Cosmology

When a source of light moves relative to an observer, the observed wavelength and frequency change. This is the Doppler effect — the same physics that raises the pitch of an approaching siren and lowers it as the siren passes, but for light instead of sound.

• A receding source (moving away) stretches its light to longer wavelengths — a shift towards the red end of the spectrum, called a redshift.
• An approaching source compresses its light to shorter wavelengths — a shift towards the blue end, called a blueshift.

We detect this using the spectral lines of elements. Hydrogen (and other elements) absorb or emit light at precise, well-known wavelengths measured in the laboratory (the rest wavelengths). When we observe the same lines from a distant galaxy, the whole pattern is shifted. Comparing the shifted position with the laboratory position tells us the galaxy's motion.

For speeds much less than the speed of light ($v \ll c$), the fractional shift equals the ratio of the source's speed to the speed of light:

$z = \frac{\Delta\lambda}{\lambda} = \frac{\Delta f}{f} = \frac{v}{c}$

Here $z$ is the redshift (a dimensionless ratio), $\Delta\lambda$ is the change in wavelength (observed minus rest), $\lambda$ is the rest wavelength, $v$ is the recessional velocity and $c = 3.00\times10^{8}\ \text{m s}^{-1}$. Rearranged, a galaxy's recessional velocity is simply $v = zc$.

Because $z = v/c$, a galaxy's spectral shift is a direct speedometer. The relation is an approximation valid only for $v \ll c$ — fine for the galaxy speeds in Edexcel IAL questions, but it breaks down as $v$ approaches $c$.

In the 1920s Edwin Hubble found that the light from almost every distant galaxy is red-shifted — they are nearly all receding from us. More strikingly, he found a pattern: the more distant a galaxy, the faster it recedes. This is Hubble's law:

$v = H_0 d$

where $v$ is the recessional velocity, $d$ is the distance to the galaxy, and $H_0$ is the Hubble constant. Plotting $v$ (y-axis) against $d$ (x-axis) for many galaxies gives a straight line through the origin; comparing $v = H_0 d$ with $y = mx$ shows that the gradient of this line is $H_0$.

The straight line through the origin is important: it shows velocity is directly proportional to distance. To find $H_0$ from real data you draw a large gradient triangle, calculate $\Delta v / \Delta d$, and take the unit straight from the axes: velocity ($\text{km s}^{-1}$) per distance (Mpc), giving $H_0 \approx 70\ \text{km s}^{-1}\text{ Mpc}^{-1}$.

Why this means the universe is expanding. It is not that we sit at a special centre with everything fleeing us. Rather, space itself is expanding, and this carries the galaxies apart. A helpful picture is dots drawn on the surface of a balloon: as the balloon inflates, every dot moves away from every other dot, and dots that are further apart separate faster — exactly Hubble's law. From any dot's viewpoint, all the others recede.

Not every galaxy recedes: a few nearby ones, most famously Andromeda, are blue-shifted and approaching. Within a gravitationally bound group of galaxies, local gravitational attraction can overcome the general cosmological expansion at short range.

If galaxies have been receding steadily ever since everything was together, we can estimate how long ago that was. If a galaxy at distance $d$ has always moved at constant velocity $v = H_0 d$, then the time since it was at the origin is

$t = \frac{d}{v} = \frac{d}{H_0 d} = \frac{1}{H_0}.$

So the age of the universe is estimated as:

$t \approx \frac{1}{H_0}$

The crucial step: convert $H_0$ to SI units first. $H_0$ is quoted as $\text{km s}^{-1}\text{ Mpc}^{-1}$, which is not a plain $\text{s}^{-1}$, so $1/H_0$ is only a time once $H_0$ is in $\text{s}^{-1}$. Convert the kilometres to metres and the megaparsecs to metres ($1\ \text{Mpc} = 3.09\times10^{22}\ \text{m}$):

$H_0 = \frac{70\times10^{3}\ \text{m s}^{-1}}{3.09\times10^{22}\ \text{m}} = 2.27\times10^{-18}\ \text{s}^{-1}.$

Then:

$t \approx \frac{1}{H_0} = \frac{1}{2.27\times10^{-18}} = 4.41\times10^{17}\ \text{s} \approx 1.4\times10^{10}\ \text{years}.$

That is roughly 14 billion years — reassuringly close to the modern value.

The key assumption. This estimate treats the expansion rate as constant throughout the whole history of the universe (constant $H_0$). In reality the rate has changed: gravity has slowed the expansion, while more recently dark energy has accelerated it. So $t = 1/H_0$ is an order-of-magnitude estimate, not an exact age. Whenever a question asks you to comment on the age, name this constant-rate assumption explicitly.

Running the expansion backwards in time, everything converges: the universe becomes ever hotter and denser until, about 14 billion years ago, it began from an extremely hot, dense state. This is the Big Bang theory — the universe has been expanding and cooling ever since. The Big Bang was not an explosion into space; it was the beginning of the expansion of space itself.

Two great pillars of observational evidence support it, and crucially they are independent of each other:

1. The redshift of galaxies (Hubble's law). Almost all galaxies are red-shifted, so they are receding, and more distant ones recede faster ($v = H_0 d$). Rewind this expansion and everything meets at a single origin — exactly what the Big Bang predicts.

2. The cosmic microwave background (CMB) radiation. The early universe was filled with intense, high-temperature radiation. As space expanded, this radiation was stretched (red-shifted) to much longer wavelengths and cooled. Today it reaches us uniformly from every direction as microwaves with a near-perfect black-body spectrum at about $2.7\ \text{K}$. Its peak wavelength (from Wien's law) is around $1\ \text{mm}$ — in the microwave region, hence the name. The Big Bang predicted such a cooled thermal remnant before it was discovered, making the CMB compelling evidence.

Together, the recession of galaxies and the thermal CMB give two very different lines of evidence — one kinematic, one about the leftover heat — that both point to a hot, dense beginning followed by expansion and cooling.

Will the universe expand forever, or will gravity eventually halt and reverse the expansion? The answer is a tug-of-war between the outward expansion and the inward pull of gravity, and it is decided by the universe's mean density $\rho$ compared with a critical density $\rho_c$.

There are three possible outcomes (Edexcel IAL treats these qualitatively — you describe them, you do not calculate $\rho_c$):

• Closed universe ($\rho > \rho_c$): there is enough mass for gravity to stop the expansion and pull everything back together. The universe eventually re-collapses (sometimes pictured as a 'Big Crunch').
• Open universe ($\rho < \rho_c$): gravity is too weak to halt the expansion, so the universe expands forever.
• Flat universe ($\rho = \rho_c$): the exact borderline — the universe expands forever, but the expansion rate slows towards zero.

The role of dark matter. Measuring $\rho$ is hard, and there is a surprise: galaxies rotate, and move within clusters, far faster than the gravity of their visible (luminous) matter could sustain. To hold these fast-moving objects in place there must be a great deal of extra mass that exerts gravity but emits no detectable light — this is called dark matter. It substantially raises the mean density $\rho$, and so it is crucial in deciding whether $\rho$ exceeds, equals or falls short of $\rho_c$ — that is, in deciding the ultimate fate of the universe. (Awareness of dark matter and critical density is all that is required at this level.)