Question

The fundamental observation underlying the Big Bang theory of cosmology is Edwin Hubble's 1929 discovery that the arrangement of galaxies throughout space is expanding. Like the photons of the cosmic microwave background, the light from distant galaxies is stretched to longer wavelengths by the expansion of the universe. This is not a Doppler shift: Except for their local motions around each other, the galaxies are essentially at rest in space; it is space itself that expands. The ratio of the wavelength of light received at Earth from a galaxy, \(\lambda_{\text {rec }}\), to its wavelength at emission, \(\lambda_{\text {emit }}\), is equal to the ratio of the scale factor (radius of curvature) \(a\) of the universe at reception to its value at emission. The redshift, \(z\), of the light-which is what Hubble could measure \(-\) is defined by \(1+z=\lambda_{\text {rec }} / \lambda_{\text {emit }}=a_{\text {rec }} / a_{\text {emit }}\). a) Hubble's Law states that the redshift, \(z\), of light from a galaxy is proportional to the galaxy's distance from Earth (for reasonably nearby galaxies): \(z \cong c^{-1} H \Delta s,\) where \(c\) is the vacuum speed of light, \(H\) is the Hubble constant, and \(\Delta s\) is the distance of the galaxy from Earth. Derive this law from the relationships described in the problem statement, and determine the Hubble constant in terms of the scale- factor function \(a(t)\). b) If the Hubble constant currently has the value \(H_{0}=72(\mathrm{~km} / \mathrm{s}) / \mathrm{Mpc},\) how far away is a galaxy whose light has the redshift \(z=0.10 ?\) (The megaparsec \((\mathrm{Mpc})\) is a unit of length equal to \(3.26 \cdot 10^{6}\) light-years. For comparison, the Great Nebula in Andromeda is approximately 0.60 Mpc from Earth.)

Short answer

Answer: The approximate distance to a galaxy with a redshift of 0.10 is 416.67 Mpc.

Step-by-step solution

Deriving Hubble's Law

Assuming that the wavelength of light emitted from the galaxy is \(\lambda_{\text {emit }}\) and the wavelength of light received at the Earth is \(\lambda_{\text {rec }}\). According to the problem statement, we have: $$ 1+z=\frac{\lambda_{\text {rec }}}{\lambda_{\text {emit }}}=\frac{a_{\text {rec}}}{a_{\text {emit}}} $$ Now, consider two nearby points in space. At the time of emission, the distance between them is \(\Delta s_{\text {emit}}\) and at the time of reception, the distance is \(\Delta s_{\text {rec}}\). According to the scale factor, we have: $$ \frac{\Delta s_{\text {rec}}}{\Delta s_{\text {emit}}}=\frac{a_{\text {rec}}}{a_{\text {emit}}} $$ In terms of redshift, we can write this as: $$ 1+z=\frac{\Delta s_{\text {rec}}}{\Delta s_{\text {emit}}} $$ Taking the time derivative of both sides, we get: $$ \frac{d(1+z)}{dt}=\frac{d(\Delta s_{\text {rec}})}{dt} $$ Since \(z\) is the redshift which Hubble could measure, the derivative of \(1+z\) in terms of \(t\) is very close to zero, so we can write: $$ 0 \approx \frac{d(\Delta s_{\text {rec}})}{dt} $$ Then, the velocity of the galaxy is: $$ v \approx \Delta s_{\text {rec}}\frac{d(1+z)}{dt} \Rightarrow v \cong c^{-1} H \Delta s $$ This is Hubble's Law.

The Hubble constant in terms of the scale-factor function \(a(t)\)

To find the Hubble constant in terms of \(a(t)\), we'll differentiate \(a(t)\) with respect to time \(t\): $$ H=\frac{\dot{a}(t)}{a(t)} $$ Where \(\dot{a}\) is the time derivative of the scale factor function.

Calculate the distance to a galaxy with a redshift of \(z=0.10\)

We're given that the value of the Hubble constant is \(H_0=72 \frac{\text{km/s}}{\text{Mpc}}\). Using Hubble's Law \(v \cong c^{-1}H\Delta s\) and the redshift value \(z=0.10\), we have: $$ \Delta s =\frac{c}{H_0}z $$ Plugging in the values of \(c=3 \times 10^5 \frac{\text{km}}{\text{s}}\), \(H_0=72\frac{\text{km/s}}{\text{Mpc}}\) and \(z=0.10\), we find the distance \(\Delta s\): $$ \Delta s = \frac{(3 \times 10^5 \frac{\text{km}}{\text{s}})}{(72\frac{\text{km/s}}{\text{Mpc}})}(0.10) \approx 416.67 \ \text{Mpc} $$ So, the distance to a galaxy with a redshift of \(z=0.10\) is approximately 416.67 Mpc.

Key concepts

Hubble's Law

Hubble's Law is a fundamental principle of astronomy that relates the expansion of the universe to the distances of galaxies from Earth. It states that the further away a galaxy is, the faster it appears to be moving away from us. This is not because galaxies are traveling through space at high speeds, but rather because space itself is expanding.

The formula for Hubble's Law is given by \(v = H_0 \times d\), where \(v\) is the galaxy's recessional velocity, \(H_0\) is the Hubble constant, and \(d\) is the distance of the galaxy from Earth. This provides us with a way to measure the rate of expansion of the universe. Understanding Hubble's Law is essential for cosmologists as it helps to estimate the age of the universe and provides evidence for the Big Bang theory.

Redshift

Redshift occurs when the wavelength of light is stretched, causing the light to appear more red than it originally was. This cosmic phenomenon is a key indicator of the universe's expansion. For example, when light from distant galaxies reaches us, we observe that their wavelengths have been lengthened, or 'redshifted,' due to the stretching of space itself.

Redshift is measured by the variable \(z\), where \(1 + z = \frac{\lambda_{\text{rec}}}{\lambda_{\text{emit}}}\), with \(\lambda_{\text{rec}}\) being the received wavelength of light, and \(\lambda_{\text{emit}}\) the emitted wavelength. The greater the redshift, the faster a galaxy is moving away, providing a direct measurement of the universe's expansion rate.

Cosmology

Cosmology is the scientific study of the large scale properties of the universe as a whole. It encompasses not just the study of the universe's structure and origin but also its ultimate fate. Cosmologists use theories like the Big Bang to explain how the universe came into being and employ mathematical models, physics, and astronomy to understand its past, present, and future.

Cosmology relies heavily on observable phenomena such as redshift and cosmic microwave background radiation to draw conclusions about the broad properties of the cosmos. These observations help in developing and refining models of the universe's expansion and provide insight into the possible existence of dark matter and dark energy.

Universe Expansion

The expansion of the universe is a critical discovery in modern astronomy, conceptualizing that the universe has been growing ever since its inception. According to this theory, distances between galaxies increase over time as space itself expands, rather than galaxies moving through space.

This phenomenon of expansion is not uniform; it's believed to be accelerating due to a mysterious force termed dark energy. The concept of universe expansion was first deduced from the observation of redshifted light from distant galaxies and is a vital piece of evidence supporting the Big Bang theory, which posits that the universe began as a singular, extremely dense and hot point roughly 13.8 billion years ago.