Viscous Cosmology: How Bulk Viscosity Changes the Birth of Primordial Black Holes ✛ ✎ ✦
Context
One of the leading candidates for dark matter is primordial black holes, hypothetically born from the collapse of overdense regions after inflationary expansion. Their prediction is linked to the names of Stephen Hawking and other pioneers. However, the standard analysis assumes an ideal cosmic fluid, whereas in the extreme conditions of the Big Bang, dissipative effects—in particular, bulk viscosity—may emerge. This work for the first time applies numerical simulation in full general relativity to determine how viscosity changes the collapse threshold and the mass spectrum of primordial black holes, which is crucial for testing the dark matter hypothesis and interpreting data from gravitational microlensing.
Methods
Within the three-dimensional BSSN formalism (Baumgarte–Shapiro–Shibata–Nakamura), the system of Einstein field equations and hydrodynamic equations with viscous pressure according to Eckart's theory was solved. The stress-energy tensor contained a term Π = −ε $ρ^{1/2}$ $Θ_f$, where ε is the dimensionless viscosity strength and $Θ_f$ is the fluid expansion. The background was described by the Friedmann equations with an effective equation of state $w_{eff}$ = w − ε√(24π). Initial overdensities were set with two curvature profiles: a sinc function (characteristic of a sharp spectral peak) and a Gaussian. Evolution was computed on a non-uniform radial grid using an explicit fourth-order Runge–Kutta method. Black hole formation was identified by the appearance of a trapped surface, and its mass by the Misner–Sharp mass. An analytical estimate of the threshold shift relied on the Jeans argument about the balance between gravitational collapse and sound pressure.
Results
It was found that the critical threshold µ_c increases with viscosity almost linearly: for ε=0.01 and w=1/3, the increase is ~0.04 (from 0.60 to 0.64 for the sinc profile, from 0.80 to 0.83 for the Gaussian). The dependence on the equation-of-state parameter remains monotonic, similar to the inviscid case. A power-law fit µ_c − µ_{c,0} = K ε^Γ yields Γ ≈ 1 and K of order unity (Table II). Near the threshold, the PBH mass follows the scaling $M_{PBH}$/$M_H$ = k |µ−µ_c|^γ. Viscosity increases the amplitude k: for example, for the sinc profile, k changes from 8.30 to 10.22, while the exponent γ decreases slightly (from 0.37 to 0.34). This result is consistent with the physical picture: dissipation slows collapse, requiring a larger initial density contrast to overcome gravitational collapse, analogous to the effect of Jeans instability in a viscous medium.
Implications
The results demonstrate for the first time that even a small bulk viscosity (ε ~ 0.01) can systematically change the formation threshold of primordial black holes and their mass function. This means that reliable predictions of PBH abundance as dark matter must account for the non-ideality of the primordial plasma. Viscosity shifts the distribution toward larger masses, which could ease tensions with observational constraints. Moreover, the work expands the toolkit of numerical cosmology, demonstrating the applicability of the BSSN formalism to dissipative media, and paves the way for more realistic modeling of extreme gravitational processes.
Future development
Future studies could incorporate second-order viscosity (Israel–Stewart theory) with a finite relaxation time, eliminating acausalities. Full 3+1 simulations with anisotropy would allow computation of spins and ellipticities of newborn black holes. The aspect of Hubble tension is also intriguing: viscosity softens the background equation of state, potentially reducing the discrepancy between local and global measurements of the Hubble constant, creating a nontrivial link between collapse microphysics and large-scale cosmology.
Impact
The study will impact the interpretation of data from missions searching for dark matter through gravitational microlensing (Subaru, OGLE, future space telescopes), as well as the analysis of the stochastic gravitational-wave background, where the PBH mass function is important. For high-energy physics, it exemplifies the significant role of dissipation in gravitational collapse, which may also be relevant in astrophysical contexts.
Next steps
Conduct simulations with a wider range of ε and w parameters, as well as with nonlinearities in the power spectrum. Develop robust schemes for viscosity evolution in arbitrary gauges, suitable for negative and large perturbations.
Key open problems
The work directly addresses the nature of dark matter: if primordial black holes explain it entirely, then the mass function, determined by the collapse threshold, must satisfy numerous astrophysical constraints. The viscous modification shifts the theoretical predictions and may improve agreement. Moreover, the study fits into the broader context of the problem of the initial conditions of the universe: non-ideal effects could have played a role in structure formation, smoothing or sharpening inhomogeneities.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work numerically models the formation of primordial black holes in the early Universe, taking into account bulk viscosity. It is shown that viscosity raises the critical collapse threshold and alters the mass of the resulting black holes, shifting the scaling relations.
Where its strength is
For the first time, the dissipative effect of bulk viscosity is included in full general relativity, providing quantitative corrections to the threshold and mass spectrum of PBHs—key parameters for connecting to dark matter.
Where it could go next
- It might be interesting to compare the obtained threshold shifts with the effect of a non-uniform equation of state, which is considered in the neighboring paper on pressure halting black hole formation.based on: How Pressure Stops Black Holes from Being Born arXiv:2606.29590
- Nearby lies an analysis of how primordial black holes affect the expansion rate of the Universe; one could check how viscous corrections to the mass function would impact those estimates.based on: Cosmic Hot Tap: How Primordial Black Holes Rewrite the Speed of the Universe arXiv:2607.18528v1
- A possible extension is to consider black hole formation not from overdense regions but in the dark star scenario, where viscosity may also play a role in the late stages of collapse.based on: Dark Stars: Cradle of Giant Black Holes arXiv:2511.08578
Worth trying
- Check whether the linear shift in the threshold persists for profiles with a broader spectrum, using the approach from the paper on pressure.
- Assess how viscous corrections to the PBH mass spectrum would alter dark matter predictions in the model from the cosmic kettle paper.
- Apply the viscosity formalism to the collapse of dark stars and compare the thresholds with those obtained for primordial perturbations.
Why this may matter
Including viscosity can noticeably shift predictions for the primordial black hole population, which is important for interpreting microlensing data and dark matter constraints.
Nearby works in our archive
- How Pressure Stops Black Holes from Being Bornsimilarity 0.70
- Cosmic Hot Tap: How Primordial Black Holes Rewrite the Speed of the Universesimilarity 0.68
- Dark Stars: Cradle of Giant Black Holessimilarity 0.67
- How the First Black Holes Are Bornsimilarity 0.67
- Black Holes Born from Nothing, Then Diesimilarity 0.67
- Black Holes from the Universe's Infancysimilarity 0.66
The archive is crowded around this work: 8 reviews nearby, closest at 0.70. This ground is well walked — and there is plenty to compare against.
Although viscosity is often associated with a slowing action, in the context of black hole birth it plays a stabilizing role: the 'thick honey' of the cosmic medium requires a greater density excess to collapse. Only in 2026 did numerical simulations enable a quantitative assessment of this effect — more than half a century after the pioneering works of Hawking and Carr.
Key numbers
- Collapse threshold increase: Δµ_c ≈ 0.04 at ε=0.01 (w=1/3)
- Effective equation of state parameter: w_eff ≈ 0.25 at w=1/3, ε=0.01
- Mass scaling amplitude (sinc): k increases from 8.3 to 10.2 at ε=0.01
- Mass scaling exponent (sinc): γ ≈ 0.37 (ε=0) and 0.34 (ε=0.01)
- Power-law index for threshold: Γ ≈ 1