REVIEW
Constraints on the density perturbation spectrum from primordial black holes
Astrophysics · Submitted 25 Apr 1997 · 21 references
Pith's verdict: A solid, citable tightening of the PBH bound on n, with a genuinely new thermal-inflation extension; the one thing to press on is the unshown COBE-to-sigma normalization.
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Referee report
The paper
The paper claims
Primordial black holes give the tightest bound yet on how blue the primordial density spectrum can be, with the spectral index capped near $n = 1.25$ once two errors in the earlier derivation are corrected.
The reading
What carries the argument
The load-bearing relation is the scaling of the smoothed density variance at horizon crossing with the black hole mass, $σ_{hor}(M) = σ_{hor}(M_{eq})(M/M_{eq})^{(1-n)/4}$, which converts a bound on the initial black hole mass fraction $β_i$ into a bound on $n$. The mass fraction follows from a gaussian probability for the smoothed density contrast to land between $1/3$ and $1$, an integral that is well approximated by $β(M) ≈ σ(M) exp(-1/(18σ^2(M)))$ because the integrand falls so steeply. Normalizing $σ$ at the present Hubble scale using the four-year COBE fit, $σ(10^{56} g) = 9.5×10^{-5}$, and then running it down to black hole masses with the scaling above, closes the chain from observation to spectral index. Thermal inflation enters by changing the relation between comoving scale and horizon mass, replacing $M/M_{eq}$ in the exponent with $M (10^3 GeV/T_{ti})^2/M_{eq}$, and by diluting any black holes formed before it.
Extended reading notes
Core claim
The central claim is that the primordial black hole abundance limits, evolved back to the time of formation and converted into a bound on the amplitude of the density perturbation at horizon crossing, cap the spectral index of the primordial spectrum at $n$ below roughly 1.25 in the standard radiation-dominated cosmology. The tightening over the earlier result of Carr, Gilbert and Lidsey comes from two corrections that push in the same direction: the variance at horizon crossing during radiation domination scales as $M^{(1-n)/4}$, not $M^{(1-n)/6}$, because the comoving mass density falls with time rather than being conserved; and the COBE normalization used earlier was too low by more than a factor of twenty, through a missing numerical prefactor and through treating the horizon-crossing amplitude $δ_H$ and the smoothed variance $σ(M)$ as interchangeable. Extending the same machinery to a cosmology containing a late period of thermal inflation weakens the bound to $n$ below about 1.3, because the second inflation dilutes the black holes relative to radiation by a factor of order $10^{12}$, and it produces a distinctive prediction: black holes cannot form with masses between about $10^{18}$ g and $10^{26}$ g, since the scales that would make them are pushed back outside the horizon and re-enter with perturbations too small to collapse. Allowing for the non-gaussianity expected in the tail of the perturbation distribution weakens the bound by at most about 0.05, a change the authors judge comparable to the shift from thermal inflation itself.
Load-bearing premise
The headline number treats the spectral index as a single constant across the roughly twelve orders of magnitude between the COBE normalization scale and the black hole formation scale, and the paper itself concedes that there is in general no reason to expect inflation to produce a power law over so wide a range, though models exist that do.
Editorial extensions
If this is right
- If the corrected bound holds, a power-law primordial spectrum with $n$ above about 1.25 is ruled out by the simple fact that the universe is not filled with the black holes it would have produced.
- The constraint reaches scales from about a megaparsec down to $10^{-16}$ Mpc, far below the scales probed by COBE and large-scale structure, so it constrains the spectrum over roughly twelve decades in scale.
- A confirmed period of thermal inflation would relax the bound to about 1.3 and would additionally forbid black hole formation in the mass window $10^{18}$ g to $10^{26}$ g.
- Non-gaussianity in the large-density tail can loosen the bound by at most about 0.05 in $n$, so it cannot rescue a strongly blue spectrum.
- If black holes leave stable Planck-mass relics, the bound on $n$ can tighten further for reheat temperatures above roughly $10^{14}$ GeV, where the relics survive dilution.
Reading between the lines
- Because the bound is quoted for a single constant $n$, a spectrum whose local slope drifts could satisfy it at black hole scales while still being steeply blue elsewhere; a running-index analysis would be the natural next step, and the paper's own numbers suggest the bound shifts by only about 0.01 from the change of slope at matter domination.
