Problem 38 - Astronomy: At Play in the Cosmos

Question

By what factor had the Universe expanded since recombination when the peak wavelength of the CMB was 0.00038 meter?

Step-by-Step Solution

Answer

The Universe has expanded by a factor of approximately 2.79.

Step 1: Understand the Problem

We need to find the factor by which the Universe has expanded since the epoch of recombination. This involves the change in the peak wavelength of the Cosmic Microwave Background (CMB) radiation, from the time of recombination to the present.

Step 2: Know the Peak Wavelength Condition

During recombination, the Universe became transparent, and the CMB was released. At that time, the peak wavelength was much shorter than it is now. We know the current peak wavelength of the CMB is approximately \(1.06 \times 10^{-3}\) meter.

Step 3: Apply the Expansion Factor

The peak wavelength of the CMB has stretched due to the expansion of the Universe. This `stretching` can be represented as \(\lambda_{now} = \lambda_{then} \times \text{Scale Factor}\), where \(\lambda_{now} = 1.06 \times 10^{-3}\) meter and initially, \(\lambda_{then} = 0.00038\) meter.

Step 4: Solve for the Scale Factor

Rearrange the equation to find the Scale Factor: \(\text{Scale Factor} = \frac{\lambda_{now}}{\lambda_{then}}\). Plug in the values: \(\text{Scale Factor} = \frac{1.06 \times 10^{-3}}{0.00038}\).

Step 5: Calculate the Scale Factor

Perform the division: \(\text{Scale Factor} = \frac{1.06 \times 10^{-3}}{3.8 \times 10^{-4}} \approx 2.78947\). Thus, the Universe has expanded by a factor of approximately \(2.78947\) since recombination.

Key Concepts

Expansion of the Universe

The concept of the expansion of the Universe is one of the central pillars in cosmology. It refers to the idea that the Universe has been growing ever since the Big Bang, causing galaxies to move away from each other. This expansion affects everything at the cosmic scale, including the light emitted by stars and galaxies.

As the Universe expands, so does the wavelength of light traveling across it, a phenomenon known as "cosmological redshift." This is helpful in understanding the changes observed in the Cosmic Microwave Background (CMB) radiation. The CMB is the afterglow of the Big Bang and serves as a snapshot of the early Universe.

If we consider the Universe's expansion, we notice that it also impacts the light and radiation emitted during critical early epochs, like the recombination epoch. Understanding this expansion helps scientists calculate how much the Universe has stretched since these early periods. In our scenario, the change in the peak wavelength of the CMB from the time of recombination to now signifies this cosmic stretching.

Recombination Epoch

The recombination epoch is a term describing a period in the early Universe when atoms began to form. During this time, about 378,000 years after the Big Bang, electrons combined with protons to create neutral hydrogen atoms. This process was crucial because it caused the Universe to transform dramatically.

Before recombination, the Universe was opaque, full of free electrons that scattered photons similar to how light is scattered in a fog. This scattering made it impossible for light to travel freely. Once recombination occurred, the fog "lifted," and the Universe became transparent.

This transparency allowed CMB radiation to travel uninterrupted, eventually reaching us here on Earth. By studying the CMB, we gain insight into the conditions of the early Universe right after recombination. Understanding the recombination epoch not only highlights a transformative period but also explains why the CMB is such a useful tool for studying the Universe.

Peak Wavelength

The term "peak wavelength" refers to the wavelength at which a body's emission spectrum is at its most intense. For the Cosmic Microwave Background, the peak wavelength is especially significant because it provides clues about the temperature of the early Universe.

Initially, the CMB's peak wavelength was approximately 0.00038 meters. As the Universe expanded, this wavelength elongated due to the stretching effect brought about by cosmic expansion. Today, it measures around 1.06 micrometers, considerably longer than during the recombination epoch.

This change in wavelength is directly tied to the Universe expanding. By calculating how much the peak wavelength has changed, we can determine the scale of the expansion. Such measurements are critical, as they aid scientists in understanding the dynamics of the Universe's growth over billions of years. The peak wavelength serves as a cosmic timekeeper, recording the expansion history of the Universe.