Cosmology
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Cosmology: Redshift, Hubble's Law & Dark Matter
Quick Summary
- Doppler shift — a moving wave source changes the wavelength/frequency an observer detects: squashed (shorter λ) in front, stretched (longer λ) behind.
- Redshift = light stretched to longer wavelength because the source (a galaxy) is receding from us. Blueshift = shorter wavelength, source approaching.
- The redshift equation links the fractional wavelength/frequency shift to the recession speed:
Δλ/λ = Δf/f = v/c. - Hubble's Law:
v ≈ H₀d— the further away a galaxy is, the faster it's moving away from us. This is the key evidence for an expanding universe. - The Hubble constant, H₀, is the gradient of a velocity-vs-distance graph for galaxies. Current best estimate ≈ 67.4 km s⁻¹ Mpc⁻¹.
- Dark matter is inferred because galaxies rotate faster at their edges than visible mass alone predicts — there must be extra, unseen mass spread through the galaxy.
Topic 1 — The Doppler Shift
Think about an ambulance siren. As it speeds towards you, the pitch sounds . As it speeds away, the pitch drops . Nothing about the siren itself has changed — the change is entirely due to the ambulance's motion relative to you, the observer. This is the Doppler effect, and it applies to any wave, including light.
Here's the mechanism: imagine the source emits wavefronts (crests) at regular time intervals while it's moving. If it's moving you, each successive wavefront is emitted from a position slightly closer to you than the last one — so the wavefronts "pile up" and arrive closer together. That means a shorter wavelength and higher frequency is observed. If the source moves , the opposite happens: wavefronts get stretched apart, giving a longer wavelength and lower frequency.
Applying this to light: redshift & blueshift
The Doppler effect works for electromagnetic radiation too. If we compare the absorption line spectrum of a distant galaxy to the spectrum of a nearby, "stationary" reference source (like our Sun), we can see whether the galaxy's lines have shifted:
- Redshift — spectral lines shift towards the red (longer wavelength) end → the galaxy is moving away from Earth.
- Blueshift — spectral lines shift towards the blue (shorter wavelength) end → the galaxy is moving towards Earth.
When astronomers looked at spectra from distant galaxies, they found the lines were consistently shifted towards red — and the further away the galaxy, the bigger the shift. That single observation is one of the strongest pieces of evidence that the universe is expanding.
A student observes that the absorption lines in the spectrum of Galaxy X are shifted towards the blue end of the spectrum compared to a laboratory reference. What does this tell us about Galaxy X's motion?
Explain, in terms of wavefronts, why the observed frequency increases when a source moves towards an observer.
Topic 2 — The Redshift Equation
Redshift isn't just a qualitative "shifted towards red" observation — we can quantify it. Redshift is defined precisely as:
For galaxies that aren't moving at a significant fraction of the speed of light (i.e. "non-relativistic"), the fractional change in wavelength equals the fractional change in frequency, which equals the ratio of the recession speed to the speed of light:
In plain words: this equation says "the shift is proportional to the speed." Double the recession speed, and you double the fractional shift in wavelength (or frequency). This lets us go the other way too — measure the shift, and calculate how fast the galaxy is receding.
Worked Example
Problem: A spectral line from a laboratory source has frequency f = 4.570 × 10¹⁴ Hz. The same line, observed from a distant galaxy, has frequency 4.547 × 10¹⁴ Hz. Find the galaxy's speed relative to Earth, and state whether it is approaching or receding.
Step 1 — List knowns:
f = 4.570 × 10¹⁴ Hz
Δf = (4.547 − 4.570) × 10¹⁴ = −2.3 × 10¹² Hz
c = 3.0 × 10⁸ m s⁻¹
Step 2 — Write the equation: Δf/f = v/c
Step 3 — Rearrange and calculate:
v = (c × Δf) / f = (3.0 × 10⁸ × 2.3 × 10¹²) / (4.570 × 10¹⁴)
v ≈ 1.5 × 10⁶ m s⁻¹
Step 4 — Direction: The observed frequency (4.547 × 10¹⁴ Hz) is than the emitted frequency (4.570 × 10¹⁴ Hz). Lower observed frequency = longer wavelength = redshifted = the galaxy is receding (moving away) from Earth.
A galaxy emits light with a laboratory wavelength of 656.3 nm. The observed wavelength from this galaxy is 660.1 nm. Calculate the recession speed of the galaxy. (c = 3.0 × 10⁸ m s⁻¹)
Topic 3 — The Hubble Equation
In 1929, Edwin Hubble made a discovery that changed cosmology forever. He didn't just find that distant galaxies were redshifted (moving away) — he found a pattern: galaxies that were showed redshifts. In other words, the more distant a galaxy, the faster it's receding.
This is Hubble's Law: Written as an equation:
In plain words: velocity is directly proportional to distance. If you plot recessional velocity (y-axis) against distance (x-axis) for lots of galaxies, you get a straight line through the origin — and the gradient of that line is the Hubble constant, H₀.
Worked Example
Problem: A distant galaxy is 20 light-years away from Earth. Use Hubble's Law to find its recession velocity. Take H₀ = 2.2 × 10⁻¹⁸ s⁻¹, and 1 ly ≈ 9.5 × 10¹⁵ m.
Step 1 — List knowns:
d = 20 light-years, H₀ = 2.2 × 10⁻¹⁸ s⁻¹
Step 2 — Convert distance to metres:
d = 20 × (9.5 × 10¹⁵) = 1.9 × 10¹⁷ m
Step 3 — Substitute into v ≈ H₀d:
v ≈ (2.2 × 10⁻¹⁸) × (1.9 × 10¹⁷)
v ≈ 0.42 m s⁻¹
Notice how tiny this speed is — that's because 20 light-years is an incredibly small distance on a cosmological scale. Hubble's Law only becomes significant (giving speeds of thousands of km s⁻¹) at truly enormous, intergalactic distances.
