<p>Partial answer; although the question is 17 days old, I hope that this guide to the dependence on <em>h</em> in cosmology can give you a good insight into this concept.</p>

<p><a href="https://www.astro.ljmu.ac.uk/~ikb/research/h-units.html" rel="nofollow noreferrer">Quick guide to <em>h</em> dependence in cosmology</a></p>

<p>which links to <a href="https://ui.adsabs.harvard.edu/abs/2013PASA...30...52C/abstract" rel="nofollow noreferrer">Damn You, Little h! (Or, Real-World Applications of the Hubble Constant Using Observed and Simulated Data)</a>. The abstract says:</p>

<blockquote>
  <p>The Hubble constant, H<sub>0</sub>, <strong>or its dimensionless equivalent, "little h"</strong>, is a fundamental cosmological property that is now known to an accuracy better than a few per cent. Despite its cosmological nature, little h commonly appears in the measured properties of individual galaxies. This can pose unique challenges for users of such data, particularly with survey data. In this paper we show how little h arises in the measurement of galaxies, how to compare like-properties from different datasets that have assumed different little h cosmologies, and how to fairly compare theoretical data with observed data, where little h can manifest in vastly different ways. This last point is particularly important when observations are used to calibrate galaxy formation models, as calibrating with the wrong (or no) little h can lead to disastrous results when the model is later converted to the correct h cosmology. We argue that in this modern age little h is an anachronism, being one of least uncertain parameters in astrophysics, and we propose that observers and theorists instead treat this uncertainty like any other. We conclude with a `cheat sheet' of nine points that should be followed when dealing with little h in data analysis.</p>
</blockquote>

CC BY-SA 4.0
2020-05-31T15:55:49.787Z
2020-05-31T21:55:24.580Z
2020-05-31T21:55:24.580Z
sbjartmar
<p>As argued in the paper you cite, you shouldn't think of <span class="math-container">$h$</span> as a <em>unit</em>, but rather as a part of the value. Although it's commonly seen, and although numerically it's not wrong, I therefore think the term &quot;<span class="math-container">$10h^{-1}\,\mathrm{Mpc}$</span>&quot; is preferred to &quot;<span class="math-container">$10\,\mathrm{Mpc}/h$</span>&quot;.</p>
<p>There's no need to express <span class="math-container">$c$</span> in terms of <span class="math-container">$h$</span>, because our lack of knowledge about the value of <span class="math-container">$h$</span> doesn't affect the value of <span class="math-container">$c$</span>. If your distance is, say, <span class="math-container">$x_0 = 10h^{-1}\,\mathrm{Mpc}$</span>, so that your inverse distance is <span class="math-container">$1/x_0 = 0.1h\,\mathrm{Mpc}^{-1}$</span>, your calculation becomes
<span class="math-container">$$
\begin{array}{rcl}
c\frac{1}{x_0} &amp; = &amp; \left[9.716\times10^{-15}\,\mathrm{Mpc}\,\mathrm{s}^{-1}\right] \times \left[0.1h\,\mathrm{Mpc}^{-1}\right] \\
      &amp; =      &amp; 9.716\times10^{-16}h\,\mathrm{s}^{-1} \\
\big( &amp; \simeq &amp; 0.7\times10^{-15}\,\mathrm{s}^{-1}.\big)
\end{array}
$$</span></p>

CC BY-SA 4.0
2022-07-06T15:35:43.247Z
2022-07-06T15:35:43.247Z
pela
<p>I am running a cosmological simulation and am having some trouble putting things into code units. The physical distance units in my simulation are in terms of <span class="math-container">$\text{Mpc/h}$</span>, where <span class="math-container">$h$</span> is the dimensionless Hubble parameter. This makes enough sense because, as noted <a href="https://arxiv.org/abs/1308.4150" rel="nofollow noreferrer">elsewhere</a>, simulations are often scale-free so it makes sense to factor out the <span class="math-container">$h$</span> dependence and make it explicit. This unit convention is causing me some confusion however. In one calculation I have to do during the simulation, I essentially (ignoring context which I can provide later) have to multiply the speed of light <span class="math-container">$c$</span> by an inverse distance <span class="math-container">$1/x_0$</span> which is given in units of <span class="math-container">$\text{Mpc/h}$</span>.</p>

<p>In order to properly have the units cancel, I first put <span class="math-container">$c$</span> in units of <span class="math-container">$\text{Mpc/s}$</span> to get 
<span class="math-container">$$
9.716 \times10^{-15} \text{Mpc/s}
$$</span> 
However, should I know factor out the <span class="math-container">$h$</span> dependence? These seems strange to me because in my mind, the value of the speed of light should not depend on the underlying cosmology I have simulated. On the other hand, I feel that I should not cancel units of <span class="math-container">$\text{Mpc}$</span> with units of <span class="math-container">$\text{Mpc/h}$</span>. To make things concrete, let's assume I have a value of <span class="math-container">$h=.7$</span>. Should I then take the quantity above and multiply it by <span class="math-container">$.7$</span> to yield 
<span class="math-container">$$
6.802 \times 10^{-15} \text{(Mpc/h)/s}
$$</span></p>

