<h3>Power spectrum</h3>

<p>The distribution of matter in the Universe is not uniform<span class="math-container">$^\dagger$</span>. Rather, it has clumped together by gravity to form the structure we see, from planets and stars, to galaxies, to groups and clusters of galaxies. Exactly <em>how</em> much the Universe clumps on the various scales can be described statistically by the <a href="https://en.wikipedia.org/wiki/Correlation_function_(astronomy)" rel="nofollow noreferrer">two-point correlation function</a> <span class="math-container">$\xi(r)$</span>, which gives the excess probability of finding a clump of matter at a certain distance <span class="math-container">$r$</span> from another clump, relative to a random, Poisson-distributed matter.</p>

<p>The Fourier transform of <span class="math-container">$\xi(\mathbf{r})$</span> is called the matter power spectrum <span class="math-container">$P(k)$</span>, where <span class="math-container">$k = 2\pi/r$</span> is the wavenumber. The power spectrum thus shows readily how much structure there is on various physical scales, i.e. a high value means much structure. It evolves (increases in amplitude) with time or, correspondingly, redshift <span class="math-container">$z$</span>, and so may be written <span class="math-container">$P(k,z)$</span>.</p>

<p>The power spectrum can be written as the product of the <a href="https://en.wikipedia.org/wiki/Primordial_fluctuations" rel="nofollow noreferrer">primordial power spectrum</a> <span class="math-container">$P_*(k) \propto k^{n_s-1}$</span> originating from <a href="https://en.wikipedia.org/wiki/Inflation_(cosmology)" rel="nofollow noreferrer">cosmic inflation</a>, and a transfer function <span class="math-container">$T(k,z)$</span> (squared) describing what happens at a later epoch. A good description is given by <a href="http://adsabs.harvard.edu/abs/1980lssu.book.....P" rel="nofollow noreferrer">Peebles (1980)</a>. </p>

<h3>The figure</h3>

<p>The figure you show is taken from <a href="https://arxiv.org/abs/astro-ph/0207047v3" rel="nofollow noreferrer">Tegmark &amp; Zaldarriaga (2002)</a>. It shows the power spectrum as a function of wavenumber, but on the top <span class="math-container">$y$</span> axis you see the corresponding spatial scale. This scale is expressed in <em>comoving megaparsecs</em> (Mpc) (where 1 parsec equals 3.26 light-years). This factors out the size of the expanding Universe — or, equivalently, the epoch at which an observation is made — such that observations at different epochs in the history of the Universe can be readily compared. A distance <span class="math-container">$r$</span> at redshift <span class="math-container">$z$</span> will expand to <span class="math-container">$r(1+z)$</span> at redshift <span class="math-container">$z=0$</span>. Using comoving coordinates, no matter at what value of <span class="math-container">$z$</span> we observe <span class="math-container">$P(k,z)$</span>, we "extrapolate" to the current power spectrum <span class="math-container">$P(k)$</span>.</p>

<p>Moreover, the distance is divided by the dimensionless Hubble constant, <span class="math-container">$h \equiv H_0\,/\,100\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1} \simeq 0.7$</span>. The reason for this is that different observations may have made different assumptions about the exact value of <span class="math-container">$H_0$</span>; in general inferred distances will scale as <span class="math-container">$H_0$</span>, so measuring distances in <span class="math-container">$h^{-1}\mathrm{Mpc}$</span> allows for direct comparison.</p>

<p>In other words, <span class="math-container">$1 h^{-1}\mathrm{Mpc} \simeq 4.7\times10^6$</span> light-years at redshift <span class="math-container">$z=0$</span>, but at redshift, say, <span class="math-container">$z=4$</span> — when the Universe was five times smaller in all directions — we had that <span class="math-container">$1 h^{-1}\mathrm{Mpc} \simeq 9.3\times10^5$</span> light-years.</p>

