Microscopic foundation of the holographic-equipartition law for the emergence and expansion of cosmic space

T. Padmanabhan (2012) postulated that cosmic expansion is the emergence of space driven by holographic equipartition: in Planck units the Hubble volume grows as $dV/dt=L_P^2(N_{\rm sur}-\epsilon N_{\rm bulk})$, where $N_{\rm sur}=A/L_P^2$ counts horizon degrees of freedom, $N_{\rm bulk}$ is fixed by equipartition at the horizon temperature $T=H/2\pi$, and $\epsilon=\pm1$ distinguishes accelerating from decelerating phases; the law reproduces standard Friedmann evolution. All of its ingredients are postulated, and the paper poses the task of deriving them from a pre-geometric microscopic theory. The open problem is to construct a concrete quantum-gravity microstructure whose coarse-grained limit yields the law, fixing the temperature and volume conventions, the matter/dark-energy degree-of-freedom relation (which may allow determining the value of the cosmological constant), the gauge-invariant perturbation theory with its thermal corrections, and quantum corrections to the discrete law in the Planck era. The problem is open.

Background

Since Jacobson (1995), a substantial research program has viewed gravity as a thermodynamic phenomenon: demanding that the relation $\delta Q=TdS$ between heat, temperature and entropy hold on every local causal horizon through every spacetime point implies Einstein's field equations, which govern the spacetime metric in general relativity. On this view spacetime resembles a fluid and the metric equations resemble the equations of gas dynamics, suggesting an underlying microscopic theory whose coarse-grained limit is classical gravity. The holographic principle - the statement that the number of fundamental degrees of freedom in a spatial region is fixed by its boundary area rather than its volume - makes this picture quantitatively sharp: one assigns to a surface of area $A$ a number of surface degrees of freedom $N_{\rm sur}=A/L_P^2$ (one per Planck area, $L_P$ being the Planck length), and in any static spacetime the equipartition law $N_{\rm bulk}=N_{\rm sur}$ holds on equipotential surfaces, where the effective bulk degrees of freedom are $N_{\rm bulk}=2|E|/k_BT$ with $E$ the energy the surface encloses and $T$ its horizon temperature. The exactly de Sitter universe (an exponentially expanding solution with constant Hubble parameter $H$) obeys the same holographic equipartition on its cosmological horizon, which radiates at the temperature $T=H/2\pi$ in natural units ($\hbar=c=k_B=1$). Padmanabhan (2012) observed that our universe, which is asymptotically de Sitter, reaches this equipartition only asymptotically, and proposed that the departure from equipartition is what drives cosmic expansion: the expansion of the universe is the emergence of space. His postulate is the rate law

$$\frac{dV}{dt}=L_P^2\,(N_{\rm sur}-\epsilon N_{\rm bulk})$$

where $V=4\pi/(3H^3)$ is the Hubble volume, $t$ is cosmic time, $N_{\rm bulk}=-\epsilon\,2(\rho+3p)V/k_BT$ counts effective bulk degrees of freedom at the horizon temperature, and $\epsilon=+1$ when $\rho+3p<0$ (accelerating phases), $\epsilon=-1$ otherwise. Substituting these definitions reproduces the standard Friedmann acceleration equation $\ddot a/a=-4\pi L_P^2(\rho+3p)/3$, so the law is equivalent to standard cosmological dynamics; in Planck units it reads as a discrete 'bit-wise' rule in which, per Planck unit of cosmic time, the emerged cosmic space grows by $N_{\rm sur}-\epsilon N_{\rm bulk}$ bits. The conceptual gain over the Friedmann equation is that expansion acquires a drive - the quest for holographic equipartition - but every ingredient of the law is postulated rather than derived: the temperature $T=H/2\pi$ is exact only for de Sitter space, and defining a horizon temperature when $H$ is time-dependent is genuinely controversial; the choice of the Hubble volume among alternative volume elements is a convention; the sign prescription $\epsilon$ is a convention inserted by hand - the paper notes that the opposite convention works equally well, since only the combination $\epsilon^2(\rho+3p)=(\rho+3p)$ enters the law; and the law acquires foundational content only if the microscopic, 'pre-geometric' degrees of freedom it counts (hypothetical 'atoms of spacetime' from which space itself emerges) can be identified in a fundamental quantum-gravity theory. The paper poses exactly this set of issues, together with three consequences that a complete microscopic theory should deliver: a relation between matter and dark-energy degrees of freedom that could fix the observed numerical value of the cosmological constant (the dimensionless $\Lambda L_P^2\approx10^{-122}$); a gauge-invariant cosmological perturbation theory (a formulation of density perturbations in variables unaffected by coordinate gauge choices) built on the perturbed equipartition equation, predicting thermal corrections at $T=H/2\pi$ to late-time perturbations; and well-motivated quantum corrections to the bit-wise rule when $N_{\rm sur}$ or $N_{\rm bulk}$ is of order unity, relevant to the Planck era and the initial big-bang singularity.

