Hubble's law and the Big Bang theory

1. Overview

The study of cosmology in the A-Level syllabus focuses on the evidence for the expansion of the Universe. By observing the redshift of light from distant galaxies, we can determine their recessional velocity. This leads to Hubble’s Law, which states that the further a galaxy is from Earth, the faster it is moving away. Tracing this expansion backward in time provides the fundamental evidence for the Big Bang theory and allows us to estimate the age of the Universe.

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Key Definitions

• Redshift ($z$): The fractional increase in the wavelength (or decrease in frequency) of electromagnetic radiation received from an astronomical source due to its motion away from the observer.
• Doppler Effect: The change in the observed frequency or wavelength of a wave when there is relative motion between the source and the observer.
• Hubble’s Law: The recessional speed $v$ of a galaxy is directly proportional to its distance $d$ from the observer.
• Hubble Constant ($H_0$): The constant of proportionality in Hubble’s Law, defined as the ratio of the recessional speed of a galaxy to its distance from Earth.
• Big Bang Theory: The cosmological model which suggests the Universe originated from an extremely hot and dense singularity (a single point) and has been expanding and cooling for billions of years.

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Content

3.1. Redshift in Emission and Absorption Spectra

Stars and galaxies emit light that passes through cooler gases in their outer atmospheres. These gases absorb specific frequencies of light, creating absorption spectra characterized by dark lines (Fraunhofer lines) at specific wavelengths.

• The Reference Point: In a laboratory on Earth, these lines appear at fixed, known wavelengths ($\lambda$) based on the atomic structure of elements like Hydrogen and Helium.
• The Observation: When we observe the spectra from distant galaxies, these same patterns of lines are present, but they are shifted toward the longer wavelength (red) end of the visible spectrum.
• The Mechanism: This shift occurs because the space through which the light travels is expanding, or the source is moving away from us. This is known as redshift.
• Blueshift: If a galaxy were moving toward Earth, the wavelengths would be compressed, shifting toward the blue end of the spectrum. This is rarely observed for distant galaxies.

3.2. The Redshift Equation

For galaxies moving at speeds $v$ that are much less than the speed of light ($v \ll c$), the redshift $z$ is calculated using the following relationship:

$$\mathbf{\frac{\Delta\lambda}{\lambda} \approx \frac{\Delta f}{f} \approx \frac{v}{c}}$$

Where:

• $\Delta\lambda = \lambda_{\text{observed}} - \lambda_{\text{source}}$ (The change in wavelength, in $\text{m}$)
• $\lambda = \lambda_{\text{source}}$ (The original wavelength emitted by the source, in $\text{m}$)
• $\Delta f = f_{\text{source}} - f_{\text{observed}}$ (The change in frequency, in $\text{Hz}$)
• $f = f_{\text{source}}$ (The original frequency emitted by the source, in $\text{Hz}$)
• $v$ = Recessional speed of the galaxy (in $\text{m s}^{-1}$)
• $c$ = Speed of light in a vacuum ($3.00 \times 10^8 \text{ m s}^{-1}$)

Important Note: In the denominator, always use the source (laboratory) wavelength/frequency, not the observed value.

3.3. Evidence for an Expanding Universe

The observation of redshift is the primary evidence for the expansion of the Universe. The logical steps are:

1. Observation: Light from almost all distant galaxies shows redshift.
2. Inference of Motion: According to the Doppler effect, a redshift indicates that the source of light is moving away from the observer.
3. Relationship with Distance: Observations show that the further away a galaxy is, the greater its redshift (and thus the greater its recessional speed).
4. Conclusion: Since galaxies are moving away from us in all directions, and those further away move faster, it is not the galaxies moving through space, but space itself that is expanding.

3.4. Hubble’s Law

Edwin Hubble quantified the relationship between a galaxy's distance and its speed. He found a linear correlation, now known as Hubble's Law.

