Cosmological black holes in the inflationary epoch

This study investigates the evolution of inflationary black holes coupled to the cosmological background via a generalized McVittie geometry within Starobinsky's $\mathcal{R}^2$ inflation model, finding that only those with a specific initial mass range can survive Hawking evaporation and avoid runaway growth to persist today as sub-solar mass objects.

The Gist

Imagine the universe as a giant, expanding balloon. Now, imagine that stuck to the surface of this balloon are tiny, invisible marbles. In standard physics, these marbles (black holes) are usually thought of as static objects that just sit there, slowly shrinking as they leak energy (Hawking radiation) or grow by eating nearby dust.

But this paper asks a "What if?" question: What if these marbles were glued to the balloon itself?

If the balloon expands, the glue stretches, and the marbles get bigger because the balloon is expanding. This is the core idea of the paper: Cosmological Coupling. The authors suggest that black holes born during the universe's explosive growth phase (Inflation) might be physically tied to the expansion of space itself.

Here is the story of their journey, told in simple terms:

1. The Setup: The Glued Marbles

The authors use a mathematical model (called the "Generalized McVittie geometry") to describe these black holes. Think of it like a rubber sheet. If you stretch the sheet, a hole in the middle doesn't just stay the same size; the hole stretches with the sheet.

• The Rule: The mass of the black hole grows in direct proportion to the size of the universe. As the universe gets bigger, the black hole gets heavier.

2. The Great Balloon Race (Inflation)

The story starts at the very beginning, during Inflation. This was a split-second where the universe expanded faster than the speed of light.

• The Danger: If a black hole starts too big, it grows so fast that it swallows the entire "visible universe" (the particle horizon) before inflation even ends. It's like a bubble of soap that expands so fast it engulfs the whole room.
• The Result: The authors calculate that for the black hole to survive without swallowing the whole universe, it must start incredibly small—roughly the mass of a large grain of sand or a small pebble.

3. The Radiation Era: The Tug-of-War

After inflation, the universe cools down and fills with a hot soup of radiation (light and particles). Now, the black hole faces a three-way tug-of-war:

1. The Stretch (Cosmological Coupling): The expanding universe tries to pull the black hole's mass up.
2. The Leak (Hawking Radiation): The black hole tries to evaporate and shrink, like a melting ice cube.
3. The Feast (Accretion): The black hole tries to eat the surrounding hot soup, growing heavier.

The Critical Moment:
The authors found a "Goldilocks" zone.

• If the black hole is too light, the "Leak" wins. It evaporates completely before the universe gets old enough to have stars.
• If the black hole is too heavy, the "Feast" wins too hard. It starts eating the radiation so fast that its mass shoots up to infinity in a mathematical explosion (a "runaway" effect), which breaks the physics of the model.
• The Sweet Spot: There is a very narrow range of starting masses where the "Stretch" and the "Feast" balance out the "Leak" just enough to let the black hole survive.

4. The Journey to Today

The paper tracks these lucky survivors through the history of the universe:

• Radiation Era: They grow a bit by eating radiation.
• Matter Era: The universe fills with stars and galaxies. The black holes stop eating as much but keep growing slowly because the universe is still stretching.
• Dark Energy Era: The universe accelerates its expansion again. The black holes get a final boost in mass.

The Final Verdict: The "Sub-Solar" Ghost

The authors conclude that if these black holes exist today, they are not the massive monsters we see in movies (like the one in Interstellar). They are tiny.

• Current Mass: They would weigh about 0.001 times the mass of our Sun.
• The Analogy: Imagine a black hole that is roughly the size of a large asteroid but has the density of a star. It's too small to be a standard star, but too heavy to be a normal asteroid.
• Why it matters: These objects are "ghosts." They are too small to be seen by telescopes, but they might be hiding in plain sight, potentially making up some of the universe's "Dark Matter" (the invisible stuff holding galaxies together).

