--- title: 'Holographic Dark Energy: Concepts & Models' url: https://www.emergentmind.com/topics/holographic-dark-energy-hde type: topic --- # Holographic Dark Energy: Concepts & Models Holographic dark energy (HDE) is a class of dynamical dark-energy models in which the dark-energy density is determined by an infrared cutoff \(L\) rather than by a strictly constant vacuum term. In its standard form, motivated by the Cohen–Kaplan–Nelson black-hole bound, the density is written as \(\rho_{de}=3c^2 M_p^2 L^{-2}\), with the cosmological dynamics fixed by the choice of \(L\), most commonly the future event horizon. The subject has developed into a broad framework encompassing event-horizon models, Hubble- and Ricci-scale variants, entropy-modified constructions, dark-sector interactions, and realizations in modified-gravity backgrounds [1612.00345]. ## 1. Holographic construction and the basic density law The foundational argument begins from the holographic principle and the requirement that the energy stored in an effective field theory inside a region of size \(L\) should not exceed the mass of a black hole of the same size. In the standard formulation, this yields \[ L^3\Lambda^4 \lesssim L M_p^2, \] so that the dark-energy density scales as \[ \rho_{de}\sim M_p^2L^{-2}. \] The conventional HDE parametrization is therefore \[ \boxed{\rho_{de}=3C^2 M_p^2 L^{-2}}, \] where the literature represented here uses the dimensionless coefficient in the forms \(C\), \(c\), or \(d\). A more general dimensional expansion, \[ \rho_{de}=C_1 M_p^4 + C_2 M_p^2 L^{-2} + C_3 L^{-4}+\cdots, \] is often invoked to emphasize that the \(M_p^4\) term is incompatible with the holographic bound, while the \(L^{-2}\) term is the dominant admissible contribution at late times [1612.00345]. Within this framework, HDE is not defined by a single equation alone but by the pair consisting of the holographic density law and the infrared scale prescription. The choice of \(L\) is therefore the essential model-defining input. This is why the same basic density relation supports sharply different cosmologies: de Sitter-like, quintessence-like, phantom-like, or effectively \(\Lambda\)CDM-like behavior can all emerge from different cutoffs or modified entropy prescriptions. ## 2. Standard HDE with the future event horizon The standard Li-type model identifies the infrared cutoff with the future event horizon, \[ L=a\int_t^\infty \frac{dt'}{a(t')}=a\int_a^\infty \frac{da'}{Ha'^2}. \] For a spatially flat FRW universe containing matter and HDE, \[ 3M_p^2H^2=\rho_m+\rho_{de}, \] and the dark-energy density fraction obeys \[ \frac{d\Omega_{de}}{dz}=-\frac{\Omega_{de}(1-\Omega_{de})}{1+z}\left(1+\frac{2}{C}\sqrt{\Omega_{de}}\right). \] The corresponding equation of state is \[ \boxed{w=-\frac13-\frac{2}{3C}\sqrt{\Omega_{de}}}. \] As \(\Omega_{de}\to 1\), the asymptotic fate is controlled by the holographic parameter: \(C=1\) gives \(w\to-1\), \(C>1\) yields \(w>-1\), and \(C<1\) gives \(w<-1\), implying a phantom-like future and, classically, a big-rip fate [1612.00345]. This event-horizon choice is the standard viable prescription because alternative simple cutoffs do not generically reproduce late-time acceleration. In particular, ordinary HDE with the Hubble cutoff \(L=H^{-1}\) does not accelerate without additional structure, whereas the Tsallis generalization changes the scaling to \[ \rho_D = B\,L^{2\delta-4}, \] so that with \(L=H^{-1}\), \[ \rho_D = B H^{-2\delta+4}, \] and the model can produce late-time acceleration even without dark-sector interaction. In that construction, \[ \omega_D=\frac{\delta-1}{(2-\delta)\Omega_D-1}, \qquad q=\frac{1}{2}\left[\frac{(1-2\delta)\Omega_D+1}{1-(2-\delta)\Omega_D}\right], \] with \(\delta=2\) reproducing an exact cosmological constant, while the accelerating branch \(\delta>1\) is classically unstable because \(v_s^2<0\) over \(0<\Omega_D<1\) [1804.02983]. The observational interpretation of the standard event-horizon parameter is correspondingly direct: \(c=1\) gives an asymptotic de Sitter-like