Thermodynamic Split Conjecture

• Thermodynamic Split Conjecture is a framework positing that black hole and cosmological horizon thermodynamics differ, as the lack of key microstate ingredients invalidates the area law in cosmological settings.
• It emphasizes that the failure of essential BKE criteria—boundary presence, a global Killing vector, and a near-horizon AdS₂ throat—in cosmological spacetimes prevents standard microstate counting and equilibrium thermodynamics.
• This conjecture leads to modified cosmological dynamics with observable implications, such as altered Friedmann equations, modified inflationary behavior, and potential resolutions to cosmic tensions.

The Thermodynamic Split Conjecture (TSC) posits that black hole horizon thermodynamics and cosmological horizon thermodynamics are generically inequivalent in any UV-complete theory of gravity. This conjecture fundamentally revises the traditional assumption that the thermodynamic properties derived in stationary black hole settings—specifically the Bekenstein–Hawking area–entropy law and the Gibbons–Hawking temperature—apply unchanged to cosmological horizons such as those in FLRW or de Sitter spacetimes. The TSC asserts that the necessary microscopic structures underpinning the black hole area law are absent in cosmological spacetimes, requiring the introduction of empirically determined, cosmology-native entropy and temperature functions and sharpening the distinction between black hole and cosmological horizon thermodynamics (Trivedi, 12 Oct 2025, Trivedi, 17 Sep 2025, Chirco et al., 2015).

1. Conceptual Foundations and BKE Criterion

The TSC is motivated by the observation that the microstate-counting arguments establishing the black hole area law in string theory and holography rely on three structural pillars: (B) an asymptotic boundary with conserved charges, (K) the presence of a global Killing horizon supporting a KMS (Gibbs) equilibrium state, and (E) a universal near-horizon AdS₂ throat enabling a regulator-independent microcanonical entropy via the quantum entropy function. In cosmological settings (e.g., FLRW or global de Sitter), all three ingredients fail:

• No asymptotic boundary (B=0): There are no global conserved charges due to the absence of spatial infinity or boundaries at null infinity.
• No global timelike Killing vector (K=0): Expanding cosmological spacetimes are nonstationary and lack global thermal equilibrium.
• No universal near-horizon throat (E=0): There is no well-defined AdS₂ region or attractor, precluding the path-integral techniques that anchor the area law.

The BKE criterion (Boundary/Killing/Extremal) formalizes this: only when B·K·E = 1 is there an observer-independent microcanonical ensemble whose logarithmic degeneracy produces the area law. In all cosmological cases, B·K·E ≠ 1, so there is no justification—within UV-complete gravity—for importing black hole entropy formulas wholesale (Trivedi, 17 Sep 2025, Trivedi, 12 Oct 2025).

2. Statistical Mechanics and Thermodynamics in Reparametrization-Invariant Systems

The TSC is rooted in the statistical mechanics of reparametrization-invariant systems. In such systems, time and energy are gauge-dependent quantities, and the definition of equilibrium thermodynamics requires particular structural splits in the phase space (Chirco et al., 2015):

• Microcanonical statistical mechanics is well-defined only when the total system admits a two-way split—with one non-interacting subsystem serving as a generalized clock. This ensures the existence of a conserved generalized energy.
• Thermodynamic equilibrium (entropy, temperature, zeroth law, additivity) is possible only when the measured subsystem further admits a split into two weakly interacting parts (i.e., the system consists of three subsystems: clock, $S^b$, $S^c$).

In this framework, the additivity of the conserved generalized energy $I = I^b + I^c$ is crucial: it enables the construction of a unique “thermal time” and a unique corresponding generalized temperature $T_I$, defined by $1/T_I = dS/dI$. These results show that neither a preferred Newtonian time nor a conventional Hamiltonian is required to formulate statistical mechanics; instead, these emerge from the structure of the constraints and system decomposition (Chirco et al., 2015).

3. Mathematical Formulation: Departure from the Area Law

In black holes, the area law associates entropy $\displaystyle S_{BH} = A/(4G_N)$ to the event horizon, and the Gibbons–Hawking temperature in de Sitter is $\displaystyle T_{GH} = H/(2\pi)$. Under the TSC, these identifications are no longer justified for cosmological horizons. Instead, entropy and temperature are replaced by empirical, model-dependent functions (Trivedi, 12 Oct 2025):

• The effective cosmological temperature is given by

$T_{eff}(H) = \chi(H) \cdot \frac{H}{2\pi}$

where $\chi(H)$ quantifies the deviation from standard thermality ($\chi=1$ recovers $T_{GH}$).

• Cosmological horizon entropy follows

$S_{cos}(H) = \alpha H^{-p}$

with $\alpha$ a dimensionful parameter and $p$ a real exponent. The black hole limit is $(\alpha = 8\pi^2 M_{Pl}^2, p=2)$, but generically $p \neq 2$, indicating non-area scaling.

These functions are not determined a priori; their values must be fixed observationally, and departures from the standard area-law exponents (especially $p \neq 2$) signal violations of the standard thermodynamic identifications in cosmology (Trivedi, 12 Oct 2025, Trivedi, 17 Sep 2025).

