MACHOs

In astronomy, MACHOs is an acronym for "massive compact halo objects", and is the only dark matter candidate that does not require "new physics" either in the form of a new kind of fundamental particle beyond the Standard Model, or a modification of the laws of gravity.

But, a variety of recent papers on the topic acknowledge that MACHOs with masses of more than the mass of our Sun, divided by 10,000,000, which is roughly half the mass of the planet Earth (i.e. a mass of not more than 2*10^23 kilograms, since the Sun has a mass of about 2*10^30 kg), are ruled out.  Other papers suggest that MACHOs that low in mass must make up significantly less than 4% of all dark matter, and if all of those conclusions are correct, dark matter is not primarily made up of MACHOs.

So, what kinds of objects could fit this definition?

Terrestrial Planets

A type of MACHO that accounted for most dark matter couldn't be terrestrial planets like Earth or Mars.

This is because terrestrial planets have a high proportion of relatively heavy elements (i.e. elements other than the hydrogen and helium that make up the bulk of most stars such as the silicon (element 14) and oxygen (element 8) that make up most of the rock in the planet Earth), because nucleosynthesis theory, which has accurately predicted the relative abundance of the chemical elements that we observe in the universe, implies that the heavy elements found in terrestrial planets have to make up only a tiny percentage of the total amount of ordinary matter in the universe.  

Even the largest planets are "gas giants" with large proportions are hydrogen, helium, methane, and the like, as opposed to solid rock which has substantial amounts of silicon, oxygen, iron, and other metals.

Baby Gas Giants

One possibility would be a "baby gas giant", with about 3-7% of the mass of the planet Uranus, that isn't even close to having enough mass to ignite the nuclear reactions that could produce a star or a brown dwarf.  But, it would have to be much larger than Earth in volume, if it was made mostly of hydrogen and helium and a smattering of other light elements.  Also, if the galaxy were teaming with these such that the lion's share of the Milky Way's mass was made of them, one would think we would have been able to observe more of them, which we have not. 

Still, because these objects don't require any "new physics" ruling them out as a possibility is particularly important.

The main disqualifier for these objects might be that their properties would probably caused them to have too strong non-gravitational interactions with other kinds of matter.

Primordial Black Holes

Another popular candidate is a "primordial black hole." It earns this name because an ordinary black hole is formed by the collapse of a star and needs to have a critical mass about three times the mass of the Sun to become a black hole. And, even if all it sucks up is cosmic background radiation and the odd bit of interstellar gas and dust, black holes of this size never emit so much Hawking radiation that they get smaller rather than getting bigger.

But, if some other unknown process that can only take place in the high energy, ultra-dense environment of the early moments after the Big Bang (hence the "primordial" designation) could overcome the three solar mass threshold to squish a smaller amount of mass into a small enough space to form a black hole, then smaller black holes might be possible and might last quite a long time, and might also slowly get smaller, losing more mass to Hawking radiation than it gains by swallowing up matter and energy that it encounters. As a recent article by a astrophysicist and physics blogger in Forbes magazine explains:

The idea is pretty simple: we know the Universe started off from a hot, dense, rapidly expanding and roughly uniform state. Wherever you were located, gravitation would try to pull nearby masses towards you, while the radiation pressure from photons would try to push those masses back apart. But if on small scales, you had regions of space that were just 68% (or more) denser than average, that radiation pressure wouldn’t matter. Instead, gravitational collapse all the way to a black hole would be inevitable. If this happened at one particular mass scale in the Universe — say at 1 kilogram masses, or 10^10 kilogram masses, or even 30 solar masses — you’d wind up with a large number of primordial black holes of that particular mass. They’d be strewn roughly evenly throughout the Universe, they’d form large, diffuse-but-clumpy halos around galaxies, and they’d be an excellent candidate for the dark matter.

I've emphasized one key phrase in this analysis, because while the other hypothetical conditions for creating primordial black holes are very plausible, this emphasized one is the weak link.

But, just for curiosity's stake, how big would a black hole with the mass of the Earth be?

The event horizon would have a radius of about 8.4 millimeters!

Certainly, unlike "baby gas giants," we could easily fail to see primordial black holes of this size with telescopes, even if there were a great many of them, and unlike the neutron star type objects discussed below, a primordial black hole would be at least very long lived and meta-stable, even if they were not actually fully stable over any possible time frame no matter how long.

Of course, there is no known process that can actually create primordial black holes, but some sort of "new physics" which could make their formation possible in the high energy environment of the early universe, is really not particularly less plausible than the "new physics" necessary to give rise to some sort of new beyond the Standard Model fundamental particle, or a modification of the general relativity that would give rise to dark matter phenomena.

