Problem 46 - An Introduction to Modern Astrophysics

Question

In the matter era, the distance to the particle horizon for a closed, one- component universe of pressureless dust is given by $$d_{h}(z)=\frac{c}{H_{0}(1+z) \sqrt{\Omega_{0}-1}} \cos ^{-1}\left[1-\frac{2\left(\Omega_{0}-1\right)}{\Omega_{0}(1+z)}\right]$$. In this problem, you will derive this expression for \(d_{h} .\) First change variables in Eq. to obtain \\[d_{h}(t)=R(t) \int_{0}^{\frac{1}{1+2}} \frac{c d R}{R(d R / d t)},\\] where the limits of integration range from \(R=0\) (at \(t=0\) ) to \(R=1 /(1+z)\) (at time \(t\) ). Then show that \\[ \left(\frac{d R}{d t}\right)^{2}=H_{0}^{2}\left(\frac{\Omega_{0}}{R}-\Omega_{0}+1\right) \\]. and make this substitution into the denominator of the integral. You may find \\[ \int \frac{d x}{\sqrt{b x-a x^{2}}}=\frac{1}{\sqrt{a}}\left[\cos ^{-1}\left(1-\frac{2 a x}{b}\right)-\frac{\pi}{2}\right] \\] to be useful.

Step-by-Step Solution

Answer

The expression for the particle horizon is derived using substitution and integration formulas.

Step 1: Rearrange the Distance Equation

We start with the substitution of variables in the distance equation. Original equation is \[ d_{h}(t)=R(t) \int_{0}^{\frac{1}{1+z}} \frac{c \ dR}{R(dR/dt)}. \] Change variables, noting that \( R \) evolves with time as \( R = 1/(1+z) \) when \( t = t \) and \( R = 0 \) when \( t = 0 \).

Step 2: Express Derivative of R

Next, we are given the expression for the derivative \( \left(\frac{dR}{dt}\right)^{2} = H_{0}^{2}\left(\frac{\Omega_{0}}{R} - \Omega_{0} + 1\right) \). This expression reflects the cosmological parameters of a closed universe with matter domination.

Step 3: Substitute Derivative into Integral

Substitute \( \frac{dR}{dt} \) in the denominator of the integral from step 1:\[ d_{h}(t) = R(t) \int_{0}^{1/(1+z)} \frac{c \ dR}{R \sqrt{H_{0}^{2}\left(\frac{\Omega_{0}}{R} - \Omega_{0} + 1\right)}}. \]

Step 4: Solve the Integral with a Helpful Formula

Use the provided integral formula \[ \int \frac{dx}{\sqrt{bx - ax^{2}}} = \frac{1}{\sqrt{a}} \left[ \cos^{-1}\left(1 - \frac{2ax}{b}\right) - \frac{\pi}{2} \right]. \] Identify compatible terms where \( b = \Omega_{0} \) and \( a = \Omega_{0} - 1 \).

Step 5: Evaluate and Simplify the Expression

Evaluate the integral \( \int_{0}^{1/(1+z)} \frac{dR}{\sqrt{\frac{\Omega_{0}}{R} - \Omega_{0} + 1}} \) using the evaluation formula and limits. Compute \[ \int_{0}^{1/(1+z)} \frac{dR}{\sqrt{\Omega_{0} R - (\Omega_{0} - 1) R^{2}}} \approx \frac{1}{H_{0}\sqrt{\Omega_{0} - 1}} \cos^{-1}\left(1 - \frac{2(\Omega_{0} - 1)}{\Omega_{0}(1+z)}\right). \]

Step 6: Write the Final Expression for Particle Horizon

Include the factor \( R(t) = 1/(1+z) \) to write the complete expression:\[ d_{h}(z) = \frac{c}{H_{0}(1+z)\sqrt{\Omega_{0} - 1}} \cos^{-1}\left(1 - \frac{2(\Omega_{0} - 1)}{\Omega_{0}(1+z)}\right). \]

Key Concepts

Closed Universe

In cosmology, a universe is termed "closed" if it has a finite volume and curves back onto itself, much like the surface of a sphere. This type of universe is one of the three possible geometric descriptions proposed by Friedmann, following Einstein's field equations in General Relativity.
In a closed universe, the density parameter \( \Omega_0 \) is greater than one. This implies that the gravitational pull is strong enough to eventually halt the expansions driven by the Big Bang, leading to a scenario where the universe would eventually collapse back onto itself. This fate is often called a "Big Crunch."
The curvature of space in a closed universe causes unique phenomena. For instance, it affects the paths of light and can lead to intriguing properties, like repeated appearances of the same object in different directions. Moreover, the conditions in a closed universe influence the calculation of cosmic distances, such as the particle horizon, which marks the limit of the observable universe.

Pressureless Dust

A universe described by "pressureless dust" is a model where matter is predominantly non-relativistic with negligible pressure compared to its energy density. This simplification helps in formulating equations that describe cosmic evolution.
In this scenario, the energy-momentum tensor, a fundamental component of general relativity equations, only involves mass density. This aligns with how galaxies and other massive structures interact gravitationally over large scales in our universe, as they have relatively low speeds compared to the speed of light.
Using the "pressureless dust" approximation in cosmological models simplifies calculations by removing pressure terms from equations, leading to clearer insights into how a universe might develop over time. This concept was utilized in solving the given problem to obtain an expression for the particle horizon by focusing on gravitational interactions alone.

Cosmology

Cosmology is the scientific study of the universe's origin, evolution, structure, and eventual fate. It combines aspects of astronomy and physics to build models that describe and predict the behavior of cosmic bodies and phenomena.
A major aspect of cosmology is understanding how different types of universes can arise based on initial conditions and physical laws, such as gravity and electrodynamics. Modern cosmological models often begin with the Big Bang, a theory describing the rapid expansion from an extremely hot and dense state.
Cosmologists use mathematical frameworks to predict the behavior of the universe, employing equations like the Friedmann equations which stem from Einstein’s general theory of relativity. These models are pivotal in determining crucial features like the particle horizon, a boundary that defines the observable extent of the universe today, and are explored with tools such as the Hubble parameter \( H_0 \) and density parameters \( \Omega_0 \).

Redshift

Redshift in cosmology refers to how the light from distant galaxies shifts to longer wavelengths as the universe expands. This shift is visible in the light's spectrum and is measured using the redshift parameter \( z \). It provides vital information on how fast galaxies are moving away and, consequently, how the universe is expanding.
The redshift phenomenon is a critical observational tool for astronomers and cosmologists. It leads to the idea of the "expanding universe" and enables calculations of cosmic distances and velocities, forming a basis for measuring astronomical phenomena.
The formula \( d_h(z) \) for the particle horizon includes the \( (1+z) \) factor as it accounts for the expansion between the light emission and present observation. As \( z \) increases, indicating older light from farther away, this parameter helps describe the scale factor of the universe at different times. Understanding redshift is thus essential for interpreting the dynamics of a universe, especially when coupled with concepts like particle horizons in a closed universe.