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Chapter 22: Problem 50

The universe today has an average density \(\rho_{0}=3 \times 10^{-28} \mathrm{kg} / \mathrm{m}^{3}\) Assuming that the average density depends on the scale factor, as \(\rho=\rho_{0} / R_{\mathrm{U}}^{3},\) what was the scale factor of the universe when its average density was about the same as Earth's atmosphere at sea level \(\left(\rho=1.23 \mathrm{kg} / \mathrm{m}^{3}\right) ?\)

Short Answer

Expert verified
The scale factor of the universe was approximately \( 2.88 \times 10^{-10} \).

Step by step solution

01

Identify Given Values

The average density of the universe today is \( \rho_0 = 3 \times 10^{-28} \text{kg/m}^3 \) and the density when it was similar to the Earth's atmospheric density at sea level is \( \rho = 1.23 \text{kg/m}^3 \).
02

Understand the Relationship Between Density and Scale Factor

The given relationship between the average density \( \rho \) and the scale factor \( R_{\text{U}} \) is \( \rho = \frac{\rho_0}{R_{\text{U}}^3} \).
03

Set Up the Equation

Substitute the given values into the relationship: \[ \rho = \frac{\rho_0}{R_{\text{U}}^3} \Rightarrow 1.23 = \frac{3 \times 10^{-28}}{R_{\text{U}}^3} \]
04

Solve for the Scale Factor \( R_{\text{U}} \)

Rearrange the equation to solve for \( R_{\text{U}} \): \[ R_{\text{U}}^3 = \frac{3 \times 10^{-28}}{1.23} \Rightarrow R_{\text{U}}^3 = 2.439 \times 10^{-28} \]
05

Calculate the Scale Factor

Take the cube root of both sides to find \( R_{\text{U}} \): \[ R_{\text{U}} = \( 2.439 \times 10^{-28} \)^{1/3} \approx 2.88 \times 10^{-10} \]

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Key Concepts

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

Scale Factor of the Universe
The scale factor of the universe is a crucial concept in cosmology. It reflects how distances within the universe change over time. We denote it by the symbol \( R_{\text{U}} \).

In simpler terms, it helps us understand how the entire universe has expanded or contracted since the Big Bang.
  • When \(R_{\text{U}} = 1\), we refer to the current size of the universe.
  • In the early universe, \( R_{\text{U}} \) was much smaller, showing that everything was much closer together.
In our exercise, we determined when the universe had the same density as Earth's atmosphere at sea level. This needed us to calculate the scale factor using given density values.
Cosmological Equations
Cosmological equations are foundational in describing the universe's evolution. They provide mathematical models to explain how different factors like density, pressure, and scale factor change with time.

A key equation we used in the exercise is the relationship between density and the scale factor:

\[ \rho = \frac{\rho_0}{R_{\text{U}}^3} \]

This states that the density \(\rho \) of the universe is inversely proportional to the cube of the scale factor \( R_{\text{U}} \). As the universe expands ( \( R_{\text{U}} \) increases), the density decreases, and vice versa.
  • This relationship is paramount in interpreting how conditions in the universe have changed over time.
  • In our specific problem, we applied this equation to find the previous state of the universe's density and scale factor.
Density and Volume Relationships
The link between density and volume is fundamental in understanding cosmology. Density (\rho ) is defined as mass per unit volume:

\[ \rho = \frac{m}{V} \]

When we discuss the universe, the average density changes as the volume (and thus the scale factor) changes.
  • As the universe expands, its volume increases, and the average density decreases.
  • Conversely, when the universe was smaller, its volume was lesser, and therefore, its density was higher.
In the exercise, knowing the density \( 1.23 \text{kg}/\text{m}^3 \) and how it relates to the today’s average density \( 3 \times 10^{-28} \text{kg}/\text{m}^3 \), allowed us to calculate how much the universe has expanded. This insight is essential to explore how the universe evolved from a dense, hot state into the vast cosmos we observe today.

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Most popular questions from this chapter

Astronomers will never directly observe the first few minutes of the universe, because a. the universe was opaque at that time. b. the universe is too large now. c. there were no particles or other matter to see. d. there were no photons.

The force that acts to slow the expansion of the universe is the ______ force. a. gravitational b. electromagnetic c. strong nuclear d. weak nuclear

What is the flatness problem, and why has it created difficulties for cosmologists?

Assume a planet's orbit is perfectly circular as it travels in the gravitational well of its star. If this were true, would the orbit's circumference be greater than, less than, or equal to \(2 \pi\) times the radius of the orbit?

When a particle and an antiparticle come together, they a. annihilate each other, releasing photons. b. create a black hole. c. release astronomical amounts of energy. d. create new particles.
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