<h3 id="photon-spheres-and-light-paths-u2m0">Photon spheres and light paths</h3>
<p>The photon sphere of a black hole (BH) is the spherical region of radius 3/2 times the Schwarzschild radius where photons may circle in an (unstable) orbit. Light passing near the photon sphere may be deflected so much as to make a full circle around the BH before escaping. As the light ray come closer, it may make two, or more, circles around the BH before escaping. Thus, in principle you may be able to see the same background object several times, the closer you look to the BH.</p>
<p><a href="https://i.sstatic.net/YDzeI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/YDzeI.png" alt="BHpaths" /></a>
<sup>Light from the background galaxy circles a black hole an increasing number of times, the closer it passes the hole, and we therefore see the same galaxy in several directions (credit: <a href="https://cosmicdawn.dk/news/dawn-student-albert-sneppen-solves-how-the-universe-is-reflected-near-black-holes/" rel="nofollow noreferrer">Peter Laursen</a>).</sup></p>
<p>For each extra circle, the factor that you have to go closer to the photon sphere is <span class="math-container">$e^{2\pi} \sim 500$</span>. In other words, to see the extra image of the background object, you must look <em>very</em> near the photon sphere. While this was known before, the calculations of
<a href="https://ui.adsabs.harvard.edu/abs/2021arXiv210704044S" rel="nofollow noreferrer">Sneppen (2021)</a> gives an exact mathematical and physical intuition of the reason behind. Moreover, the result is valid for a non-spinning BH; in the more realistic case of a spinning BH, Sneppen's result can be generalized to show that the factor decreases quite a lot, all the way down to only a factor of two for maximally-spinning BHs.</p>
<p>The existence of a photon sphere is possible only for a BH or, in exotic cases, for an ultracompact neutron star (<a href="https://ui.adsabs.harvard.edu/abs/1993ApJ...406..590N" rel="nofollow noreferrer">Nemiroff et al. 1993</a>), in which cases it is <em>inside</em> the star, and is probably not useful for photons, but perhaps for neutrinos.</p>
<h3 id="dark-matter-black-holes-hq6b">Dark matter black holes?</h3>
<p>Creating a black hole out of dark matter is difficult, but in principle not forbidden. The problem is that dark matter is collisionless, so clumping together sufficiently to create a BH would require some sort of direct collapse. However, even <em>if</em> you somehow managed to do this, you wouldn't know whether the BH was created out of dark or baryonic matter, since all information about the former state of the matter is lost (except mass, charge, and angular momentum). &quot;Normal&quot; dark matter halos, even the densest ones, are much too dilute to be even close to having a photon sphere.</p>
<p>So the answer to your question is &quot;No&quot;.</p>
<h3 id="generalizing-the-solution-to-large-radii-k17p">Generalizing the solution to large radii</h3>
<p>Fig. 4 in the paper shows the relationship between deflection angle <span class="math-container">$\phi$</span> and distance <span class="math-container">$r$</span> from the photon sphere:</p>
<p><a href="https://i.sstatic.net/ST7mX.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ST7mX.png" alt="phi" /></a></p>
<p>Sneppen's solution (dashed orange) is a linearized approximation which is only valid when you're closer to the photon sphere than roughly 0.3–1 Schwarzschild radii (where it approaches the numerical solution; blue solid). Hence it is neither applicable, nor generalizable (I think), to dark matter halos which are some million times larger than their Schwarzschild radius.</p>
<p>Instead, in this regime you'll be better off using the regular Schwarzschild metric for calculating the deflection angle of a light beam.</p>

