What's the deal with dark matter

[Snooch]

Can someone explain why we are certain there is dark matter in our universe?

I understood it like this. At some Point in our recent history we figured out, that according to our math galaxies (or sth like that) wouldn't have developed like they did or wouldn't even stay in 1 Piece since they do not have enough mass. So we came up with dark matter as a source of mass to Keep it all in place. Now since dark matter does not react with anything other than its gravitational pull u cannot testify its existence.

If that is the case "dark matter" seems like a pretty far fetched concept to me which maybe is validating bad math or sth just to make it work.

Im sorry for my bad spelling english isn't my first language and I am not very educated. I hope someone maybe takes the time to help me with this even a good link (other than Wikipedia) would help me alot.

 

[russ_watters]

Can someone explain why we are certain there is dark matter in our universe?

I understood it like this. At some Point in our recent history we figured out, that according to our math galaxies (or sth like that) wouldn't developed like they did or wouldn't even stay in 1 Piece since they do not have enough mass. So we came up with dark matter as a source of mass to Keep it all in place. Now since dark matter does not react with anything other than its gravitational pull u cannot testify its existence.

If that is the case "dark matter" seems like a pretty far fetched concept to me which maybe is validating bad math or sth just to make it work.

Your understanding of how we got here is pretty good, but you underestimate how difficult it would be for "bad math" to be the cause. GR has worked exquisitely well for pretty much every situation it has been thrown at except this, so one would have to be able to correct the "bad math" in such a way as to have no effect on all those other calculations.

 

[Snooch]

GR has worked exquisitely well for pretty much every situation it has been thrown at except this, so one would have to be able to correct the "bad math" in such a way as to have no effect on all those other calculations.

So there is no experiment at this Moment which can proof dark matter and its just an assumption based on GR?

 

[russ_watters]

So there is no experiment at this Moment which can proof dark matter and its just an assumption based on GR?

No experiment separate from GR, right.

 

[Laurie K]

Your understanding of how we got here is pretty good, but you underestimate how difficult it would be for "bad math" to be the cause. GR has worked exquisitely well for pretty much every situation it has been thrown at except this, so one would have to be able to correct the "bad math" in such a way as to have no effect on all those other calculations.

If the math error occurred in universal sum calculations and correct answers were obtained in isolation how would we know that we had an error in our universal sum calculations?

 

[Janus]

Can someone explain why we are certain there is dark matter in our universe?

I understood it like this. At some Point in our recent history we figured out, that according to our math galaxies (or sth like that) wouldn't have developed like they did or wouldn't even stay in 1 Piece since they do not have enough mass. So we came up with dark matter as a source of mass to Keep it all in place. Now since dark matter does not react with anything other than its gravitational pull u cannot testify its existence.

If that is the case "dark matter" seems like a pretty far fetched concept to me which maybe is validating bad math or sth just to make it work.

Im sorry for my bad spelling english isn't my first language and I am not very educated. I hope someone maybe takes the time to help me with this even a good link (other than Wikipedia) would help me alot.

It's a bit more complex than that. It isn't just that there should be more mass, but that mass would have to be distributed around the galaxy in a different shape than the visible matter we see. And you just can't say that the dark matter just happens to collect that way to produce the desired result. Instead you determine what properties dark matter would have to have in order to not be visible. That determines how it would tend to form around galaxies, and then you see whether that prediction matches how the dark would have to form around the galaxy to provide the correct gravity. If the two match, you have good evidence for dark matter to be the reason.
It's not as people haven't tried to explain things by assuming that our understanding of gravity needs to be modified, its just that all such attempts to come up with a modified theory have come up short in explaining what we observe and the dark matter hypothesis is a much better fit.
And "dark" matter is not that far fetched. We already know of one type of particle that matches that description. Neutrinos don't interact with light or electromagnetically at all. There are reasons why the neutrinos that we know of aren't what dark matter we are looking for is made off, but it is evidence that something else that shares some of its properties could exist.

 

[russ_watters]

If the math error occurred in universal sum calculations and correct answers were obtained in isolation how would we know that we had an error in our universal sum calculations?

I don't understand. Are you saying as opposed to an error in the theory a literal typo repeated thousands of times, identically? That's basically impossible.

 

[Laurie K]

I don't understand. Are you saying as opposed to an error in the theory a literal typo repeated thousands of times, identically? That's basically impossible.