<h3>Observational probes</h3>

<p>The power spectrum depends on various cosmological parameters, mostly on the matter density parameter <span class="math-container">$\Omega_\mathrm{m}$</span> and the expension rate <span class="math-container">$H_0$</span>, but also the baryons (<span class="math-container">$\Omega_\mathrm{b}$</span>) and dark energy (<span class="math-container">$\Omega_\Lambda$</span>).
Given these cosmological parameters, the transfer function and the resulting matter power spectrum can be calculated. In the plot, the red line shows <span class="math-container">$P(k)$</span> as calculated theoretically from <a href="https://en.wikipedia.org/wiki/Cosmological_perturbation_theory" rel="nofollow noreferrer">linear perturbation theory</a>. For details on the calculations, see <a href="https://arxiv.org/abs/astro-ph/0207047v3" rel="nofollow noreferrer">Tegmark &amp; Zaldarriaga (2002)</a>. It is compared to constraints from various observational probes of the cosmological parameters:</p>

<ul>
<li><strong>Black</strong>: Anisotropies in the <a href="https://en.wikipedia.org/wiki/Cosmic_microwave_background" rel="nofollow noreferrer">CMB</a> shows how matter clumped in the very early Universe, and its polarization pattern, caused by subsequent scattering, gives constraints on matter distribution in later epochs, on the largest scales.</li>
<li><strong>Green</strong>: Clustering of galaxies (here from the <a href="https://en.wikipedia.org/wiki/2dF_Galaxy_Redshift_Survey" rel="nofollow noreferrer">2dF survey</a>) shows how matter has formed structure on (comoving) scales roughly from a few tens of Mpc (a bit larger than the typical distance between galaxies) to several hundred Mpc (above which the Universe is homogeneous).</li>
<li><strong>Blue</strong>: Abundances of galaxy clusters. Here, the error bars reflect the spread in the literature.</li>
<li><strong>Magenta</strong>: <a href="https://en.wikipedia.org/wiki/Weak_gravitational_lensing" rel="nofollow noreferrer">Weak lensing</a> measures photons that have traveled through fluctuations in the metric caused by intervening matter; this technique probes the power spectrum on <span class="math-container">$\sim10\,\mathrm{Mpc}$</span> scales. These data are from <a href="https://arxiv.org/abs/astro-ph/0204295" rel="nofollow noreferrer">Hoekstra et al. (2002)</a>.</li>
<li><strong>Red</strong>: The smallest scales (largest wavenumbers) are probed by the <a href="https://en.wikipedia.org/wiki/Lyman-alpha_forest" rel="nofollow noreferrer">Lyman <span class="math-container">$\alpha$</span> forest</a> (LAF). Whereas all the previous techniques measure light, the LAF is the <em>absence</em> of light, namely absorption lines in a bright background source, caused by the neutral hydrogen in all the diffuse clouds and filaments of gas lying between the source and us. The background source is usually a <a href="https://en.wikipedia.org/wiki/Quasar" rel="nofollow noreferrer">quasar</a>, but may also be, e.g., a <a href="https://en.wikipedia.org/wiki/Gamma-ray_burst" rel="nofollow noreferrer">gamma-ray burst</a>. As light blueward of the quasar's own Ly<span class="math-container">$\alpha$</span> emission line travels through the Universe, it is gradually redshifted toward the rest wavelength of Ly<span class="math-container">$\alpha$</span> (1216 Å), so whenever a parcel of hydrogen gas is present along the line of sight, it will cause an absorption line at the corresponding wavelength, creating a "forest" of absorption lines in the blue part of the quasar's spectrum. These data are (primarily) from <a href="https://arxiv.org/abs/astro-ph/0012324" rel="nofollow noreferrer">Croft et al. (2002)</a>.</li>
</ul>

<hr>

<p><span class="math-container">$^\dagger$</span><sub>This is fortunate, since otherwise you wouldn't be here to ask the question.</sub></p>

CC BY-SA 4.0
2019-05-16T07:40:52.483Z
wonderful answer; talk about due diligence ;-)
uhoh
CC BY-SA 4.0
2019-05-13T12:55:54.590Z
2019-05-15T11:30:21.850Z
2019-05-15T11:30:21.850Z
pela
<p>I have a picture of a power spectrum of matter in the Universe. What is the connection to Lyman alpha? Please describe it for me (horizontal and vertical and shape).</p>

