Cosmology

Cosmology is the study of the origin, structure, and evolution of the universe as a whole. At A-Level, you need to understand the observational evidence for the Big Bang model, Hubble's law, the cosmic microwave background radiation (CMBR), and what these tell us about the age and fate of the universe.

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The Expanding Universe

In 1929, Edwin Hubble made one of the most important discoveries in the history of science: distant galaxies are moving away from us, and the further away they are, the faster they recede. The universe is expanding.

Redshift

The key evidence comes from redshift — the stretching of light from distant galaxies to longer (redder) wavelengths. When a galaxy moves away from us, the light it emits is Doppler shifted towards the red end of the spectrum.

The redshift z is defined as:

$z = \frac{\Delta \lambda}{\lambda_0} = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}}$z=λ0​Δλ​=λemitted​λobserved​−λemitted​​

For velocities much less than the speed of light:

$z \approx \frac{v}{c}$z≈cv​

where v is the recession velocity and c is the speed of light.

Worked Example: Calculating Recession Velocity from Redshift

A galaxy shows a hydrogen spectral line at 660.2 nm instead of the laboratory value of 656.3 nm. Calculate the galaxy's recession velocity.

$z = \frac{660.2 - 656.3}{656.3} = \frac{3.9}{656.3} = 5.94 \times 10^{-3}$z=656.3660.2−656.3​=656.33.9​=5.94×10−3

$v = zc = 5.94 \times 10^{-3} \times 3.0 \times 10^8 = 1.78 \times 10^6 \text{ m s}^{-1} \approx 1780 \text{ km s}^{-1}$v=zc=5.94×10−3×3.0×108=1.78×106 m s−1≈1780 km s−1

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Hubble's Law

Hubble discovered that recession velocity is proportional to distance:

$v = H_0 d$v=H0​d

where:

• v is the recession velocity (km s⁻¹ or m s⁻¹)
• d is the distance to the galaxy (Mpc or m)
• H₀ is the Hubble constant — the constant of proportionality

The current best estimate of the Hubble constant is approximately:

$H_0 \approx 67-73 \text{ km s}^{-1} \text{ Mpc}^{-1}$H0​≈67−73 km s−1 Mpc−1

(The exact value is still debated — this is the "Hubble tension." For exam purposes, use the value given in the question.)

flowchart LR
    subgraph Hubble["Hubble's Law: v = H₀d"]
        direction TB
        A["Plot recession velocity v\nvs distance d for many galaxies"]
        B["Result: straight line through origin"]
        C["Gradient = H₀ (Hubble constant)"]
        D["Implies: universe is expanding uniformly"]
    end
    subgraph Evidence["Observational Evidence"]
        E["Redshift of spectral lines\nfrom distant galaxies"]
        F["Greater distance → greater redshift"]
        G["All directions show same expansion"]
    end
    Evidence --> Hubble

Worked Example: Distance from Hubble's Law

A galaxy has a recession velocity of 15,000 km s⁻¹. Using H₀ = 70 km s⁻¹ Mpc⁻¹, find its distance.

$d = \frac{v}{H_0} = \frac{15000}{70} = 214 \text{ Mpc}$d=H0​v​=7015000​=214 Mpc

Converting: 214 × 3.09 × 10²² = 6.61 × 10²⁴ m (about 700 million light-years).

Worked Example: Complete Calculation from Spectral Line to Distance

A distant galaxy shows the calcium K line at 401.8 nm (rest wavelength 393.4 nm). Using H₀ = 68 km s⁻¹ Mpc⁻¹, find the distance to this galaxy.

Step 1: Redshift: $z = \frac{401.8 - 393.4}{393.4} = \frac{8.4}{393.4} = 0.02136$z=393.4401.8−393.4​=393.48.4​=0.02136

Step 2: Recession velocity: $v = zc = 0.02136 \times 3.0 \times 10^5 = 6408 \text{ km s}^{-1}$v=zc=0.02136×3.0×105=6408 km s−1

Step 3: Distance: $d = \frac{v}{H_0} = \frac{6408}{68} = 94.2 \text{ Mpc}$d=H0​v​=686408​=94.2 Mpc

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The Age of the Universe

If the universe has been expanding at a constant rate, then the age of the universe can be estimated from the Hubble constant. Consider a galaxy at distance d receding at velocity v = H₀d. The time to travel that distance at constant speed is:

