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 the space itself that expands. The ratio of the wavelength of light \(\lambda_{\text {rec }}\) Earth receives from a galaxy to its wavelength \(\lambda_{\text {emit }}\) at emission is equal to the ratio of the scale factor (e.g., 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 us (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. Derive this law from the first relationships stated in the problem, and determine the Hubble constant in terms of the scale-factor function \(a(t)\). b) If the present Hubble constant has the value \(H_{0}=72(\mathrm{~km} / \mathrm{s}) / \mathrm{Mpc},\) how far away is a galaxy, the light from which has 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 us.)

Short answer

Short Answer Question: Derive Hubble's Law and use it to find the distance of a galaxy with a redshift of 0.10, given the Hubble constant as \(H_0 = 72(\mathrm{~km} / \mathrm{s}) / \mathrm{Mpc}\). Answer: Hubble's Law can be derived as \(z \cong c^{-1} H \Delta s\). Using Hubble's Law, the distance of a galaxy with redshift \(z=0.10\) and Hubble constant \(H_0 = 72(\mathrm{~km} / \mathrm{s}) / \mathrm{Mpc}\) is approximately 430 Mpc away.

Step-by-step solution

Recall the definition of redshift: \(1+z=\lambda_{\text {rec }} / \lambda_{\text {emit}}=a_{\text {rec }} / a_{\text {emit }}\). Let's rewrite this in terms of the change in scale factor, \(\Delta a = a_{\text {rec }} - a_{\text {emit }}\): $$1+z = \frac{a_{\text {rec }}}{a_{\text {emit }}} = \frac{\Delta a + a_{\text {emit }}}{a_{\text {emit }}}$$ Now, let's consider the cosmic time intervals at which the light is emitted and received, \(\Delta t_{\text {emit}}\) and \(\Delta t_{\text {rec}}\) respectively, and their ratio: $$\frac{\Delta t_{\text {emit}}}{\Delta t_{\text {rec}}} = \frac{a_{\text {emit }}}{a_{\text {rec }}}$$ Combining with the redshift equation, $$z = \frac{c \Delta t_{\text {rec}}}{\Delta t_{\text {emit}}} - 1$$ For galaxies at small distances, the cosmic time intervals are nearly the same, so we can approximate \(\Delta t_{\text {rec}} \approx \Delta t_{\text {emit}}\). This leads to: $$z \approx c^{-1} \Delta s$$ By definition, \(\Delta s(t) = a(t)\Delta x\), where \(\Delta x\) is the comoving distance between two objects. Taking a time derivative, we have: $$\frac{d\Delta s}{dt} = \frac{da}{dt}\Delta x$$ Defining the Hubble constant as: $$H(t) = \frac{1}{a}\frac{da}{dt}$$ We can then write the time derivative of distance as: $$\frac{d\Delta s}{dt} = H(t)a\Delta x = H(t)\Delta s$$ Therefore, for small distances, we have derived Hubble's Law: $$z \cong c^{-1} H \Delta s$$ Using this result, the Hubble constant is given by: $$H(t) = \frac{1}{a}\frac{da}{dt}$$ #b) Distance of a galaxy with redshift z=0.10#

For this part, we will use Hubble's Law and the given value of the Hubble constant to find the distance of the galaxy: 1. Rearrange Hubble's Law for distance: \(\Delta s = c z / H_0\) 2. Plug in the given values: \(z = 0.10\), \(H_0 = 72(\mathrm{~km} / \mathrm{s}) / \mathrm{Mpc}\) 3. Convert the Hubble constant to proper units by multiplying with \([\dfrac{3.26\times 10^6 ( \text{light-years})}{\text{Mpc}}]\) and \( [\dfrac{9.461\times 10^{12}( \text{km})}{\text{light year}}]\): $$H_0 = 72\dfrac{\text{km s}^{-1}}{\text{Mpc}}\times \dfrac{3.26\times 10^{6} \text{light-years}}{\text{Mpc}}\times \dfrac{9.461\times 10^{12} \text{km}}{\text{light year}}=2.25\times 10^{-18} \dfrac{\text{km}}{\text{km s}}$$ 4. Calculate the distance: $$\Delta s = \dfrac{c z}{H_0} = \dfrac{(3 \times 10^{5} \mathrm{~km / s})( 0.10)}{2.25 \times 10^{-18} \mathrm{~km^{-1} s}}= 1.33 \times 10^{25} \mathrm{~km}$$ 5. Convert the distance to Mpc: $$\Delta s = \dfrac{1.33 \times 10^{25} \mathrm{~km}}{3.09 \times 10^{19} \mathrm{~km / Mpc}}= 430 \mathrm{~Mpc}$$ A galaxy with redshift \(z=0.10\) is approximately 430 Mpc away.

Key concepts

Big Bang Theory

The Big Bang Theory presents the origin story of our universe. It is a cosmological model that explains how the universe expanded from a high-density, high-temperature initial state. According to this theory, about 13.8 billion years ago, the universe was a singularity, containing all of the mass and space-time of the Universe within an infinitely small volume. From this singularity, the universe has been expanding to its current form, filled with galaxies, stars, planets, and other astronomical objects.

This expansion is supported by several lines of evidence, including the cosmic microwave background, the abundance of light elements, and the redshift of galaxies, which is a cornerstone of Hubble's Law. As our understanding of physics grows, so too does our knowledge of the early universe, though many questions about the Big Bang remain unanswered.

Cosmic Microwave Background

The cosmic microwave background (CMB) is a key piece of evidence for the Big Bang Theory. It is the afterglow radiation left over from the moment the universe became transparent to radiation, approximately 380,000 years after the Big Bang. When scientists observe the CMB, they are looking at a 'snapshot' of the oldest light in the universe.

The CMB was discovered by accident in 1965 by Arno Penzias and Robert Wilson. This discovery has provided a critical test for cosmological theories by offering a view of the universe when it was very young. The uniformity of the CMB across the sky requires explanation, and slight variations or 'anisotropies' in temperature provide information about the early universe, such as the distribution of mass and the nature of cosmic inflation.

Redshift

The redshift is a phenomenon observed in the light coming from distant galaxies or celestial objects. It refers to the shift of spectral lines toward the red end of the visible light spectrum. In the context of the universe's expansion, redshift occurs because the wavelengths of light are stretched as the space through which they travel expands.

Hubble's observation that distant galaxies exhibit redshift led to the formulation of the Hubble's Law, illustrating the universe's expansion. The higher the redshift, the faster the object is moving away from us, and by inference, the further away it is. Redshift is not due to the Doppler effect, which is a shift in frequency due to the relative motion of source and observer, but rather a result of the expansion of space itself.

Universe Expansion

Universe expansion is the increase in distance between any two given gravitationally unbound parts of the universe with time. It is a fundamental observation that stems from the Big Bang Theory and is evidenced by the redshift of galaxies. Edwin Hubble was the first to empirically codify this phenomenon in what we now call Hubble's Law. Hubble's Law states that the further away a galaxy is, the faster it appears to be moving away from Earth.

This expansion isn't due to galaxies physically moving through space away from each other, but rather due to the space between galaxies stretching and expanding. The rate of this expansion is given by the Hubble constant, which has been a critical factor in determining the age, size, and ultimate fate of the universe. Current models of the universe suggest that this rate of expansion is accelerating, a discovery that was honored with the Nobel Prize in Physics in 2011.