No, I'm saying that the error may be that the universal sum of the individual parts is incorrect even though the calculations for each individual part may well be correct.

 

[russ_watters]

No, I'm saying that the error may be that the universal sum of the individual parts is incorrect even though the calculations for each individual part may well be correct.

That is, in essence, what the problem is. I don't see a good reason to consider that a "math error" instead of "we're missing some parts".

 

[OCR]

No, I'm saying that the error may be that the universal sum of the individual parts is incorrect even though the calculations for each individual part may well be correct.

That almost sounds like the definition of emergence... ?

Although strong emergence is logically possible, it is uncomfortably like magic. How does an irreducible but supervenient downward causal power arise, since by definition it cannot be due to the aggregation of the micro-level potentialities? Such causal powers would be quite unlike anything within our scientific ken. This not only indicates how they will discomfort reasonable forms of materialism. Their mysteriousness will only heighten the traditional worry that emergence entails illegitimately getting something from nothing.

 

[Laurie K]

That almost sounds like the definition of emergence... ?

A model of the universe that puts the boundary at our observation limit (visible universe has an observational data limit) and excludes everything that could physically exist beyond that boundary may very well have sum problems as you can never actually know if your boundary is correct or not.

 

[Adrian59]

The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts.--Bertrand Russell.

Very true but why could this not apply to dark matter itself?

 

[Adrian59]

That is, in essence, what the problem is. I don't see a good reason to consider that a "math error" instead of "we're missing some parts".

Reading between the lines, would "math error" be compatible with MOND and of course "we're missing some parts" be compatible with dark matter? So you appear to be agreeing with one side of the argument, that the only way to solve this problem is with dark matter.

 

[r8chard]

In my opinion, what we need right now is patience. We do not have the answers we want. The few verified answers we have from experiments contradicts our speculations. We are befuddled as what to do next. How shall we invent, design, develop, trouble-shoot the next generation of experiments? Can the sources of funding the technology be convinced to invest in an uncertain outcome?

 

[Adrian59]

In my opinion, what we need right now is patience. We do not have the answers we want. The few verified answers we have from experiments contradicts our speculations. We are befuddled as what to do next. How shall we invent, design, develop, trouble-shoot the next generation of experiments? Can the sources of funding the technology be convinced to invest in an uncertain outcome?

I agree with your post and of course there are ongoing experiments to try to directly detect dark matter, some of which will only be completed in a few years from now. However, I note theorists are moving away from dark matter candidates that can be directly detected in any way and that raises serious philosophical questions. On a technical matter, I find it difficult to see how some of the putative candidates, for example neutrinos or axions, can be responsible for dark matter because of their negligible mass. To my thinking there needs to be a quite astonishing number density of these particles to be responsible for 85% of the universe's mass.

 

[f todd baker]

In the second half or the 19th century electromagnetism was thought to be perfect. It correctly described the results of numerous experiments, much as GR today describes correctly the results of numerous experiments. There were a few issues which were a bit perplexing and so it was necessary to hypothesize a medium through which electromagnetic waves could travel. Bit of a problem detecting the luminiferous aether though, eh? Of course, it is really, really difficult to observe something which was eventually found to not exist! In spite of difficulties with moving forward with gravitation theory, there is still the question as to whether we understand gravity itself as well as we think we do; I occasionally hear reputable cosmologists (which I am most certainly not) acknowledge this possibility. Just sayin'!

 

[stefan r]

...Can the sources of funding the technology be convinced to invest in an uncertain outcome?

Not likely. More and better telescopes make pretty pictures. The pictures have independent value even if they do not generate new physics . Like any natural phenomenon the rotation of the galaxy should be explained long before we try changing it. It is hard to argue there is any urgency there. People who are professionally teaching as well as many amateurs attack this question without sending anyone a bill.

 

[George Jones]

A short, recent summary in Nature:

https://www.nature.com/news/dark-matter-hunt-fails-to-find-the-elusive-particles-1.22970

 

[Drakkith]

In spite of difficulties with moving forward with gravitation theory, there is still the question as to whether we understand gravity itself as well as we think we do; I occasionally hear reputable cosmologists (which I am most certainly not) acknowledge this possibility. Just sayin'!

Scientists are well aware of this question and are doing their best to answer it. It hasn't been forgotten.

