Evidence for the Big Bang, Hubble's law, the age of the universe and cosmic microwave background radiation
AQA A-Level Physics — Astrophysics
💥State the key evidence for the Big Bang model
🌡️Explain the cosmic microwave background radiation (CMBR)
📐Use Hubble's law v = H₀d to estimate the age of the universe
⏱️Calculate the age: T ≈ 1/H₀
🔭Explain Olbers' paradox and its resolution
🌐Describe the scale factor and the expansion of space
The Big Bang Model
Big Bang: The current scientific model for the origin of the universe, in which all matter, energy, space and time began from an extremely hot, dense singularity approximately 13.8 billion years ago and has been expanding ever since.
The Big Bang is supported by three main pieces of evidence:
Recession of galaxies (Hubble's law): All distant galaxies are moving away from us, with recession velocity proportional to distance. This implies the universe is expanding — winding back gives a starting point.
Cosmic Microwave Background Radiation (CMBR): A uniform, almost isotropic microwave radiation filling the universe, corresponding to a blackbody at ~2.7 K. This is the cooled remnant of the hot radiation from the early universe (~380,000 years after the Big Bang).
Abundance of light elements: About 75% hydrogen and 25% helium (by mass) in the universe, consistent with predictions from Big Bang nucleosynthesis (the conditions in the first few minutes after the Big Bang).
The CMBR was predicted by Gamow, Alpher and Herman in 1948 and accidentally discovered by Penzias and Wilson in 1965. It is one of the strongest pieces of evidence for the Big Bang — there is no other known mechanism that could produce such a uniform microwave background.
Age of the Universe from Hubble's Law
Hubble's Law: v = H₀ d (recession velocity is proportional to distance)
If a galaxy is at distance d and receding at velocity v = H₀d, and if the expansion rate has been approximately constant, then the time since the galaxy was at the same location as us (i.e. the age of the universe) is:
T = 1/H₀ = 1/(2.2 × 10⁻¹⁸) = 4.5 × 10¹⁷ s ≈ 14 × 10⁹ years ≈ 14 billion years
This is an estimate because it assumes constant expansion rate. In reality, the expansion was decelerated by gravity in the early universe and is now being accelerated by dark energy. The actual age (from detailed cosmological models and CMBR data) is ≈ 13.8 billion years — close to the simple 1/H₀ estimate.
In the AQA exam, use H₀ in s⁻¹ (not km s⁻¹ Mpc⁻¹) when calculating the age. The conversion is: 70 km s⁻¹ Mpc⁻¹ = 2.27 × 10⁻¹⁸ s⁻¹.
Cosmic Microwave Background Radiation (CMBR)
In the early universe (first ~380,000 years), it was too hot for neutral atoms to exist — the universe was an opaque plasma of protons and electrons. When the temperature dropped to ~3000 K, electrons and protons combined (recombination) to form neutral hydrogen. The universe became transparent and the radiation was released.
This radiation has since cooled as the universe expanded. Today it corresponds to a blackbody spectrum at T ≈ 2.725 K, peaking in the microwave region.
The CMBR is almost perfectly uniform in all directions (isotropic) to 1 part in 100,000. Tiny temperature fluctuations in the CMBR correspond to density fluctuations that grew into the galaxies and clusters we see today. Mapping these fluctuations (by WMAP and Planck satellites) gives precise values for cosmological parameters including the age of the universe.
Olbers' Paradox
Olbers' Paradox: If the universe is infinite, static and eternal, the night sky should be uniformly bright — every line of sight should eventually hit a star. But the night sky is dark. Why?
Resolution (Big Bang model): the night sky is dark because:
The universe has a finite age (~13.8 Gyr) — light from sufficiently distant stars has not had time to reach us yet (observable universe has a horizon).
The universe is expanding — distant light is redshifted out of the visible spectrum into the infrared and microwave, reducing the brightness.
Stars have finite lifetimes — not all stars have been shining forever.
The resolution of Olbers' paradox supports both the finite age and the expansion of the universe — both key features of the Big Bang model.
The Hubble constant is measured as H₀ = 2.3 × 10⁻¹⁸ s⁻¹. Estimate the age of the universe in years.
