Suppose that there are \(n(z)=n_{0}(1+z)^{3}\) damped Ly \(\alpha\) clouds per \(\mathrm{Mpc}^{3}\) at redshift \(z\), each with cross-sectional area \(\sigma\). Explain why we expect to see through \(n(z) \sigma l\) of them along a length \(l\) of the path toward the quasar. Use Equation \(8.47\) to show that between \(z\) and \(z+\Delta z\) the path \(\Delta l=c \Delta z /[H(z)(1+z)]\), so the number per unit redshift is $$ \frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} \Delta z=\frac{n(z) \sigma c \Delta z}{H(z)(1+z)} \equiv \frac{n_{0} \sigma c}{H_{0}} \Delta z \frac{\mathrm{d} X(z)}{\mathrm{d} z}, $$ where (using Equation \(8.26\) for the second equality) $$ \frac{\mathrm{d} X(z)}{\mathrm{d} z}=\frac{H_{0}(1+z)^{2}}{H(z)}=\frac{(1+z)^{2}}{\sqrt{\Omega_{\mathrm{m}}(1+z)^{3}+\left(1-\Omega_{\text {tot }}\right)(1+z)^{2}+\Omega_{\Lambda}}} . $$ Show that, if the Universe is flat and \(\Omega_{10 t}=1\), then at early times $$ X(z)=\frac{2}{3 \sqrt{\Omega_{\mathrm{m}}}}\left[(1+z)^{3 / 2}-1\right] \text { while }(1+z)^{3} \gg \Omega_{\Lambda} / \Omega_{\mathrm{m}} . $$ Locally we find \(\mathrm{d} \mathcal{N} / \mathrm{d} z \approx 0.045\); if the cross-section \(\sigma\) does not change, show that we expect \(\mathrm{d} \mathcal{N} / \mathrm{d} z \approx 0.16\) at \(z=3\). At \(z=5\) we observe \(\mathrm{d} \mathcal{N} / \mathrm{d} z \approx 0.4\). Show that this is roughly twice what we expect if \(\sigma\) is constant: this result indicates that there were more absorbing clouds, or each was larger.

Step by step solution

01

Understanding the number of clouds along the path

Given the number density of damped Lyα clouds at redshift z is given by \(n(z)=n_{0}(1+z)^{3}\), the number of clouds we expect to see through along a path of length l towards the quasar can be obtained by multiplying the number density with the cross-sectional area \(\sigma\) and the path length \(l\). Thus, the expression becomes \(n(z) \sigma l\).

02

Path length in terms of redshift

Using Equation (8.47), the path length between z and z+\(\Delta z\) is given by \[\Delta l = \frac{c \Delta z}{H(z)(1+z)}\]where c is the speed of light and \(H(z)\) is the Hubble parameter at redshift z.

03

Number per unit redshift

To find the number of clouds per unit redshift, we use the path length derived in the previous step. The number of clouds in a redshift interval \(\Delta z\) is:\[\frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} \Delta z = \frac{n(z) \sigma c \Delta z}{H(z)(1+z)}\]Substituting \(n(z) = n_{0}(1+z)^{3}\) into the equation, we get:\[\frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} \Delta z = \frac{n_{0} \sigma c (1+z)^{3} \Delta z}{H(z)(1+z)} = \frac{n_{0} \sigma c \Delta z}{H(z)} (1+z)^{2}\]By definition:\[\frac{n_{0} \sigma c \Delta z}{H_{0}} \frac{(1+z)^{2}}{H(z)/H_{0}} = \frac{n_{0} \sigma c}{H_{0}} \frac{\mathrm{d} X(z)}{\mathrm{d} z} \Delta z\]