Summary in a Nutshell

This paper suggests that if black holes were born in the very first second of the universe and were "glued" to the fabric of space, they wouldn't have died out. Instead, the expansion of the universe would have fed them just enough to keep them alive, but not so much that they became giants.

Today, they would be tiny, invisible, asteroid-mass black holes, weighing about a thousandth of our Sun, drifting through the cosmos as a hidden population of survivors from the Big Bang.

Technical Summary

Here is a detailed technical summary of the paper "Cosmological black holes in inflation era" by Milos Ertola Urtubey and Daniela Pérez.

1. Problem Statement

The paper investigates the theoretical evolution of black holes (BHs) that may have existed during the cosmic inflationary epoch and survived to the present day. Standard General Relativity (GR) describes black holes in asymptotically flat spacetimes, but this framework is inadequate for objects embedded in a dynamical cosmological background.

The core problem addressed is how the cosmological coupling of a black hole to the expanding universe affects its mass and horizon size over time. Specifically, the authors examine whether such black holes can survive the competing effects of:

1. Cosmological coupling: The dynamical embedding of the BH in an expanding background, which may cause the mass to scale with the scale factor.
2. Hawking radiation: Mass loss due to quantum evaporation.
3. Radiation accretion: Mass gain from the surrounding cosmological fluid (specifically during the radiation-dominated era).

The goal is to determine the allowed initial mass range for these primordial black holes (PBHs) such that they do not evaporate completely, do not grow uncontrollably (runaway growth), and do not exceed the particle horizon (which would imply the universe itself is inside a black hole).

2. Methodology

A. Theoretical Framework

• Metric: The authors utilize the generalized McVittie spacetime, a solution to Einstein's field equations describing a black hole coupled to a cosmological background.
• Mass Definition: They adopt the choice where the gravitational mass $M(t)$ is proportional to the cosmic scale factor $a(t)$:
  $$M(t) = m_0 a(t)$$
  Here, $m_0$ is a constant mass parameter. This implies the black hole's mass grows as the universe expands, distinct from standard accretion.
• Horizon Definition: The black hole event horizon radius is defined as $R_{BH}(t) = 2GM(t)/c^2$.
• Cosmological Model: The background evolution is modeled using Starobinsky $R^2$ inflation (a specific $f(R)$ gravity model consistent with Planck data), transitioning through radiation, matter, and dark energy-dominated eras.

B. Evolution Equations

The total rate of change of the black hole mass is modeled by the sum of three contributions:
$$\frac{dM}{dt} = \underbrace{\gamma w \dot{a}(t)}_{\text{Cosmological Coupling}} \underbrace{- \frac{A(M)}{M(t)^2}}_{\text{Hawking Radiation}} + \underbrace{\frac{27\pi G^2}{c^3}\rho_{rad}(t)M(t)^2}_{\text{Radiation Accretion}}$$

• Cosmological Coupling: Driven by the expansion rate $\dot{a}(t)$.
• Hawking Radiation: Modeled using standard semi-classical formulas where $A(M)$ is a mass-dependent coefficient.
• Accretion: Modeled using the Zel'dovich–Novikov prescription (effective for a Schwarzschild BH in a thermal bath) as a phenomenological estimate for the maximum impact of radiation accretion.

C. Constraints and Bounds

The authors derive bounds on the dimensionless mass parameter $\gamma$ (where $m_0 = \gamma w$, with $w=1$ g) by enforcing physical consistency:

1. Upper Bound (Horizon Constraint): The black hole event horizon $R_{BH}$ must always remain smaller than the particle horizon $d_h$ to ensure the universe is not inside a black hole.
2. Upper Bound (Runaway Growth): During the radiation era, accretion can cause mass divergence ($M \to \infty$). The initial mass must be low enough to prevent this singularity before the end of the radiation era.
3. Lower Bound (Survival): The initial mass must be high enough to prevent complete evaporation via Hawking radiation before the end of inflation.

3. Key Contributions

• Unified Evolution Model: The paper provides a comprehensive numerical and analytical treatment of BH evolution across four distinct cosmic epochs (Inflation, Radiation, Matter, Dark Energy) within a single dynamical framework.
• Refined Mass Constraints: Unlike standard PBH studies that often neglect cosmological coupling or treat accretion simplistically, this work integrates the specific growth mechanism of the generalized McVittie metric with Hawking evaporation and radiation accretion.
• Identification of a "Survival Window": The authors identify a narrow, specific range of initial masses formed during inflation that allows a black hole to survive to the present day.
• Quantification of Runaway Growth: The study explicitly calculates the critical mass threshold at the end of inflation above which radiation accretion leads to a finite-time mass divergence, effectively ruling out larger primordial black holes in this specific coupling model.

4. Key Results

A. Upper Mass Limits

• Horizon Constraint: To ensure $R_{BH} < d_h$ during inflation, the mass parameter is bounded by $\gamma \leq \gamma_1 \approx 6.34 \times 10^{58}$.
• Accretion Constraint: To prevent runaway mass growth during the radiation era, a stricter bound is found: $\gamma \leq \gamma_3 \approx 2.07 \times 10^{30}$.
  • This corresponds to a maximum mass at the end of inflation: $M(t_f) \approx 1062.35$ g.
  • If the mass exceeds this, the black hole undergoes catastrophic growth due to radiation accretion.

B. Lower Mass Limit

• Evaporation Constraint: To survive Hawking evaporation during inflation, the initial mass at the onset of inflation ($t_i$) must satisfy:
  $$M(t_i) \geq M_{min} \approx 253.7 m_{Planck} \approx 5.52 \times 10^{-3} \text{ g}$$
  Black holes with lower initial masses evaporate completely before inflation ends.

C. Present-Day Mass

For black holes satisfying the survival window ($5.52 \times 10^{-3} \text{ g} \leq M(t_i) \leq 5.52 \times 10^{-3} \text{ g}$at formation, evolving to$M(t_f) \approx 1062$g), the mass at the present epoch ($t_0$) is calculated to be:
$$M(t_0) \simeq 1.043 \times 10^{-3} M_{\odot}$$
This places the surviving population in the sub-solar mass regime (roughly $10^{-3}$ solar masses).

Note: The paper also notes that for $\gamma < \gamma_3$, present-day masses could be significantly smaller (down to asteroid masses, $\sim 10^{-11} M_{\odot}$), potentially fitting the "asteroid-mass window" for dark matter candidates.

5. Significance and Implications

• Challenge to Standard PBH Scenarios: In standard isolated PBH models, objects with masses $M \lesssim 5 \times 10^{14}$ g are expected to have evaporated by today. This paper demonstrates that cosmological coupling can counterbalance Hawking evaporation, allowing much lighter primordial black holes to survive.
• Dark Matter Candidates: The derived mass range ($10^{-11} M_{\odot}$to$10^{-3} M_{\odot}$) overlaps with mass windows often discussed for primordial black holes as dark matter candidates. However, the authors caution that their results establish a survival window based on dynamics, not a specific prediction of abundance.
• Model Dependence: The results are highly dependent on the specific choice of the generalized McVittie metric (where $M \propto a(t)$) and the phenomenological accretion prescription. The authors acknowledge that a fully self-consistent relativistic treatment of accretion in a dynamical spacetime is required for definitive conclusions.
• Physical Consistency: The work highlights the necessity of ensuring that local objects (black holes) do not dynamically overtake global causal structures (particle horizons) in expanding universes, providing a rigorous check on cosmological black hole models.

In conclusion, the paper establishes that if primordial black holes exist and are coupled to the cosmic expansion as described by the generalized McVittie geometry, they must have formed within a very specific mass range to survive to the present day, potentially resulting in a population of sub-solar mass black holes that could contribute to the dark matter budget.