1T = 1/H₀ = 1/(2.3 × 10⁻¹⁸) = 4.35 × 10¹⁷ s
2Convert to years: 1 year = 365.25 × 24 × 3600 = 3.156 × 10⁷ s
3T = 4.35 × 10¹⁷ / 3.156 × 10⁷ = 1.38 × 10¹⁰ years ≈ 13.8 billion years
Age ≈ 13.8 × 10⁹ years = 13.8 Gyr
A galaxy is observed to have a recession velocity of 3.0 × 10⁶ m s⁻¹. Using H₀ = 2.2 × 10⁻¹⁸ s⁻¹, find the distance to the galaxy in light-years.
1v = H₀d → d = v/H₀ = (3.0 × 10⁶)/(2.2 × 10⁻¹⁸) = 1.36 × 10²⁴ m
21 light-year = 9.46 × 10¹⁵ m
3d = 1.36 × 10²⁴ / 9.46 × 10¹⁵ = 1.44 × 10⁸ light-years = 144 million light-years
Distance ≈ 1.44 × 10⁸ light-years ≈ 144 Mly
Explain why the CMBR has a temperature of ~2.7 K today rather than the ~3000 K it had at recombination.
1At recombination (~380,000 years after Big Bang), the universe had temperature ~3000 K and the radiation had wavelength λ ∝ 1/T (Wien's law).
2As the universe expanded, space itself stretched. The wavelengths of all photons were stretched (cosmological redshift) by the same factor as the scale factor of the universe.
3Temperature ∝ 1/λ ∝ 1/scale factor. The universe has expanded by a factor of ~1100 since recombination, so T has decreased by factor 1100: 3000/1100 ≈ 2.7 K.
The CMBR photons have been redshifted by the expansion of space — their wavelengths increased by factor ~1100, reducing equivalent temperature from 3000 K to ~2.7 K.
State three pieces of observational evidence that support the Big Bang model and explain what each shows.
1Recession of galaxies / Hubble's law (v = H₀d): all galaxies recede — the universe is expanding, implying a beginning.
2CMBR at 2.7 K: uniform microwave background in all directions — the cooled remnant of hot early universe radiation, released at recombination ~380,000 years post-Big Bang.
3Abundance of light elements (~25% He, ~75% H by mass): matches Big Bang nucleosynthesis predictions — conditions in the first few minutes were hot and dense enough to fuse hydrogen into helium at exactly these proportions.
Three pieces: (1) galaxy recession → expanding universe; (2) CMBR → hot early universe; (3) He/H abundance → Big Bang nucleosynthesis
1. The age of the universe is estimated using T ≈ 1/H₀. If H₀ doubles, what happens to the estimated age?
T = 1/H₀. If H₀ doubles, T = 1/(2H₀) = (1/H₀)/2 — the estimated age halves.
2. What is the temperature of the cosmic microwave background radiation (CMBR) today?
The CMBR corresponds to a blackbody at ~2.725 K. It was ~3000 K at recombination but has cooled by a factor of ~1100 due to the expansion of the universe.
3. Which observation provides evidence for the Big Bang by showing the correct proportions of light elements formed in the first few minutes?
Big Bang nucleosynthesis in the first ~3 minutes produced mainly hydrogen (~75%) and helium-4 (~25%), with traces of deuterium and lithium — matching observed abundances in the oldest stars and gas clouds.
4. Olbers' paradox asks why the night sky is dark. Which of the following is the correct resolution based on the Big Bang model?
The finite age (~13.8 Gyr) means light from beyond a certain distance hasn't reached us yet. Expansion also redshifts distant light out of the visible range, reducing the sky's brightness. The assumption of an infinite, static, eternal universe in Olbers' original paradox is incorrect.
5. Calculate the distance (in m) to a galaxy receding at 1% of the speed of light, given H₀ = 2.2 × 10⁻¹⁸ s⁻¹.
1. Two measurements of H₀ give different values: survey A gives H₀ = 67 km s⁻¹ Mpc⁻¹ (from CMBR data), survey B gives H₀ = 73 km s⁻¹ Mpc⁻¹ (from Type Ia supernovae). Convert both to s⁻¹ (1 Mpc = 3.086 × 10²² m) and calculate the corresponding age estimates. Discuss why the "Hubble tension" matters.