$t = \frac{d}{v} = \frac{d}{H_0 d} = \frac{1}{H_0}$t=vd​=H0​dd​=H0​1​

So the age of the universe is approximately 1/H₀. Converting the Hubble constant to SI units:

$H_0 = 70 \text{ km s}^{-1} \text{ Mpc}^{-1} = \frac{70 \times 10^3}{3.09 \times 10^{22}} = 2.27 \times 10^{-18} \text{ s}^{-1}$H0​=70 km s−1 Mpc−1=3.09×102270×103​=2.27×10−18 s−1

$t = \frac{1}{H_0} = \frac{1}{2.27 \times 10^{-18}} = 4.41 \times 10^{17} \text{ s}$t=H0​1​=2.27×10−181​=4.41×1017 s

Converting to years: 4.41 × 10¹⁷ / (3.156 × 10⁷) = 14.0 billion years

This is remarkably close to the accepted age of 13.8 billion years (refined by considering that the expansion rate has changed over time).

Exam tip: The age estimate 1/H₀ assumes constant expansion rate (no acceleration or deceleration). The actual expansion has changed over time, so 1/H₀ is only an approximation.

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The Big Bang Model

The standard model of cosmology is the Big Bang — the universe began from an extremely hot, dense state approximately 13.8 billion years ago and has been expanding and cooling ever since.

The Big Bang model makes specific, testable predictions:

Prediction	Observation	Status
Universe is expanding	Hubble's law / galactic redshift	Confirmed
Remnant radiation from early universe	Cosmic microwave background (CMBR)	Confirmed (1965)
Specific ratio of light elements	H: 75%, He: 25% by mass	Confirmed
Universe was hotter/denser in the past	Redshift of CMBR; distant galaxies look younger	Confirmed

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Cosmic Microwave Background Radiation (CMBR)

The CMBR is the afterglow of the Big Bang — electromagnetic radiation that fills the entire universe. It was accidentally discovered in 1965 by Arno Penzias and Robert Wilson using a radio antenna at Bell Labs.

Origin of the CMBR

About 380,000 years after the Big Bang, the universe cooled to about 3,000 K — cool enough for electrons and protons to combine into neutral hydrogen atoms. Before this moment (the epoch of recombination), the universe was an opaque plasma that constantly scattered photons. After recombination, the universe became transparent and the photons were free to travel.

Those photons have been travelling through space ever since. As the universe expanded, the wavelength of this radiation was stretched (redshifted) by a factor of about 1,100. What was originally visible/infrared radiation at ~3,000 K is now microwave radiation corresponding to a temperature of about 2.725 K.

Properties of the CMBR

Property	Value/Description
Temperature	2.725 K (±0.001 K)
Peak wavelength	~1.1 mm (microwave)
Spectrum	Perfect black body
Isotropy	Uniform to 1 part in 100,000
Tiny fluctuations	~10⁻⁵ K variations (seeds of galaxies)

Worked Example: Peak Wavelength of the CMBR

Using Wien's law with T = 2.725 K:

$\lambda_{\text{max}} = \frac{2.898 \times 10^{-3}}{2.725} = 1.063 \times 10^{-3} \text{ m} = 1.063 \text{ mm}$λmax​=2.7252.898×10−3​=1.063×10−3 m=1.063 mm

This is in the microwave region of the electromagnetic spectrum, confirming why it is called the cosmic microwave background.

Why the CMBR Supports the Big Bang

The CMBR's properties are powerful evidence:

• Its existence was predicted by the Big Bang model before it was discovered
• Its perfect black body spectrum at 2.725 K matches the predicted cooled remnant of the hot early universe
• Its near-perfect uniformity (isotropy) indicates the early universe was extremely homogeneous
• The tiny fluctuations (anisotropies) are the right size to have grown into the galaxies and galaxy clusters we see today
• No other model adequately explains a uniform microwave background filling all of space

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The Fate of the Universe

The future of the universe depends on the balance between the kinetic energy of expansion and the gravitational potential energy pulling everything back together. This balance depends on the average density of the universe.

