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<!DOCTYPE html>

<head>

<title>Make a plot with both redshift and universe age axes using astropy.cosmology — Redshift Plot</title>

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<h1>Make a plot with both redshift and universe age axes using astropy.cosmology</h1>

<div>

<h2> Contents </h2>

</div>

<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#authors">Authors</a></li>

<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#learning-goals">Learning Goals</a></li>

<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#keywords">Keywords</a></li>

<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#summary">Summary</a></li>

<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#exercise">Exercise</a></li>

</ul>

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<h1>Make a plot with both redshift and universe age axes using astropy.cosmology<a class="headerlink" href="#make-a-plot-with-both-redshift-and-universe-age-axes-using-astropy-cosmology" title="Link to this heading">#</a></h1>

<h2>Authors<a class="headerlink" href="#authors" title="Link to this heading">#</a></h2>

<p>Neil Crighton, Stephanie T. Douglas</p>

</section>

<h2>Learning Goals<a class="headerlink" href="#learning-goals" title="Link to this heading">#</a></h2>

<li><p>Plot relationships using <code class="docutils literal notranslate"><span class="pre">matplotlib</span></code></p></li>

<li><p>Add a second axis to a <code class="docutils literal notranslate"><span class="pre">matplotlib</span></code> plot</p></li>

<li><p>Relate distance, redshift, and age for two different types of cosmology using <code class="docutils literal notranslate"><span class="pre">astropy.cosmology</span></code></p></li>

</ul>

</section>

<h2>Keywords<a class="headerlink" href="#keywords" title="Link to this heading">#</a></h2>

<p>units, physics, cosmology, matplotlib</p>

</section>

<h2>Summary<a class="headerlink" href="#summary" title="Link to this heading">#</a></h2>

<p>Each redshift corresponds to an age of the universe, so if you’re plotting some quantity against redshift, it’s often useful show the universe age too. The relationship between the two changes depending the type of cosmology you assume, which is where <code class="docutils literal notranslate"><span class="pre">astropy.cosmology</span></code> comes in. In this tutorial we’ll show how to use the tools in <code class="docutils literal notranslate"><span class="pre">astropy.cosmology</span></code> to make a plot like this:</p>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Set up matplotlib</span>

<span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>

<span class="o">%</span><span class="k">matplotlib</span> inline

</pre></div>

</div>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span><span class="w"> </span><span class="nn">IPython.display</span><span class="w"> </span><span class="kn">import</span> <span class="n">Image</span>

<span class="n">Image</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s2">"ang_dist.png"</span><span class="p">,</span> <span class="n">width</span><span class="o">=</span><span class="mi">500</span><span class="p">)</span>

</pre></div>

</div>

</a>

</div>

<p>We start with a cosmology object. We will make a flat cosmology (which means that the curvature density <span class="math notranslate nohighlight">$\Omega_k=0$</span>) with a hubble parameter of <span class="math notranslate nohighlight">$70$</span> km/s/Mpc and matter density <span class="math notranslate nohighlight">$\Omega_M=0.3$</span> at redshift 0. The <code class="docutils literal notranslate"><span class="pre">FlatLambdaCDM</span></code> cosmology then automatically infers that the dark energy density <span class="math notranslate nohighlight">$\Omega_\Lambda$</span> must <span class="math notranslate nohighlight">$=0.7$</span>, because <span class="math notranslate nohighlight">$\Omega_M + \Omega_\Lambda + \Omega_k = 1$</span>.</p>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span><span class="w"> </span><span class="nn">astropy.cosmology</span><span class="w"> </span><span class="kn">import</span> <span class="n">FlatLambdaCDM</span>

<span class="kn">import</span><span class="w"> </span><span class="nn">astropy.units</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">u</span>

<span class="c1"># In this case we just need to define the matter density</span>

<span class="c1"># and hubble parameter at z=0.</span>

<span class="c1"># Note the default units for the hubble parameter H0 are km/s/Mpc.</span>

<span class="c1"># We will pass in a `Quantity` object with the units specified.</span>

<span class="n">cosmo</span> <span class="o">=</span> <span class="n">FlatLambdaCDM</span><span class="p">(</span><span class="n">H0</span><span class="o">=</span><span class="mi">70</span> <span class="o">*</span> <span class="n">u</span><span class="o">.</span><span class="n">km</span> <span class="o">/</span> <span class="n">u</span><span class="o">.</span><span class="n">s</span> <span class="o">/</span> <span class="n">u</span><span class="o">.</span><span class="n">Mpc</span><span class="p">,</span> <span class="n">Om0</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>