The Equation: $$\mathbf{v = H_0 d}$$

Where:

• $v$ = Recessional speed of the galaxy (in $\text{m s}^{-1}$)
• $d$ = Distance of the galaxy from Earth (in $\text{m}$)
• $H_0$ = The Hubble constant (in $\text{s}^{-1}$)

Graphical Representation: If you plot a graph of Recessional Speed ($v$) on the y-axis against Distance ($d$) on the x-axis:

• The relationship is a straight line passing through the origin.
• The gradient of the line represents the Hubble Constant ($H_0$).

3.5. The Big Bang Theory and the Age of the Universe

Hubble’s Law provides a mathematical foundation for the Big Bang theory. If the Universe is currently expanding, then tracing the motion of galaxies backward in time implies that all matter and energy were once concentrated at a single point (a singularity).

Deriving the Age of the Universe ($T$):

1. Consider a galaxy at distance $d$ moving at a constant speed $v$.
2. The time $T$ it has been traveling since it was at the same location as Earth (the start of the Big Bang) is: $$T = \frac{d}{v}$$
3. From Hubble’s Law, we know $v = H_0 d$, which can be rearranged to: $$\frac{d}{v} = \frac{1}{H_0}$$
4. Therefore, the age of the Universe $T$ is approximately the reciprocal of the Hubble constant: $$\mathbf{T \approx \frac{1}{H_0}}$$

Current Estimates: Using the SI value for $H_0 \approx 2.3 \times 10^{-18} \text{ s}^{-1}$: $$T = \frac{1}{2.3 \times 10^{-18}} \approx 4.35 \times 10^{17} \text{ s}$$ Converting to years: $$T \approx \frac{4.35 \times 10^{17}}{365.25 \times 24 \times 3600} \approx 1.38 \times 10^{10} \text{ years (13.8 billion years)}$$

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Key Equations

Equation	Description	Data Sheet?
$\mathbf{\frac{\Delta\lambda}{\lambda} \approx \frac{v}{c}}$	Redshift equation for non-relativistic speeds ($v \ll c$).	Yes
$\mathbf{v = H_0 d}$	Hubble’s Law relating speed and distance.	Yes
$\mathbf{T \approx \frac{1}{H_0}}$	Estimate for the age of the Universe.	No (Must derive)
$\mathbf{c = f \lambda}$	Wave equation (used to convert between $f$ and $\lambda$).	Yes

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5. Worked Examples

Worked Example 1 — Calculating Recessional Speed

A specific absorption line of Calcium is observed in the laboratory at a wavelength of $393.4 \text{ nm}$. When analyzing the spectrum of a distant galaxy, this same line is found at a wavelength of $401.8 \text{ nm}$. Calculate the recessional speed of the galaxy.

Step 1: Identify the known variables.

• $\lambda_{\text{source}} = 393.4 \text{ nm} = 393.4 \times 10^{-9} \text{ m}$
• $\lambda_{\text{observed}} = 401.8 \text{ nm} = 401.8 \times 10^{-9} \text{ m}$
• $c = 3.00 \times 10^8 \text{ m s}^{-1}$

Step 2: Calculate the change in wavelength ($\Delta\lambda$). $$\Delta\lambda = 401.8 - 393.4 = 8.4 \text{ nm} = 8.4 \times 10^{-9} \text{ m}$$

Step 3: Use the redshift equation to find $v$. $$\frac{\Delta\lambda}{\lambda} = \frac{v}{c}$$ $$v = \frac{\Delta\lambda}{\lambda} \times c$$ $$v = \frac{8.4 \times 10^{-9}}{393.4 \times 10^{-9}} \times 3.00 \times 10^8$$ $$v = 0.02135 \times 3.00 \times 10^8$$ $$v = 6.41 \times 10^6 \text{ m s}^{-1}$$

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Worked Example 2 — Determining Distance from Redshift

A galaxy has a measured redshift $z = 0.050$. Using a Hubble constant of $H_0 = 2.3 \times 10^{-18} \text{ s}^{-1}$, calculate the distance to this galaxy in meters.

