Introduction

In our universe, the cosmic microwave background was formed approximately 400,000 years after the Big Bang. It was hot, but within a few million years after the Big Bang, it would no longer have consisted significantly of visible light. The first stars formed about 100 million years later, give or take, as larger structures were slowly beginning to form.

In my universe, I'd like to see if I can create a period of overlap, where the first stars form while the CMB is still hot enough to be visible to the naked human eye, and exists at wavelengths suitable for chlorophyll-based photosynthesis. I've found by playing around with the Saha equation that I can keep the CMB hot and visible for a few million years more, but only if I increase the baryon density by a lot.

Therefore, I want to see if I can change my universe's parameters to instead accelerate star formation by a factor of 100 or so. I'm not going to change most fundamental constants like the speed of light; that tends to cause issues later on. The parameters I'm willing to change are the various density parameters for photons, baryonic matter, dark matter, and dark energy: $\Omega_{\gamma}$, $\Omega_M$, $\Omega_D$, and $\Omega_{\Lambda}$. These evolve over time; today, $\Omega_{\Lambda,0}=0.692$, $\Omega_{D,0}=0.258$, $\Omega_{M,0}=0.048$ and $\Omega_{\gamma,0}\approx0$. As the first structures were forming, however, the universe would have been matter-dominated (that is to say, $\Omega_M,\Omega_D\gg\Omega_{\gamma},\Omega_{\Lambda}$).

Structure formation and star formation

Given what I know about star formation in the early universe (see e.g. 1 2, for more information), I think we can break the process down into a couple key stages:

1. Small density fluctuations grow as gravitational instabilities cause perturbations to collapse. These form small dark matter halos rich in primordial gas.
2. This gas cools mainly after molecular hydrogen forms because much of the gas should exist at temperatures less than $\sim10^4\text{ K}$ - the threshold where atomic cooling is important.
3. When clumps in a gas cloud are massive enough (i.e. reach the Jeans mass), they can collapse to form stars, just as they do today.

If I could affect any of the three stages - halo collapse, cooling, or protostellar collapse - I might be able to achieve what I want. The problem is, I don't know how changing my parameters would affect the relevant timescales - if at all.

Existing work

I've done a basic literature search on theoretical work on early structure formation. Much of the existing results are based on numerical simulations (e.g. Abel et al. 2000, Bromm et al. 1999). They assume a universe dominated (at the time) by cold dark matter, i.e. with $\Omega_D\approx0.95$ and $\Omega_M\approx0.05$. Using a couple of different numerical methods, they studied the evolution of clumps through collapse. As it is beyond me to reproduce the simulations, I can't even speculate on how they would behave differently in another universe.

If there are analytical approximations for the timescales involved, I can't find them. I suspect that there's something out there, but I don't know where it is (cosmology is not exactly an area of expertise of mine).

The question

Let's say I want stars to form within the first 2 million years after the Big Bang. What combination of the cosmological parameters ($\Omega_{\gamma}$, $\Omega_M$, $\Omega_D$, and $\Omega_{\Lambda}$) is needed to cause this? (I assume, that $\Omega_M$ and $\Omega_D$ are the ones I should be focusing on.) By simply adjusting the contributions of different types of matter and energy, can I make star formation in this universe begin earlier than it did in ours?

Requirements

I have a couple of requirements:

• The universe needs to be stable, and should eventually evolve to become what it is today: expanding at an accelerated rate and dominated by dark energy.
• Fundamental constants not derived from the density parameters should not change. For instance, increasing the speed of light, lowering the mass of an electron or increasing the gravitational constant are forbidden. I don't want to run into any unfortunate paradoxes or contradictions.
• Please note the hard-science tag on the question. Ideally, an answer would be backed up by either analytical or numerical results. I'm not asking anyone to run simulations . . . but if you did, that could be amazingly helpful.

Notes

The question's remained unanswered for a while. Aside from the fact that simulations of subhalo collapse might be necessary to address the problem in detail, I think the question could be difficult to answer given our current knowledge of the physics behind it all. There are a few possible sticking points:

• I recently got to talk with an astrochemist about Population III star formation in general; it turns out that rate coefficients for the molecular hydrogen cooling reactions are not precisely known.
• There are still some discrepancies between different simulations of halo collapse/early structure formation.
• We don't have a lot of information about Population III stars.

Putting this all together, my question might remain unanswered for a while, but I'm okay with that. If you know of new developments (or old ones) that make this question answerable, and you can apply those properly, please do write an answer. But if we can only speculate - well, I'd rather wait until we can do more than speculate.