</pre></div>

</div>

<p>Note that we could instead use one of the built-in cosmologies, like <code class="docutils literal notranslate"><span class="pre">WMAP9</span></code> or <code class="docutils literal notranslate"><span class="pre">Planck13</span></code>, in which case we would just redefine the <code class="docutils literal notranslate"><span class="pre">cosmo</span></code> variable.</p>

<p>Now we need an example quantity to plot versus redshift. Let’s use the angular diameter distance, which is the physical transverse distance (the size of a galaxy, say) corresponding to a fixed angular separation on the sky. To calculate the angular diameter distance for a range of redshifts:</p>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span><span class="w"> </span><span class="nn">numpy</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">np</span>

<span class="n">zvals</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">)</span>

<span class="n">dist</span> <span class="o">=</span> <span class="n">cosmo</span><span class="o">.</span><span class="n">angular_diameter_distance</span><span class="p">(</span><span class="n">zvals</span><span class="p">)</span>

</pre></div>

</div>

<p>Note that we passed an array of redshifts to <code class="docutils literal notranslate"><span class="pre">cosmo.angular_diameter_distance</span></code> and it produced a corresponding array of distance values, one for each redshift. Let’s plot them:</p>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>

<span class="n">ax</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">add_subplot</span><span class="p">(</span><span class="mi">111</span><span class="p">)</span>

</pre></div>

</div>

<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[<matplotlib.lines.Line2D at 0x7f1b93ab41a0>]

</pre></div>

</div>

</div>

<p>To check the units of the angular diameter distance, take a look at the unit attribute:</p>

</pre></div>

</div>

\[\mathrm{Mpc}\]</div>

</div>

<p>Now let’s put some age labels on the top axis. We’re going to pick a series of round age values where we want to place axis ticks. You may need to tweak these depending on your redshift range to get nice, evenly spaced ticks.</p>

<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">ages</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mi">13</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">,</span> <span class="mi">6</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mf">1.5</span><span class="p">,</span> <span class="mf">1.2</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span> <span class="o">*</span> <span class="n">u</span><span class="o">.</span><span class="n">Gyr</span>

</pre></div>

</div>

<p>To link the redshift and age axes, we have to find the redshift corresponding to each age. The function <code class="docutils literal notranslate"><span class="pre">z_at_value</span></code> does this for us.</p>

<span class="n">ageticks</span> <span class="o">=</span> <span class="p">[</span><span class="n">z_at_value</span><span class="p">(</span><span class="n">cosmo</span><span class="o">.</span><span class="n">age</span><span class="p">,</span> <span class="n">age</span><span class="p">)</span> <span class="k">for</span> <span class="n">age</span> <span class="ow">in</span> <span class="n">ages</span><span class="p">]</span>

</pre></div>

</div>

<p>Now we make the second axes, and set the tick positions using these values.</p>

<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">ageticks</span><span class="p">)</span>

</pre></div>

</div>

<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[<matplotlib.axis.XTick at 0x7f1b93b7c560>,

<matplotlib.axis.XTick at 0x7f1b941a5dc0>,

<matplotlib.axis.XTick at 0x7f1b93b0e930>,

<matplotlib.axis.XTick at 0x7f1b939a0290>,

<matplotlib.axis.XTick at 0x7f1b939a0ce0>,

<matplotlib.axis.XTick at 0x7f1b939a1670>,

<matplotlib.axis.XTick at 0x7f1b93b7f5f0>,

<matplotlib.axis.XTick at 0x7f1b939a1e80>,

<matplotlib.axis.XTick at 0x7f1b939a2870>,

<matplotlib.axis.XTick at 0x7f1b939a3260>,

<matplotlib.axis.XTick at 0x7f1b939a3c20>]

</pre></div>

</div>

</div>

<p>We have ticks on the top axis at the correct ages, but they’re labelled with the redshift, not the age. We can fix this by setting the tick labels by hand.</p>

<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">ageticks</span><span class="p">)</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticklabels</span><span class="p">([</span><span class="s2">"</span><span class="si">{:g}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">age</span><span class="p">)</span> <span class="k">for</span> <span class="n">age</span> <span class="ow">in</span> <span class="n">ages</span><span class="o">.</span><span class="n">value</span><span class="p">])</span>

