Problem 34 - University Physics with Modern Physics

Question

(a) According to the Hubble law, what is the distance \(r\) from us for galaxies that are receding from us with a speed \(c ?\) (b) Explain why the distance calculated in part (a) is the size of our observable universe (ignoring any slowing of the expansion of the universe due to gravitational attraction).

Step-by-Step Solution

Answer

(a) The distance is \( r = \frac{c}{H_0} \). (b) This distance defines the edge of the observable universe, as light from beyond cannot reach us.

Step 1: Understanding Hubble's Law

The Hubble Law describes the relationship between the speed at which a galaxy is receding from us and its distance from us. Mathematically, it is expressed as \( v = H_0 r \), where \( v \) is the speed of the galaxy, \( H_0 \) is the Hubble constant, and \( r \) is the distance to the galaxy.

Step 2: Identify Given Values

In this problem, the speed \( v \) is given as the speed of light \( c \). This represents the fastest speed at which any object can travel, which is approximately \( 3 \times 10^8 \) meters per second.

Step 3: Solve for Distance

Using Hubble's Law \( v = H_0 r \), we can solve for \( r \) by substituting \( v = c \). This gives us \( r = \frac{c}{H_0} \). This equation allows us to calculate how far away a galaxy would be if it were receding at the speed of light.

Step 4: Interpret the Result

The distance \( r = \frac{c}{H_0} \) represents the maximum theoretical distance we can observe in our universe. This is because galaxies at this distance are receding at the speed of light, meaning any light it emits now will never reach us, thus marking the edge of our observable universe.

Step 5: Answer the Second Part

The distance calculated, \( r = \frac{c}{H_0} \), is significant because it bounds the observable universe. Objects further away than this distance recede faster than the speed at which their light can reach us, making them unobservable.

Key Concepts

Hubble constant

The Hubble Constant, symbolized as \( H_0 \), is a critical figure in cosmology. It relates the speed of a distant galaxy to its distance from us.

Imagine \( H_0 \) as a way to measure the universe's expansion rate. It tells us how fast galaxies are moving away from us.

• A higher Hubble Constant means a faster expansion rate.
• Unit of \( H_0 \): typically km/s/Mpc, where Mpc stands for megaparsecs, a unit of distance.

A simplified version of Hubble's Law is \( v = H_0 \times r \), where \( v \) is velocity and \( r \) is the distance. This equation defines that the farther away a galaxy is, the faster it appears to be moving away, which is because of the universe's expansion.

Understanding \( H_0 \) helps us comprehend not just how fast things are moving now, but it also gives us clues about the universe's past and future.

observable universe

The observable universe refers to the part of the universe that we can see or observe from Earth.

The light from galaxies within this cosmos can reach us because they aren't moving away faster than the light can traverse the distance. Therefore, it becomes a finite bubble around us, defined by the furthest distance light could have traveled since the Big Bang.

• Bounded by a maximum distance based on light speed \( c \) and the Hubble constant \( H_0 \).
• Everything beyond this bubble is currently unobservable because light has not had the time to travel to us yet.

The formula \( r = \frac{c}{H_0} \) calculates this boundary. Any galaxy at this distance is receding at the speed of light, defining the horizon of our observable universe.

cosmological expansion

Cosmological expansion describes the stretching of space over time.

This concept was deduced from observing distant galaxies and noticing that they are all moving away from us.

• The idea that space itself is expanding comes from Einstein's general theory of relativity.
• Every point in the universe appears to be moving away from each other, not just galaxies from us.

This expansion impacts the observable universe by increasing the distance to galaxies over time, implying that parts of the universe that are observable today might not be in the future as structures move beyond our observational boundaries. Cosmological expansion is not about galaxies moving through space, but space itself expanding. This means even if a galaxy is stationary relative to its local space, it gets further from us as the universe expands. Understanding this helps clarify why the universe looks the way it does today.