Conversion: H₀ (s⁻¹) = H₀ (km s⁻¹ Mpc⁻¹) × 1000 m/km / (3.086×10²² m/Mpc). Survey A: H₀ = 67×1000/3.086×10²² = 6.7×10⁴/3.086×10²² = 2.172×10⁻¹⁸ s⁻¹. T_A = 1/2.172×10⁻¹⁸ = 4.60×10¹⁷ s = 4.60×10¹⁷/3.156×10⁷ = 14.6 Gyr. Survey B: H₀ = 73×1000/3.086×10²² = 7.3×10⁴/3.086×10²² = 2.365×10⁻¹⁸ s⁻¹. T_B = 1/2.365×10⁻¹⁸ = 4.23×10¹⁷ s = 13.4 Gyr. The "Hubble tension" (the ≈5σ discrepancy between these two methods) is a major unsolved problem. If real (not just systematic error), it might indicate new physics — a dynamical dark energy, early dark energy affecting recombination, or new relativistic particles. It has implications for our entire understanding of cosmological evolution.
2. The CMBR has wavelength at its peak of ~1.9 mm. (a) Use Wien's displacement law (λ_max T = 2.898 × 10⁻³ m K) to find the temperature. (b) At recombination, the CMBR had T ≈ 3000 K. Calculate the factor by which the universe has expanded since recombination.
(a) T = 2.898×10⁻³ / λ_max = 2.898×10⁻³ / 1.9×10⁻³ = 1.525 K ≈ 1.5 K. (Note: the actual CMBR peak is at λ ≈ 1.06 mm for T = 2.725 K using λ_max = 2.898×10⁻³/2.725 = 1.064×10⁻³ m. If the question gives 1.9 mm, T = 1.5 K — but the exact peak depends on the frequency/wavelength version of Wien's law. For AQA purposes, T ≈ 2.7 K.) (b) Since T ∝ 1/scale factor a: a_now/a_then = T_recombination/T_now = 3000/2.725 ≈ 1101. The universe has expanded by a factor of ~1100 since recombination (about 380,000 years after the Big Bang).
3. Explain how the discovery of the accelerating expansion of the universe (from Type Ia supernova observations in 1998) changed our understanding of the age estimate T ≈ 1/H₀, and what it implies about the composition of the universe.
The simple estimate T ≈ 1/H₀ assumes the expansion rate has been constant throughout the universe's history. If expansion is decelerating (as expected from gravity alone), galaxies were moving faster in the past — the universe would be younger than 1/H₀. If expansion is accelerating (as found from Type Ia supernovae observations by Riess and Perlmutter et al. in 1998), the universe would be older than 1/H₀. The discovery showed that distant Type Ia supernovae were fainter than expected — they were further away than a constant-expansion model predicted, implying accelerating expansion. This required a new component: dark energy (cosmological constant Λ), which acts as a repulsive form of energy. The current model (ΛCDM) gives the universe composed of ~68% dark energy, ~27% dark matter, ~5% ordinary matter. In ΛCDM, the true age (~13.8 Gyr) is slightly larger than the naive 1/H₀ estimate because dark energy has only recently come to dominate — in the past, the universe was matter-dominated and decelerating, then dark energy took over and began accelerating the expansion.
Observational Activity Estimating H₀ from Galaxy Recession Data
Objective: Use provided data on galaxy distances and recession velocities to plot a Hubble diagram and estimate H₀.
Background
The Hubble constant cannot be measured directly in a school laboratory (distances involved are billions of light-years). However, students can analyse real astronomical data sets.
Method
Use provided data: recession velocities (from redshift z: v = zc for z << 1) and distances (from standard candles such as Type Ia supernovae or Cepheid variables).
Plot recession velocity v (y-axis, km s⁻¹) against distance d (x-axis, Mpc).
Draw the best-fit straight line through the origin. The gradient = H₀ (in km s⁻¹ Mpc⁻¹).
Convert H₀ to SI units (s⁻¹) and calculate the estimated age T = 1/H₀.
Sample Data
Galaxy
Distance / Mpc
v / km s⁻¹
A
50
3400
B
120
8100
C
200
13500
D
350
24000
E
500
33500
Best-fit gradient ≈ 67 km s⁻¹ Mpc⁻¹ → H₀ = 67 km s⁻¹ Mpc⁻¹ = 2.17 × 10⁻¹⁸ s⁻¹ → T ≈ 14.6 Gyr.
Discussion Points
Why is a best-fit line through the origin more appropriate than a general linear fit?
What are the main uncertainties in measuring extragalactic distances?
How does the scatter in the Hubble diagram relate to peculiar velocities of galaxies?