Problem Statement

Construct a concrete microscopic (pre-geometric) quantum-gravity theory - that is, specify its fundamental degrees of freedom and their dynamics - whose coarse-grained, long-wavelength limit reproduces Padmanabhan's holographic-equipartition expansion law

$$\frac{dV}{dt}=L_P^2\,(N_{\rm sur}-\epsilon N_{\rm bulk})$$

for an isotropic and homogeneous universe, with the currently postulated ingredients derived rather than inserted. The construction must yield, within the same theory: (i) the operational horizon temperature for time-dependent $H(t)$, determining which of the alternatives discussed in the literature is correct and whether and why it reduces to $T=H/2\pi$, and the emergence of the Hubble volume as the relevant volume element; (ii) the microscopic relation between matter degrees of freedom and dark-energy degrees of freedom, identified by Padmanabhan (2012) as a primary challenge for any pre-geometric model, which may allow determining the numerical value of the cosmological constant (dimensionless $\Lambda L_P^2\approx10^{-122}$); (iii) a gauge-invariant formulation of cosmological perturbation theory based on the perturbed equipartition equation, computing the predicted thermal corrections at $T=H/2\pi$ to late-time perturbations; and (iv) well-motivated quantum corrections to the discrete bit-wise law when $N_{\rm sur}$ or $N_{\rm bulk}$ is of order unity, with implications for the Planck-era universe and the initial singularity. These sub-questions share one scientific objective - grounding the postulated emergence law in microscopic physics - and are mutually linked: (ii)-(iv) are consequences and consistency tests of the construction that yields the law itself.

References

1. T. Padmanabhan, 'Emergence and Expansion of Cosmic Space as due to the Quest for Holographic Equipartition' (2012), arXiv:1206.4916, DOI: 10.48550/arXiv.1206.4916, https://arxiv.org/abs/1206.4916 (no journal reference)
2. Ted Jacobson, 'Thermodynamics of Spacetime: The Einstein Equation of State' (1995), Physical Review Letters 75, 1260-1263, arXiv:gr-qc/9504004, DOI: 10.1103/PhysRevLett.75.1260, https://arxiv.org/abs/gr-qc/9504004
3. Ke Yang, Yu-Xiao Liu, Yong-Qiang Wang, 'Emergence of Cosmic Space and the Generalized Holographic Equipartition' (2012), Physical Review D 86, 104013, arXiv:1207.3515, DOI: 10.1103/PhysRevD.86.104013, https://arxiv.org/abs/1207.3515
4. Sumanta Chakraborty, T. Padmanabhan, 'Evolution of Spacetime arises due to the Departure from Holographic Equipartition in all Lanczos-Lovelock Theories of Gravity' (2014), Physical Review D 90, 124017, arXiv:1408.4679, DOI: 10.1103/PhysRevD.90.124017, https://arxiv.org/abs/1408.4679
5. T. Padmanabhan, 'Emergent Gravity Paradigm: Recent Progress' (2015), Modern Physics Letters A 30, 1540007, arXiv:1410.6285, DOI: 10.1142/S0217732315400076, https://arxiv.org/abs/1410.6285
6. T. Padmanabhan, 'Distribution Function of the Atoms of Spacetime and the Nature of Gravity' (2015), Entropy 17, 7420-7452, arXiv:1508.06286, DOI: 10.3390/e17117420, https://arxiv.org/abs/1508.06286
7. T. Padmanabhan, Hamsa Padmanabhan, 'Cosmological Constant from the Emergent Gravity Perspective' (2014), International Journal of Modern Physics D 23, 1430011, arXiv:1404.2284, DOI: 10.1142/S0218271814300110, https://arxiv.org/abs/1404.2284
8. T. Padmanabhan, 'The Atoms of Space, Gravity and the Cosmological Constant' (2016), International Journal of Modern Physics D 25, 1630020, arXiv:1603.08658, DOI: 10.1142/S0218271816300202, https://arxiv.org/abs/1603.08658
9. T. Padmanabhan, Hamsa Padmanabhan, 'Cosmic Information, the Cosmological Constant and the Amplitude of primordial perturbations' (2017), Physics Letters B 773, 81-85, DOI: 10.1016/j.physletb.2017.07.066, arXiv:1703.06144, https://arxiv.org/abs/1703.06144
10. T. Padmanabhan, 'The atoms of spacetime and the cosmological constant' (2017), Journal of Physics: Conference Series 880, 012008, arXiv:1702.06136, DOI: 10.1088/1742-6596/880/1/012008, https://arxiv.org/abs/1702.06136
11. T. Padmanabhan, Sumanta Chakraborty, 'Microscopic origin of Einstein's field equations and the raison d'être for a positive cosmological constant' (2022), Physics Letters B 824, 136828, arXiv:2112.09446, DOI: 10.1016/j.physletb.2021.136828, https://arxiv.org/abs/2112.09446
12. Nobuyoshi Komatsu, 'Generalized thermodynamic constraints on holographic-principle-based cosmological scenarios' (2019), Physical Review D 99, 043523, arXiv:1810.11138, DOI: 10.1103/PhysRevD.99.043523, https://arxiv.org/abs/1810.11138
13. P. B. Krishna, Titus K. Mathew, 'Does holographic equipartition demand a pure cosmological constant?' (2020), Modern Physics Letters A 35, 2050334, arXiv:1905.03529, DOI: 10.1142/S0217732320503344, https://arxiv.org/abs/1905.03529
14. Fei-Quan Tu, Yi-Xin Chen, Qi-Hong Huang, 'Cosmological scenario based on particle creation and holographic equipartition' (2020), Chinese Physics C 44, 065103, arXiv:1908.11729, DOI: 10.1088/1674-1137/44/6/065103, https://arxiv.org/abs/1908.11729
15. M. Muhsinath, V. T. Hassan Basari, Titus K. Mathew, 'Modified expansion law with Kodama-Hayward temperature for the horizon' (2023), General Relativity and Gravitation 55, 43, arXiv:2211.01739, DOI: 10.1007/s10714-023-03091-x, https://arxiv.org/abs/2211.01739
16. Kimet Jusufi, Ahmad Sheykhi, 'Entropic corrections to Friedmann equations and bouncing universe due to the zero-point length' (2023), Physics Letters B 836, 137621, arXiv:2210.01584, DOI: 10.1016/j.physletb.2022.137621, https://arxiv.org/abs/2210.01584