<p>The first dark matter halos typically originated in the redshift range 30-70, at a time of 30-100 million years. This is based on assuming that the initial variations in the density of the universe at small scales, for which we have no observational data, are comparable to those at large scales, for which we have ample observational data (e.g. the cosmic microwave background and large-scale structure). The particular time is solely a function of how long it took density fluctuations to grow in amplitude from their minuscule initial values of about one part in <span class="math-container">$10^{4}$</span> up to the <span class="math-container">$\mathcal{O}(1)$</span> values required for regions of excess density to collapse into halos.</p>
<p>More specifically, I did some quick calculations using some <a href="https://lesgourg.github.io/class_public/class.html" rel="nofollow noreferrer">CLASS</a>-generated power spectra I had lying around. Suppose the minimum halo mass is around an earth mass (a typical assumption for cold dark matter, but the precise value doesn't affect the result much, as long as it's small). Then <span class="math-container">$1\sigma$</span> density excesses reach the &quot;spherical collapse&quot; threshold around redshift 15. That means that under the assumption of spherical symmetry, that is the time when they would collapse to form halos. Similarly, <span class="math-container">$2\sigma$</span> excesses collapse around redshift 33, <span class="math-container">$3\sigma$</span> excesses around redshift 52, and <span class="math-container">$4\sigma$</span> excesses around redshift 72.</p>
<p>Note that the story is different for halos large enough to form galaxies inside them. If I repeat the calculation for a mass scale of (let's say) <span class="math-container">$10^7$</span> solar masses, the result is that <span class="math-container">$1\sigma$</span> density excesses at this mass scale collapse around redshift 4, <span class="math-container">$2\sigma$</span> around redshift 10, <span class="math-container">$3\sigma$</span> around redshift 15, and <span class="math-container">$4\sigma$</span> around redshift 20.</p>
<p>Also, different assumptions about the amplitudes of initial density variations at small scales can change the formation time of the smallest halos. These initial fluctuations are believed to have been seeded during inflation, and inflationary physics are pretty poorly constrained. If you make the initial variations more extreme, you can make your cosmology form dark matter halos as early as you want. Or can you? During the radiation epoch (before a redshift of about 3400 or a time of about 52000 years), there are no peculiar gravitational forces. The radiation's gravitational influence dominates, and its pressure keeps it homogeneous. You can't really make a gravitationally bound dark matter halo where radiation dominates.</p>

