[Paper Review] Semianalytic Analysis of Primordial Black Hole Formation During a First-order QCD Phase Transition

Christian Y. Cardall, George M. Fuller|arXiv (Cornell University)|Jan 13, 1998

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates the formation of primordial black holes (PBHs) during a first-order quantum chromodynamics (QCD) phase transition, using a semianalytic model to assess whether such PBHs could explain observed microlensing events. It finds that while a sharp peak in the PBH mass function arises, it corresponds to horizon masses from an earlier epoch, requiring a finely tuned blue-tilted primordial power spectrum (n ≈ 1.37–1.42) for cosmologically significant PBH production.

ABSTRACT

It has recently been suggested that cosmologically significant numbers of black holes could form during a first-order QCD phase transition. Further, it has been asserted that these black holes would have masses corresponding naturally to the inferred mass ($\sim 1 M_{\odot}$) of the MACHOs responsible for the observed gravitational microlensing events. In this model, the underlying spectrum of primordial density perturbations provides the fluctuations that give rise to black holes at the epoch of the QCD transition. We employ a simplified model to estimate the reduction in the critical overdensity of a horizon-sized primordial perturbation required for collapse to a black hole. We find that a first-order QCD transition does indeed produce a sharp peak in the black hole mass spectrum, but that this peak corresponds to the horizon mass at an epoch somewhat earlier than the cosmological transition itself. Assuming a COBE normalized primordial density perturbation spectrum with constant spectral index, for the black holes so produced to be cosmologically significant would require an extremely finely tuned ``blue'' primordial density perturbation spectrum. Specifically, in the context of our simplified model, a spectral index in the range $n=1.37-1.42$ corresponds to the range $Ω\sim 10^{-5}-10^3$ of the black hole contribution to the present-day density parameter.

Motivation & Objective

• To assess whether primordial black holes (PBHs) could form during a first-order QCD phase transition and account for MACHO microlensing events.
• To determine the conditions under which such PBHs could constitute a significant fraction of the cosmic density parameter Ω.
• To evaluate the role of primordial density perturbations in triggering PBH formation at the QCD transition epoch.
• To analyze how the critical overdensity for black hole collapse is modified by the dynamics of a first-order phase transition.

Proposed method

• A simplified semianalytic model is used to estimate the reduction in the critical overdensity required for horizon-sized perturbations to collapse into black holes during a first-order QCD phase transition.
• The model incorporates the effects of a phase transition front on the local energy density and gravitational collapse threshold.
• The critical overdensity is calculated using a modified version of the spherical collapse criterion, adjusted for the presence of a phase boundary.
• The primordial power spectrum is assumed to be scale-invariant (n=1) initially, with a range of spectral indices explored to assess PBH yield.
• The mass function of PBHs is derived from the distribution of overdensities at the time of the QCD transition, using the horizon mass at that epoch.
• The contribution of PBHs to the present-day density parameter Ω is computed as a function of the primordial spectral index n.

Experimental results

Research questions

• RQ1 Can a first-order QCD phase transition produce a significant population of primordial black holes with masses near 1 M☉, as inferred from microlensing observations?
• RQ2 What is the effective mass scale of PBHs formed during the QCD transition, and how does it compare to the horizon mass at the transition epoch?
• RQ3 How sensitive is the PBH mass function to the spectral index of the primordial density perturbation spectrum?
• RQ4 What level of primordial power spectrum tuning (i.e., spectral index) is required for PBHs to contribute significantly to the cosmic density parameter Ω?
• RQ5 Does the presence of a first-order phase transition reduce the critical overdensity threshold for PBH formation compared to standard collapse models?

Key findings

• A first-order QCD phase transition produces a sharp peak in the primordial black hole mass function, but this peak corresponds to the horizon mass at an epoch earlier than the QCD transition itself.
• The peak mass is determined by the horizon size at the time of maximum compression during the phase transition, not at the transition epoch.
• For PBHs to be cosmologically significant (Ω ~ 10⁻⁵ to 10³), the primordial power spectrum must have a spectral index in the narrow range n = 1.37–1.42.
• This required spectral index represents a strong deviation from scale invariance (n=1), indicating a finely tuned 'blue' primordial spectrum.
• The model shows that the phase transition reduces the critical overdensity for collapse, but not sufficiently to allow significant PBH formation without extreme spectral tuning.
• The results imply that PBHs formed during the QCD transition are unlikely to account for the observed MACHO microlensing events unless the primordial spectrum is highly non-Gaussian or finely tuned.