Not likely. More and better telescopes make pretty pictures. The pictures have independent value even if they do not generate new physics .

There are many possible ways other than telescopes to search for dark matter. I believe we've set up various particle detectors in different areas of the world to see if we can find dark matter or other new types of particles. The Large Underground Xenon experiment was/is one of them.

Reading between the lines, would "math error" be compatible with MOND and of course "we're missing some parts" be compatible with dark matter?

I would think a math error would be incompatible with everything since it's just an error. The "we're missing something" would seem to be compatible with all theories, since we almost certainly have to be missing some pieces if we are observing phenomena that aren't predicted or explained by current theory.

 

[ahiddenvariable]

We observe that the visible matter of galaxies is moving in ways that cannot be attributed to only the gravitational forces associated with this visible matter and conclude that there must be additional gravitational forces caused by matter we can't see. As I understand it, gravitational forces are associated with warping in space-time, according to GR. A question arises, is there some way to cause space-time warping - and hence, forces that attract matter - other than by the presence of massive bodies?

 

[rootone]

A question arises, is there some way to cause space-time warping - and hence, forces that attract matter - other than by the presence of massive bodies?

Nothing that we know of, that is why it is called 'dark'.

 

[Adrian59]

We observe that the visible matter of galaxies is moving in ways that cannot be attributed to only the gravitational forces associated with this visible matter and conclude that there must be additional gravitational forces caused by matter we can't see. As I understand it, gravitational forces are associated with warping in space-time, according to GR. A question arises, is there some way to cause space-time warping - and hence, forces that attract matter - other than by the presence of massive bodies?

I think the short answer is no, within current accepted theory. The most concise representation of General Relativity is,
Gμν = - κ Tμν
where Tμν is the energy-momentum tensor and by implication since mass is energy, it accounts for mass as well. Gμν is the Einstein curvature term and κ is Einstein's constant of gravity which is directly related to Newton's constant G. So curvature of space is directly linked the amount of mass and energy in a locality. Because of the relation E = mc^2, mass is by far the largest contributor to the curvature of space.

However, the point you make at the start of your post, 'We observe that the visible matter of galaxies is moving in ways that cannot be attributed to only the gravitational forces associated with this visible matter', is not backed up by all observational data!

 

[stefan r]

... gravitational forces are associated with warping in space-time, according to GR. A question arises, is there some way to cause space-time warping - and hence, forces that attract matter - other than by the presence of massive bodies?

Nothing that we know of, that is why it is called 'dark'.

Are you sure? Light going into a black hole increases the gravity of the hole and therefore "space time warping" [link is not peer reviewed, arXiv.org]. Photons of light do not have mass.

This is not useful information for dark matter. Dark matter is clearly not made of photons.

 

[SKHanson57]

Reading between the lines, would "math error" be compatible with MOND and of course "we're missing some parts" be compatible with dark matter? So you appear to be agreeing with one side of the argument, that the only way to solve this problem is with dark matter.

Everyone is going to laugh about my opinion in that if we could calculate the frame drag of the galaxy using Earth's frame drag as a representation and apply that figure with regards to the amount of visible matter contained within our galaxy would we get a figure that is close to what we have calculated for the amount of dark matter needed? Maybe I’m in left field on this or not I don’t know.

 

[Janus]

Everyone is going to laugh about my opinion in that if we could calculate the frame drag of the galaxy using Earth's frame drag as a representation and apply that figure with regards to the amount of visible matter contained within our galaxy would we get a figure that is close to what we have calculated for the amount of dark matter needed? Maybe I’m in left field on this or not I don’t know.

Frame dragging is just too tiny an effect compared to what would be needed.

 

[ohwilleke]

It's not as people haven't tried to explain things by assuming that our understanding of gravity needs to be modified, its just that all such attempts to come up with a modified theory have come up short in explaining what we observe and the dark matter hypothesis is a much better fit.

This is not actually true.

In fact, all straightforward applications of the dark matter hypothesis, which looked promising at first, have pretty much been ruled out by observational evidence. The plain vanilla model in which there is a single type of thermal dark matter with a mass O(1-100) GeV has been ruled out by observation for almost a decade. And, there are, in fact, modified gravity theories (although not the most well known example of the genre called MOND) that do fit the data better than any of the extant dark matter theories (see, e.g., Moffat's MOG theory), and do so with fewer free parameters in their models, although few modified gravity theories have been tested as rigorously and by as large a group of investigators as the leading dark matter theories have been.

