<p>Different combinations of cosmological parameters make different predictions for the detailed shape of the distance modulus versus redshift plot.</p>
<p>Meanwhile, the distance modulus of the supernovae is estimated from their peak apparent magnitude and an estimate of their peak absolute magnitude based on their light curve data, and their redshifts are measured quite precisely from spectra.</p>
<p>The various model curves are then compared with the data to see which one provides the best fit. Unfortunately, this fitting process is not well illustrated in the SCP Union 2.1 paper, but below is a plot from a much earlier paper (with sparser data), with &quot;effective peak magnitude&quot; playing the same role as distance modulus, showing multiple, labelled model curves.</p>
<p><a href="https://i.sstatic.net/ykexPrw0.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ykexPrw0.gif" alt="Hubble diagram with multiple cosmological models" /></a></p>
<p>The sensitivity to late-time acceleration, as <span class="math-container">$\Omega_\Lambda$</span> becomes greater than <span class="math-container">$\Omega_M$</span>, is because in an accelerated expansion , at a given distance (in the past) the supernovae would be receding slower than expected from a more uniform or decelerating expansion. The comparison is most sensitive to <span class="math-container">$\Omega_\Lambda - \Omega_M$</span>, the difference in the accelerating and decelerating terms. You can see in the plot above that curves with <span class="math-container">$\Omega_\Lambda - \Omega_M &lt;0$</span> (decelerating universes, broadly speaking) are lower than curves with <span class="math-container">$\Omega_\Lambda - \Omega_M &gt;0$</span> (accelerating universes).</p>
<p>The plot below then illustrates how well the cosmological parameters are constrained from the data. The supernova data alone provide strong constraints only on <span class="math-container">$\Omega_\Lambda - \Omega_M$</span> but do indicate that the universe must be accelerating<span class="math-container">$^{\dagger}$</span> and that <span class="math-container">$\Omega_\Lambda&gt;0.4$</span>, whilst the cosmic microwave background gives strong complementary constraints on <span class="math-container">$\Omega_\Lambda + \Omega_M$</span>. The combination, and with the baryon acoustic oscillation data, suggest a geometrically flat, accelerating universe.</p>
<p><a href="https://i.sstatic.net/yygIO20w.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yygIO20w.png" alt="Omega lambda versus Omega m constraints" /></a></p>
<p><span class="math-container">$\dagger$</span> The &quot;<a href="https://en.wikipedia.org/wiki/Deceleration_parameter" rel="nofollow noreferrer">deceleration parameter</a>&quot; is <span class="math-container">$q \simeq 0.5\Omega_M - \Omega_\Lambda$</span>, which would be negative in an accelerating expansion. Looking at the possibilities enclosed by the blue ellipse in the second plot, you can convince yourself that <span class="math-container">$q$</span> must be less than zero and is about <span class="math-container">$q=-0.55$</span> for the best-fit concordance model.</p>

CC BY-SA 4.0
2024-12-21T07:15:54.493Z
Thank you very much for your effort trying to answer my question but I did realize that I miss an important base to understand the universe expansion acceleration : cosmological parameters which I do not have a clue of what it is :(
I guess I have to ask that question first... 
(just in case : no way to explain universe expansion acceleration without all those ΩM−ΩΛ "cosmological parameters" ? :) )
Fabien
CC BY-SA 4.0
2024-12-20T07:21:53.127Z
2024-12-20T10:03:40.320Z
2024-12-20T10:03:40.320Z
ProfRob
<p>Is it possible only from the average distance modulus and redshift Z in the set of data :</p>
<p><a href="https://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt" rel="nofollow noreferrer">https://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt</a></p>
<p>Which can be represented by :</p>
<p><a href="https://i.sstatic.net/v8XcWjco.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/v8XcWjco.png" alt="enter image description here" /></a></p>
<p>To conclude that the expansion of the universe is accelerated ?</p>
<p>And above all, if yes, how ? With what calculation ?</p>

CC BY-SA 4.0
2024-12-16T21:31:01.007Z
Have you looked at the paper from which the data came?
ProfRob
CC BY-SA 4.0
2024-12-16T21:57:54.133Z
Sadly it's very dense and exhaustive, it's hard for me to "extract" the essential step from it... Is there a way to resume how we deduce the type of expansion to some simple & essential step of calculation ?
Fabien
CC BY-SA 4.0
2024-12-16T22:05:06.077Z
Yes. Plot the solid line in your graph for different combinations of cosmological parameters. The one that fits the data best has $\Omega_\Lambda>0$. Whether this conclusively demonstrates acceleration will be discussed in the paper. Other evidence should be considered; principally the cosmic microwave background.
ProfRob
CC BY-SA 4.0
2024-12-17T00:34:22.947Z
You can just note the Hubble constant's existence and definition as $v=\frac{\text{d}x}{\text{d}t}=Hx$ and solve to find that $x$ is exponential as a function of time, i.e. accelerating.
controlgroup
CC BY-SA 4.0
2024-12-20T07:00:03.393Z
@controlgroup that isn't correct. The "x" that's plotted is not proper distance and you would need to know the cosmological parameters to find it.
ProfRob
CC BY-SA 4.0
2024-12-20T13:26:32.287Z
My point is that you don’t need to graph it to know that expansion is exponential.
controlgroup
<universe><expansion>
Is it possible to conclude the expansion is accelerated only from distance modulus vs redshift
Fabien
# Is it possible to conclude the expansion is accelerated only from distance modulus vs redshift