<p><a href="https://i.sstatic.net/8PF1r.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8PF1r.jpg" alt="enter image description here"></a></p>

CC BY-SA 4.0
2019-05-12T09:27:27.080Z
I think you have *more than just a picture*. It probably comes with some additional information of some kind. Why don't you add some context; like a figure caption, the name of the source where you got it, anything?
uhoh
CC BY-SA 4.0
2019-05-12T09:57:15.983Z
Also what is your prior research on this topic. We're not a solve-my-homework network here. The graphics will have a source, if you read that source you should already be able to learn a lot about what the graphic says. After you've read that, you can happily ask what you didn't understand.
AtmosphericPrisonEscape
CC BY-SA 4.0
2019-05-12T11:13:41.317Z
It would be helpful to know more about what exactly you don't know. Do you know what a power spectrum is? Do you know what Lyman alpha is? The figure you show is not a "power spectrum in Lyman alpha for a quasar", but the red data points do have something to do with quasars.
pela
CC BY-SA 4.0
2019-05-12T14:21:11.177Z
This figure is quite the puzzler. I can't even guess what the units mean. $h$ can't be the Planck constant if Mpc is megaparsec (it doesn't result in length or inverse length), so I'm stumped. @pela have you seen units like these for wavelength or wavenumber? I've added the `identify-this-object` in lieu of a `puzzler` tag.
uhoh
CC BY-SA 4.0
2019-05-12T14:21:29.537Z
@AtmosphericPrisonEscape ditto?
uhoh
CC BY-SA 4.0
2019-05-12T14:53:05.597Z
related: [What is the Lyman Alpha forest Used For?](https://astronomy.stackexchange.com/q/658/7982) and also (perhaps) [Has the Lyman-alpha forest ever been used to test Arp's theory?](https://astronomy.stackexchange.com/q/20804/7982)
uhoh
CC BY-SA 4.0
2019-05-13T05:13:08.783Z
I presume h is the Hubble constant in units of something like 70 km/s / Mpc, and it is needed to convert the spectrum to wavelength in the frame of the emitter.
Ken G
CC BY-SA 4.0
2019-05-13T05:21:06.823Z
Actually, the "power spectrum" looks like fluctuations in the early universe, not anything that directly has to do with quasars.
Ken G
CC BY-SA 4.0
2019-05-13T06:22:35.170Z
Hi, I know that power spectrum is for identifiation of fluctuation in univers. but i dont know that picture how does do it?
Sara
CC BY-SA 4.0
2019-05-13T07:16:06.707Z
@uhoh Yes, $h$ is the dimensionless Hubble constant, as Ken G says. The figure is from Tegmark & Zaldarriaga 02. I can write an answer, but a bit more detail would be nice.
pela
CC BY-SA 4.0
2019-05-13T07:19:01.547Z
@KenG okay that makes a lot more sense, thanks!
uhoh
CC BY-SA 4.0
2019-05-13T07:21:04.587Z
@pela I understand what you mean. It looks like the OP was here an hour ago but doesn't understand that they should edit the question and add more information.
uhoh
CC BY-SA 4.0
2019-05-13T07:22:06.907Z
Hello @user28004 please go back and **edit your question and add more information** about the figure. Where it came from, what you think it means. If you look at other questions in this site you can see they are much more complete. Thanks!
uhoh
<cosmology><identify-this-object>
Power spectrum of matter in the Universe
Sara
# Power spectrum of matter in the Universe

(Asked by: username_0 on 2019-05-12)

I have a picture of a power spectrum of matter in the Universe. What is the connection to Lyman alpha? Please describe it for me (horizontal and vertical and shape).