future, \(c>1\) keeps \(w>-1\), and \(c<1\) drives the equation of state across the phantom divide \(w=-1\), implying big-rip evolution [2411.08639]. ## 3. Generalized HDE families and cutoff prescriptions A useful modern classification divides HDE into four categories: models with other characteristic length scales, models with extended Hubble scales, models with dark-sector interaction, and models with modified black-hole entropy. This classification captures most of the active model space studied with contemporary cosmological data [2412.09064]. Several representative constructions illustrate how broad the framework has become: | Category | Representative definition | Representative implication | |---|---|---| | Other characteristic length scale | \(\rho_{de}=3d^2 M_p^2\eta^{-2}\), \(\eta=\int_0^t dt'/a(t')\) | Large inflation-generated \(\eta_{\rm prim}\) makes \(\eta\)HDE behave nearly like a cosmological constant | | Extended Hubble scale | \(L=(\alpha H^2+\beta \dot H)^{-1/2}\) | GO-HDE can yield analytic \(H(z)\), \(w(z)\), and \(q(z)\) in DGP cosmology | | Modified entropy | \(\rho_D=BL^{2\delta-4}\), \(\rho_{de}=3c^2M_{\rm eff}^2L^{\delta-2}\), or \(\rho_{de}=3M_P^2b^2(L^{-2}+\Lambda)\) | Late acceleration is possible, and phantom crossing may occur depending on entropy parameters and coupling | | Generalized cutoff | \(L_{\rm IR}=L_{\rm IR}(L_p,\dot L_p,\ddot L_p,\ldots,L_f,\dot L_f,\ldots,a)\) | Tsallis, Rényi, Sharma–Mittal, Quintessence, and Ricci DE can be rewritten as generalized HDE | The total-comoving-horizon model, or \(\eta\)HDE, uses \[ \eta=\int_0^t \frac{dt'}{a(t')}=\int_0^a \frac{da'}{a'H'a'} \] and \[ \rho_{de}=3d^2M_p^2\eta^{-2}. \] Its defining physical feature is the inflation-generated primordial contribution \[ \eta=\eta_{\rm prim}+\tilde\eta, \] with \(\eta_{\rm prim}\) so large that \(\rho_{de}\simeq 3d^2M_p^2\eta_{\rm prim}^{-2}\) over much of cosmic history. In this sense, \(\eta\)HDE behaves almost like a cosmological constant while retaining holographic origin. The reported best fit, \[ d=209.86,\qquad \Omega_{m0}=0.270,\qquad \chi^2_{\min}=542.915, \] is essentially degenerate with the quoted \(\Lambda\)CDM value \(\chi^2_{\Lambda{\rm CDM}}=542.919\) [1202.4228]. Extended-Hubble constructions are exemplified by the Granda–Oliveros (GO) cutoff, \[ L=(\alpha H^2+\beta \dot H)^{-1/2}. \] In DGP braneworld cosmology with late-time dark-energy domination, this gives \[ \rho_D=3(\alpha H^2+\beta\dot H), \] and leads to analytic background evolution, \[ H(z)=\frac{\epsilon}{r_c(1-\alpha)}\left[1+(1+z)^{\frac{\alpha-1}{\beta}}\right], \] together with explicit \(w_D(z)\) and \[ q(z)=-1-\frac{1-\alpha}{\beta\left[1+(1+z)^{\frac{1-\alpha}{\beta}}\right]}. \] For suitable \((\alpha,\beta)\), the transition from deceleration to acceleration occurs over \(0.4\lesssim z\lesssim 1\), and the squared sound speed can remain positive for \(\alpha>1,\beta>0\) or \(\alpha<1,\beta<0\), unlike the instability usually found for GO-HDE in standard cosmology [1506.01676]. Entropy-modified HDE forms alter the density law itself. Tsallis HDE uses \[ S_\delta=\gamma A^\delta,\qquad \rho_D=BL^{2\delta-4}, \] while Barrow-modified cosmology changes both the HDE density and the Friedmann equation, \[ H^{2-\delta}=\frac{1}{3M_{\rm eff}^2}(\rho_m+\rho_{de}),\qquad \rho_{de}=3c^2M_{\rm eff}^2L^{\delta-2}. \] With the future event horizon, the Barrow-HDE equation of state is \[ w_{de}=-\frac{1+\delta}{3}+\frac{\delta-2}{3}\left(\frac{\Omega_{de}}{c^2}\right)^{\frac{1}{2-\delta}}, \] and interaction can drive \(w_{de}\) across the phantom line. A distinct modification based on the AdS black-hole mass bound gives \[ \rho_{de}=3M_P^2 b^2\left(\frac{1}{L^2}+\Lambda\right), \] so that a constant term appears directly from the AdS sector and late-time acceleration can be obtained even with the Hubble or particle horizon, without using the future event horizon as a causal boundary condition [2211.00088; 2201.09715]. At the most general level, generalized HDE allows \[ L_{\rm IR}=L_{\rm IR}(L_p,\dot