Scenario	Density	Expansion fate	Geometry
Open	ρ < ρ_c	Expands forever (decelerating)	Hyperbolic
Flat	ρ = ρ_c	Expands forever (just barely)	Flat (Euclidean)
Closed	ρ > ρ_c	Eventually recollapses (Big Crunch)	Spherical

The critical density is:

$\rho_c = \frac{3H_0^2}{8\pi G}$ρc​=8πG3H02​​

For H₀ = 2.27 × 10⁻¹⁸ s⁻¹:

$\rho_c = \frac{3 \times (2.27 \times 10^{-18})^2}{8\pi \times 6.674 \times 10^{-11}} = \frac{1.546 \times 10^{-35}}{1.675 \times 10^{-9}} = 9.23 \times 10^{-27} \text{ kg m}^{-3}$ρc​=8π×6.674×10−113×(2.27×10−18)2​=1.675×10−91.546×10−35​=9.23×10−27 kg m−3

This is extraordinarily low — about 5 hydrogen atoms per cubic metre. Observations suggest the actual density is very close to the critical value.

Dark Energy and Accelerating Expansion

In 1998, observations of distant Type Ia supernovae revealed something unexpected: the expansion of the universe is accelerating. This implies the existence of a mysterious dark energy that acts as a repulsive force on cosmic scales.

Current observations suggest the universe is composed of:

• ~5% ordinary (baryonic) matter
• ~27% dark matter
• ~68% dark energy

Dark energy dominates the energy budget of the universe and drives the accelerating expansion. With dark energy, the universe will expand forever at an ever-increasing rate, regardless of the matter density.

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Olbers' Paradox

A simple but profound question: why is the sky dark at night? If the universe were infinite, eternal, and static, every line of sight would eventually hit a star, and the entire sky should be as bright as the surface of a star.

The Big Bang model resolves this paradox:

• The universe has a finite age (~13.8 billion years), so we can only see light from a limited volume
• The expansion of the universe redshifts light from distant sources, reducing its intensity
• Very distant stars have not existed long enough for their light to reach us

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Summary

• Hubble's law: v = H₀d — recession velocity is proportional to distance
• Redshift z = Δλ/λ₀ ≈ v/c provides evidence of expansion
• Age of universe ≈ 1/H₀ ≈ 14 billion years (approximate)
• The Big Bang model: the universe began hot and dense and has been expanding and cooling
• The CMBR (2.725 K, microwave, near-perfect black body) is the afterglow of the Big Bang
• The CMBR's existence, spectrum, and isotropy strongly support the Big Bang model
• The fate depends on density vs critical density, but dark energy causes accelerating expansion
• Olbers' paradox is resolved by the finite age and expansion of the universe

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A-Level Deep Dive: Cosmology

Spec mapping

Edexcel 9PH0 specification Topic 12 — Space covers stellar evolution, the Hertzsprung–Russell diagram, the use of standard candles to determine distance, the Doppler effect applied to galactic spectra, Hubble's law and the expansion of the universe, the Big Bang model, the cosmic microwave background radiation (CMBR), the role of critical density in determining the universe's fate, and contemporary evidence for dark matter and dark energy (refer to the official specification document for exact wording). The "Cosmology" lesson sits in the final stretch of Topic 12 and is examined chiefly on Paper 2 (9PH0/02) within the long-answer extended-response questions, but synoptic links pull it back into Paper 1 (gravitational fields, energy) and Paper 3 (data analysis, graphical work). The Edexcel formula booklet provides $v = H_0 d$v=H0​d and $z \approx \Delta\lambda / \lambda \approx v/c$z≈Δλ/λ≈v/c for non-relativistic recession; learners must memorise the units of $H_0$H0​ (km s⁻¹ Mpc⁻¹) and the conversion to SI for age-of-universe estimates.

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Worked example with full mark scheme

Question (8 marks):

A spectral line of hydrogen has rest wavelength $\lambda_0 = 656.3$λ0​=656.3 nm. When observed in the spectrum of a distant galaxy, the same line appears at $\lambda = 689.1$λ=689.1 nm.

(a) Calculate the redshift $z$z and hence the recession velocity of the galaxy. (3)

(b) Use Hubble's law with $H_0 = 70$H0​=70 km s⁻¹ Mpc⁻¹ to estimate the distance to the galaxy in Mpc and in metres (1 Mpc $\approx 3.09 \times 10^{22}$≈3.09×1022 m). (3)

(c) Use $H_0$H0​ to estimate the age of the universe in seconds and in years. (2)

Solution with mark scheme:

(a) Step 1 — compute the redshift.