This is not to say that an entirely satisfactory and well tested solution on any front exists. But, a lot of the data points which folk wisdom assumes destroyed modified gravity theories (e.g. the Bullet Cluster) do no such thing. Indeed, data points like the Bullet Cluster actually do more harm to dark matter particle theories than to modified gravity theories (many of which can accommodate this observation).

Also, just to be clear, there is really no reasonable doubt that phenomena usually attributed to dark matter, that can not be explained with GR (at least as currently interpreted and applied*) and ordinary matter, exist and are pervasive. The phenomena attributed to dark matter can only be explained with some sort of new physics that either involves beyond the Standard Model particles, or forces that have effects different from GR as currently interpreted and applied, or both. These phenomena are by far the most compelling direct observational evidence that the "Core Theory" of GR plus the Standard Model is not complete and that New Physics are necessary to explain what is observed. (In contrast, "dark energy" phenomena can be completely explained to the limits of experimental observation with the cosmological constant of conventional GR.)

* There are a couple of promising gravitation based theories that claim that they do not actually modify GR but involve a means of applying GR-like concepts different than the way that the vast majority of researchers in the field apply GR to the analysis of complex systems operationally.

 

[rootone]

I'm still not convinced that MACHOs are ruled out.
Just recently we observe a compact object from deep space flying past the solar system.

 

[SKHanson57]

Why not calculate the frame drag associated with the black hole (or holes) in the center of a galaxy based on its mass and revolution speed (Earth's frame drag has been observed). Then determine both the value and relative position of the wake created by the frame drag/revolutions and plug that into a computer model to see if that can fill the void that is suggesting dark matter is needed. I believe some kind of estimate can be derived using the dynamics associated with a so called perfect liquid (or what I would call cold plasma)?

 

[Janus]

Why not calculate the frame drag associated with the black hole (or holes) in the center of a galaxy based on its mass and revolution speed (Earth's frame drag has been observed). Then determine both the value and relative position of the wake created by the frame drag/revolutions and plug that into a computer model to see if that can fill the void that is suggesting dark matter is needed. I believe some kind of estimate can be derived using the dynamics associated with a so called perfect liquid (or what I would call cold plasma)?

Doing so for the black hole of a similar mass to the one at the center our galaxy and figuring the framing dragging effect at, say, 50,000 ly from the center, it works out to being the equivalent of an additional 7.4e-54 km/sec. Besides, frame dragging falls off with distance from the mass, so any effect it would have would be stronger near the BH than it is further, But stellar speeds nearer the center of the galaxies aren't the problem, it's the ones on the outskirts.

 

[stefan r]

I'm still not convinced that MACHOs are ruled out.
Just recently we observe a compact object from deep space flying past the solar system.

Does anyone claim MACHOs do not exist? Someone could claim MACHOs are 80% of the Milky Ways mass, or 1%, or 0.001%. If it is 0.001% as comets/asteroids that would be 10⁹ solar mass. Something like 10²⁶ comets. Finding one of them will not prove much.

1/'Oumuamua was not a halo object.

 

[ohwilleke]

@rootone

I'm still not convinced that MACHOs are ruled out.

Allow me to remind you of some of the pertinent evidence which rules out MACHOs.

As noted by stefan r above, there is no doubt that some MACHO candidates exist (although primordial black holes have not yet been observed and there is good reason to doubt that they exist), but there simply are enough of them and they aren't in the right places to account for a meaningful share of dark matter phenomena. The smallest primordial black holes are impossible:

Depending on the model, primordial black holes could have initial masses ranging from 10⁻⁸ kg (the so-called Planck relics) to more than thousands of solar masses. However primordial black holes with a mass lower than 10¹¹ kg would have evaporated (due to Hawking radiation) in a time much shorter than the age of the Universe, and so cannot have survived in the present Universe.

The constraints on non-primordial black hole (PBH) MACHOs (terrestrial planets, baby or ordinary gas giants, neutron stars, cannibalized white dwarfs that become helium or diamond planets) are severe and these candidates have been ruled out for many years. Indeed, massive compact halo objects (MACHOs) are pretty much ruled out, in general, as dark matter candidates (citations in the original omitted and paragraph breaks added in the quotation below):

A MACHO may be detected when it passes in front of or nearly in front of a star and the MACHO's gravity bends the light, causing the star to appear brighter in an example of gravitational lensing known as gravitational microlensing. Several groups have searched for MACHOs by searching for the microlensing amplification of light. These groups have ruled out dark matter being explained by MACHOs with mass in the range 1×10−8 solar masses (0.3 lunar masses) to 100 solar masses.

