# Could PBH’s have created all Dark Matter?

My final paper for this class was to represent the theory that explains that although Dark Matter can be created by a Primordial Black Hole that the amount of Dark Matter in the Universe is far more than the mathematics can support; until recently. The Double-Inflation theory extends the possibility of what happens in the volatile scenario and accounts for a significant increase in Dark Matter creation. The shortcomings of several aspects of the previously leading theory are identified and redefined to explain what we now believe to be the case.

This class is part of the Natural Science program from the University of Oxford. The nature of the assignment was to write it as one would explain it to peers, which would be master’s level asto-physicists and astro-biologists, but I tend to take it a bit further. Generally, I try to write at a basic level and the metric for success to me is that my son can understand the content; he is 14 at this stage and that precludes some details in the math, but not the spirit of the math, it’s previously defined constraints, or how we now have found to overcome them.

## Could PBH’s have created all Dark Matter?

Below is an overview describing the findings that are published in “Double Inflation as a single origin of PBHs for all dark matter and LIGO”[1]. The material in and of itself is a compelling argument illustrating how much we understand about the Universe. What is equally compelling, to me, is how much we still do not know, and how significant this time in research is continually becoming as we continue to refine methods and learn more.

What do we see when we look up into space? Mostly ‘nothing’ compared to the things that our eyes can see like planets, stars, galaxies, or other things that exist in the visible spectrum.

Although space appears largely ‘empty’ to our eyes we know that dark matter can be repeatedly tested and confirmed by at least observing gravitational effects.

The idea of understanding the origin of dark matter seems likely to be a critical key in understanding our universe better, since we know little of it now we must wholeheartedly admit that we do not know what makes up over 90% of the Universe.

Quite possibly, the explanation of dark matter creation lies within the secrets bound in the primordial black holes (PBH’s); which, for now, we also do not completely understand.

Math has supported that PBH’s could be the source for some dark matter, but the volume of dark matter in its entirety is so vast that there is a significant gap between what we can calculate was created from PBH’s and the reality of dark matter density.

One attempt at reducing this gap is the idea that the process used to support the dark matter was not representative of the natural occurrence. There is a concept of inflation since we expect the process would have been subject to heat, and molecular acceleration in ways that we know that other ‘matter’ is created like stars; but perhaps there was more going on during the inflation process than initially presumed.

The idea of a more complex inflation theory has been shown to account for constraints in the single inflation model and suggests support for all dark matter.

As the paper explains, “In general, any realistic inflation model results in the extended mass function, not a monochromatic mass function. Extended mass functions are constrained more severely than monochromatic ones.”

Hence the theory of Dark Matter being created from PBH’s by way of a double-inflation, which focuses primarily on the PBH’s that were created in the radiation dominated era and where the curvature perturbations follow the Gaussian distribution.

To understand the testable math behind the theory we must still establish some boundaries. As you may expect, we are not going to be able to rely on a complete set of reliable constants as could be provided in laboratories or would be available to us calculating against a uniform Universe.

The approach to test the theory against a more realistic set of criteria the scientists divided the components into two groups by mass.

Within each mass group there are some constraints that had bound previous testing, and it is important to understand these limitations. The definition of these limitations alone accounts for a bold step in the direction of understanding. The constraints, as presented, explain why the mathematical support that existed before was not able to account for the mass of dark matter.

The first group is defined as *O* (10^{-14 }**– **10^{-10})*M*⊙ and this mass has two documented limitations that have led to inaccurate calculations:

- Gravitational microlensing, and
- Sublunar mass from observation of white dwarfs

The effect of gravitational microlensing cannot be said plainer than in the paper, which is, “The gravitational microlensing occurs when the lens objects pass through our line of sight to background stars and is observed as the temporary amplification of the light of the background stars.” Considering that the tools we use to gather such data perform high-cadence sampling we need to consider the previously discounted wave effect. Previous theories that identified the wave effect excluded it from the calculations without an understanding that once the wavelength of light is larger than the Schwarzschild radius of the lensing object that the light waves should be calculated because the geometric optics approximation becomes invalid.