- The predicted missing-mass window is a clean signature: if thermal inflation is real, no black hole of mass between $10^{18}$ g and $10^{26}$ g should ever be identified, so any future detection in that range would count against the scenario rather than against the spectral bound.
- Since the tightest constraint comes from the deuterium-destruction limit near $10^{11}$ g, the bound is sensitive to the assumed evaporation physics and to the effective degrees of freedom at evaporation; a revised evaporation spectrum would move the quoted number more than the non-gaussianity correction does.
- The same machinery applies to any late entropy dump, not just thermal inflation: any episode that dilutes black holes after formation weakens the spectral bound by a computable factor and opens a corresponding mass gap.
Original abstract
We re-examine the constraints on the density perturbation spectrum, including its spectral index $n$, from the production of primordial black holes. The standard cosmology, where the Universe is radiation dominated from the end of inflation up until the recent past, was studied by Carr, Gilbert and Lidsey; we correct two errors in their derivation and find a significantly stronger constraint than they did, $n \lesssim 1.25$ rather than their 1.5. We then consider an alternative cosmology in which a second period of inflation, known as thermal inflation and designed to solve additional relic over-density problems, occurs at a lower energy scale than the main inflationary period. In that case, the constraint weakens to $n \lesssim 1.3$, and thermal inflation also leads to a `missing mass' range, $10^{18} g \lesssim M \lesssim 10^{26} g$, in which primordial black holes cannot form. Finally, we discuss the effect of allowing for the expected non-gaussianity in the density perturbations predicted by Bullock and Primack, which can weaken the constraints further by up to 0.05.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We re-examine the constraints on the density perturbation spectrum, including its spectral index $n$, from the production of primordial black holes. The standard cosmology, where the Universe is radiation dominated from the end of inflation up until the recent past, was studied by Carr, Gilbert and Lidsey; we correct two errors in their derivation and find a significantly stronger constraint than they did, $n \lesssim 1.25$ rather than their 1.5. We then consider an alternative cosmology in which a second period of inflation, known as thermal inflation and designed to solve additional relic over-density problems, occurs at a lower energy scale than the main inflationary period. In that case, the constraint weakens to $n \lesssim 1.3$, and thermal inflation also leads to a `missing mass' range, $10^{18} g \lesssim M \lesssim 10^{26} g$, in which primordial black holes cannot form. Finally, we discuss the effect of allowing for the expected non-gaussianity in the density perturbations predicted by Bullock and Primack, which can weaken the constraints further by up to 0.05.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We re-examine the constraints on the density perturbation spectrum, including its spectral index $n$, from the production of primordial black holes. | contribution 0.93 | 0.84 | 0.08 | asserted 0.52 |
| s1 | The standard cosmology, where the Universe is radiation dominated from the end of inflation up until the recent past, was studied by Carr, Gilbert and Lidsey; we correct two errors in their derivation and find a significantly stronger constraint than they did, $n \lesssim 1.25$ rather than their 1.5. | result 0.75 | 0.91 | 0.41 | asserted 1.00 |
| s2 | We then consider an alternative cosmology in which a second period of inflation, known as thermal inflation and designed to solve additional relic over-density problems, occurs at a lower energy scale than the main inflationary period. | contribution 0.82 | 0.38 | 0.04 | asserted 0.86 |
| s3 | In that case, the constraint weakens to $n \lesssim 1.3$, and thermal inflation also leads to a `missing mass' range, $10^{18} g \lesssim M \lesssim 10^{26} g$, in which primordial black holes cannot form. | result 1.00 | 0.87 | 0.18 | asserted 0.99 |
| s4 | Finally, we discuss the effect of allowing for the expected non-gaussianity in the density perturbations predicted by Bullock and Primack, which can weaken the constraints further by up to 0.05. | result 0.80 | 0.53 | 0.08 | hedged possibility 0.72 |
What was examined, and what was not
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Figures
Figures reproduced from arXiv:astro-ph/9704251 by the authors.
References
How the literature uses it
50 arXiv papers cite this in 80 sentences
Papers citing it · 55 sentences without a confident label
Contests it 1 paper
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An updated version of the constraints was later provided by Carr et al. [ref] , although this contained some errors which were corrected by Green and Liddle [ref] .
