<p>Yes, you multiply those integrals by the Hubble distance. It's like a cosmological base distance.</p>
<p>You generally can't calculate those integrals by algebra, you have to use a numerical method, like Simpson's rule. The tricky part is choosing a set of <span class="math-container">$\Omega$</span> (unitless) density parameters to plug into this equation:</p>
<p><span class="math-container">$$E(z) = \sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}$$</span></p>
<p>Note that <span class="math-container">$\Omega_m = \Omega_b + \Omega_c$</span>, where b is baryonic matter and c is cold dark matter. The radiation density term <span class="math-container">$\Omega_r$</span> is really the relativistic particle density, since it incorporates the photon density and the (hot) neutrino density. But its value is quite small compared to the other terms, so it's only significant with very large redshift <span class="math-container">$z$</span>.</p>
<p>Also,</p>
<p><span class="math-container">$$\Omega_r + \Omega_m + \Omega_k + \Omega_\Lambda = \Omega_{Total} = \Omega$$</span></p>
<p>We're <em>pretty</em> sure that <span class="math-container">$\Omega_{Total} = 1$</span>, and that's why that Wikipedia page on distance measures calculates the curvature term <span class="math-container">$\Omega_k$</span> using</p>
<p><span class="math-container">$$\Omega_k = 1 - \Omega_r - \Omega_m 
- \Omega_\Lambda$$</span></p>
<p>To obtain actual values for those <span class="math-container">$\Omega$</span> density parameters, you have two main options, the <a href="https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe" rel="nofollow noreferrer">WMAP</a> data, and the data from the <a href="https://en.wikipedia.org/wiki/Planck_(spacecraft)" rel="nofollow noreferrer">Planck</a> collaboration. Each of those sources provides results derived from several data sets, interpreted according to various cosmological models.</p>
<p>Here's the relevant link for  <a href="https://lambda.gsfc.nasa.gov/product/map/dr5/parameters_info.cfm" rel="nofollow noreferrer">WMAP</a>. Click the &quot;View Matrix&quot; link near the top of the page, which takes you to a page of ~50 sets of parameters.</p>
<p>There's more recent data from the <a href="https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Cosmological_Parameters" rel="nofollow noreferrer">Planck collaboration</a>, but there are various disagreements with the WMAP data. Notably, they disagree on the value of the Hubble distance. Also, you won't find a value for radiation density <span class="math-container">$\Omega_r$</span> in the Planck data, but the answers <a href="https://physics.stackexchange.com/q/94181/123208">here</a> explain how it can be calculated from other tabulated values. The simplest way is to use</p>
<p><span class="math-container">$$\Omega_r = \Omega_m / (z_{eq}+1)$$</span></p>

CC BY-SA 4.0
2021-02-06T16:49:06.880Z
2021-02-06T16:49:06.880Z
PM 2Ring
<p>According to Wikipedia, <a href="https://en.wikipedia.org/wiki/Distance_measures_(cosmology)" rel="nofollow noreferrer">Distance measures (cosmology)</a>,</p>
<p>Comoving distance:
<span class="math-container">$${\displaystyle d_{C}(z)=d_{H}\int _{0}^{z}{\frac {dz'}{E(z')}}}$$</span></p>
<p>Light-travel distance:
<span class="math-container">$${\displaystyle d_{T}(z)=d_{H}\int _{0}^{z}{\frac {dz'}{(1+z')E(z')}}}$$</span></p>
<p>What is with the 'Hubble distance' (<span class="math-container">$d_H$</span>) before the integration symbol?  Do you multiply it by the result of integration?</p>
<p>Also, how do you take the derivative of <span class="math-container">$z$</span>, <span class="math-container">$z'$</span>, if <span class="math-container">$z$</span> is a constant? <span class="math-container">$z$</span> is the (current) redshift for that particular object, right?</p>
<p>I am confused....</p>
<p>From 'Distance measures (cosmology)' on Wikipedia:</p>
<p><img src="https://i.sstatic.net/g90Fa.png" alt="Wikipedia excerpt" /></p>
<p>Is there an example on the web somewhere of these two things, comoving and travel distance, being calculated?  That might help....</p>

CC BY-SA 4.0
2021-02-05T00:24:39.270Z
Thank you for the helpful edit.... I am both stupid and lazy when it comes to LaTeX and MathJax... I am just starting to learn more MathJax, and refresh my LaTeX....
Kurt Hikes
CC BY-SA 4.0
2021-02-05T05:38:12.323Z
In those equations, $z'$ is *not* the derivative of $z$. $z'$ is the variable that we're doing the integration over, and $z$ is a particular value of that variable. Here's a simple example (using a familiar function) to illustrate this concept. The volume of a sphere of radius $r$ is $V = \frac43\pi r^3$, which we can get by integrating the surface area, $$V = \int_0^r 4\pi r'^2 dr'$$ So $r'$ is the variable radius, and $r$ is a particular radius of a sphere whose volume we want to calculate.
PM 2Ring
CC BY-SA 4.0
2021-02-05T20:27:19.247Z
Thanks!  I just want to find a specific example(s) somewhere, or maybe a 'comoving distance calculator' or 'travel-distance calculator',.... There are several 'redshift to parsecs' calculators on the web....
Kurt Hikes
<cosmology><distances><redshift><cosmological-horizon>
How do you calculate comoving distance and light's travel distance? According to the formulae below?
Kurt Hikes
# How do you calculate comoving distance and light's travel distance? According to the formulae below?

