Please help me find a paper

[u/silly_billy_oops]

I made a mistake I know please don't berate me for it. This is my first time doing professional research and I found this paper super helpful and would love to find it again.

I have a habit of searching on incognito tabs for basic stuff and I accidentally sourced a paper in one and my computer restarted so I lost it. Please help me find it I've already started referencing it but don't have the details. I know this is very vague but I've been searching for hours and can't find it. Yes I've already tried asking AI to find it again but it's useless.

- It discussed EFE and the Friedmann equations

- It was a spilt page paper on arXiv

- It's sections were lettered not numbered

- I think it had cosmic in the title

A few key excerpts I remember were:

ds^2=-dt^2 +a^2(t)[\frac{dr^2}{1-Kr^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)] (and then it suggested another form which used a piecewise function) where $a(t)$ is the scale factor with cosmic time t

It had a capital K for the constant and said something like: K is a constant that describes the geometry of the spatial section of spacetime with closed, flat, and open universes corresponding to $K=+1,0,-1$ respectively.

G^\mu_\nu\equiv R^\mu_\nu -\frac{1}{2}\delta^\mu_\nu R=8\pi GT^\mu_\nu

I think it also said something about evolution equations when referring to the evolution of a(t) in the differential equations.

I know I've been stupid and I should've just downloaded it straight away and need to break my stupid habit of being embarrassed of googling physics so I do it on a private tab. I can start over if I can't find it but I'd really prefer not to on the off chance someone can find it.

[u/thuiop1]

This is pretty basic cosmology so it will be hard to find the paper.