CC BY-SA 4.0
2023-01-10T21:40:50.277Z
Thank you very much for making such a clear and precise reply. I understand it now.
user9867
CC BY-SA 4.0
2023-01-11T06:26:50.483Z
What are "excesses" and "excesses collapse"? And what is the standard deviation meaning here?
Daddy Kropotkin
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2023-01-11T07:55:01.163Z
@DaddyKropotkin Density excesses, that is, regions with higher density than the cosmological average. Due to gravity, they gradually pull in surrounding material, becoming ever denser until they collapse. $\sigma$ is the rms (root mean squared) deviation from the mean of the density field, i.e. the standard deviation of the Gaussian distribution. The initial cosmic density field is known to be a [Gaussian random field](https://en.wikipedia.org/wiki/Gaussian_random_field) at large scales, and we assume it's also Gaussian at small scales. The simplest inflation models produce such a field.
Sten
CC BY-SA 4.0
2023-01-11T07:58:55.570Z
To add to that (due to word limit), there are technical definitions for when collapse happens, but it's basically when the density blows up.
Sten
CC BY-SA 4.0
2023-01-11T10:45:57.953Z
Don't you have to assume some volume in order for a question like "When did the first halo collapse?" to make sense? The number of halos scale with volume, so in principle if you take a large enough volume, you could have large halos at even higher redshifts. I think $z\sim30$ sounds a bit low. Much too low, in fact. If I calculate a halo mass function at z = 30, I get billions of Earth-mass halos per Mpc.
pela
CC BY-SA 4.0
2023-01-11T11:01:33.600Z
@pela That's right. Smaller scales collapse before larger scales, so I interpreted the question more in the sense of "when did the smallest halos form" than "when did the most extreme outlier halo form". I'm just giving a statistical sense for when typical peaks in the density field at the smallest scales collapsed. Outliers collapsed much earlier, as detailed in the second paragraph.
Sten
CC BY-SA 4.0
2023-01-11T11:19:29.643Z
Right, okay, but… I still don't understand how your result is compatible with this: Using a rather standard (Tinker 2008) HMF, at z = 30 I get that there should be one halo of M ~ 3e5 Msun or higher, per Mpc (i.e. if I calculate dN(>M)/dM; the cumulative HMF).
pela
CC BY-SA 4.0
2023-01-11T11:20:57.287Z
(and also how, at z ~ 20, we could already have full-blown, massive galaxies)
pela
CC BY-SA 4.0
2023-01-11T11:29:57.570Z
If I smooth on a 3e5 Msun scale, I find that $4\sigma$ density excesses collapse around redshift 30. In a Mpc^3 region, there are 130000 3e5 solar mass regions. $>4\sigma$ has about 1/30000 probability. So we expect about 4 such regions within a Mpc^3 volume. Tinker's effective collapse threshold is a bit higher than spherical collapse, which probably explains why his mass function would predict fewer halos.
Sten
CC BY-SA 4.0
2023-01-11T11:33:02.197Z
Regarding galaxies at $z\sim 20$, they must have formed from outliers! As I noted, at the $10^7$ solar mass scale, $4\sigma$ density excesses meet the collapse threshold around $z\sim 20$.
Sten
CC BY-SA 4.0
2023-01-11T11:38:05.783Z
Okay, I guess my error was not to appreciate how far there is from "typical" to "at least one". Thanks!
pela
CC BY-SA 4.0
2023-01-11T11:42:09.780Z
Thanks for the comments! I reworded the first sentence a bit to reflect this discussion.
Sten
CC BY-SA 4.0
2023-01-10T21:21:43.027Z
2023-01-11T11:38:57.323Z
2023-01-11T11:38:57.323Z
Sten
<p>I know that these dark matter halos should have been created in an early universe because during the formation of galaxies, the baryonic matter was too hot to form gravitationally self-bound objects and should have required the cdm to form these structures. Did they form in an era dominated by non-relativistic matter or by radiation?</p>
<p>Thanks for your response.</p>

<cosmology><universe><dark-matter>
When did the first cold dark matter halos begin to originate?
user9867
# When did the first cold dark matter halos begin to originate?

(Asked by: username_0 on 2023-01-10)

I know that these dark matter halos should have been created in an early universe because during the formation of galaxies, the baryonic matter was too hot to form gravitationally self-bound objects and should have required the cdm to form these structures. Did they form in an era dominated by non-relativistic matter or by radiation?

Thanks for your response.

**Tags:** cosmology, universe, dark-matter

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## 1 answers

### Answer by username_1 on 2023-01-10. Score: 4 | Accepted answer

The first dark matter halos typically originated in the redshift range 30-70, at a time of 30-100 million years. This is based on assuming that the initial variations in the density of the universe at small scales, for which we have no observational data, are comparable to those at large scales, for which we have ample observational data (e.g. the cosmic microwave background and large-scale structure). The particular time is solely a function of how long it took density fluctuations to grow in amplitude from their minuscule initial values of about one part in $10^{4}$ up to the $\mathcal{O}(1)$ values required for regions of excess density to collapse into halos.

More specifically, I did some quick calculations using some [CLASS](https://lesgourg.github.io/class_public/class.html)-generated power spectra I had lying around. Suppose the minimum halo mass is around an earth mass (a typical assumption for cold dark matter, but the precise value doesn't affect the result much, as long as it's small). Then $1\sigma$ density excesses reach the "spherical collapse" threshold around redshift 15. That means that under the assumption of spherical symmetry, that is the time when they would collapse to form halos. Similarly, $2\sigma$ excesses collapse around redshift 33, $3\sigma$ excesses around redshift 52, and $4\sigma$ excesses around redshift 72.

