<p>There are different concepts of distance, which all obtain the same measure in a non-expanding flat space, i.e. in everyday live and the local universe, but differ in an expanding universe. The <strong>luminosity distance</strong> <span class="math-container">$D_L$</span> is defined as that inferred from the apparent brightness of a source with known absolute brightness, while the <strong>angular-diameter distance</strong> <span class="math-container">$D_A$</span> is defined as that inferred from the angular size of an object with known physical size. In addition, there is the <strong>co-moving distance</strong> (<em>reception distance</em>) <span class="math-container">$D_c$</span> and <strong>proper distance</strong> <span class="math-container">$D_p$</span> (<em>emission distance</em> in the figure below).</p>
<p><a href="https://i.sstatic.net/kvU8IXb8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/kvU8IXb8.png" alt="enter image description here" /></a></p>
<p>The co-moving or reception distance can be computed from
<span class="math-container">$$
D_c = \int\mathrm{d} x = \int_{t(z)}^{t_0}\frac{c\mathrm{d}t}{a(t)} = \int_0^z\frac{c\mathrm{d}z}{H(z)}
$$</span>
where (from the Friedman equation)
<span class="math-container">$$
H^2(z) = H_0^2 \left[\Omega_{\mathrm{r}}(1+z)^4 + \Omega_{\mathrm{m}}(1+z)^3 + \Omega_\kappa(1+z)^2 + \Omega_\Lambda\right].
$$</span>
Here <span class="math-container">$\Omega_{\mathrm{r}}$</span>, <span class="math-container">$\Omega_{\mathrm{m}}$</span>, <span class="math-container">$\Omega_\Lambda$</span> are the density parameters for radiation, matter, and dark energy, while <span class="math-container">$\Omega_\kappa=1-\Omega_{\mathrm{r}}-\Omega_{\mathrm{m}}-\Omega_\Lambda$</span> is the curvature term. The currently favoured cosmology has <span class="math-container">$\Omega_{\mathrm{r}}=8.4\cdot10^{-5}$</span>, <span class="math-container">$\Omega_{\mathrm{m}}=0.309$</span>, <span class="math-container">$\Omega_\Lambda=0.691$</span>, <span class="math-container">$\Omega_\kappa=0$</span>, and <span class="math-container">$H_0=67.7\,$</span>km/s/Mpc (see <a href="https://en.wikipedia.org/wiki/Lambda-CDM_model" rel="nofollow noreferrer">here</a>).</p>
<p>The proper or emission distance <span class="math-container">$D_p$</span> follows from
<span class="math-container">$$
D_p = a(t) D_c = \frac{D_c}{1+z}
$$</span>
and for a flat universe (as ours seems to be, since <span class="math-container">$\Omega_\kappa=0$</span>) <span class="math-container">$D_A=D_p$</span>, while
<span class="math-container">$$
D_L = (1+z)^2D_A
$$</span>
in any case. Note that <span class="math-container">$D_p(z)$</span> is non-monotonic: the angular-diameter distance becomes maximal for some redshift. For the currently favoured cosmology, this occurs for <span class="math-container">$z\approx1.5$</span> with <span class="math-container">$D_{A,\max} \approx 0.4 c/H_0 \sim 10^{13}$</span>pc ('maximum distance of lightcone' in the figure). Objects at yet larger redshift appear larger and hence at smaller <span class="math-container">$D_A$</span>.</p>

CC BY-SA 4.0
2024-07-16T18:41:28.477Z
2024-07-16T19:24:23.297Z
2024-07-16T19:24:23.297Z
Walter
<p>Given some redshift z, how do we find the distance using the cosmological red shift formula (and presumably the Friedmann equations)? a = 1/(1+z), and we can assume that omega_0 = 1. I know a reasonable answer can be achieved using Hubbles law, but was looking for the specific method going via a. Thank you! (all the necessary cosmological parameters can be assumed to be known..)</p>

CC BY-SA 4.0
2024-07-16T18:20:54.737Z
Which distance are you talking about? The luminosity distance, angular-diameter distance, co-moving distance, or proper distance?
Walter
<cosmology><redshift>
How can you calculate the distance to an object knowing its cosmological redshift? (without Hubbles law)
n014
# How can you calculate the distance to an object knowing its cosmological redshift? (without Hubbles law)

(Asked by: username_0 on 2024-07-16)

Given some redshift z, how do we find the distance using the cosmological red shift formula (and presumably the Friedmann equations)? a = 1/(1+z), and we can assume that omega\_0 = 1. I know a reasonable answer can be achieved using Hubbles law, but was looking for the specific method going via a. Thank you! (all the necessary cosmological parameters can be assumed to be known..)

**Tags:** cosmology, redshift

> **Comments:**
> **username_1**: Which distance are you talking about? The luminosity distance, angular-diameter distance, co-moving distance, or proper distance?  

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## 1 answers

### Answer by username_1 on 2024-07-16. Score: 5 | Accepted answer

There are different concepts of distance, which all obtain the same measure in a non-expanding flat space, i.e. in everyday live and the local universe, but differ in an expanding universe. The **luminosity distance** $D\_L$ is defined as that inferred from the apparent brightness of a source with known absolute brightness, while the **angular-diameter distance** $D\_A$ is defined as that inferred from the angular size of an object with known physical size. In addition, there is the **co-moving distance** (*reception distance*) $D\_c$ and **proper distance** $D\_p$ (*emission distance* in the figure below).

[![enter image description here](https://i.sstatic.net/kvU8IXb8.png)](https://i.sstatic.net/kvU8IXb8.png)

The co-moving or reception distance can be computed from
$$
D\_c = \int\mathrm{d} x = \int\_{t(z)}^{t\_0}\frac{c\mathrm{d}t}{a(t)} = \int\_0^z\frac{c\mathrm{d}z}{H(z)}
$$
where (from the Friedman equation)
$$
H^2(z) = H\_0^2 \left[\Omega\_{\mathrm{r}}(1+z)^4 + \Omega\_{\mathrm{m}}(1+z)^3 + \Omega\_\kappa(1+z)^2 + \Omega\_\Lambda\right].
$$
Here $\Omega\_{\mathrm{r}}$, $\Omega\_{\mathrm{m}}$, $\Omega\_\Lambda$ are the density parameters for radiation, matter, and dark energy, while $\Omega\_\kappa=1-\Omega\_{\mathrm{r}}-\Omega\_{\mathrm{m}}-\Omega\_\Lambda$ is the curvature term. The currently favoured cosmology has $\Omega\_{\mathrm{r}}=8.4\cdot10^{-5}$, $\Omega\_{\mathrm{m}}=0.309$, $\Omega\_\Lambda=0.691$, $\Omega\_\kappa=0$, and $H\_0=67.7\,$km/s/Mpc (see [here](https://en.wikipedia.org/wiki/Lambda-CDM_model)).

The proper or emission distance $D\_p$ follows from
$$
D\_p = a(t) D\_c = \frac{D\_c}{1+z}
$$
and for a flat universe (as ours seems to be, since $\Omega\_\kappa=0$) $D\_A=D\_p$, while
$$
D\_L = (1+z)^2D\_A
$$
in any case. Note that $D\_p(z)$ is non-monotonic: the angular-diameter distance becomes maximal for some redshift. For the currently favoured cosmology, this occurs for $z\approx1.5$ with $D\_{A,\max} \approx 0.4 c/H\_0 \sim 10^{13}$pc ('maximum distance of lightcone' in the figure). Objects at yet larger redshift appear larger and hence at smaller $D\_A$.


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