New cosmological constraints on primordial black holes · Introduction · contest 0.76
Uses its method or tool 6 papers
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We follow the `peaks theory' method counting the number density of peaks above a given criterion [ref] to compute the PBH mass fraction $β_M_PBH$ of the universe and the PBH dark matter abundance $f_PBH≡Ω_PBH / Ω_DM$ (see also [ref] ):
Primordial Black Holes in Higgs-$R^2$ Inflation as a Whole Dark Matter · PBH abundance as dark matter · 1.00
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Using the usual relations between the mass variance at the PBH formation time $σ_H(t_form)$ and the same quantity today $σ_H(t_0)$ [ref] , [equation] where $M_H(t)$ is the Hubble mass at time $t$ and $t_eq$ is the equilibrium time, it is possible to set an upper value on $β$ which can be expressed as [equation] where $δ_min≈0.3$ is the minimum density contrast required to form a PBH.
Primordial Black Holes · Cosmological consequences · 0.97
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The mass fraction $β_i$ can be found assuming a Gaussian perturbation spectrum of fluctuations that leads to black hole formation with (see [ref] ) [equation] where $T_form$ is the temperature when the primordial black holes were formed, while $T_0$ is the temperature today.
Constraints on dark matter particles charged under a hidden gauge group from primordial black holes · Constraints · 0.82
Uses its data or values 4 papers
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The set of observational constraints on the abundance of PBHs is listed in Table I and has been used to bound the mean amplitude of $δ$ defined by for distinct cosmologies [ref] .
The effect of non-Gaussian curvature perturbations on the formation of primordial black holes · Constraints on non-Gaussian perturbations in the PBH range · 0.86
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The present limit at the scale of PBH/UCMH formation is $n≲1.25$ [ref] .
Gamma rays from ultracompact primordial dark matter minihalos · Introduction · 0.89
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The value of $n$ frequently used in this context is between $1.23$ and $1.31$ [ref] .
Constraints on dark matter particles charged under a hidden gauge group from primordial black holes · Constraints · 0.95
Compares with it 2 papers
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The limit derived in the previous subsection leads to $n<1.20$ in the mSUGRA case whereas the usually derived limits range between 1.23 and 1.31 [ref] .
Primordial Black Holes · Cosmological consequences · 0.93
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This value is higher that the value $1.25$ found in [ref] because here we have assumed that the mass of the collapsed region to form PBH is only 20% of the entire energy density inside the particle horizon.
Running-Mass Inflation Model and Primordial Black Holes · Summary and Conclusions · 0.99
Cites it as support 1 paper
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If the early universe included a black hole dominated era, such a scenario is strongly excluded (see also Refs. [ref] ).
Dark Radiation and Superheavy Dark Matter from Black Hole Domination · Primordial Black Holes in the Presence of Large Hidden Sectors · 0.97
Cites it as background 11 papers
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Various aspects of PBH physics, mechanisms of their formation, evolution and effects are discussed in [ref] particularly specifying PBH formation and effects in braneworld cosmology [ref] , on inflationary preheating [ref] , formation of PBHs in QCD phase transition [ref] , properties of superhorizon BHs [ref] , role of PBHs in baryosynthesis [ref] , effects of PBH evaporation in the early Universe and in modern cosmic ray, neutrino and gamma fluxes [ref] , in creation of hypothetical particles [ref] , PBH clustering and creation of supermassive BHs [ref] , effects in cosmic rays and colliders from PBHs in low scale gravity models [ref] .
Primordial Black Holes · Introduction · 0.98
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Following the conventional arguments in Refs. [ref] , suppose a spherically symmetric over-dense region enters the cosmological horizon in the radiation-dominated era.
Simple Cosmological Solution to the Higgs Instability Problem in the Chaotic Inflation and Formation of Primordial Black Holes · Primordial Black Hole as Dark Matter · 0.83
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Processes induced by inflation gained much interest with a localized enhancement in the curvature power spectrum [ref] , in correlation with copious primordial black hole (PBH) production [ref] .[ For a recent review on PBHs see [ref] and references within. ]
The Inflaton that Could : Primordial Black Holes and Second Order Gravitational Waves from Tachyonic Instability induced in Higgs-$R^2$ Inflation · Introduction · 0.84
Discussion
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