(Asked by: username_0 on 2021-02-04)

According to Wikipedia, [Distance measures (cosmology)](https://en.wikipedia.org/wiki/Distance_measures_(cosmology)),

Comoving distance:
$${\displaystyle d\_{C}(z)=d\_{H}\int \_{0}^{z}{\frac {dz'}{E(z')}}}$$

Light-travel distance:
$${\displaystyle d\_{T}(z)=d\_{H}\int \_{0}^{z}{\frac {dz'}{(1+z')E(z')}}}$$

What is with the 'Hubble distance' ($d\_H$) before the integration symbol? Do you multiply it by the result of integration?

Also, how do you take the derivative of $z$, $z'$, if $z$ is a constant? $z$ is the (current) redshift for that particular object, right?

I am confused....

From 'Distance measures (cosmology)' on Wikipedia:

![Wikipedia excerpt](https://i.sstatic.net/g90Fa.png)

Is there an example on the web somewhere of these two things, comoving and travel distance, being calculated? That might help....

**Tags:** cosmology, distances, redshift, cosmological-horizon
> **Comments:**
> **username_0**: Thank you for the helpful edit.... I am both stupid and lazy when it comes to LaTeX and MathJax... I am just starting to learn more MathJax, and refresh my LaTeX....  
> **username_1**: In those equations, $z'$ is *not* the derivative of $z$. $z'$ is the variable that we're doing the integration over, and $z$ is a particular value of that variable. Here's a simple example (using a familiar function) to illustrate this concept. The volume of a sphere of radius $r$ is $V = \frac43\pi r^3$, which we can get by integrating the surface area, $$V = \int_0^r 4\pi r'^2 dr'$$ So $r'$ is the variable radius, and $r$ is a particular radius of a sphere whose volume we want to calculate.  
> **username_0**: Thanks!  I just want to find a specific example(s) somewhere, or maybe a 'comoving distance calculator' or 'travel-distance calculator',.... There are several 'redshift to parsecs' calculators on the web....  

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## 1 answers

### Answer by username_1 on 2021-02-06. Score: 2 | Accepted answer

Yes, you multiply those integrals by the Hubble distance. It's like a cosmological base distance.

You generally can't calculate those integrals by algebra, you have to use a numerical method, like Simpson's rule. The tricky part is choosing a set of $\Omega$ (unitless) density parameters to plug into this equation:

$$E(z) = \sqrt{\Omega\_r(1+z)^4 + \Omega\_m(1+z)^3 + \Omega\_k(1+z)^2 + \Omega\_\Lambda}$$

Note that $\Omega\_m = \Omega\_b + \Omega\_c$, where b is baryonic matter and c is cold dark matter. The radiation density term $\Omega\_r$ is really the relativistic particle density, since it incorporates the photon density and the (hot) neutrino density. But its value is quite small compared to the other terms, so it's only significant with very large redshift $z$.

Also,

$$\Omega\_r + \Omega\_m + \Omega\_k + \Omega\_\Lambda = \Omega\_{Total} = \Omega$$

We're *pretty* sure that $\Omega\_{Total} = 1$, and that's why that Wikipedia page on distance measures calculates the curvature term $\Omega\_k$ using

$$\Omega\_k = 1 - \Omega\_r - \Omega\_m
- \Omega\_\Lambda$$

To obtain actual values for those $\Omega$ density parameters, you have two main options, the [WMAP](https://en.wikipedia.org/wiki/Wilkinson_Microwave_Anisotropy_Probe) data, and the data from the [Planck](https://en.wikipedia.org/wiki/Planck_(spacecraft)) collaboration. Each of those sources provides results derived from several data sets, interpreted according to various cosmological models.

Here's the relevant link for [WMAP](https://lambda.gsfc.nasa.gov/product/map/dr5/parameters_info.cfm). Click the "View Matrix" link near the top of the page, which takes you to a page of ~50 sets of parameters.

There's more recent data from the [Planck collaboration](https://wiki.cosmos.esa.int/planck-legacy-archive/index.php/Cosmological_Parameters), but there are various disagreements with the WMAP data. Notably, they disagree on the value of the Hubble distance. Also, you won't find a value for radiation density $\Omega\_r$ in the Planck data, but the answers [here](https://physics.stackexchange.com/q/94181/123208) explain how it can be calculated from other tabulated values. The simplest way is to use

$$\Omega\_r = \Omega\_m / (z\_{eq}+1)$$


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