Note that the story is different for halos large enough to form galaxies inside them. If I repeat the calculation for a mass scale of (let's say) $10^7$ solar masses, the result is that $1\sigma$ density excesses at this mass scale collapse around redshift 4, $2\sigma$ around redshift 10, $3\sigma$ around redshift 15, and $4\sigma$ around redshift 20.

Also, different assumptions about the amplitudes of initial density variations at small scales can change the formation time of the smallest halos. These initial fluctuations are believed to have been seeded during inflation, and inflationary physics are pretty poorly constrained. If you make the initial variations more extreme, you can make your cosmology form dark matter halos as early as you want. Or can you? During the radiation epoch (before a redshift of about 3400 or a time of about 52000 years), there are no peculiar gravitational forces. The radiation's gravitational influence dominates, and its pressure keeps it homogeneous. You can't really make a gravitationally bound dark matter halo where radiation dominates.

> **Comments:**
> **username_0**: Thank you very much for making such a clear and precise reply. I understand it now.  
> **username_2**: What are "excesses" and "excesses collapse"? And what is the standard deviation meaning here?  
> **username_1**: @DaddyKropotkin Density excesses, that is, regions with higher density than the cosmological average. Due to gravity, they gradually pull in surrounding material, becoming ever denser until they collapse. $\sigma$ is the rms (root mean squared) deviation from the mean of the density field, i.e. the standard deviation of the Gaussian distribution. The initial cosmic density field is known to be a [Gaussian random field](https://en.wikipedia.org/wiki/Gaussian_random_field) at large scales, and we assume it's also Gaussian at small scales. The simplest inflation models produce such a field.  
> **username_1**: To add to that (due to word limit), there are technical definitions for when collapse happens, but it's basically when the density blows up.  
> **username_3**: Don't you have to assume some volume in order for a question like "When did the first halo collapse?" to make sense? The number of halos scale with volume, so in principle if you take a large enough volume, you could have large halos at even higher redshifts. I think $z\sim30$ sounds a bit low. Much too low, in fact. If I calculate a halo mass function at z = 30, I get billions of Earth-mass halos per Mpc.  
> **username_1**: @pela That's right. Smaller scales collapse before larger scales, so I interpreted the question more in the sense of "when did the smallest halos form" than "when did the most extreme outlier halo form". I'm just giving a statistical sense for when typical peaks in the density field at the smallest scales collapsed. Outliers collapsed much earlier, as detailed in the second paragraph.  
> **username_3**: Right, okay, but… I still don't understand how your result is compatible with this: Using a rather standard (Tinker 2008) HMF, at z = 30 I get that there should be one halo of M ~ 3e5 Msun or higher, per Mpc (i.e. if I calculate dN(>M)/dM; the cumulative HMF).  
> **username_3**: (and also how, at z ~ 20, we could already have full-blown, massive galaxies)  
> **username_1**: If I smooth on a 3e5 Msun scale, I find that $4\sigma$ density excesses collapse around redshift 30. In a Mpc^3 region, there are 130000 3e5 solar mass regions. $>4\sigma$ has about 1/30000 probability. So we expect about 4 such regions within a Mpc^3 volume. Tinker's effective collapse threshold is a bit higher than spherical collapse, which probably explains why his mass function would predict fewer halos.  
> **username_1**: Regarding galaxies at $z\sim 20$, they must have formed from outliers! As I noted, at the $10^7$ solar mass scale, $4\sigma$ density excesses meet the collapse threshold around $z\sim 20$.  
> **username_3**: Okay, I guess my error was not to appreciate how far there is from "typical" to "at least one". Thanks!  
> **username_1**: Thanks for the comments! I reworded the first sentence a bit to reflect this discussion.  

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