[![enter image description here](https://i.sstatic.net/8PF1r.jpg)](https://i.sstatic.net/8PF1r.jpg)

**Tags:** cosmology, identify-this-object

> **Comments:**
> **username_1**: I think you have *more than just a picture*. It probably comes with some additional information of some kind. Why don't you add some context; like a figure caption, the name of the source where you got it, anything?  
> **username_2**: Also what is your prior research on this topic. We're not a solve-my-homework network here. The graphics will have a source, if you read that source you should already be able to learn a lot about what the graphic says. After you've read that, you can happily ask what you didn't understand.  
> **username_3**: It would be helpful to know more about what exactly you don't know. Do you know what a power spectrum is? Do you know what Lyman alpha is? The figure you show is not a "power spectrum in Lyman alpha for a quasar", but the red data points do have something to do with quasars.  
> **username_1**: This figure is quite the puzzler. I can't even guess what the units mean. $h$ can't be the Planck constant if Mpc is megaparsec (it doesn't result in length or inverse length), so I'm stumped. @username_3 have you seen units like these for wavelength or wavenumber? I've added the `identify-this-object` in lieu of a `puzzler` tag.  
> **username_1**: @username_2 ditto?  
> **username_1**: related: [What is the Lyman Alpha forest Used For?](https://astronomy.stackexchange.com/q/658/7982) and also (perhaps) [Has the Lyman-alpha forest ever been used to test Arp's theory?](https://astronomy.stackexchange.com/q/20804/7982)  
> **username_4**: I presume h is the Hubble constant in units of something like 70 km/s / Mpc, and it is needed to convert the spectrum to wavelength in the frame of the emitter.  
> **username_4**: Actually, the "power spectrum" looks like fluctuations in the early universe, not anything that directly has to do with quasars.  
> **username_0**: Hi, I know that power spectrum is for identifiation of fluctuation in univers. but i dont know that picture how does do it?  
> **username_3**: @username_1 Yes, $h$ is the dimensionless Hubble constant, as username_4 says. The figure is from Tegmark & Zaldarriaga 02. I can write an answer, but a bit more detail would be nice.  
> **username_1**: @KenG okay that makes a lot more sense, thanks!  
> **username_1**: @username_3 I understand what you mean. It looks like the OP was here an hour ago but doesn't understand that they should edit the question and add more information.  
> **username_1**: Hello @user28004 please go back and **edit your question and add more information** about the figure. Where it came from, what you think it means. If you look at other questions in this site you can see they are much more complete. Thanks!  

---

## 1 answers

### Answer by username_3 on 2019-05-13. Score: 5 | Accepted answer

### Power spectrum

The distribution of matter in the Universe is not uniform$^\dagger$. Rather, it has clumped together by gravity to form the structure we see, from planets and stars, to galaxies, to groups and clusters of galaxies. Exactly *how* much the Universe clumps on the various scales can be described statistically by the [two-point correlation function](https://en.wikipedia.org/wiki/Correlation_function_(astronomy)) $\xi(r)$, which gives the excess probability of finding a clump of matter at a certain distance $r$ from another clump, relative to a random, Poisson-distributed matter.

The Fourier transform of $\xi(\mathbf{r})$ is called the matter power spectrum $P(k)$, where $k = 2\pi/r$ is the wavenumber. The power spectrum thus shows readily how much structure there is on various physical scales, i.e. a high value means much structure. It evolves (increases in amplitude) with time or, correspondingly, redshift $z$, and so may be written $P(k,z)$.

The power spectrum can be written as the product of the [primordial power spectrum](https://en.wikipedia.org/wiki/Primordial_fluctuations) $P\_\*(k) \propto k^{n\_s-1}$ originating from [cosmic inflation](https://en.wikipedia.org/wiki/Inflation_(cosmology)), and a transfer function $T(k,z)$ (squared) describing what happens at a later epoch. A good description is given by [Peebles (1980)](http://adsabs.harvard.edu/abs/1980lssu.book.....P).

### The figure

The figure you show is taken from [Tegmark & Zaldarriaga (2002)](https://arxiv.org/abs/astro-ph/0207047v3). It shows the power spectrum as a function of wavenumber, but on the top $y$ axis you see the corresponding spatial scale. This scale is expressed in *comoving megaparsecs* (Mpc) (where 1 parsec equals 3.26 light-years). This factors out the size of the expanding Universe — or, equivalently, the epoch at which an observation is made — such that observations at different epochs in the history of the Universe can be readily compared. A distance $r$ at redshift $z$ will expand to $r(1+z)$ at redshift $z=0$. Using comoving coordinates, no matter at what value of $z$ we observe $P(k,z)$, we "extrapolate" to the current power spectrum $P(k)$.