A graph of recessional velocity against distance for a set of galaxies is a straight line through the origin, reaching v = 15,000 km s⁻¹ at d = 220 Mpc. Determine the Hubble constant from this graph, with units.
Using H₀ = 67.4 km s⁻¹ Mpc⁻¹, calculate the distance to a galaxy that has a measured recessional velocity of 3,370 km s⁻¹.
Topic 4 — The Hubble Constant
Rearranging Hubble's Law gives us a direct formula for the Hubble constant itself:
H₀ has been estimated using data from thousands of galaxies, plotting v against d and finding the gradient of the best-fit line. The most recent, highly precise estimate — based on observations of the Cosmic Microwave Background (CMB) by the Planck satellite — is:
Notice the ± 0.5 — this uncertainty exists because measuring the exact distance to a galaxy is genuinely hard. Astronomers rely on several distance-measuring techniques (e.g. standard candles like Cepheid variable stars or Type Ia supernovae), each of which carries its own random and systematic errors. Different measurement methods sometimes even give slightly different values of H₀ — this mismatch is known in cosmology as the "Hubble tension," and it's an active area of research.
Worked Example — reading H₀ off a graph
Problem: A graph plots recessional velocity v (km s⁻¹) against distance d (Mpc) for a set of galaxies. The best-fit straight line passes through the origin and through the point (305 Mpc, 20,000 km s⁻¹). Find H₀.
Step 1 — Recall the law: v ≈ H₀d, so H₀ = gradient of the v–d graph.
Step 2 — Read off two points on the line:
(x₁, y₁) = (0, 0)
(x₂, y₂) = (305, 20,000)
Step 3 — Calculate the gradient:
H₀ = (y₂ − y₁)/(x₂ − x₁) = (20,000 − 0)/(305 − 0)
H₀ ≈ 66 km s⁻¹ Mpc⁻¹
This is very close to the accepted Planck value of 67.4 — a nice sanity check that the graph-reading method works.
Explain why cosmologists cannot state the Hubble constant with perfect precision, and give one source of error involved.
Topic 5 — Dark Matter
Here's a puzzle. In our solar system, planets further from the Sun orbit — Neptune crawls along compared to Mercury. That's exactly what Newtonian gravity predicts: gravitational field strength weakens with distance, so orbital speed should fall off the further out you go, roughly following v ∝ 1/√r.
So when astronomers measured the orbital velocities of stars and gas at different distances from the centre of spiral galaxies, they expected the same pattern: fast near the centre (where visible mass is concentrated), slowing down further out. But that's not what they found.
Instead of falling off, the rotation curve stays roughly flat — stars way out at the edge of a galaxy orbit almost as fast as stars near the centre. Based on the mass we can actually (stars, gas, dust — all concentrated near the centre), this shouldn't happen. There must be extra mass out there that we can't see, contributing extra gravitational pull to keep those outer stars moving fast without flying off into space.
Because dark matter doesn't interact with light at all, you can never see it directly through a telescope — no matter how powerful. We only know it's there because of its gravitational effects:
- Galaxy rotation curves — the flat rotation curve problem described above.
- Gravitational lensing — massive unseen matter bends the path of starlight passing near it, distorting the images of galaxies behind it, more than the visible mass alone could explain.
Current estimates suggest dark matter makes up around 27% of all the mass-energy in the universe — dwarfing the roughly 5% made up of "ordinary" matter (the atoms making up stars, planets, and us).
Explain why the observed rotation curves of galaxies provide evidence for the existence of dark matter.
Give two reasons why dark matter cannot be detected directly using telescopes, and state one method used to detect it indirectly.
What to Memorise
| Term / Formula | Meaning |
|---|---|
| Doppler effect | Apparent change in wavelength/frequency of waves due to relative motion between source and observer |
| Redshift | Fractional increase in wavelength (decrease in frequency) — source receding from observer |
| Blueshift | Source approaching the observer — wavelength decreases, frequency increases |
| Δλ/λ = Δf/f = v/c | Doppler redshift equation (non-relativistic galaxies) |
| v ≈ H₀d | Hubble's Law — recessional velocity is proportional to distance |
| H₀ = v/d | Hubble constant — gradient of a v–d graph, ≈ 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ (Planck 2020) |
| Dark matter | Unseen matter that doesn't emit/absorb EM radiation; detected via gravitational effects; ~27% of universe's mass |
| Flat rotation curve | Orbital velocity of stars stays roughly constant with distance from galactic centre — key dark matter evidence |
| Gravitational lensing | Bending of light around massive (including dark) matter — another way to detect dark matter |
Concepts Checklist
Exam Tips
- Using the exact defined phrasing for redshift, the Doppler effect, and dark matter
- Showing full working when rearranging v ≈ H₀d or Δλ/λ = v/c — don't skip steps
- Correctly identifying direction of motion (towards/away) from the sign or size of a frequency/wavelength shift
- Explaining dark matter in terms of the between observed and predicted rotation velocities, not just "we can't see it"
- Calculate recession velocity from a given wavelength/frequency shift
- Read H₀ off a v–d graph and state its units
- Use H₀ and a measured velocity (or distance) to find the missing quantity
- Explain, using rotation curve data, why dark matter is proposed to exist
- Define redshift/blueshift/Doppler effect precisely, using the command word "define" or "state"
- Applying this to light: redshift & blueshift
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