Robustness of apparent-horizon area quantization and entropy-area scaling from quantum cosmology beyond the closed-FLRW radiation/dust minisuperspace

Rashki and Jalalzadeh (Physical Review D 91, 023501 (2015)) applied deformation (phase-space) quantization to the closed Friedmann-Lemaitre-Robertson-Walker minisuperspace filled with radiation or dust, showing that the phase-space average of the apparent-horizon area is quantized in Planck units and that the total entropy obeys sub-linear entropy-area power laws, $S\simeq(\langle A_{\rm ah}\rangle/4G)^{3/4}$ for radiation and $S\simeq(\langle A_{\rm ah}\rangle/4G)^{2/3}$ for dust - the radiation law matching 't Hooft's holographic bound for a blackbody gas, and the dust law being the authors' parallel construction in the same sub-linear form. The derivation, however, relies on the integrability of the simplest isotropic minisuperspace and on heuristic entropy identifications. The open problem is to determine whether area quantization and these entropy-area exponents persist in anisotropic (Bianchi) minisuperspaces, for other perfect fluids besides radiation and dust, and in modified-gravity (string-cosmology or $f(R)$) minisuperspaces - that is, whether the cosmological holographic principle as a manifestation of quantization is generic or an artifact of the model.

Background

The holographic principle, first proposed by 't Hooft, asserts that the maximum number of independent degrees of freedom contained in a spatial region scales with its boundary area rather than its volume. In his founding essay on dimensional reduction in quantum gravity, 't Hooft also gave a sharper thermodynamic version of this counting: a blackbody gas confined to a region bounded by area $A$ and pushed to the Schwarzschild limit (total energy of order the linear size) carries entropy $S<C_5 Z^{1/4}A^{3/4}$, where $C_5$ is a constant of order unity and $Z$ counts particle species below the ambient temperature - a sub-linear power of the area, well below the black-hole value $S=A/4G$. For cosmology the analogous question is whether the entropy content of the universe is bounded by, or scales with, a horizon area. Fischler and Susskind proposed a cosmological holographic principle (CHP) requiring that the entropy within the particle horizon not exceed the horizon area in Planck units; Easther and Lowe reformulated the requirement using the second law of thermodynamics; and Bak and Rey ('Cosmic Holography') formulated it on the apparent horizon. In a Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime with Hubble rate $H$ and scale factor $a$, the apparent horizon is the sphere of radius $R_{\rm ah}=(H^2+1/a^2)^{-1/2}$; for a closed ($k=+1$) universe it encloses the entire observable universe.

On the quantum side, canonical quantum cosmology truncates general relativity to a minisuperspace - a finite-dimensional phase space of homogeneous degrees of freedom, for closed FLRW essentially the scale factor and its momentum plus matter variables - and quantizes the Hamiltonian constraint into a Wheeler-DeWitt (WDW) equation. Deformation (phase-space) quantization is a formulation of quantum mechanics that avoids wave functions and operator ordering on configuration space: observables remain classical phase-space functions, operator products are replaced by the associative but noncommutative Moyal star product $f\star_m g$, states are Wigner quasi-probability distributions $W(x,\Pi)$, and expectation values become phase-space averages $\langle A\rangle=\int W\star_m A\,dx\,d\Pi$. Introduced into quantum cosmology by Calzetta and Hu, this formalism sidesteps the interpretational difficulties of a wave function of the universe.