L_p,\ddot L_p,\ldots,L_f,\dot L_f,\ldots,a), \] and reinterprets Tsallis, Rényi, Sharma–Mittal, Quintessence, and Ricci dark energy as members of a common holographic family. This suggests a formal equivalence class structure in which the same dark-energy dynamics can be encoded in different effective cutoff functionals [2105.08438]. ## 4. Asymptotic dynamics, rip classification, and cosmic fate The future dynamics of HDE are usually discussed in terms of big rip, little rip, and pseudo-rip evolution. For GO-type models and modified Friedmann backgrounds, a unified criterion has been formulated by rewriting the evolution equation as \[ \dot H=g(H), \qquad \int \frac{dH}{g(H)}=\int dt. \] The classification is then determined by the convergence properties of the integral: a big rip occurs when \(H\to\infty\) at finite future time, a little rip when \(H\to\infty\) only as \(t\to\infty\), and a pseudo-rip when \(H\) approaches a finite asymptotic value \(H_f\) as \(t\to\infty\) [2404.08912]. For the simplest HDE with GO cutoff, \[ \rho_\Lambda=3c^2L^{-2},\qquad L=(\alpha H^2+\beta \dot H)^{-1/2}, \] substitution yields \[ H^2=c^2(\alpha H^2+\beta\dot H), \] and the resulting integral shows that \(H\) diverges at finite time. In this formulation, the standard GO-HDE model produces a big rip for all values of \(\alpha\), \(\beta\), and \(c\). The same asymptotic bias persists across several generalized densities: Tsallis HDE with \[ \rho_\Lambda=3c^2L^{-(4-2\sigma)} \] admits a big rip for \(1\le \sigma<2\); Barrow HDE with \[ \rho_\Lambda=3c^2L^{\Delta-2} \] again yields a big rip because the effective exponent satisfies \(21\) anomaly, but they did strongly alleviate the Planck–HST tension: the residual \(\chi^2\) for Planck+WP+HST falls from \(7.8\) in \(\Lambda\)CDM to \(0.3\) in HDE [1305.5302]. The Hubble-tension application sharpened this point. A later analysis with Planck 2018 CMB and BAO data found \[ H_0=71.54\pm1.78\ \mathrm{km\,s^{-1}\,Mpc^{-1}}, \] already consistent with the local R19 determination at the \(1.4\sigma\) level, while the combined fit Planck+BAO12+R19 gave \[ H_0=73.12\pm1.14\ \mathrm{km\,s^{-1}\,Mpc^{-1}},\qquad c=0.51\pm0.02. \] The physical mechanism was identified as a transition of the HDE equation of state from \(w>-1\) at earlier times to \(w<-1\) at later times, producing “delayed acceleration” and preserving the distance to last scattering while increasing the inferred present-day expansion rate [2003.03602]. DESI-era analyses have made the dataset dependence more explicit. Using DESI 2024 BAO with SN compilations, ordinary HDE yields \[ c=1.001^{+0.083}_{-0.110}\quad(\text{DESI+PantheonPlus}), \] \[ c=1.150^{+0.121}_{-0.190}\quad(\text{DESI+Union3}), \] \[ c=1.106^{+0.096}_{-0.130}\quad(\text{DESI+DESY5}), \] so late-time data alone lean toward \(c\gtrsim1\). Once CMB is added, the parameter shifts decisively below unity: \[ c=0.673\pm0.023,\quad 0.642\pm0.028,\quad 0.701\pm0.024 \] for CMB+DESI combined with PantheonPlus, Union3, and DESY5, respectively, and the \(c<1\) conclusion is described as stronger than \(10\sigma\). In the interacting model with \[ Q=\beta H\rho_{de}, \] the inferred values remain above unity even with CMB, \[ c=1.12\pm0.17,\quad 1.20\pm0.20,\quad 1.07\pm0.16, \] while the coupling is positive, \[ \beta=0.55^{+0.19}_{-0.11},\quad 0.55^{+0.21}_{-0.10},\quad 0.56^{+0.18}_{-0.11}, \] at more than \(2\sigma\). In that sign convention, \(\beta>0\) means dark matter decays into dark energy, and the ePPF treatment is used to evolve perturbations consistently. Bayesian evidence leaves HDE and IHDE statistically comparable to \(\Lambda\)CDM for DESI BAO+SN alone, but renders both decisively less favored once CMB is included [2411.08639]. A broader comparison across seven HDE models—OHDE, RDE, GRDE, IHDE1, IHDE2, THDE, and BHDE—found that \(\Lambda\)CDM remains the most competitive model, that RDE is ruled out, and that interacting HDE performs worst among the viable categories. The study further reported that the future-event-horizon models tend to be