CC BY-SA 4.0
2021-07-27T11:56:16.760Z
Can we say that every black hole contains at least traces (as it would be spoken in chemistry) of dark matter? I mean, while DM is certainly extremely diluted, there should be some falling in a BH. Is this sensible?
Alchimista
CC BY-SA 4.0
2021-07-27T13:46:28.817Z
@Alchimista Yes, since dark matter is ubiquitous, black holes will also attract that. But black holes form in heavily baryon-dominated regions, so the dark matter fraction will be correspondingly small. Plus, as I write, once matter crosses the horizon, it no longer really makes sense to distinguish between baryonic and dark matter contents.
pela
CC BY-SA 4.0
2021-07-27T13:52:38.347Z
@Alex Gravitational lensing is caused by _matter_, dark or baryonic. Clusters and galaxies are dominated by dark matter, but you can also have lensing caused by stars or even planets, which are baryonic. Lensing wasn't the original reason for introducing the concept, though — rather it was kinematic discrepancies between observed velocities of galaxies/stars and velocities expected from luminous matter. But Sneppen's solution has to do with orbits _very_ close to the photon sphere (and hence quite close to the Schwarzschild radius); this doesn't really have anything to do with…
pela
CC BY-SA 4.0
2021-07-27T13:54:51.263Z
…dark matter halos, which are many orders of magnitude larger than their Schwarzschild radii. For instance, if you have a galaxy cluster of $M=10^{14}\,M_\odot$, its Schwarzschild radius is ~10 pc, but lensing happens on scales of _millions_ of parsec. In other words, you'd have to compress a whole galaxy cluster down to a size of ~10 pc, in order for Sneppen's solution to be relevant.
pela
CC BY-SA 4.0
2021-07-27T13:59:54.337Z
Could the phenomenons of gravitational lensing (caused by dark matter) and photon sphere be both described within the framework of the GENERALIZED (over work of Albert Sneppen) mathematical apparatus?
Alex
CC BY-SA 4.0
2021-07-27T14:04:03.163Z
@Alex No, because Sneppen calculates the deflection angle _very_ close to a point mass. Gravitational lensing of (extended) dark matter halos happens on much, much larger scales. In order for light to circle a point mass just _once_, you need to be roughly 0.1 Schwarzschild radii from the photon sphere.
pela
CC BY-SA 4.0
2021-07-27T14:08:36.787Z
@Alex Okay, you edited your question to concern a "generalized apparatus". I can't really answer that, I suppose it depends on your definition of "generalized". I mean, if you have a general solution for all distances from the Schwarzschild radius to infinity, then I guess so…
pela
CC BY-SA 4.0
2021-07-27T14:16:22.607Z
Yes I exactly meant to generalize for all distances - thanks!!!
Alex
CC BY-SA 4.0
2021-07-27T15:36:32.873Z
@Alex You're welcome, but thinking more about it, I don't think it makes sense to generalize the result to larger distances. Or rather, it is already well-known: Sneppen's result tells you the deflection angle as a function of distance from the BH, and specifically the distance you have to go closer to the BH to see light circling around the BH an increasing number of times. Going farther away than needed to make light circle _once_ just means that you have the deflection already known from the standard Schwarzschild metric.
pela
CC BY-SA 4.0
2021-07-27T16:12:11.577Z
I think that generalizations are always good to make. They could cover situations, which are currently not in the trend to consider but may become of interest in the future.
Alex
CC BY-SA 4.0
2021-07-27T19:34:51.777Z
@Alex I agree, but Sneppen's solution isn't the basis for such a generalization. If you look at Fig. 4 you'll see that his (linearized) solution only approaches the numerical solution once the distance to the point mass is $(r-3/2)/r_\mathrm{s} \lesssim 0.3$.
pela
CC BY-SA 4.0
2021-07-31T01:20:25.897Z
fyi I've reused the illustration in [How well conserved is etendue in extreme gravitational lensing scenarios?](https://astronomy.stackexchange.com/q/45056/7982)
uhoh
CC BY-SA 4.0
2021-08-10T12:27:30.010Z
Yes, thanks very much.
Alex
CC BY-SA 4.0
2021-08-10T12:32:10.497Z
Would you look into https://physics.stackexchange.com/questions/657861/is-below-description-of-the-difference-between-laws-and-principles-of-physics-co may be you will vote to reopen and answer it...
Alex
CC BY-SA 4.0
2021-08-10T13:28:30.323Z
I voted to open it, but (without going into too much detail) I think I agree with most of the comments/answer, that law and principle are used somewhat interchangeably. At least in physics, I think, while in the field of mathematics there may be a more rigorous definition. To me (an astrophysicist), "law" _usually_ denotes the mathematical description of a principle, or relation, or theory. But not always… I don't think I'm comfortable writing a sensible answer here, sorry…
pela
CC BY-SA 4.0
2021-08-18T17:15:34.310Z
Shure, this is very detailed and complete answer - thanks!
Alex
CC BY-SA 4.0
2021-09-29T12:53:58.880Z
@Alex Would you mind accepting it then? Also for future readers to see that you, as the OP, regards this an answering your question? :)
pela
CC BY-SA 4.0
2021-09-29T15:13:15.440Z
I don't see the formal option for accepting answer on my Android mobile phone - sorry!
Alex
CC BY-SA 4.0
2021-09-29T19:35:03.607Z
@Alex Hmm, that's odd, but no worries :)
pela
CC BY-SA 4.0
2021-07-27T07:36:54.213Z
2021-07-27T19:48:07.167Z
2021-07-27T19:48:07.167Z
pela
<p>Gravitational lensing <em>has</em> been used to look for and rule out massive compact objects (MACHOS) as the source of dark matter in our galaxy. The headline result is that MACHOS in the range <span class="math-container">$6\times 10^{-8} &lt; M/M_\odot &lt;15$</span> were ruled out (<a href="https://arxiv.org/abs/astro-ph/0607207" rel="nofollow noreferrer">Tisserand et al. 2008</a>).</p>
<p>I don't see any plausible reason why the dark matter responsible (in aggregate) for the large scale gravitational lensing in galaxy clusters should be any different. The extreme lensing effects caused by black holes only occur for light travelling very close to the &quot;photon sphere&quot;, and thus affects only a tiny fraction of light traversing a massive cluster of galaxies.</p>
<p>Secondly, whilst the dark matter could be <em>primordial</em> black holes (with masses outside the range quoted above), it cannot be in the form of neutron stars, white dwarfs, or black holes formed as a result of stellar evolution. This is &quot;baryonic&quot; dark matter and about 80% of the dark matter needs to be in a non-baryonic form.</p>