Rashki and Jalalzadeh applied it to the closed FLRW model whose matter is a noninteracting radiation or dust fluid. The Moyal-deformed WDW equation reduces to a harmonic-oscillator-type problem with exact Laguerre-polynomial Wigner functions and spectrum $E_n=\omega(n+1/2)$, where $\omega$ is the oscillator frequency of the reduced model. For radiation, the photon number is quantized, $N_\gamma=n+1/2$, hence so is the total entropy, $S_\gamma\propto(n+1/2)^{3/4}$, and the phase-space average of the apparent-horizon area is quantized in Planck units, $\langle A_{\rm ah}\rangle=(L_p^2/4\pi)(1+n^2/(n+1/2))$ with $L_p$ the reduced Planck length; at large $n$ these combine to $S_\gamma\simeq g^{1/4}(\langle A_{\rm ah}\rangle/4G)^{3/4}$, where $g$ is the effective number of internal degrees of freedom of the radiation - the $Z^{1/4}A^{3/4}$ form of 't Hooft's radiation bound, with $Z\to g$. For dust, the total mass is quantized, $M=\sqrt{24\pi^2(n+1/2)}\,M_p$ with $M_p$ the Planck mass, the averaged area is again quantized, $\langle A_{\rm ah}\rangle=(5L_p^2/6\pi)(n+1/2)$, and - using the ideal-gas estimate $S\simeq N=M/m$ (where $M$ is the total mass, $m$ the particle mass and $N$ the particle number) together with a horizon-uncertainty heuristic $L_{\rm ah}\simeq\sqrt{N}/m$ (where $L_{\rm ah}=\sqrt{\langle A_{\rm ah}\rangle/4\pi}$ is the apparent-horizon radius defined through the averaged area) that fixes the particle-mass scaling - one obtains $S_{\rm dust}\simeq(\langle A_{\rm ah}\rangle/4G)^{2/3}$. The authors concluded that the CHP in radiation- and dust-dominated quantum universes emerges as a manifestation of quantization, with an exponent that apparently depends on the matter content (1 for black holes, 3/4 for radiation, 2/3 for dust).

Everything in this derivation is confined to the two-dimensional isotropic minisuperspace with noninteracting radiation or dust, and the entropy entering the scaling is itself fixed by heuristic identifications between minisuperspace integrals and thermodynamic quantities. The authors therefore list, as the natural extensions needed to test the CHP, anisotropic Bianchi cosmological models (the homogeneous but spatially anisotropic cosmologies, classified by Bianchi's scheme), other perfect fluids besides radiation and dust, and modified theories of gravity such as string cosmology and $f(R)$ theories (gravity theories whose action is a general function of the Ricci curvature scalar $R$). Whether area quantization and the sub-linear entropy-area exponents survive any of these enlargements is not known.

Problem Statement

Determine whether the results of Rashki and Jalalzadeh - quantization of the phase-space average of the apparent-horizon area in units of the Planck area and sub-linear entropy-area power-law scaling of the total entropy - persist outside the closed-FLRW radiation/dust minisuperspace. The single overarching objective is to test whether the cosmological holographic principle, understood as a manifestation of quantization, is a generic feature of quantum cosmology or an artifact of the simplest minisuperspace model; the requested model-class enlargements are the instruments of that test and are to be judged jointly against it. Determine whether area quantization and the 't Hooft-form sub-linear entropy-area scaling persist for (i) anisotropic (Bianchi) minisuperspaces - the homogeneous but spatially anisotropic cosmologies; (ii) other perfect fluids besides radiation and dust; and (iii) modified-gravity minisuperspaces such as string cosmology or $f(R)$ theories (gravity theories whose action is a general function of the Ricci scalar). Each enlargement concerns the class as a whole; a single confirming instance does not establish persistence. The entropy-area scaling exponent and its domain of validity are to be determined for a completely specified model (canonical variables, Hamiltonian constraint, Weyl correspondence or factor ordering, matter model and entropy definition), recovering the radiation exponent $3/4$ and the dust exponent $2/3$ of the closed-FLRW model where applicable. Whether the persistence of the sub-linear scaling rests on the heuristic identifications used in the source - the entropy estimate $S\simeq M/m$ and the horizon-uncertainty relation $L_{\rm ah}\simeq\sqrt{N}/m$, where $M$ is the total dust mass, $m$ the dust particle mass, $N=M/m$ the particle number, and $L_{\rm ah}=\sqrt{\langle A_{\rm ah}\rangle/4\pi}$ the apparent-horizon radius defined through the averaged area - is part of the question, as is whether these identifications can be replaced by principled thermodynamic definitions.

