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In the very early universe, tiny fluctuations created slight over- and under-densities in mass/energy. As far as I understand, if a region becomes sufficiently over-dense, a primordial black hole (PBH) would form. This might be very very unlikely, but it should have some non-zero probability everywhere in space.

So my question is:

A) Do we have any sense of how many of these PBHs to expect?

B) The power spectrum was essentially flat back then, which I believe means that the size of fluctuations were scale-invariant (the average over-densities on one scale are twice as big as those at half the scale). Does this mean that PBHs are equally likely to form of any size ??

C) Does inflation have anything to say about this?

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  • $\begingroup$ It is unlikely that there is a simple dependence between the PBH spectrum and the CMB spectrum. The former depends on the model of inflation (arxiv.org/pdf/1801.05235.pdf), and the CMB spectrum is almost thermal. $\endgroup$ – atarasenko Jan 28 at 9:47
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We can investigate the PBH abundance constraints for a large number of possibilities, for instance, evaporation, entropy, relic particles, nucleosynthesis, gamma rays etc. 

These constraints applied to different kind of PBH masses, that range from $10^4$g to $10^{45}$g, depending on the constraints that we are talking about. 

Here is a graph that explains these constraints in detail.

Table. 1: Summary of constraints on the initial PBH abundance, $β(M_{PBH})$.

enter image description here

A) Do we have any sense of how many of these PBHs to expect?

Yes, we have. Using these constraints we can determine their effect on the power spectrum. The amplitude of the primordial power spectrum on cosmological scales is measured to be $P_r(k ≈ 10^{−3}Mpc^{−1}) ≈ 10^{-10}$ while the PBH limits are of order $P_R < 10^{−2}−10^{−1}$ on scales $k∼10^{−2}−10^{23}Mpc^{−1}$.

As we can see the PBH numbers are very low, that we can even neglect their formation. However, this is true for if the power spectrum were a pure power law.

C) Does inflation have anything to say about this?

I'll say my answer in other way around. Any inflation model that produces a large number of PBH on the small scales should be abandoned.

I recommend these articles that explain PBH density constraints and their effect on the power spectrum in detail.

https://arxiv.org/abs/1403.1198

https://arxiv.org/abs/astro-ph/9705166

https://arxiv.org/abs/0912.5297

The graph is taken from (https://arxiv.org/abs/0903.3184) I highly recommend this article to read/study.

 

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    $\begingroup$ These are some helpful pointers to the observational literature, but it would be nice to have more of a self-contained physics description within the answer itself. The OP is asking physics questions, not just asking for upper numerical bounds based on observation. $\endgroup$ – Ben Crowell Feb 2 at 16:59
  • $\begingroup$ Well upper numerical bounds can give an average number of the expected PBH in the early universe. I dont have enough math to understand the inflationary part. But I think the articles explains all the answer of his questions in details. $\endgroup$ – Reign Feb 2 at 17:43
  • $\begingroup$ @BenCrowell If you want you can write an answer related to these articles ? $\endgroup$ – Reign Feb 2 at 18:26
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A emphasized in Reign's answer, a variety of different models of PBH formation have been considered. Different models make different predictions. For the sake of giving a concise answer, only one of the simplest models will be considered here. Non-trivial derivations will be omitted, but I'll outline some key equations, define the quantities that they involve, and provide some numerical estimates from the literature. The model considered here uses these assumptions (among others):

  • The mechanism for PBH formation is the collapse of sufficiently large-amplitude density fluctuations, which are deviations from perfect homogeneity in the density of matter/radiation in the early universe.

  • The power spectrum of density fluctuations is assumed to be $P(k)\propto k^n$ for some spectral index $n$.

  • PHBs formation will only be considered after inflation has ended, because "the ones generated before inflation [are] diluted to negligible density" (page 4 in [$1$]).

  • Only non-rotating black holes are considered. According to the first page in [2], PBHs are expected to have nearly zero rotation. This probably doesn't affect the results much; I'm mainly mentioning it for fun.

In units $G=c=1$, a black hole of mass $M$ has Schwarzschild radius $R\sim M$. In a back-of-the-envelope sense, the "density" of a black hole is $M/R^3$. For a black hole to form as a result of an inhomogeneity in the cosmological density, $M/R^3$ must be comparable to the cosmological density. After inflation, the cosmological density is close to the critical density $\rho\sim H^2$ where $H$ is the Hubble "constant." Therefore, in terms of the Hubble time $t=1/H$, we have $$ \frac{M}{R^3}\sim \frac{1}{t^2}, \tag{1} $$ and then using $R\sim M$ gives $$ M\sim t. \tag{2} $$ According to equation (1.1) in [$1$], the numerical relationship is (after restoring factors of $c$ and $G$) $$ M\sim\frac{c^3 t}{G}\sim 10^{12}\text{ kg }\cdot\frac{t}{10^{-23}\text{ sec}}. \tag{3} $$ This says that smaller PBHs form earlier, and larger PBHs form later. By the way, according to page 1 in [4], PBHs with mass $M\lesssim 10^{12}$ kg would have evaporated by now, but those with mass $M\gtrsim 10^{12}$ kg should still be here. PBHs formed at $t=1$ second would have a mass comparable to a supermassive black hole at the center of a galaxy.