CC BY-SA 4.0
2021-07-27T16:17:50.810Z
2021-07-27T16:17:50.810Z
ProfRob
<p><a href="https://www.nature.com/articles/s41598-021-93595-w" rel="nofollow noreferrer">Albert Sneppen's publication
&quot;Divergent Reflections around the Photon Sphere of a Black Hole&quot;</a> provides the math related to the issue which in common language could be described as &quot;how the universe is reflected near black holes&quot;.</p>
<p>Black holes are famously compact objects where the paths of light are drastically bent by the curvature of space-time. While light itself cannot escape the central mass at the event horizon, at further distances light may orbit the black hole. This phenomenon may enable a distant observer to see multiple versions of the same object. While this has been known for years, only now do theoretical physicists have an exact mathematical solution developed by Albert Sneppen.</p>
<p>Here is the link to the announcement re the first experimental observation of this phenomenon:
<a href="https://www.esa.int/ESA_Multimedia/Images/2021/07/XMM-Newton_sees_light_echo_from_behind_a_black_hole" rel="nofollow noreferrer">https://www.esa.int/ESA_Multimedia/Images/2021/07/XMM-Newton_sees_light_echo_from_behind_a_black_hole</a></p>
<p>I asked Albert Sneppen the following question: &quot;Could the math you have developed be applied to the analysis of gravitational lensing caused by the dark matter if it exists in Universe in the form of extreme dense clumps which could be considered as 'photon spheres'?&quot;</p>
<p>Here is the answer I have received:
&quot;The short answer is yes. The evaluating near the photon-sphere is generally valid. You would not need a black hole, any sufficiently dense object, such as a neutron star or clumped dark matter might have the same mathematical prescription.&quot;</p>
<p>So my astronomy stack exchange question is: &quot;could this effect help to find out via optical observations whether dark matter exists in the Universe in the form of sufficiently dense objects?&quot;</p>

