Cosmological Budget of Entropy from Merging Black Holes

TL;DR

This work revises the cosmological entropy budget by quantifying Bekenstein–Hawking entropy for stellar-origin BHs and their binary mergers across cosmic time, incorporating population-synthesis mappings, LVK BBH posteriors, and NR-based remnant fits. It demonstrates that BBH mergers can dominate the entropy production long before the CMB does, revealing a thermodynamic crossover near $z\approx12.6$, and shows that even a small PBH fraction can impose an entropy floor in the Dark Ages with PBH mergers potentially ruling entropy growth at very high redshift. The analysis highlights a strong thermodynamic asymmetry: mergers are energetically inefficient yet entropy-rich, and presents a volume-normalized retrospective entropy density to compare entropy accumulation against cosmic volume. These results refine our understanding of the universe’s thermodynamic history and motivate future space- and ground-based gravitational-wave observatories to map entropy production at high redshift. $S$-scaling with mass and the role of mergers in the early thermodynamic state have implications for galaxy formation scenarios and the interpretation of high-redshift JWST observations.

Abstract

Black holes contain more entropy than any other component of the observable universe. Gravitational-wave observations from LIGO and Virgo have shown evidence of a previously unknown black hole mass range, which provides new information to update the entropy budget. Increases in entropy due to binary black hole mergers, as implied in the second law of thermodynamics, should also be added to the budget. In this study, we update the cosmological entropy budget for black holes in the stellar to lite-intermediate-mass range $(5-300~M_\odot)$, originating from either supernovae or binary mergers, by utilizing a suite of population synthesis models and phenomenological fits derived from numerical relativity. We report three new insights: Firstly, the cumulative entropy from merging black holes surpasses the total entropy from cosmic microwave background photons around the onset of the Over-massive Black Hole Galaxy phase at $z\sim 12$, suggesting that mergers played a more significant role in shaping the thermodynamic state of the early universe than relic radiation. Secondly, if primordial black holes constitute a nonzero fraction of dark matter, their early binary mergers establish an ``entropy floor" in the Dark Ages and can dominate the cumulative merger-generated entropy history even for small abundances. Thirdly, by computing the cosmological density parameters, we highlight the thermodynamic asymmetry in black hole mergers, where the production of gravitational-wave energy is inefficient compared to the immense generation of Bekenstein-Hawking entropy.

Figures (7)

• Figure 1: Cosmological Binary Black Hole Entropy Inventory: the total accumulated entropy (Static Budget) from different black hole populations as a function of black hole mass. The stellar–origin black holes (grey histogram) include black holes formed via supernova and direct collapse. The population of merging black holes is decomposed into pre-merger components (light blue) and post-merger remnants (dark blue), both contributing in the range $5-200~M_\odot$. The total entropy budget from merging binaries is shown as a thick gray band. The right panel zooms in on the entropy profile of merging black holes in the range $8-70~M_\odot$ to highlight the detailed structure.
• Figure 2: Entropy Production in Individual GW Events: grey bars show the correlation between the redshift of the binary and the change in entropy during BBH mergers for 168 BBH events reported by LVK. The bar represents $z$ and $\Delta~S$ with 90% confidence interval. Certain GW events are highlighted in color. The purple-shaded region corresponds to the dark energy–dominated era ($z \le 0.5$), while the yellow-shaded region represents the matter-dominated era ($z > 0.5$).
• Figure 3: Dynamic Entropy Production: The blue curve (left axis) shows the GWTC-4 inferred number density of BBH mergers per Mpc$^{-3}$ per logarithmic mass interval. The red curve (right axis) shows the corresponding mass distribution of entropy generated ($\Delta S$) per $\log(M)$. We color-code the regions for PISN mass gap (yellow; $45\text{--}130~M_\odot$), while the remaining mass range (light shading) denotes the stellar-collapse regime outside the gap. Annotations report the integrated $\Sigma~\Delta S$ contributed by the two mass regimes.
• Figure 4: Entropy Production in Mass-Spin Space: the total entropy change $\Delta S$ from stellar BBH mergers across all redshift $z \in [0,20]$ in the mass ($M_\mathrm{tot}$) - spin ($a_1$) parameter space.
• Figure 5: Cosmic History of Merger-Generated Entropy: total entropy (left y axis & blue curve) and entropy density (right y axis & red curve) from merging BBHs in the comoving frame across $z \in [0.01, 20]$, as a function of the redshift $z$ (bottom x axis) and of the look-back time $t_{lb}$ (top x axis). The dark energy-dominated and matter-dominated eras are shaded purple and yellow, respectively.