-ph.CO].[37] D. J. Eisenstein et al. [SDSS], “Detection of the Baryon Acoustic Peak in the Large-Scale Correla-tion Function of SDSS Luminous Red Galaxies,” Astrophys. J. (2005), 560-574 [arXiv:astro-ph/0501171 [astro-ph]].[38] A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae for an ac-celerating universe and a cosmological constant,” Astron. J. (1998), 1009-1038 [arXiv:astro-ph/9805201 [astro-ph]].[39] S. Perlmutter et al. [Supernova Cosmology Project], “Measurements of Ω and Λ from 42 highredshift supernovae,” Astrophys. J. (1999), 565-586 [arXiv:astro-ph/9812133 [astro-ph]].[40] E. Di Valentino, A. Melchiorri and J. Silk, “Cosmic Discordance: Planck and luminosity distancedata exclude LCDM,” Astrophys. J. Lett. (2021) no.1, L9 [arXiv:2003.04935 [astro-ph.CO]].[41] E. M¨ortsell and S. Dhawan, “Does the Hubble constant tension call for new physics?,” JCAP (2018), 025 [arXiv:1801.07260 [astro-ph.CO]].[42] P. Lemos, E. Lee, G. Efstathiou and S. Gratton, “Model independent H ( z ) reconstruction usingthe cosmic inverse distance ladder,” Mon. Not. Roy. Astron. Soc. (2019) no.4, 4803-4810[arXiv:1806.06781 [astro-ph.CO]].[43] L. Knox and M. Millea, “Hubble constant hunter’s guide,” Phys. Rev. D (2020) no.4, 043533[arXiv:1908.03663 [astro-ph.CO]].[44] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, “Large Magellanic Cloud CepheidStandards Provide a 1% Foundation for the Determination of the Hubble Constant and StrongerEvidence for Physics beyond ΛCDM,” Astrophys. J. (2019) no.1, 85 [arXiv:1903.07603 [astro-ph.CO]]. 1145] A. Aghamousa et al. [DESI], “The DESI Experiment Part I: Science,Targeting, and Survey Design,”[arXiv:1611.00036 [astro-ph.IM]].[46] D. Scolnic et al. , “The Complete Light-curve Sample of Spectroscopically Conﬁrmed SNe Ia fromPan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,” Astrophys.J. (2018) no.2, 101 [arXiv:1710.00845 [astro-ph.CO]].[47] S. D. H. Hsu, “Entropy bounds and dark energy,” Phys. Lett. B (2004), 13-16 [arXiv:hep-th/0403052 [hep-th]].[48] H. L. Li, J. F. Zhang, L. Feng and X. Zhang, “Reexploration of interacting holographic darkenergy model: Cases of interaction term excluding the Hubble parameter,” Eur. Phys. J. C (2017) no.12, 907 [arXiv:1711.06159 [astro-ph.CO]].[49] A. G. Riess, S. A. Rodney, D. M. Scolnic, D. L. Shafer, L. G. Strolger, H. C. Ferguson, M. Postman,O. Graur, D. Maoz and S. W. Jha, et al. “Type Ia Supernova Distances at Redshift > (2018) no.2, 126 [arXiv:1710.00844 [astro-ph.CO]].[50] D. Camarena and V. Marra, “Local determination of the Hubble constant and the decelerationparameter,” Phys. Rev. Res. , no.1, 013028 (2020) [arXiv:1906.11814 [astro-ph.CO]].[51] S. Alam et al. [BOSS], “The clustering of galaxies in the completed SDSS-III Baryon OscillationSpectroscopic Survey: cosmological analysis of the DR12 galaxy sample,” Mon. Not. Roy. Astron.Soc. , no.3, 2617-2652 (2017) [arXiv:1607.03155 [astro-ph.CO]].[52] F. Beutler, C. Blake, M. Colless, D. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saundersand F. Watson, “The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local HubbleConstant,” Mon. Not. Roy. Astron. Soc. (2011), 3017-3032 [arXiv:1106.3366 [astro-ph.CO]].[53] A. J. Ross, L. Samushia, C. Howlett, W. J. Percival, A. Burden and M. Manera, “The clusteringof the SDSS DR7 main Galaxy sample ? I. A 4 per cent distance measure at z = 0 . (2015) no.1, 835-847 [arXiv:1409.3242 [astro-ph.CO]].[54] V. de Sainte Agathe, C. Balland, H. du Mas des Bourboux, N. G. Busca, M. Blomqvist, J. Guy,J. Rich, A. Font-Ribera, M. M. Pieri and J. E. Bautista, et al. “Baryon acoustic oscillations at z= 2.34 from the correlations of Ly α absorption in eBOSS DR14,” Astron. Astrophys. (2019),A85 [arXiv:1904.03400 [astro-ph.CO]].[55] M. Blomqvist, H. du Mas des Bourboux, N. G. Busca, V. de Sainte Agathe, J. Rich, C. Balland,J. E. Bautista, K. Dawson, A. Font-Ribera and J. Guy, et al. “Baryon acoustic oscillations from thecross-correlation of Ly α absorption and quasars in eBOSS DR14,” Astron. Astrophys. (2019),A86 [arXiv:1904.03430 [astro-ph.CO]].[56] N. J. Secrest, S. von Hausegger, M. Rameez, R. Mohayaee, S. Sarkar and J. Colin, “A Test of theCosmological Principle with Quasars,” [arXiv:2009.14826 [astro-ph.CO]].[57] R. G. Cai, Z. K. Guo, L. Li, S. J. Wang and W. W. Yu, “Chameleon dark energy can resolve theHubble tension,” [arXiv:2102.02020 [astro-ph.CO]].1258] Maurice H P M van Putten “Alleviating tension in ΛCDM and the local distance ladder from ﬁrstprinciples with no free parameters,” MNRAS Volume 491, Issue 1, January 2020, Pages L6-L10[59] E. ´O Colg´ain, “A hint of matter underdensity at low z?,” JCAP09