CC BY-SA 4.0
2021-07-25T18:23:49.500Z
No, because this is describing effects on the scale of the photon sphere, which is much too small to be resolved for any but the nearest supermassive black holes.
Peter Erwin
CC BY-SA 4.0
2021-07-25T18:25:56.940Z
Also, we've already ruled out black holes and other compact objects as DM candidates for almost all possible mass values. See, e.g., https://physics.stackexchange.com/questions/403025/could-dark-matter-be-made-of-black-holes and references therein.
Peter Erwin
CC BY-SA 4.0
2021-07-25T18:49:41.413Z
@Peter Erwin What data confirms your statement that there are absolutely no in the Universe sufficiently large and sufficiently dense objects (other than black holes, which were ruled out) comprised of the dark matter ?
Alex
CC BY-SA 4.0
2021-07-25T19:32:57.753Z
There is strong evidence for non baryonic dark matter as the gravitational mass density exceeds the baryon density. Dark matter also helps grow large scale structure. With these observations, it wouldn't make sense to have any baryonic matter be a dark matter candidate, or at least the dominant form of it. Also, lensing observations have ruled out MACHOS.
Astroturf
CC BY-SA 4.0
2021-07-25T20:45:03.643Z
@Alex By "comprised of the dark matter" do you mean *made up of dark matter* or "which *are* the dark matter"?
Peter Erwin
CC BY-SA 4.0
2021-07-25T22:08:21.027Z
@Peter Erwin I think that at the current state of knowledge (with regards to details of how dark matter is structured and how dark matter causes gravitational lensing) the difference between two suggested by you meanings is not definable  - if I am wrong please tell me your understanding how this difference of meanings could matter.
Alex
CC BY-SA 4.0
2021-07-25T22:27:19.007Z
@Alex The "MACHO" hypothesis was that dark matter was made up of compact, massive objects like neutron stars, faded white dwarfs, brown dwarfs, and black holes. (I.e., they *are* the dark matter.) This turned out not to work, because all those objects (except *maybe* the BHs) would be composed of baryons -- and the universe has about five times as much mass in dark matter as in all the baryons formed in the Big Bang.
Peter Erwin
CC BY-SA 4.0
2021-07-26T14:10:21.093Z
@Peter Erwin The hypothesis that the dark matter's elementary particles could be either WIMPs or Axions at the current state of knowledge is merely just that - THE HYPOTHESIS...                                                      
"There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy."
Alex
CC BY-SA 4.0
2021-07-26T19:24:10.880Z
The gold standard of science is to have predictive power. Second best is to have a hypothesis that explains current observations. It doesn't seem photon spheres do either.
Astroturf
CC BY-SA 4.0
2021-07-27T08:30:12.383Z
See https://astronomy.stackexchange.com/questions/24139/how-is-dark-matter-different-from-ordinary-visible-matter https://astronomy.stackexchange.com/questions/44713/can-dark-matter-be-normal-matter-after-all https://astronomy.stackexchange.com/questions/10384/why-arent-brown-dwarfs-the-dark-matter
ProfRob
<dark-matter>
Could dark matter exist in the Universe in the form of sufficiently dense objects?
Alex
# Could dark matter exist in the Universe in the form of sufficiently dense objects?

(Asked by: username_0 on 2021-07-25)

[Albert Sneppen's publication
"Divergent Reflections around the Photon Sphere of a Black Hole"](https://www.nature.com/articles/s41598-021-93595-w) provides the math related to the issue which in common language could be described as "how the universe is reflected near black holes".