04

Using the second equality

Using Equation (8.26):\[\frac{\mathrm{d} X(z)}{\mathrm{d} z} = \frac{H_{0}(1+z)^{2}}{H(z)} \]For a flat Universe (\(\Omega_{tot} = 1\)), H(z) is given by:\[H(z) = H_{0} \sqrt{\Omega_{m}(1+z)^{3} + \Omega_{\Lambda}} \approx H_{0} \sqrt{\Omega_{m}(1+z)^{3}} \quad \text{when} \quad (1+z)^{3} \gg \frac{\Omega_{\Lambda}}{\Omega_{m}}.\]Hence:\[\frac{\mathrm{d} X(z)}{\mathrm{d} z} = \frac{H_{0}(1+z)^{2}}{H_{0} \sqrt{\Omega_{m}(1+z)^{3}}} = \frac{(1+z)^{2}}{(1+z)^{3/2}\sqrt{\Omega_{m}}} = \frac{(1+z)^{1/2}}{\sqrt{\Omega_{m}}}.\]

05

Expression for X(z) at early times

Integrating \(\frac{\mathrm{d} X(z)}{\mathrm{d} z}\), we get:\[X(z) = \int_{0}^{z} \frac{(1+z')^{1/2}}{\sqrt{\Omega_{m}}} \mathrm{d}z' = \frac{2}{3\sqrt{\Omega_{m}}} [(1+z)^{3/2} - 1].\]

06

Locally expected number evolution

Locally, \(\frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} \approx 0.045\). With \(n_{0}\sigma\) constant, at \(z=3\):\[\frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} = \frac{n_{0} \sigma c}{H_{0}} \frac{(1+3)^{3/2}}{\sqrt{\Omega_{m}}} \approx 0.16.\]

07

Effect of higher redshift

At \(z=5\), we observe \(\frac{\mathrm{d} \mathcal{N}}{\mathrm{d} z} \approx 0.4\), which is twice the expected value. Hence, either there were more clouds or \(\sigma\) was larger in the past.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Redshift

Redshift is a crucial concept in understanding the universe. It describes how the wavelength of light stretches as objects move apart. Imagine a fire truck siren. As it drives away, the sound's pitch lowers.
Similarly, when an object in space, like a galaxy, moves away, its light shifts to longer wavelengths, getting redder. This phenomenon is called redshift. In astronomy, we use 'z' to represent redshift.
The redshift z tells us by how much the universe has expanded since the light started its journey. For example, a redshift of z=2 means the universe has expanded to three times its size over that time period. Redshift helps us study the object's velocity and distance. It also provides clues about the universe's expansion rate and history.

Hubble parameter

The Hubble parameter, often denoted as H(z), plays a key role in cosmology. It measures the universe's expansion rate at a specific time. The parameter depends on the redshift z. In simpler terms, it tells us how fast galaxies are moving apart at different points in time.
The Hubble parameter today, known as H0 or the Hubble constant, represents the current expansion rate. For different redshifts, H(z) varies and can be calculated using cosmic parameters:

• \(H(z) = H_0 \sqrt{\Omega_m(1+z)^3 + \Omega_{\Lambda}}\)\

Here, \(\Omega_m\) is the matter density parameter and \(\Omega_{\Lambda}\) is the dark energy density parameter. As z increases, we delve into the universe's past.
For instance, the Hubble parameter helps us understand how the universe expanded faster in its early stages compared to now. Understanding H(z) is vital for interpreting galaxy movement and the universe's size at different epochs.

Cross-sectional area

The cross-sectional area \(\sigma\) is a crucial concept when studying damped Ly\(\alpha\) clouds. It represents the effective area a cloud presents for blocking or absorbing light that passes through it. Imagine a cloud as a circle in the sky. The cross-sectional area is like cutting the cloud along a flat plane and measuring the surface of that slice.
In the context of the exercise, \(\sigma\) affects how many clouds intercept light along the path to a distant quasar. If each cloud has a large cross-sectional area, we're more likely to observe many of them. The clouds' number density and their cross-sectional areas help determine the expected number of clouds:

• n(z) \(\sigma\) l

When \(\sigma\) remains constant over time, the expected number of clouds we detect directly reflects changes in the universe's expansion, represented by redshift and the Hubble parameter. However, variations in \(\sigma\) imply changes in cloud size or structure over time.