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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 eye. 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 really 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 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 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 tag on the question.

Notes

As of June 5th, this question's over one month old and still unanswered. Aside from the fact that simulations of subhalo collapse might be necessary to address the problem in detail, I think the question could be unanswerable given our current knowledge of the physics behind it all:

  • 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.


Details regarding the Saha equation

This section is not particularly relevant to my thoughts on collapse, but I'm including it anyway for the sake of completeness.

The CMB is emitted over a short period of time, during recombination. We can approximate it, though, as being released when the ionization fraction $X_e=0.1$. The Saha equation should be satisfied: $$\frac{1-X_e}{X_e^2}=\frac{4\sqrt{2}\zeta(3)}{\sqrt{\pi}}\eta\left(\frac{k_BT}{m_ec^2}\right)\exp\left(\frac{13.6\text{ eV}}{k_BT}\right)$$ where $T$ is temperature and $\eta$ measures the ratio of baryons to photons: $$\eta\equiv\frac{N_b}{N_{\gamma}}=2.75\times10^{-8}\Omega_bh^2$$ with $\Omega_b$ the baryon density (essentially $\Omega_M$) and $h$ the Hubble constant in units of $100\text{ km/s/Mpc}$. $\Omega_bh^2=0.020$, with some uncertainty. Putting this together, I find a temperature of $k_BT\approx0.29\text{ eV}$, or $T\approx3300\text{ K}$. As $T$ scales with time as $T\propto t^{-2/3}$, the CMB should reach the Draper point and stop being visible in about 2.7 million years.

It turns out that I can extend this to 5 million years if I choose $T\approx4000\text{ K}$ and $k_BT\approx0.38\text{ eV}$. However, for this to work, I need $\eta$ to be increased by a factor of 20,000, which seems much too high. This is why I think adjusting the CMB temperature alone won't be very successful.

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This question asks for hard science. All answers to this question should be backed up by equations, empirical evidence, scientific papers, other citations, etc. Answers that do not satisfy this requirement might be removed. See the tag description for more information.

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    $\begingroup$ A world building question which needs knife and fork to be savored... $\endgroup$ – L.Dutch May 1 at 15:15
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    $\begingroup$ @L.Dutch I believe the term you're looking for is schmoo. $\endgroup$ – Rob May 1 at 20:52
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    $\begingroup$ @user10915156 Two things: First, I'm actually interested in conditions in the early universe; for instance, I'm exploring whether the CMB at that era could contribute to the irradiation of young planets, and therefore their subsequent evolution. Having star formation start $\sim1\text{ Myr}$ early affects things like recombination. Second, I'm not writing a story at the moment; I'm just building the world. The story may come later. I'm not at the writing stage. $\endgroup$ – HDE 226868 May 2 at 21:56
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    $\begingroup$ Why are people voting to close this question? The hard-science basically nullifies voting POB because every answer must completely justify itself (no opinions allowed, judging the best answer is very straightforward). As for "too broad," that's only true to people who don't understand the math. A question shouldn't be closed simply because you can't answer it (which I can't, HDE 226868's completely in another league when it comes to celestial mechanics). I'm voting to keep this one open. It's a prime example of a worldbuilding question. $\endgroup$ – JBH May 3 at 22:24
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    $\begingroup$ @user10915156, per our help center, "World building includes geography, culture and creatures for the world, not to mention magic and planetary physics, in short, everything from the physics underlying your reality to the entire universe you want to build." A "world" can be as small as a community and as large as the universe itself. Don't let the word "world" fool you. HDE 226868's asking about a rule and/or system of his world with no storybuilding clogging up the question. It's a perfect example of a good question. $\endgroup$ – JBH May 3 at 22:26

protected by L.Dutch May 15 at 19:29

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