Black holes are famously compact objects where the paths of light are drastically bent by the curvature of space-time. While light itself cannot escape the central mass at the event horizon, at further distances light may orbit the black hole. This phenomenon may enable a distant observer to see multiple versions of the same object. While this has been known for years, only now do theoretical physicists have an exact mathematical solution developed by Albert Sneppen.

Here is the link to the announcement re the first experimental observation of this phenomenon:
<https://www.esa.int/ESA_Multimedia/Images/2021/07/XMM-Newton_sees_light_echo_from_behind_a_black_hole>

I asked Albert Sneppen the following question: "Could the math you have developed be applied to the analysis of gravitational lensing caused by the dark matter if it exists in Universe in the form of extreme dense clumps which could be considered as 'photon spheres'?"

Here is the answer I have received:
"The short answer is yes. The evaluating near the photon-sphere is generally valid. You would not need a black hole, any sufficiently dense object, such as a neutron star or clumped dark matter might have the same mathematical prescription."

So my astronomy stack exchange question is: "could this effect help to find out via optical observations whether dark matter exists in the Universe in the form of sufficiently dense objects?"

**Tags:** dark-matter
> **Comments:**
> **username_1**: No, because this is describing effects on the scale of the photon sphere, which is much too small to be resolved for any but the nearest supermassive black holes.  
> **username_1**: Also, we've already ruled out black holes and other compact objects as DM candidates for almost all possible mass values. See, e.g., https://physics.stackexchange.com/questions/403025/could-dark-matter-be-made-of-black-holes and references therein.  
> **username_0**: @Peter Erwin What data confirms your statement that there are absolutely no in the Universe sufficiently large and sufficiently dense objects (other than black holes, which were ruled out) comprised of the dark matter ?  
> **username_2**: There is strong evidence for non baryonic dark matter as the gravitational mass density exceeds the baryon density. Dark matter also helps grow large scale structure. With these observations, it wouldn't make sense to have any baryonic matter be a dark matter candidate, or at least the dominant form of it. Also, lensing observations have ruled out MACHOS.  
> **username_1**: @Alex By "comprised of the dark matter" do you mean *made up of dark matter* or "which *are* the dark matter"?  
> **username_0**: @Peter Erwin I think that at the current state of knowledge (with regards to details of how dark matter is structured and how dark matter causes gravitational lensing) the difference between two suggested by you meanings is not definable  - if I am wrong please tell me your understanding how this difference of meanings could matter.  
> **username_1**: @Alex The "MACHO" hypothesis was that dark matter was made up of compact, massive objects like neutron stars, faded white dwarfs, brown dwarfs, and black holes. (I.e., they *are* the dark matter.) This turned out not to work, because all those objects (except *maybe* the BHs) would be composed of baryons -- and the universe has about five times as much mass in dark matter as in all the baryons formed in the Big Bang.  
> **username_0**: @Peter Erwin The hypothesis that the dark matter's elementary particles could be either WIMPs or Axions at the current state of knowledge is merely just that - THE HYPOTHESIS...                                                       "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy."  
> **username_2**: The gold standard of science is to have predictive power. Second best is to have a hypothesis that explains current observations. It doesn't seem photon spheres do either.  
> **username_3**: See https://astronomy.stackexchange.com/questions/24139/how-is-dark-matter-different-from-ordinary-visible-matter https://astronomy.stackexchange.com/questions/44713/can-dark-matter-be-normal-matter-after-all https://astronomy.stackexchange.com/questions/10384/why-arent-brown-dwarfs-the-dark-matter  

---

## 2 answers

### Answer by username_4 on 2021-07-27. Score: 7

### Photon spheres and light paths

The photon sphere of a black hole (BH) is the spherical region of radius 3/2 times the Schwarzschild radius where photons may circle in an (unstable) orbit. Light passing near the photon sphere may be deflected so much as to make a full circle around the BH before escaping. As the light ray come closer, it may make two, or more, circles around the BH before escaping. Thus, in principle you may be able to see the same background object several times, the closer you look to the BH.