One group, the MACHO collaboration, claims to have found enough microlensing to predict the existence of many MACHOs with mass of about 0.5 solar masses, enough to make up perhaps 20% of the dark matter in the galaxy. This suggests that MACHOs could be white dwarfs or red dwarfs which have similar masses. However, red and white dwarfs are not completely dark; they do emit some light, and so can be searched for with the Hubble Telescope and with proper motion surveys. These searches have ruled out the possibility that these objects make up a significant fraction of dark matter in our galaxy.

Another group, the EROS2 collaboration, does not confirm the signal claims by the MACHO group. They did not find enough microlensing effect with a sensitivity higher by a factor 2.

Observations using the Hubble Space Telescope's NICMOS instrument showed that less than one percent of the halo mass is composed of red dwarfs. This corresponds to a negligible fraction of the dark matter halo mass.

Therefore, the missing mass problem is not solved by MACHOs. . . .

Theoretical work simultaneously also showed that ancient MACHOs are not likely to account for the large amounts of dark matter now thought to be present in the universe. The Big Bang as it is currently understood could not have produced enough baryons and still be consistent with the observed elemental abundances, including the abundance of deuterium. Furthermore, separate observations of baryon acoustic oscillations, both in the cosmic microwave background and large-scale structure of galaxies, set limits on the ratio of baryons to the total amount of matter. These observations show that a large fraction of non-baryonic matter is necessary regardless of the presence or absence of MACHOs.

There have been arguments that PBHs are not subject to some of the exclusions above applicable to other kinds of MACHOs. But, a the MACHO exclusion chart below (which is a few years out of date (with the left side showing the ratio of MACHOs to dark matter and the horizontal axis showing MACHO mass) is as follows:

:

In terms of MACHOs are a dark matter candidate, only the very top part matters, since 10^-2 means just 1% of dark matter can be accounted for by MACHOs of that type. You really need to be above 10^-1 (i.e. 10%) to be considered as a significant source of dark matter phenomena.

As this chart shows, there are pretty significant observational constraints on the size of primordial black hole dark matter, however (relying mostly on this source). The sweet spot is 10^22 kilograms, which is a bit less than the mass of the Moon (which is 7*10^22 kilograms), plus or minus, which would imply a typical primordial black hole with an event horizon radius of about 0.1 millimeters.

Another paper on PBHs was even less bullish:

We investigate constraints on primordial black holes (PBHs) as dark matter candidates that arise from their capture by neutron stars (NSs). If a PBH is captured by a NS, the star is accreted onto the PBH and gets destroyed in a very short time. Thus, mere observations of NSs put limits on the abundance of PBHs. High DM densities and low velocities are required to constrain the fraction of PBHs in DM. Such conditions may be realized in the cores of globular clusters if the latter are of a primordial origin. Assuming that cores of globular clusters possesses the DM densities exceeding several hundred GeV/cm3 would imply that PBHs are excluded as comprising all of the dark matter in the mass range 3×10¹⁸g≲mBH≲10²⁴g. At the DM density of 2×10³ GeV/cm3 that has been found in simulations in the corresponding models, less than 5% of the DM may consist of PBH for these PBH masses.

Fabio Capela, et. al, "Constraints on primordial black holes as dark matter candidates from capture by neutron stars." Phys. Rev. D 87, 123524 (2013) (link is to open access pre-print version conformed to final print version).

Since the chart above was made the observational constraints on PBHs have further tightened. For example:

We model the accretion of gas on to a population of massive primordial black holes in the Milky Way, and compare the predicted radio and X-ray emission with observational data. We show that under conservative assumptions on the accretion process, the possibility that O(10)M⊙ primordial black holes can account for all of the dark matter in the Milky Way is excluded at 4σ by a comparison with the VLA radio catalog at 1.4 GHz, and at more than 5σ by a comparison with the NuSTAR X-ray catalog (10 - 40 keV). We also propose a new strategy to identify such a population of primordial black holes with more sensitive future radio and X-ray surveys.