Step 1: Understand redshift $z$. Redshift $z$ is defined as $\frac{\Delta\lambda}{\lambda}$. Therefore, $z = \frac{v}{c}$.

Step 2: Calculate the recessional speed $v$. $$v = z \times c$$ $$v = 0.050 \times 3.00 \times 10^8 = 1.50 \times 10^7 \text{ m s}^{-1}$$

Step 3: Use Hubble’s Law to find distance $d$. $$v = H_0 d \implies d = \frac{v}{H_0}$$ $$d = \frac{1.50 \times 10^7}{2.3 \times 10^{-18}}$$ $$d = 6.52 \times 10^{24} \text{ m}$$

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Worked Example 3 — Age of the Universe

If the Hubble constant is determined to be $70 \text{ km s}^{-1} \text{ Mpc}^{-1}$, and $1 \text{ Mpc} = 3.09 \times 10^{22} \text{ m}$, calculate the age of the Universe in seconds.

Step 1: Convert $H_0$ to SI units ($\text{s}^{-1}$). $$H_0 = \frac{70 \text{ km s}^{-1}}{1 \text{ Mpc}}$$ $$H_0 = \frac{70 \times 10^3 \text{ m s}^{-1}}{3.09 \times 10^{22} \text{ m}}$$ $$H_0 = 2.265 \times 10^{-18} \text{ s}^{-1}$$

Step 2: Use the age formula. $$T = \frac{1}{H_0}$$ $$T = \frac{1}{2.265 \times 10^{-18}}$$ $$T = 4.41 \times 10^{17} \text{ s}$$

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Common Mistakes to Avoid

• ❌ Wrong: Using the observed wavelength in the denominator of the redshift equation ($\frac{\Delta\lambda}{\lambda_{\text{obs}}}$).
• ✅ Right: Always use the emitted (laboratory) wavelength in the denominator ($\frac{\Delta\lambda}{\lambda_{\text{source}}}$).
• ❌ Wrong: Forgetting to convert units like $\text{km s}^{-1}$ or light-years into SI units ($\text{m s}^{-1}$ and $\text{m}$) before using Hubble's Law.
• ✅ Right: Ensure all values are in meters and seconds to match the SI unit of $H_0$ ($\text{s}^{-1}$).
• ❌ Wrong: Stating that redshift is caused by galaxies turning red.
• ✅ Right: Redshift is a shift in the position of spectral lines toward the red end of the spectrum; it describes a change in wavelength, not necessarily the perceived color of the object.
• ❌ Wrong: Assuming $v = H_0 d$ applies to very nearby objects (like the Moon or Andromeda).
• ✅ Right: Hubble's Law applies to distant galaxies where the expansion of space dominates over local gravitational effects.

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Exam Tips

1. The "Why" of the Big Bang: If an exam question asks how Hubble's Law supports the Big Bang theory, you must mention:
   • Galaxies are moving away from each other (expansion).
   • The speed of recession is proportional to distance.
   • Tracing this back in time implies a single point of origin (singularity).
2. Unit Conversions: You may be given $H_0$ in non-SI units. Always check the units on the axes of a graph. If the gradient is $H_0$, and $v$ is in $\text{km s}^{-1}$ while $d$ is in $\text{Mpc}$, your $H_0$ will not be in $\text{s}^{-1}$ until you convert $\text{km} \rightarrow \text{m}$ and $\text{Mpc} \rightarrow \text{m}$.
3. Precision: When calculating the age of the Universe, use the value of $H_0$ provided in the question. Small changes in $H_0$ lead to large changes in the estimated age.
4. Absorption Lines: Remember that we use absorption lines because they provide a "barcode" that is identical for the same elements everywhere in the Universe, giving us a reliable reference for $\lambda_{\text{source}}$.