[![BHpaths](https://i.sstatic.net/YDzeI.png)](https://i.sstatic.net/YDzeI.png)
Light from the background galaxy circles a black hole an increasing number of times, the closer it passes the hole, and we therefore see the same galaxy in several directions (credit: [Peter Laursen](https://cosmicdawn.dk/news/dawn-student-albert-sneppen-solves-how-the-universe-is-reflected-near-black-holes/)).

For each extra circle, the factor that you have to go closer to the photon sphere is $e^{2\pi} \sim 500$. In other words, to see the extra image of the background object, you must look *very* near the photon sphere. While this was known before, the calculations of
[Sneppen (2021)](https://ui.adsabs.harvard.edu/abs/2021arXiv210704044S) gives an exact mathematical and physical intuition of the reason behind. Moreover, the result is valid for a non-spinning BH; in the more realistic case of a spinning BH, Sneppen's result can be generalized to show that the factor decreases quite a lot, all the way down to only a factor of two for maximally-spinning BHs.

The existence of a photon sphere is possible only for a BH or, in exotic cases, for an ultracompact neutron star ([Nemiroff et al. 1993](https://ui.adsabs.harvard.edu/abs/1993ApJ...406..590N)), in which cases it is *inside* the star, and is probably not useful for photons, but perhaps for neutrinos.

### Dark matter black holes?

Creating a black hole out of dark matter is difficult, but in principle not forbidden. The problem is that dark matter is collisionless, so clumping together sufficiently to create a BH would require some sort of direct collapse. However, even *if* you somehow managed to do this, you wouldn't know whether the BH was created out of dark or baryonic matter, since all information about the former state of the matter is lost (except mass, charge, and angular momentum). "Normal" dark matter halos, even the densest ones, are much too dilute to be even close to having a photon sphere.

So the answer to your question is "No".

### Generalizing the solution to large radii

Fig. 4 in the paper shows the relationship between deflection angle $\phi$ and distance $r$ from the photon sphere:

[![phi](https://i.sstatic.net/ST7mX.png)](https://i.sstatic.net/ST7mX.png)

Sneppen's solution (dashed orange) is a linearized approximation which is only valid when you're closer to the photon sphere than roughly 0.3–1 Schwarzschild radii (where it approaches the numerical solution; blue solid). Hence it is neither applicable, nor generalizable (I think), to dark matter halos which are some million times larger than their Schwarzschild radius.

Instead, in this regime you'll be better off using the regular Schwarzschild metric for calculating the deflection angle of a light beam.