4. Cosmological Dynamics and Observational Consequences

The TSC modifies the dynamical equations of cosmology via generalized thermodynamic relations at the apparent horizon. Using a Clausius-like law, one obtains a modified Friedmann equation:

$H^2 = \frac{8\pi G}{3} \rho + F(H; \alpha,p,\chi)$

where $F \rightarrow 0$ in the black hole + Gibbons–Hawking limit, and for small $\chi$- and $p$-deformations

$$F(H) = \epsilon_b \left(\frac{H}{H_b}\right)^m H_0^2$$

with $$m \approx p,\, |\epsilon_b| \lesssim \mathcal{O}(10^{-2})$$.

Consequences arise for several critical cosmological phenomena (Trivedi, 12 Oct 2025):

• Eternal inflation: The diffusion coefficient for long-wavelength inflaton modes is $D_{eff}(H) = \beta(H) H^3/(8\pi^2)$, modifying the quantum–classical barrier and making the onset of eternal inflation empirically sensitive to $\beta(H)$.
• Vacuum tunneling: Tunneling rates $\Gamma_{HM}$ depend on entropy differences $\Delta S_{cos}$, with $p<2$ suppressing and $p>2$ enhancing tunneling rates compared to area law expectations.
• Quantum breaking timescales: The quantum breaking timescale $t_Q \sim S_{cos}(H)/H$ departs from $t_Q \sim M_{Pl}^2/H^3$, with implications for the persistence of semiclassicality.
• Primordial black holes (PBH): The fluctuation power spectrum is rescaled by $\beta(H)$, strongly affecting PBH production rates and rendering them sensitive probes of the TSC.

Small, TSC-motivated deformations at early ($z \sim 10^3$) and late ($z \lesssim 1$) cosmic times yield shifts in the sound horizon and Hubble parameter, generating potential resolutions to the $H_0$ (Hubble) and $S_8$ (density fluctuation amplitude) cosmological tensions (Trivedi, 12 Oct 2025).

5. Frameworks, Tests, and Falsifiability

To establish or falsify the TSC, both theoretical and observational avenues are delineated (Trivedi, 17 Sep 2025):

• Theoretical falsification requires constructing cosmological microstate ensembles or holographic duals that recover the area law (with B=K=E=1). This would entail, for example, constructing cosmological microstate geometries with degeneracies $\exp \left(A/(4G_N)\right)$ or establishing global Gibbs/KMS states and regulated entropy functional integrals, as in the black hole case.
• Observational scaling tests compare “data-native” entropy proxies $S_{LHS}(z)$ (such as information capacity, Shannon entropy or compression length of tomographic maps) against the Bekenstein–Hawking prediction $S_{RHS}(z) \propto H(z)^{-2}$. Fitting the scaling exponent $\beta$ in $S_{LHS}(z) \propto H(z)^{-\beta}$: if $\beta = 2$ is statistically excluded, it indicates a breakdown of the area law and thus supports the TSC.

A summary of the principal BKE-criteria:

Setting	Boundary (B)	Gibbs/KMS (K)	Near-horizon (E)	Area Law Applies?
Black Hole	1	1	1	Yes
Cosmological Horizon	0	0	0	No (per TSC)

6. Implications and Broader Impact

Acceptance of the TSC would have wide-ranging consequences:

• Any cosmological derivation or model reliant on the black hole area law—such as entropic-force scenarios, emergent-gravity interpretations of Friedmann equations, or holographic dark energy models—lacks direct justification if the TSC holds (Trivedi, 17 Sep 2025).
• It suggests many entropy-area-based dark energy and modified gravity proposals may reside in the Swampland of inconsistent low-energy theories if they rest on invalid area law assumptions.
• The TSC motivates new approaches to horizon thermodynamics and cosmological entropy using cosmology-native analogues or quasilocal constructions (e.g., Kodama-vector-based charges, non-equilibrium statistical frameworks). This includes potential extensions via extremal entropy functionals on local screens, leading to generalized area terms plus subleading corrections, and greater attention to signatures such as the statistics of vacuum decay remnants or the late-time behavior of cosmological entanglement entropy.

A plausible implication is that precision cosmology, particularly from next-generation 21 cm intensity mapping, CMB, BAO, and large-scale structure surveys, can empirically constrain or validate cosmological modifications predicted by the TSC, providing an interface between quantum gravity and observational data (Trivedi, 12 Oct 2025).

7. Outlook and Research Directions

Open directions for further investigation include:

• Extending ergodic theorems and statistical frameworks to generally covariant field theories and quantum gravity, addressing local gauge symmetries beyond the toy model systems (Chirco et al., 2015).
• Identifying realistic “clock” systems in full general relativity and elucidating their physical implementation in cosmological or quantum gravitational settings.
• Developing a program for a UV-complete statistical mechanics of cosmological horizons by constructing entropy functionals based on local, gauge-invariant data (e.g., dynamical holographic screens).
• Exploring observational ramifications of TSC-driven modifications through growth-rate measurements, PBH abundance constraints, and potential late-time relic signatures.

Systematic confirmation of empirical departures from $(\alpha=8\pi^2 M_{Pl}^2, p=2, \chi=1)$ would uniquely identify cosmological horizon thermodynamics as a probe of quantum gravity distinct from the black hole case, redefining foundational thermodynamic constructs for the universe (Trivedi, 12 Oct 2025, Trivedi, 17 Sep 2025, Chirco et al., 2015).