phantom-like for BAO+CMB, while BAO+CMB+PantheonPlus pulls their equation of state much closer to \(-1\). THDE and BHDE were the best-performing modified-entropy cases, with THDE especially close to \(\Lambda\)CDM behavior [2412.09064]. More flexible reconstructions modify the standard verdict. A spline-nodal generalized HDE model with scale-factor-dependent exponent \(f(a)\) found strong statistical evidence against standard HDE with \(f=-2\), while a three-node reconstruction improved the fit over \(\Lambda\)CDM by \[ \Delta\chi^2_{\Lambda{\rm CDM},3\text{-node}}=12.60 \] for DESI+SH0ES+Union3 and \[ \Delta\chi^2_{\Lambda{\rm CDM},3\text{-node}}=10.50 \] for DESI+SH0ES+Pantheon+, although the Bayes factor remained only comparable because of the complexity penalty. In this reconstruction, the one-node case effectively reduces to \(\Lambda\)CDM via \(f\to0\), whereas higher-node models allow quintessence-to-phantom transitions and raise \(H_0\) toward SH0ES values [2507.22292]. ## 6. Foundational issues, thermodynamics, and reinterpretations The physical origin of the HDE density formula has been debated since the model’s inception. One critical line of work argues that three common explanations are unsatisfactory: the Cohen UV/IR bound gives the correct scaling but not a convincing physical vacuum origin across cosmic epochs; Thomas’ bulk-holography argument and Ng’s spacetime-foam argument reproduce \(\rho_{de}\propto M_{\rm Pl}^2L^{-2}\) but overpredict the energy. A distinct proposal instead attributes HDE to quantum fluctuations of spacetime itself, limited by the event horizon, and obtains \[ \rho_{\rm de}=3d^2M_{\rm Pl}^2L^{-2}, \qquad d\approx \frac{\sqrt7}{2}\approx 0.886, \] which was presented as compatible with then-current observational fits. This suggests a different microscopic interpretation of the same macroscopic density law, though the proposal remains explicitly conjectural [1001.4567]. A separate critique concerns horizon thermodynamics. For Li-type HDE with \(L=d_E\), equilibration with the cosmic horizon was argued to require \[ d_E^2H\gtrsim \frac12, \] which is satisfied only near the onset of dark-energy domination, not during the radiation era or earlier. The entropy hierarchy is then \[ L M_{\rm Pl}\ll L^{3/2}M_{\rm Pl}^{3/2}\ll L^2M_{\rm Pl}^2, \] with entanglement entropy \[ S_{\rm ent}\sim L M_{\rm Pl} \] far too small to bridge the gap between HDE entropy and black-hole entropy. A large-\(N\) rescue requires \[ N\sim L^2M_{\rm Pl}^2, \] which is about \(10^{122}\) today and was argued to be phenomenologically untenable. The same analysis also identified a negative-entropy problem for non-phantom HDE with \(c>1\), while the de Sitter limit \(c=1\) was singled out as thermodynamically cleaner in the far future [1003.4363]. These foundational tensions coexist with a broad formal unification program. Generalized HDE, by allowing \(L_{\rm IR}\) to depend on particle and future horizons and their derivatives, can reproduce Tsallis, Rényi, Sharma–Mittal, Quintessence, Ricci, modified Ricci, and related dark-energy models within a single holographic functional language. This suggests that the primary open question may not be whether a given model is “holographic” in a formal sense, but which cutoff prescription, entropy functional, and gravitational background—if any—capture the underlying microphysics in a non-contrived way [2105.08438]. In that sense, HDE is best understood not as one model but as a family of UV/IR-inspired dark-energy theories whose viability depends on the infrared scale choice, the entropy law, the interaction sector, and the background gravitational dynamics. The empirical record presently favors \(\Lambda\)CDM overall, but it does not eliminate holographic constructions; rather, it strongly disfavors the simplest realizations while leaving room for structured generalizations, especially those that deviate from the rigid \(L^{-2}\) law only in carefully constrained ways [2412.09064]. Source: https://www.emergentmind.com/topics/holographic-dark-energy-hde