> **Comments:**
> **username_5**: Can we say that every black hole contains at least traces (as it would be spoken in chemistry) of dark matter? I mean, while DM is certainly extremely diluted, there should be some falling in a BH. Is this sensible?  
> **username_4**: @Alchimista Yes, since dark matter is ubiquitous, black holes will also attract that. But black holes form in heavily baryon-dominated regions, so the dark matter fraction will be correspondingly small. Plus, as I write, once matter crosses the horizon, it no longer really makes sense to distinguish between baryonic and dark matter contents.  
> **username_4**: @Alex Gravitational lensing is caused by _matter_, dark or baryonic. Clusters and galaxies are dominated by dark matter, but you can also have lensing caused by stars or even planets, which are baryonic. Lensing wasn't the original reason for introducing the concept, though — rather it was kinematic discrepancies between observed velocities of galaxies/stars and velocities expected from luminous matter. But Sneppen's solution has to do with orbits _very_ close to the photon sphere (and hence quite close to the Schwarzschild radius); this doesn't really have anything to do with…  
> **username_4**: …dark matter halos, which are many orders of magnitude larger than their Schwarzschild radii. For instance, if you have a galaxy cluster of $M=10^{14}\,M_\odot$, its Schwarzschild radius is ~10 pc, but lensing happens on scales of _millions_ of parsec. In other words, you'd have to compress a whole galaxy cluster down to a size of ~10 pc, in order for Sneppen's solution to be relevant.  
> **username_0**: Could the phenomenons of gravitational lensing (caused by dark matter) and photon sphere be both described within the framework of the GENERALIZED (over work of Albert Sneppen) mathematical apparatus?  
> **username_4**: @Alex No, because Sneppen calculates the deflection angle _very_ close to a point mass. Gravitational lensing of (extended) dark matter halos happens on much, much larger scales. In order for light to circle a point mass just _once_, you need to be roughly 0.1 Schwarzschild radii from the photon sphere.  
> **username_4**: @Alex Okay, you edited your question to concern a "generalized apparatus". I can't really answer that, I suppose it depends on your definition of "generalized". I mean, if you have a general solution for all distances from the Schwarzschild radius to infinity, then I guess so…  
> **username_0**: Yes I exactly meant to generalize for all distances - thanks!!!  
> **username_4**: @Alex You're welcome, but thinking more about it, I don't think it makes sense to generalize the result to larger distances. Or rather, it is already well-known: Sneppen's result tells you the deflection angle as a function of distance from the BH, and specifically the distance you have to go closer to the BH to see light circling around the BH an increasing number of times. Going farther away than needed to make light circle _once_ just means that you have the deflection already known from the standard Schwarzschild metric.  
> **username_0**: I think that generalizations are always good to make. They could cover situations, which are currently not in the trend to consider but may become of interest in the future.  
> **username_4**: @Alex I agree, but Sneppen's solution isn't the basis for such a generalization. If you look at Fig. 4 you'll see that his (linearized) solution only approaches the numerical solution once the distance to the point mass is $(r-3/2)/r_\mathrm{s} \lesssim 0.3$.  
> **username_6**: fyi I've reused the illustration in [How well conserved is etendue in extreme gravitational lensing scenarios?](https://astronomy.stackexchange.com/q/45056/7982)  
> **username_0**: Yes, thanks very much.  
> **username_0**: Would you look into https://physics.stackexchange.com/questions/657861/is-below-description-of-the-difference-between-laws-and-principles-of-physics-co may be you will vote to reopen and answer it...  
> **username_4**: I voted to open it, but (without going into too much detail) I think I agree with most of the comments/answer, that law and principle are used somewhat interchangeably. At least in physics, I think, while in the field of mathematics there may be a more rigorous definition. To me (an astrophysicist), "law" _usually_ denotes the mathematical description of a principle, or relation, or theory. But not always… I don't think I'm comfortable writing a sensible answer here, sorry…  
> **username_0**: Shure, this is very detailed and complete answer - thanks!  
> **username_4**: @Alex Would you mind accepting it then? Also for future readers to see that you, as the OP, regards this an answering your question? :)  
> **username_0**: I don't see the formal option for accepting answer on my Android mobile phone - sorry!  
> **username_4**: @Alex Hmm, that's odd, but no worries :)  

---

### Answer by username_3 on 2021-07-27. Score: 4

Gravitational lensing *has* been used to look for and rule out massive compact objects (MACHOS) as the source of dark matter in our galaxy. The headline result is that MACHOS in the range $6\times 10^{-8} < M/M\_\odot <15$ were ruled out ([Tisserand et al. 2008](https://arxiv.org/abs/astro-ph/0607207)).

I don't see any plausible reason why the dark matter responsible (in aggregate) for the large scale gravitational lensing in galaxy clusters should be any different. The extreme lensing effects caused by black holes only occur for light travelling very close to the "photon sphere", and thus affects only a tiny fraction of light traversing a massive cluster of galaxies.

Secondly, whilst the dark matter could be *primordial* black holes (with masses outside the range quoted above), it cannot be in the form of neutron stars, white dwarfs, or black holes formed as a result of stellar evolution. This is "baryonic" dark matter and about 80% of the dark matter needs to be in a non-baryonic form.


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