</pre></div>

</div>

<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>[Text(0.03426542424730703, 1, '13'),

Text(0.30529969870539037, 1, '10'),

Text(0.56227619972859, 1, '8'),

Text(0.939431308199015, 1, '6'),

Text(1.2114538180402321, 1, '5'),

Text(1.5867361300254537, 1, '4'),

Text(2.153211270001333, 1, '3'),

Text(3.150381753523927, 1, '2'),

Text(4.035713900198318, 1, '1.5'),

Text(4.847670846184226, 1, '1.2'),

Text(5.606047443188419, 1, '1')]

</pre></div>

</div>

</div>

<p>We need to make sure the top and bottom axes have the same redshift limits. They may not line up properly in the above plot, for example, depending on your setup (the age of the universe should be ~13 Gyr at z=0).</p>

<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">ageticks</span><span class="p">)</span>

</pre></div>

</div>

<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>(0.0, 5.9)

</pre></div>

</div>

</div>

<p>We’re almost done. We just need to label all the axes, and add some minor ticks. Let’s also tweak the y axis limits to avoid putting labels right near the top of the plot.</p>

<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">ageticks</span><span class="p">)</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Time since Big Bang (Gyr)"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Redshift"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Angular diameter distance (Mpc)"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">minorticks_on</span><span class="p">()</span>

</pre></div>

</div>

</div>

<p>Now for comparison, let’s add the angular diameter distance for a different cosmology, from the Planck 2013 results. And then finally, we save the figure to a png file.</p>

<span class="n">dist2</span> <span class="o">=</span> <span class="n">Planck13</span><span class="o">.</span><span class="n">angular_diameter_distance</span><span class="p">(</span><span class="n">zvals</span><span class="p">)</span>

<span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">))</span>

<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">zvals</span><span class="p">,</span> <span class="n">dist2</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Planck 2013"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">zvals</span><span class="p">,</span> <span class="n">dist</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$h=0.7,\ \Omega_M=0.3,\ \Omega_\Lambda=0.7$"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">frameon</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">loc</span><span class="o">=</span><span class="s2">"lower right"</span><span class="p">)</span>

<span class="n">ax2</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">twiny</span><span class="p">()</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xticks</span><span class="p">(</span><span class="n">ageticks</span><span class="p">)</span>

<span class="n">ax2</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Time since Big Bang (Gyr)"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"Redshift"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"Angular diameter distance (Mpc)"</span><span class="p">)</span>

<span class="n">ax</span><span class="o">.</span><span class="n">minorticks_on</span><span class="p">()</span>

<span class="n">fig</span><span class="o">.</span><span class="n">savefig</span><span class="p">(</span><span class="s2">"ang_dist.png"</span><span class="p">,</span> <span class="n">dpi</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">bbox_inches</span><span class="o">=</span><span class="s2">"tight"</span><span class="p">)</span>

</pre></div>

</div>

<div class="output stderr highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span><>:8: SyntaxWarning: invalid escape sequence '\ '

<>:8: SyntaxWarning: invalid escape sequence '\ '

/tmp/ipykernel_2656/1680874098.py:8: SyntaxWarning: invalid escape sequence '\ '

ax.plot(zvals, dist, label="$h=0.7,\ \Omega_M=0.3,\ \Omega_\Lambda=0.7$")

</pre></div>

</div>

</div>

<p><code class="docutils literal notranslate"><span class="pre">bbox_inches='tight'</span></code> automatically trims any whitespace from around the plot edges.</p>

</section>

<h2>Exercise<a class="headerlink" href="#exercise" title="Link to this heading">#</a></h2>

<p>Well, almost done. Notice that we calculated the times on the upper axis using the original cosmology, not the new cosmology based on the Planck 2013 results. So strictly speaking, this axis applies only to the original cosmology, although the difference between the two is small. As an exercise, you can try plot two different upper axes, slightly offset from each other, to show the times corresponding to each cosmology. Take a look at the first answer to <a class="reference external" href="http://stackoverflow.com/questions/7733693/matplotlib-overlay-plots-with-different-scales">this question on Stack Overflow</a> for some hints on how to go about this.</p>

</section>

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repo: "astropy-learn/tutorial--redshift-plot",

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