<p>and use that result in my calculations? I think this situation confuses me because it doesn't involve measurements, where it is clear how <span class="math-container">$h$</span> can enter, and it involves a constant of nature, which should be independent of the assumed cosmology. </p>

<p>Let me know if more info is needed.</p>

<cosmology><n-body-simulations><hubble-constant><units>
'Little h' usage in cosmological simulations
gabe
# 'Little h' usage in cosmological simulations

(Asked by: username_0 on 2020-05-14)

I am running a cosmological simulation and am having some trouble putting things into code units. The physical distance units in my simulation are in terms of $\text{Mpc/h}$, where $h$ is the dimensionless Hubble parameter. This makes enough sense because, as noted [elsewhere](https://arxiv.org/abs/1308.4150), simulations are often scale-free so it makes sense to factor out the $h$ dependence and make it explicit. This unit convention is causing me some confusion however. In one calculation I have to do during the simulation, I essentially (ignoring context which I can provide later) have to multiply the speed of light $c$ by an inverse distance $1/x\_0$ which is given in units of $\text{Mpc/h}$.

In order to properly have the units cancel, I first put $c$ in units of $\text{Mpc/s}$ to get
$$
9.716 \times10^{-15} \text{Mpc/s}
$$
However, should I know factor out the $h$ dependence? These seems strange to me because in my mind, the value of the speed of light should not depend on the underlying cosmology I have simulated. On the other hand, I feel that I should not cancel units of $\text{Mpc}$ with units of $\text{Mpc/h}$. To make things concrete, let's assume I have a value of $h=.7$. Should I then take the quantity above and multiply it by $.7$ to yield
$$
6.802 \times 10^{-15} \text{(Mpc/h)/s}
$$

and use that result in my calculations? I think this situation confuses me because it doesn't involve measurements, where it is clear how $h$ can enter, and it involves a constant of nature, which should be independent of the assumed cosmology.

Let me know if more info is needed.

**Tags:** cosmology, n-body-simulations, hubble-constant, units


---

## 2 answers

### Answer by username_1 on 2020-05-31. Score: 1

Partial answer; although the question is 17 days old, I hope that this guide to the dependence on *h* in cosmology can give you a good insight into this concept.

[Quick guide to *h* dependence in cosmology](https://www.astro.ljmu.ac.uk/~ikb/research/h-units.html)

which links to [Damn You, Little h! (Or, Real-World Applications of the Hubble Constant Using Observed and Simulated Data)](https://ui.adsabs.harvard.edu/abs/2013PASA...30...52C/abstract). The abstract says:

> The Hubble constant, H0, **or its dimensionless equivalent, "little h"**, is a fundamental cosmological property that is now known to an accuracy better than a few per cent. Despite its cosmological nature, little h commonly appears in the measured properties of individual galaxies. This can pose unique challenges for users of such data, particularly with survey data. In this paper we show how little h arises in the measurement of galaxies, how to compare like-properties from different datasets that have assumed different little h cosmologies, and how to fairly compare theoretical data with observed data, where little h can manifest in vastly different ways. This last point is particularly important when observations are used to calibrate galaxy formation models, as calibrating with the wrong (or no) little h can lead to disastrous results when the model is later converted to the correct h cosmology. We argue that in this modern age little h is an anachronism, being one of least uncertain parameters in astrophysics, and we propose that observers and theorists instead treat this uncertainty like any other. We conclude with a `cheat sheet' of nine points that should be followed when dealing with little h in data analysis.


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### Answer by username_2 on 2022-07-06. Score: 0

As argued in the paper you cite, you shouldn't think of $h$ as a *unit*, but rather as a part of the value. Although it's commonly seen, and although numerically it's not wrong, I therefore think the term "$10h^{-1}\,\mathrm{Mpc}$" is preferred to "$10\,\mathrm{Mpc}/h$".

There's no need to express $c$ in terms of $h$, because our lack of knowledge about the value of $h$ doesn't affect the value of $c$. If your distance is, say, $x\_0 = 10h^{-1}\,\mathrm{Mpc}$, so that your inverse distance is $1/x\_0 = 0.1h\,\mathrm{Mpc}^{-1}$, your calculation becomes
$$
\begin{array}{rcl}
c\frac{1}{x\_0} & = & \left[9.716\times10^{-15}\,\mathrm{Mpc}\,\mathrm{s}^{-1}\right] \times \left[0.1h\,\mathrm{Mpc}^{-1}\right] \\
& = & 9.716\times10^{-16}h\,\mathrm{s}^{-1} \\
\big( & \simeq & 0.7\times10^{-15}\,\mathrm{s}^{-1}.\big)
\end{array}
$$


---

