General Relativity "doesn't quite work" without dark matter?

Question

I am taking an astrophysics course, and my astro professor said that "we need to introduce dark matter because Einstein's general relativity doesn't quite work without it".

I wanted to clarify/confirm with you experts 2 points based on my previous study of GR:

1. It's not that GR doesn't work, it is that dark matter is needed when using a $k = 0$ FLRW model of the universe. So, it's the FLRW assumption that is problematic.

2. Other, inhomogeneous models like Lemaitre-Tolman-Bondi or Swiss-cheese models are not well tested on cosmological scales.

Would love to hear thoughts on the above 2 points.

Answer 1

You're right about 1, sort of. GR works if you assume dark matter exists. Phrased alternatively, if you assume GR is correct, then you need dark matter. However it's worth pointing out that evidence for dark matter comes from more angles than just cosmology - so even if the FLRW universe is problematic, it does not mean there's no dark matter.

About 2, the key question as with all science is whether the models match observations. Since you're asking about inhomogeneous models, the question becomes, is the observable universe homogeneous & isotropic (which are the key assumptions that go into FLRW)? The answer is yes to a high degree. The canonical example for isotropy is the CMB, which is very nearly completely isotropic; see also this paper. Homogeneity can also be tested, and the universe appears to be highly homogeneous (example). This is why we believe in FLRW.

However, it's also worth pointing out that the current Standard Model of Cosmology is showing cracks, especially the Hubble tension but also a few other observations (like S8 tension). There are a myriad possibilities to resolve these problems, and one of them is to give up on the homogeneous & isotropic FLRW universe. Certainly people are thinking about it (example). The ending of that story is yet to be written.

Answer 2

I very much agree with your first point.

Many people don’t understand general relativity, and this is very prominent unfortunately in the astrophysics community, that mostly always sees things from a Newtonian perspective.

When people talk about GR working/not working, they mean solutions of the field equations that are then tested against observations. Einstein’s equations themselves simply relate curvature of spacetime to the energy-momentum tensor.

The current standard model of cosmology relies on spatial homogeneity and isotropy, which yields solutions of the EFE that have 6 Killing vectors/a six-dimensional isometry group. In addition, they also consider spatially flat models. To make “these” models fit with observations, they then introduce dark matter/energy.

I don’t believe a high degree of isotropy indicates isotropy. Something is either isotopic or it isn’t. Further, even from an astrophysical perspective, it is clear that the existence of black holes in the universe implies that the universe is inhomogeneous. There was a lot of work done on the Swiss Cheese models that were shown to challenge the dark energy assumptions. This also leads to the important matching / boundary conditions of how to join a S-metric universe with FLRW.

Answer 3

I think the point that your professor is getting at is not at all concerning the validity of the general theory but touches instead on the fact that neglecting to include "dark matter" into one's standard application of GR to the problem at hand produces less satisfactory results.

Essentially, researchers found that when they included greater amounts of matter into the stress-energy tensor, the results of theory better matched the observations. The point to take away from the professor's comment is that dark matter is needed within the paradigm of standard general relativity.

P.S. It is worth noting that the idea of a flat universe is not very consistent with a universe that contains any kind of matter, dark or otherwise, as matter and energy content are responsible for inducing curvature into the theory. Technically, a flat cosmological model is an absolute vacuum.

Answer 4

Dark matter is more about the galactic rotation curves (so more a local issue). Whereas at a cosmological level its "dark energy", aka the cosmological constant, that is also the one that enters as a free parameter in GR models derived from Einstein equations under simplifying assumptions. Please correct me if I m wrong.