Now, 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 even less than 8.4 millimeters.

Notably, medium sized black holes of the type observed merging by LIGO last year (in results announced earlier this year), while notable because they represent black holes bigger than those created by a single stellar collapse, but much smaller than the supermassive black holes at the centers of galaxies, which we had almost never observed before LIGO, can be ruled out as sources for more than about 0.1% of all dark matter by observations of the cosmic background radiation of the universe.

In the chart above, even a 10^-1 proportion would account for only 10% of dark matter, which would still require that the lion's share of dark matter be of some other non-MACHO type. 

And, the narrow mass range of the "sweet spot" would also require that the process which gave rise to the primordial black holes that make up dark matter only took place in a single, very narrow mass range. This is a real problem because none of the processes commonly proposed to give rise to primordial black holes in the universe can naturally and easily explain why this particular sweet spot (actually quite a bit heavier, because a primordial black hole of this size would have shrunk slightly over the last 13.6 billion years due to Hawking radiation in an amount that could be determined quite precisely), and not also other very different masses, would give rise to primordial black holes.

It probably isn't impossible to come up with a theory that could explain why this would happen, particularly because there is no definitive theory of quantum gravity or extremely high energy physics in place to place firm boundaries on the physics at these energy scales which are far beyond anything we have ever observed in a laboratory or a telescope or ever will at any time in the future. But, it would be quite a trick to make this result fall plausibly and naturally out of the theory.

It is also worth observing that the reason that there is a "sweet spot" mass for primordial black holes has almost nothing to do with any fundamental physics related to that particular mass range. It is simply a function of the fact that this particular mass range is the hardest one for us to observe with early 21st century technology. It is driven primarily by instrumentation and engineering considerations and not by anything fundamental about the physics.

Assuming that there is likely to be primordial black hole dark matter in the mass range simply because it is the only mass range left that hasn't been ruled out is basically "God of the gaps" thinking. There is really no fundamental reason, other than mere coincidence, for that mass range to be special.

Of course, up close, direct detection of a primordial black hole would be easy. An Earth strength gravitational field would deflect the path of objects in its vicinity despite the fact that there is no visible matter in that very easy to localize vicinity. And, since they purportedly make up so much of the matter in the galaxy, we can calculate to considerable precision how many of them ought to be within any particular distance of Earth, which would probably be closer than the nearest star. So, if this is the source of dark matter and we continue to explore our little corner of the galaxy with deep space probes, sooner or later this (or micro-lensing) should reveal candidates for this, and once you found a candidate, sending a probe to confirm its character as a primordial black hole wouldn't be beyond our engineering abilities.

Overall, primordial black hole dark matter is a long shot, but it is probably the most plausible of the MACHO candidates.

Neutron Stars

An object with the hypothetical density of a neutron star would have a radius on the order of about 3.2 meters. But, neutron stars in actuality have masses of about 1.4 to 3 solar masses.  At the low end, this is about 10,000,000 times heavier than the maximum empirically observed threshold mass for MACHOs that account for all dark matter.

The gravity necessary for neutron stars to have their extreme density demands that they have large masses.  The math just doesn't work out for neutron stars this small to be stable objects.

Cannibalized White Dwarfs

An object with the hypothetical density of a white dwarf would have a radius on the order of about 320 meters.  White dwarfs in actuality have masses from just below the minimum mass for a neutron star to as little as about 0.17 times the mass of the Sun.  Typically they have 0.5 to 0.7 times the mass of the Sun and a volume from slightly smaller than the planet Earth to about twice the volume of the planet Earth.  Again, the lower boundary on a white dwarf's mass is about 1,000,000 times heavier than the maximum empirically observed threshold mass for MACHOs that account for all dark matter.

A white dwarf can't form from the collapse of a star much smaller than half of the mass of the sun, because that is the minimum size for gravity to cause the nuclear fusion, that defines a body to be a star, to take place.  White dwarfs of less than this mass are believed to arise from the collapse of a smallish star followed by loss of mass in a binary system to another star or black hole or perhaps even to a large planetary companion. 

In theory, if a white dwarf loses so much mass that it becomes a planetary mass object small enough to fall below the Earth mass MACHO threshold, the resulting object would be a "helium planet" or perhaps even a "diamond planet."

But, while such planet sized remnants of white dwarfs could meet the size threshold for MACHO theories, there is simply no plausible mechanism by which there could be so many of them that almost 90% of the mass in the universe is made of these objects.