$z = \frac{\Delta\lambda}{\lambda_0} = \frac{689.1 - 656.3}{656.3} = \frac{32.8}{656.3} \approx 0.0500$z=λ0​Δλ​=656.3689.1−656.3​=656.332.8​≈0.0500

M1 — correct redshift formula and substitution.

Step 2 — recession velocity from $z \approx v/c$z≈v/c.

$v \approx zc = 0.0500 \times 3.00 \times 10^8 = 1.50 \times 10^7 \text{ m s}^{-1}$v≈zc=0.0500×3.00×108=1.50×107 m s−1

A1 — correct numerical value with units.

A1 — comment that the non-relativistic approximation is acceptable because $z \ll 1$z≪1.

(b) Step 1 — apply Hubble's law.

Convert $v$v to km s⁻¹: $v = 1.50 \times 10^4$v=1.50×104 km s⁻¹.

$d = \frac{v}{H_0} = \frac{1.50 \times 10^4}{70} \approx 214 \text{ Mpc}$d=H0​v​=701.50×104​≈214 Mpc

M1 — correct rearrangement and consistent units (km s⁻¹ with km s⁻¹ Mpc⁻¹).

A1 — $d \approx 214$d≈214 Mpc.

A1 — converting: $d \approx 214 \times 3.09 \times 10^{22} \approx 6.6 \times 10^{24}$d≈214×3.09×1022≈6.6×1024 m.

(c) Step 1 — convert $H_0$H0​ to SI (s⁻¹).

$H_0 = \frac{70 \times 10^3 \text{ m s}^{-1}}{3.09 \times 10^{22} \text{ m}} \approx 2.27 \times 10^{-18} \text{ s}^{-1}$H0​=3.09×1022 m70×103 m s−1​≈2.27×10−18 s−1

M1 — correct unit conversion.

Step 2 — age estimate.

$t \approx \frac{1}{H_0} \approx 4.41 \times 10^{17} \text{ s} \approx 1.4 \times 10^{10} \text{ years}$t≈H0​1​≈4.41×1017 s≈1.4×1010 years

A1 — answer ~14 billion years.

Total: 8 marks.

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Specimen question modelled on the Edexcel 9PH0 Paper 2 format

Question (6 marks): Astronomers observe a Type Ia supernova in a distant galaxy. The supernova's apparent magnitude allows the distance to be calculated as $d = 150$d=150 Mpc. The hydrogen-α line in the galaxy's spectrum is redshifted from 656.3 nm to 679.3 nm.

(a) Show that this observation is consistent with Hubble's law and estimate $H_0$H0​. (4)

(b) Explain briefly how Type Ia supernovae act as standard candles. (2)

Mark scheme decomposition by AO:

(a)

• M1 (AO2.1) — calculating redshift $z = 23.0/656.3 \approx 0.0350$z=23.0/656.3≈0.0350.
• M1 (AO1.2) — $v \approx zc \approx 1.05 \times 10^7$v≈zc≈1.05×107 m s⁻¹ $= 1.05 \times 10^4$=1.05×104 km s⁻¹.
• M1 (AO2.1) — applying $H_0 = v/d = 1.05 \times 10^4 / 150$H0​=v/d=1.05×104/150.
• A1 (AO2.2) — $H_0 \approx 70$H0​≈70 km s⁻¹ Mpc⁻¹, consistent with accepted values.

(b)

• B1 (AO1.1) — Type Ia supernovae have a known peak luminosity (white-dwarf accretion to the Chandrasekhar limit).
• B1 (AO1.2) — comparing apparent and absolute brightness gives distance via the inverse-square law.

Total: 6 marks split AO1 = 3, AO2 = 3. Edexcel typically rewards consistent unit handling in (a) — AO2 marks are awarded for "showing the link" between the data and Hubble's law, not just for arithmetic.

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Synoptic links

Connects to:

1. Doppler effect (Topic 5 — Waves): the cosmological redshift formula $z = \Delta\lambda/\lambda$z=Δλ/λ is mathematically identical to the non-relativistic Doppler formula for a receding source. Conceptually, however, cosmological redshift arises from the expansion of space itself, not motion through space — a distinction examiners reward when stated carefully.

2. Gravitational redshift (Topic 12 — Space): photons climbing out of a gravitational potential well lose energy and shift to longer wavelengths. The same equation $\Delta f / f = -\Delta\Phi / c^2$Δf/f=−ΔΦ/c2 governs both the gravitational redshift of light from a white dwarf and (at higher order) cosmological time dilation.