Is there a theory of "dark mass" similar to but different from existing theories of "dark matter"? I.e. gravity without matter?

I’m curious to know if any mainstream theoretical physicists have theorized or spoken about the possibility of a theory of dark matter that doesn’t involve any actual matter like hypothesized MACHOs or WIMPs.

The thought experiment this leads me to is the classical "bowling ball on a rubber sheet" demonstration of how matter curves spacetime through mass, with a black hole being better represented by e.g. a hook in the bottom of the sheet that is then pulled an infinite distance away.

In essence, my question is, is it possible for there to be similar "dimples" in the fabric that have mass, but without hypothetical particles like MACHOs, WIMPs, or other types of matter? Such as a source of energy that curves spacetime?

I’ve been looking for any examples of such a theory but have been unable to find anything, so I either lack the right terminology for my search, or there’s some reason that such an idea has been considered and discarded without any fanfare.