(Asked by: username_0 on 2024-12-16)

Is it possible only from the average distance modulus and redshift Z in the set of data :

<https://supernova.lbl.gov/Union/figures/SCPUnion2.1_mu_vs_z.txt>

Which can be represented by :

[![enter image description here](https://i.sstatic.net/v8XcWjco.png)](https://i.sstatic.net/v8XcWjco.png)

To conclude that the expansion of the universe is accelerated ?

And above all, if yes, how ? With what calculation ?

**Tags:** universe, expansion
> **Comments:**
> **username_1**: Have you looked at the paper from which the data came?  
> **username_0**: Sadly it's very dense and exhaustive, it's hard for me to "extract" the essential step from it... Is there a way to resume how we deduce the type of expansion to some simple & essential step of calculation ?  
> **username_1**: Yes. Plot the solid line in your graph for different combinations of cosmological parameters. The one that fits the data best has $\Omega_\Lambda>0$. Whether this conclusively demonstrates acceleration will be discussed in the paper. Other evidence should be considered; principally the cosmic microwave background.  
> **username_2**: You can just note the Hubble constant's existence and definition as $v=\frac{\text{d}x}{\text{d}t}=Hx$ and solve to find that $x$ is exponential as a function of time, i.e. accelerating.  
> **username_1**: @controlgroup that isn't correct. The "x" that's plotted is not proper distance and you would need to know the cosmological parameters to find it.  
> **username_2**: My point is that you don’t need to graph it to know that expansion is exponential.  

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## 1 answers

### Answer by username_1 on 2024-12-20. Score: 2 | Accepted answer

Different combinations of cosmological parameters make different predictions for the detailed shape of the distance modulus versus redshift plot.

Meanwhile, the distance modulus of the supernovae is estimated from their peak apparent magnitude and an estimate of their peak absolute magnitude based on their light curve data, and their redshifts are measured quite precisely from spectra.

The various model curves are then compared with the data to see which one provides the best fit. Unfortunately, this fitting process is not well illustrated in the SCP Union 2.1 paper, but below is a plot from a much earlier paper (with sparser data), with "effective peak magnitude" playing the same role as distance modulus, showing multiple, labelled model curves.

[![Hubble diagram with multiple cosmological models](https://i.sstatic.net/ykexPrw0.gif)](https://i.sstatic.net/ykexPrw0.gif)

The sensitivity to late-time acceleration, as $\Omega\_\Lambda$ becomes greater than $\Omega\_M$, is because in an accelerated expansion , at a given distance (in the past) the supernovae would be receding slower than expected from a more uniform or decelerating expansion. The comparison is most sensitive to $\Omega\_\Lambda - \Omega\_M$, the difference in the accelerating and decelerating terms. You can see in the plot above that curves with $\Omega\_\Lambda - \Omega\_M <0$ (decelerating universes, broadly speaking) are lower than curves with $\Omega\_\Lambda - \Omega\_M >0$ (accelerating universes).

The plot below then illustrates how well the cosmological parameters are constrained from the data. The supernova data alone provide strong constraints only on $\Omega\_\Lambda - \Omega\_M$ but do indicate that the universe must be accelerating$^{\dagger}$ and that $\Omega\_\Lambda>0.4$, whilst the cosmic microwave background gives strong complementary constraints on $\Omega\_\Lambda + \Omega\_M$. The combination, and with the baryon acoustic oscillation data, suggest a geometrically flat, accelerating universe.

[![Omega lambda versus Omega m constraints](https://i.sstatic.net/yygIO20w.png)](https://i.sstatic.net/yygIO20w.png)

$\dagger$ The "[deceleration parameter](https://en.wikipedia.org/wiki/Deceleration_parameter)" is $q \simeq 0.5\Omega\_M - \Omega\_\Lambda$, which would be negative in an accelerating expansion. Looking at the possibilities enclosed by the blue ellipse in the second plot, you can convince yourself that $q$ must be less than zero and is about $q=-0.55$ for the best-fit concordance model.

> **Comments:**
> **username_0**: Thank you very much for your effort trying to answer my question but I did realize that I miss an important base to understand the universe expansion acceleration : cosmological parameters which I do not have a clue of what it is :( I guess I have to ask that question first...  (just in case : no way to explain universe expansion acceleration without all those ΩM−ΩΛ "cosmological parameters" ? :) )  

---