Daniele Gaggero, et al., "Searching for Primordial Black Holes in the radio and X-ray sky" (Pre-Print December 1, 2016).

Another recent paper constraining MACHOs as cluster dark matter (keeping in mind that galaxy clusters are inferred to have much more dark matter proportionately than any kind of galaxy) finds:

Galaxy clusters, employed by Zwicky to demonstrate the existence of dark matter, pose new stringent tests. If merging clusters demonstrate that dark matter is self-interacting with cross section σ/m∼2 cm2/gr, MACHOs, primordial black holes and light axions that build MACHOs are ruled out as cluster dark matter. Recent strong lensing and X-ray gas data of the quite relaxed and quite spherical cluster A1835 allow to test the cases of dark matter with Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac distribution, next to Navarro-Frenck-White profiles. Fits to all these profiles are formally rejected at over 5σ, except in the fermionic situation.

Theodorus Maria Nieuwenhuizen "Subjecting dark matter candidates to the cluster test" (October 3, 2017) (omitting from the abstract conclusions about sterile neutrino dark matter candidates).

It also turns out that the model used in many PBH exclusion analyses is actually too lenient and that more realistic modeling makes the exclusions even tighter.

 

[Adrian59]

I would think a math error would be incompatible with everything since it's just an error. The "we're missing something" would seem to be compatible with all theories, since we almost certainly have to be missing some pieces if we are observing phenomena that aren't predicted or explained by current theory.

Whether I have read this right but I was regarding 'math error' as using the wrong or incomplete math rather than an error like 2+2=5!

 

[Jacinta]

I guess it's darker than light matter? Sounds like something out a sci-fi film.

 

[Buzz Bloom]

The sweet spot is 10^22 kilograms, which is a bit less than the mass of the Moon (which is 7*10^22 kilograms), plus or minus, which would imply a typical primordial black hole with an event horizon radius of about 0.1 millimeters.

It also turns out that the model used in many PBH exclusion analyses is actually too lenient and that more realistic modeling makes the exclusions even tighter.

Hi ohwilleke:
I confess that the links you provided are too technical for me to follow in detail, but I think I get the gist which I summarize as follows.

PBHs can only make up a significant fraction of DM if they are very small, i.e., less massive than the moon.​

From https://en.wikipedia.org/wiki/Hawking_radiation

However, since the universe contains the cosmic microwave background radiation, in order for the black hole to dissipate, it must have a temperature greater than that of the present-day blackbody radiation of the universe of 2.7 K = 2.3×10−4 eV. This implies that M must be less than 0.8% of the mass of the Earth[21] – approximately the mass of the Moon.​

If I am understanding all this correctly, if a sufficient number of moon size or smaller PBHs exist, they could not be detected by observing their Hawking radiation evaporation. I am not knowledgeable enough to read the diagram in #31 to estimate the number of possible PBHs that would be sufficiently larger than the moon to have their evaporation radiation detectable. Can you help me with that?

Regards,
Buzz

 

[ohwilleke]

Nobody directly observes Hawking radiation. One uses its predicted evaporation rate and the age of the universe to determine how big PBHs created in the Big Bang era would be today. This sets a lower bound on the size of PBHs that still exist today.

You put an upper bound on the size of PBHs mostly based upon the failure of microlensing observations to see light bending in ways consistent with masses of a certain size or larger. If PBHs were much larger than the Moon in mass, we would see microlensing caused by PBHs all over the place with telescopes.

We know the total inferred amount of DM, for example, in the Milky Way, from star dynamics and gravitational lensing. And for any given PBH mass, you can determine the number of PBHs per unit volume that have to exist to produce that quantity of DM. You can then determine how common PBH scale gravitational lensing should be per area viewed with telescopes, and if the observed amount of lensing is much lower than it should be given this analysis, then you know that the PBH hypothesis to explain DM is wrong.

In reality, of course, PBH DM would not all have exactly the same mass. And, if you make reasonable estimates of the distribution of PBH DM around a mean value as one of the links I reference does, it turns out that the exclusion of PBH DM is stronger than it would be if there was a uniform PBH mass, because the heavier than average PBHs would have a stronger signal which makes it easier to detect, than if every PBH were exactly the same mass, because it makes some lower resolution lensing measurements more useful.

The radius of a PBH for a given mass is determined using GR.