Moreover, the distance is divided by the dimensionless Hubble constant, $h \equiv H\_0\,/\,100\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1} \simeq 0.7$. The reason for this is that different observations may have made different assumptions about the exact value of $H\_0$; in general inferred distances will scale as $H\_0$, so measuring distances in $h^{-1}\mathrm{Mpc}$ allows for direct comparison.

In other words, $1 h^{-1}\mathrm{Mpc} \simeq 4.7\times10^6$ light-years at redshift $z=0$, but at redshift, say, $z=4$ — when the Universe was five times smaller in all directions — we had that $1 h^{-1}\mathrm{Mpc} \simeq 9.3\times10^5$ light-years.

### Observational probes

The power spectrum depends on various cosmological parameters, mostly on the matter density parameter $\Omega\_\mathrm{m}$ and the expension rate $H\_0$, but also the baryons ($\Omega\_\mathrm{b}$) and dark energy ($\Omega\_\Lambda$).
Given these cosmological parameters, the transfer function and the resulting matter power spectrum can be calculated. In the plot, the red line shows $P(k)$ as calculated theoretically from [linear perturbation theory](https://en.wikipedia.org/wiki/Cosmological_perturbation_theory). For details on the calculations, see [Tegmark & Zaldarriaga (2002)](https://arxiv.org/abs/astro-ph/0207047v3). It is compared to constraints from various observational probes of the cosmological parameters:

* **Black**: Anisotropies in the [CMB](https://en.wikipedia.org/wiki/Cosmic_microwave_background) shows how matter clumped in the very early Universe, and its polarization pattern, caused by subsequent scattering, gives constraints on matter distribution in later epochs, on the largest scales.
* **Green**: Clustering of galaxies (here from the [2dF survey](https://en.wikipedia.org/wiki/2dF_Galaxy_Redshift_Survey)) shows how matter has formed structure on (comoving) scales roughly from a few tens of Mpc (a bit larger than the typical distance between galaxies) to several hundred Mpc (above which the Universe is homogeneous).
* **Blue**: Abundances of galaxy clusters. Here, the error bars reflect the spread in the literature.
* **Magenta**: [Weak lensing](https://en.wikipedia.org/wiki/Weak_gravitational_lensing) measures photons that have traveled through fluctuations in the metric caused by intervening matter; this technique probes the power spectrum on $\sim10\,\mathrm{Mpc}$ scales. These data are from [Hoekstra et al. (2002)](https://arxiv.org/abs/astro-ph/0204295).
* **Red**: The smallest scales (largest wavenumbers) are probed by the [Lyman $\alpha$ forest](https://en.wikipedia.org/wiki/Lyman-alpha_forest) (LAF). Whereas all the previous techniques measure light, the LAF is the *absence* of light, namely absorption lines in a bright background source, caused by the neutral hydrogen in all the diffuse clouds and filaments of gas lying between the source and us. The background source is usually a [quasar](https://en.wikipedia.org/wiki/Quasar), but may also be, e.g., a [gamma-ray burst](https://en.wikipedia.org/wiki/Gamma-ray_burst). As light blueward of the quasar's own Ly$\alpha$ emission line travels through the Universe, it is gradually redshifted toward the rest wavelength of Ly$\alpha$ (1216 Å), so whenever a parcel of hydrogen gas is present along the line of sight, it will cause an absorption line at the corresponding wavelength, creating a "forest" of absorption lines in the blue part of the quasar's spectrum. These data are (primarily) from [Croft et al. (2002)](https://arxiv.org/abs/astro-ph/0012324).

---

$^\dagger$This is fortunate, since otherwise you wouldn't be here to ask the question.

> **Comments:**
> **username_1**: wonderful answer; talk about due diligence ;-)  

---

