Problem 37 - Astronomy: At Play in the Cosmos

Question

What value would you expect for the peak wavelength of the CMB if the Universe had expanded by a factor of 800 since recombination?

Step-by-Step Solution

Answer

The peak wavelength of the CMB would be 800 times longer than at recombination.

Step 1: Understand the Initial State

At the time of recombination, the cosmic microwave background (CMB) had a peak wavelength much shorter than what we observe today, due to the higher energy levels intrinsic to that era.

Step 2: Learn About Expansion Effects

The Universe has expanded, causing the wavelength of the CMB to stretch (redshift). This relationship is described by the formula \( \lambda_0 = z \cdot \lambda \), where \( z \) is the expansion factor, \( \lambda \) is the original peak wavelength, and \( \lambda_0 \) is the current peak wavelength.

Step 3: Calculate the New Peak Wavelength

Given an expansion factor \( z = 800 \), the peak wavelength after expansion is \( \lambda_0 = 800 \cdot \lambda \). Since original peak wavelengths are typically on the order of micrometers at recombination, substituting conservatively estimates the new peak wavelength. Exact preceding values are not needed for this exercise.

Key Concepts

Recombination Era

The recombination era is a significant period in the early Universe, occurring around 378,000 years after the Big Bang. During this time, the temperature dropped sufficiently for protons and electrons to combine into neutral hydrogen atoms. This combination allowed the Universe, which was previously opaque, to become transparent to light for the first time. This light is what we currently detect as the Cosmic Microwave Background (CMB).

Key points during this era include:

• The universe's temperature was approximately 3000 Kelvin, conducive for the electrons to recombine with nuclei without being immediately ionized.
• This era marked the decoupling of matter and radiation, enabling photons to travel freely through the Universe.
• The peak wavelength of CMB was much shorter due to the high-energy state of photons at that time.

Universal Expansion

Universal expansion refers to the process where the Universe has been stretching or expanding since the Big Bang. This expansion is driven by dark energy, causing galaxies to move away from each other. The expansion impacts everything in the Universe, including the cosmic microwave background (CMB).

Effects of Expansion:

• As space expands, the wavelengths of the photons in the CMB also stretch, making them appear longer and shifting toward the red end of the spectrum, a process known as redshift.
• The rate of expansion can be described by the scale factor or expansion factor, which quantifies how much the Universe has grown at any given time.
• This expansion is a key factor in explaining the observable dynamics of the Universe and its structure today.

Redshift

Redshift is a term used in astrophysics to describe how light or other forms of electromagnetic radiation from objects in space shift to longer wavelengths as they move away from us. This shift makes objects appear redder, hence the name "redshift." In the context of the cosmic microwave background (CMB), redshift is a result of the expanding Universe.

Here’s what happens:

• The further away a galaxy or cosmic object is, the faster it seems to move away from us due to universal expansion, leading to a larger redshift.
• CMB photons have undergone significant redshift from their original wavelengths at the recombination era to the much longer wavelengths we observe today.
• This redshift helps scientists calculate the expansion factor of the Universe, which in the problem was given as a factor of 800.

The measurement of redshift provides insights into the history and scale of the Universe's expansion, offering clues about its age, size, and future fate.