The second group is defined as *O* (10)*M*⊙ and this mass has several documented limitations that have led to inaccurate calculations including effects from:

- Radio and X-ray accretion, and
- Ultra-faint dwarf galaxies

“Radio and X-ray from accretion review has set a new upper limit on the abundance of primordial black holes (PBH) based on existing X-ray data.”[2] Interstellar medium will interact with PBH’s and result in significant fluctuations of X-ray photons. The fluctuations contribute to the observed number-density of compact X-ray objects in galaxies. The range of mass subject to these fluctuations ranges from a few* M*⊙ to 2 x 10^{7}*M*⊙.

As explained by Timothy D. Brandt’s paper[3], ultra-faint dwarf galaxies provide strong constraints on massive compact halo objects (MACHOs) of gsim5 *M*⊙ as the main component of dark matter. What has been observed is that the dwarf galaxy cluster is dynamically heated by the dark matter, the heat increase can be significant enough that the cluster grows and speeds up until its host galaxy consumes the dissolved remains. The stars in these galaxies are subject to the same heat, and that we can observe at least 10 examples of such galaxies there are independent limits on MACHO dark matter of masses gsim10 *M*⊙. “Both Eri II’s cluster and the compact ultra-faint dwarfs are characterized by stellar masses of just a few thousand *M*⊙ and half-light radii of 13 pc (for the cluster) and ~30 pc (for the ultra-faint dwarfs). These systems close the ~20–100 *M*⊙ window of allowed MACHO dark matter and combine with existing constraints from microlensing, wide binaries, and disk kinematics to rule out dark matter composed entirely of MACHOs from ~10−7 *M*⊙ up to arbitrarily high masses.”[4]

Considering these shortcomings brings us to the concrete inflation model where we have a pre-inflation process that nurtures the proper inflation steps. Our new equation accounts for a stabilization term, *V*stb 1, to represent the pre-inflation stabilizing at the origin.

Next, the inflation, catalyzed by the pre-inflation, the inflation oscillates and functions as the Universe would be expected to do when dominated by matter.

Further, the new-inflation begins when the inflation energy becomes smaller than the new-inflation energy scale; this occurs due to a combination of the inflations reduction during decay and the new-inflation’s influence from our expanding Universe.

The mechanics behind the process are first, the inflection point of the new-inflation potential, and second, the Hubble Induced mass during the oscillation.

From Fig. 1 and Fig. 2, we see that the perturbation curvature peaks sharply which is characteristic of the double inflation model.

What the paper has clarified is that PBH’s around the sublunar mass can explain the existence of all dark matter. It appears that the double inflation model can “simultaneously explain PBHs as DM and PBHs as the BHs detected by LIGO.”[5] The two peak curvature perturbations (as represented in Figure 2) have been confirmed in multiple test.

We have clearly taken amazing steps forward since our ancient ancestors first looked up to the sky. Remembering archaic theories such as flat earth, the earth being the universe center, and even that we have come to be upon a tortoise back! Because of what we know, it is very easy to become a bit complacent and justified in our progress.

However, I submit to you that this time in history is not merely exciting because of what we have learned, but because we have learned how much we have yet to learn. The prospects of further exploration and ability to collect and organize data pave an unprecedented and very exciting future of discovery for us all. Not unlike the rolling boil of water in a kettle, or perhaps a flywheel, every molecular unit of work is significant and provides a contribution to the ultimate acceleration and future state.

We live in exciting times, and with no small thanks to those who have come before us and those who are continually working towards improving theories.

[1] This paper was chosen from the provided list and the original can be found online in the Cornell University library.

[2] Y. Inoue and A. Kusenko, JCAP 1710, 034 (2017), arXiv:1705.00791 [astro-ph.CO].

[3] T. D. Brandt, Astrophys. J. 824, L31 (2016), arXiv:1605.03665[astro-ph.GA].

[4] T. D. Brandt, Astrophys. J. 824, L31 (2016), arXiv:1605.03665[astro-ph.GA].

[5] This paper was chosen from the provided list and the original can be found online in the Cornell University library.

The paper that this work summarizes is Dark-Matter-origin-from-PBHs-via-Double-Inflation

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Fair warning, the scope of my paper was to ‘support the math’ in 1500 characters which is entirely different than detailing all of the mathematical calculations. If you are interested in more of the math please enjoy the paper below: