Investigate the evidence for the Big Bang theory and the early evolution of the universe, including cosmic microwave background radiation, abundance of light elements, and Hubble's law v = H_0 d

What this dot point is asking

NESA wants you to summarise the three primary observational pillars of the Big Bang model: Hubble's law as evidence for the expansion of space, the cosmic microwave background as the cooled relic of the hot early universe, and the abundance of light elements as evidence for primordial nucleosynthesis in the first few minutes. You should be able to use Hubble's law numerically, and to give a coherent timeline of the early universe.

The answer

Hubble's law and the expanding universe

Edwin Hubble (1929) measured distances to galaxies using Cepheid variable stars and found that the redshifts of their spectra (taken to be Doppler shifts of recession) were proportional to those distances:

where $H_0$ is the Hubble constant, today measured at around 70 km/s/Mpc (about $2.3 \times 10^{-18}$ s$^{-1}$).

Two important consequences:

• Uniform expansion. A linear $v$ vs $d$ relation is exactly what every observer in a uniformly expanding space sees. It does not single out our galaxy as the centre; every observer everywhere sees the same law.
• Hubble time as an age estimate. Running the expansion backward at constant rate gives a "Hubble time" $t = 1/H_0 \approx 14$ billion years as a rough age of the universe.

The interpretation is that the galaxies are not flying apart through static space; the space between them is itself expanding, and the redshift is a cosmological redshift (the wavelength stretches with space).

The cosmic microwave background

Penzias and Wilson (1964) discovered an isotropic microwave hiss in their antenna that could not be attributed to instrument noise or known sources. The spectrum measured precisely by the COBE satellite (1989) is the most perfect blackbody known in nature, with $T = 2.725$ K.

This is exactly the prediction of the Big Bang model:

• For the first 380000 years the universe was hot, dense, and opaque (a plasma of nuclei and electrons that scattered photons).
• As the universe expanded and cooled below about 3000 K, electrons combined with nuclei (recombination), the universe became transparent and the photons streamed freely.
• Those photons have been redshifted by the subsequent expansion of space, cooling them from 3000 K to 2.7 K today.

Tiny temperature fluctuations (one part in $10^5$) carry the imprint of the density variations that grew into galaxies. WMAP and Planck measured them precisely; they match simulations of a hot Big Bang universe with about 5% ordinary matter, 27% dark matter and 68% dark energy.

Primordial nucleosynthesis

Between about 1 second and 3 minutes after the Big Bang, the universe was at a temperature comparable to nuclear binding energies. Free protons and neutrons combined to form light nuclei:

• about 75% of the mass remained free protons (hydrogen-1),
• about 25% became helium-4,
• traces of deuterium, helium-3 and lithium-7 formed.

After about 3 minutes the universe had cooled enough that further fusion stopped. No heavier elements were made at this stage; carbon, oxygen, iron and all the rest required stars.

The predicted abundances depend on a single parameter (the baryon-to-photon ratio) and match the observed abundances of these light elements in pristine intergalactic gas. This is independent evidence for a hot dense early universe, complementing the CMB.

Timeline of the early universe

A standard summary:

• $10^{-43}$ s (Planck time): the laws of physics as we know them begin to apply. Earlier is outside accepted theory.
• $10^{-36}$ s to $10^{-32}$ s: rapid exponential inflation enlarges the universe by a factor of about $10^{26}$, smoothing it and stretching tiny quantum fluctuations into the seeds of structure.
• End of inflation to 10 microseconds: quark-gluon plasma cools into protons and neutrons.
• 1 second to 3 minutes: primordial nucleosynthesis produces hydrogen and helium in roughly the observed ratio.
• 380000 years: recombination releases the photons that we now see as the CMB.
• 100 million to 1 billion years: first stars form. They live and die rapidly, producing the first elements heavier than helium (lithium, carbon, oxygen, iron) by stellar nucleosynthesis and supernovae.
• 13.8 billion years (today): continuing expansion, accelerating under dark energy.