To begin addressing part A of the question, let $\beta(M)$ denote the total mass of all the PBHs with individual masses $\geq M$ divided by the total mass-energy available in the universe at the time of PBH formation. The quantity $\beta(M)$ can be related to the power spectrum of density fluctuations, based on this principle from page 2 in [3]:

In order for a PBH be formed, a collapsing overdense region must be large enough to overcome the pressure force resisting its collapse as it falls within its Schwarzschild radius. ... This requires that its radius exceeds the Jeans length...

According to equation (10) in [3], the result is $$ \beta(M)\approx \sigma(M)\exp\left(-\frac{1}{18\sigma^2(M)}\right) \tag{4} $$ where $\sigma^2(M)$ is given an integral over wavenumbers, weighted by the power spectrum $P(k)$ of density fluctuations and with an $M$-dependent cutoff, with $$ P(k)\propto k^n \tag{4b} $$ where $n\gtrsim 1$ is the "spectral index." Equation (14) in [3] gives $$ \sigma(M)\propto \left(\frac{1}{M}\right)^{(n-1)/4}. \tag{4c} $$ Page 3 in [3] says:

Strictly speaking, this [quantity $\beta(M)$] is the mass fraction in black holes of mass greater than $M$, but in practice $\beta(M)$ is such a rapidly falling function that these can be taken to all have the same mass $M$.

Equation (4) addresses part B of the question within the simple model being considered here. It says that the number of PBHs formed is a rapidly decreasing function of their mass. Some parametric predictions are shown in figure 4 in [5], reproduced here for convenience:

enter image description here

According to page 3 in [$1$], "one of the strongest [observational] constraints on $\beta$ over all mass ranges" is $$ \beta(10^{12}\text{ kg})\lesssim 10^{-26}, \tag{5} $$ and figure 6 in [$1$] shows some observational constraints for other (small) values of $M$. (The relationship between the quantity $\beta'$ used in figure 6 and the quantity $\beta(M)$ defined above is shown in equation (2.7) in [$1$]. They are numerically similar.)

By the way, according to page 11 in [6],

The power spectrum on cosmological scales, $k \sim 10^{-3} - 1\text{ Mpc}^{-1}$, is accurately measured by Cosmic Microwave Background... Cosmological observations probe a fairly limited region of the inflaton potential. [In contrast,] the PBH constraints on the power spectrum are fairly weak... However they apply over a very wide range of scales, $k \sim 10^{-2} - 10^{23}\text{ Mpc}^{-1}$...

The point of showing this excerpt is to emphasize that the comparison between PBH-density predictions and observational constraints are often used to constrain the model's parameters, rather than using independent constraints on those parameters ot predict the PBH-density. So the "predictions" reviewed here should be interpreted with that in mind. Also, regarding the question about scale invariance (part B), page 12 in [6] says this:

if the power spectrum of the density perturbations were exactly scale invariant... then the abundance of PBHs would be completely negligible, $\beta(M)\sim \exp(-10^8)$, since on cosmological scales the PBH mass variance is measured to be of order $10^{-5}$ ...

This is why the results reviewed above consider values of the spectral index (that is, the exponent in $P(k)\propto k^n$) that deviate from strict scale-invariant, and this is why $\sigma(M)$ and therefore $\beta(M)$ ends up depending on $M$.

The next step is to relate $\beta(M)$ to the modern density of PBHs having mass $\geq M$. This will be denoted $\Omega_\text{PBH}$ and expressed in units of the modern critical density $\Omega_c$. According to equation (1.4) in [$1$], the relationship for PBHs formed in the radiation-dominated era is $$ \Omega_\text{PBH}\cong(1+z)\Omega_c \beta(M) \sim 10^{24} \beta(M)\sqrt{\frac{1\text{ kg}}{M}} \tag{6} $$ where $z$ is the redshift. This is for black holes that have not yet decayed ($M\gtrsim 10^{12}$ kg). This addresses part A of the question within the simple model being considered here.

Regarding part C of the question, since PBH formation was assumed here to occur only after inflation, the role of inflation is indirect: it serves to ensure that the cosmological density is close to the critical density that was used to get equation (1), and it also affects the spectrum of density fluctuations, as stated on page 1 in [3]:

In the "standard" cosmology, the universe has been radiation dominated ever since the end of the reheating period after a phase of inflation at extremely high energies, which was responsible for the generation of density perturbations.


References:

[$1$] "New cosmological constraints on primordial black holes," https://arxiv.org/abs/0912.5297

[2] "Steepest growth of the power spectrum and primordial black holes," https://arxiv.org/abs/1811.11158

[3] "Constraints on the density perturbation spectrum from primordial black holes," https://arxiv.org/abs/astro-ph/9704251

[4] "Primordial black holes and early cosmology," https://arxiv.org/abs/astro-ph/9710235

[5] "Calculating the mass fraction of primordial black holes," https://arxiv.org/abs/1405.7023

[6] "Primordial Black Holes: sirens of the early Universe," https://arxiv.org/abs/1403.1198

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