References

1. M. Rashki and S. Jalalzadeh, Holography from quantum cosmology, Physical Review D 91, 023501 (2015). DOI: 10.1103/PhysRevD.91.023501; arXiv:1412.3950, https://arxiv.org/abs/1412.3950
2. G. 't Hooft, Dimensional Reduction in Quantum Gravity (1993). arXiv:gr-qc/9310026, https://arxiv.org/abs/gr-qc/9310026
3. W. Fischler and L. Susskind, Holography and Cosmology (1998). arXiv:hep-th/9806039, https://arxiv.org/abs/hep-th/9806039
4. R. Easther and D. Lowe, Holography, Cosmology, and the Second Law of Thermodynamics, Physical Review Letters 82, 4967 (1999). DOI: 10.1103/PhysRevLett.82.4967, https://doi.org/10.1103/PhysRevLett.82.4967
5. D. Bak and S.-J. Rey, Cosmic holography, Classical and Quantum Gravity 17, L83 (2000). DOI: 10.1088/0264-9381/17/15/101; arXiv:hep-th/9902173, https://arxiv.org/abs/hep-th/9902173
6. E. Calzetta and B. L. Hu, Wigner distribution function and phase-space formulation of quantum cosmology, Physical Review D 40, 380 (1989). DOI: 10.1103/PhysRevD.40.380, https://doi.org/10.1103/PhysRevD.40.380
7. S. Jalalzadeh, S. M. M. Rasouli, and P. V. Moniz, Quantum cosmology, minimal length, and holography, Physical Review D 90, 023541 (2014). DOI: 10.1103/PhysRevD.90.023541; arXiv:1403.1419, https://arxiv.org/abs/1403.1419
8. M. Rashki and S. Jalalzadeh, The quantum state of the universe from deformation quantization and classical-quantum correlation, General Relativity and Gravitation 49, 14 (2017). DOI: 10.1007/s10714-016-2178-3; arXiv:1612.03047, https://arxiv.org/abs/1612.03047
9. M. Fathi, S. Jalalzadeh, and P. V. Moniz, Classical universe emerging from quantum cosmology without horizon and flatness problems, The European Physical Journal C 76, 527 (2016). DOI: 10.1140/epjc/s10052-016-4373-5; arXiv:1609.04488, https://arxiv.org/abs/1609.04488
10. S. Jalalzadeh, M. Rashki, and S. Abarghouei Nejad, Classical universe arising from quantum cosmology, Physics of the Dark Universe 30, 100741 (2020). DOI: 10.1016/j.dark.2020.100741; arXiv:2001.00556, https://arxiv.org/abs/2001.00556
11. S. Jalalzadeh, H. Moradpour, and P. V. Moniz, Modified cosmology from quantum deformed entropy, Physics of the Dark Universe 42, 101320 (2023). DOI: 10.1016/j.dark.2023.101320; arXiv:2308.12089, https://arxiv.org/abs/2308.12089
12. M. Tavayef, A. Sheykhi, K. Bamba, and H. Moradpour, Tsallis holographic dark energy, Physics Letters B 781, 195 (2018). DOI: 10.1016/j.physletb.2018.04.001, https://doi.org/10.1016/j.physletb.2018.04.001
13. P. F. da Silva Junior, S. Jalalzadeh, and H. Moradpour, Fractional entropy of the Brown-Kuchař dust in fractional anti-de Sitter quantum gravity (2025). arXiv:2501.01244, https://arxiv.org/abs/2501.01244
14. C. J. Fewster and H. Sahlmann, Phase space quantization and loop quantum cosmology: a Wigner function for the Bohr-compactified real line, Classical and Quantum Gravity 25, 225015 (2008). DOI: 10.1088/0264-9381/25/22/225015, https://doi.org/10.1088/0264-9381/25/22/225015