The Big Bang model does not describe "what came before"; it describes the evolution of the universe from a hot dense state of which we have direct evidence (the CMB and primordial element abundances).

Try it: Doppler shift calculator to estimate recession velocities of distant galaxies from observed redshifts of spectral lines.

Worked example: distance to a galaxy

A galaxy shows the H$\alpha$ line at 670 nm instead of its rest wavelength 656.3 nm. Estimate the recession velocity and the distance to the galaxy, using $H_0 = 70$ km/s/Mpc.

Doppler (non-relativistic, valid here):

$v = c \, \Delta \lambda / \lambda_0 = 3.00 \times 10^8 \times (670 - 656.3) / 656.3 = 3.00 \times 10^8 \times 0.0209 = 6.26 \times 10^6$ m/s = 6260 km/s.

Hubble:

$d = v / H_0 = 6260 / 70 = 89$ Mpc, about 290 million light-years.

Examples in context

Example 1. Cosmic microwave background detection at the Murchison Widefield Array. The CMB is a near-perfect blackbody at $T = 2.725 \text{ K}$. Wien's law: $\lambda_{\max} = 2.898 \times 10^{-3} / 2.725 = 1.064 \times 10^{-3} \text{ m} = 1.064 \text{ mm}$ (microwave). The MWA in WA observes at $80$ to $300 \text{ MHz}$, looking for the redshifted $21 \text{ cm}$ hydrogen line from $z \sim 10$ (cosmic dawn). Photon energy at peak: $E = h c / \lambda = 1.87 \times 10^{-22} \text{ J} = 1.17 \times 10^{-3} \text{ eV}$. The CMB photon-to-baryon ratio of $\sim 10^9$ matches primordial nucleosynthesis predictions, anchoring the Big Bang timeline at $13.8 \pm 0.05 \text{ Gyr}$.

Example 2. Primordial helium abundance measured at Mt Stromlo. Big Bang Nucleosynthesis predicts $^4$He mass fraction $Y_p = 0.247$. Mt Stromlo Observatory's High Efficiency and Resolution Multi-Element Spectrograph measures HeII/HI line ratios in low-metallicity blue compact galaxies. Result: $Y_p = 0.246 \pm 0.003$, matching prediction to better than $2\%$. This is one of the three pillars (with CMB and Hubble expansion) confirming the Big Bang. Light element abundances ($H$, $He$, $Li$) require nucleosynthesis from $\sim 100 \text{ s}$ to $\sim 20 \text{ min}$ after the Big Bang, at temperatures $T \sim 10^9 \text{ K}$ and densities allowing only the first few nuclear reactions.

Try this

Q1. State Hubble's law and define each symbol. [2 marks]

• Cue. $v = H_0 d$; $v$ recession velocity (km/s), $d$ distance (Mpc), $H_0$ Hubble constant ($\sim 70 \text{ km/s/Mpc}$).

Q2. A galaxy is observed at a distance of $200 \text{ Mpc}$. Calculate its recession velocity using $H_0 = 70 \text{ km/s/Mpc}$. [2 marks]

• Cue. $v = 70 \times 200 = 14{,}000 \text{ km/s}$.

Q3. The cosmic microwave background is one piece of evidence for the Big Bang. (a) Calculate the peak wavelength of a blackbody at $T = 2.725 \text{ K}$. (b) State two other pieces of evidence supporting the Big Bang. (c) Explain why the universe must have been hotter and denser in the past. [2+2+2 marks]

• Cue. (a) $\lambda_{\max} = 2.898 \times 10^{-3} / 2.725 = 1.06 \text{ mm}$. (b) Hubble's law (expansion); primordial $^4$He abundance $\sim 24\%$. (c) Reversing expansion increases density; cooling required to allow neutral atom formation by 380,000 yr.