The Expanding Universe

By the end of this reading, you will be able to:

• Explain

  why redshift alone is not enough — cosmology needs distances too

• Trace

  the completed distance ladder, from parallax to standard candles to the Hubble flow

• Use

  Hubble’s law $v = H_0 d$ to find a recession speed, and $1/H_0$ to estimate the age of the universe

• Distinguish

  cosmological redshift from ordinary Doppler motion, reading it as the scale-factor stretch

• Explain

  why recession faster than light is allowed, and why $v = H_0 d$ is a low-redshift law

• Use

  deep fields as evidence that looking far away means looking back in time

Concept Throughline

The universe is not simply a stage where galaxies move. The stage itself changes. Light crossing that changing stage carries a record of how far, how fast, and how long ago.

This reading takes up the question Module 2 opened with — how far away is that object? — and follows it all the way to the edge of the observable universe. The method is unchanged: observe → model → infer. We observe redshifts and standard-candle brightnesses. We model light crossing expanding spacetime. Then we infer distances, the expansion rate, and a first estimate of the age of the universe.

The surprise is what the model demands. When we extend the simple law that organizes the data, it predicts galaxies receding faster than light — and that turns out to be allowed, because it is space itself that expands.

Spacetime

The combined description of space and time. In cosmology, expanding spacetime means the scale factor changes with time, stretching large-scale distances and the wavelengths of traveling light.

Scale factor

A quantity, written $a(t)$, that describes the relative size of the universe at a given cosmic time. Normalizing today to $a_0 = 1$, a past epoch with $a = 1/2$ is when the universe was half its present scale.

Part 1: Distance First, Then Expansion

Imagine observing a galaxy spectrum and seeing that its lines are redshifted. That tells you the light has been stretched. But by itself, it does not yet tell you the expansion history of the universe. To discover a pattern, you need distances.

This is why the distance ladder matters. Nearby methods calibrate farther-reaching methods. Parallax calibrates local stars. Cepheids extend the ladder because their pulsation periods reveal their luminosities. Supernovae extend it farther still because they can be seen across enormous distances.

Distance ladder

A chain of distance-measurement methods in which nearby calibrated methods support more distant methods — parallax calibrates Cepheids, Cepheids calibrate supernovae, and so on out to cosmological distances.

A Cepheid’s brightness rises and falls in a regular pattern. The directly measured observable is time: the pulsation period. The model is the period-luminosity relation: longer-period Cepheids are more luminous. Once we know luminosity and measure flux, the inverse-square law gives distance.

This is a beautiful example of astronomical inference. A clock-like variation becomes a luminosity. A luminosity plus a flux becomes a distance. A distance plus a redshift becomes a point on the expansion diagram.

Supernova host galaxies help extend the same logic to larger distances. The important idea is not that every standard candle is identical in every detail. The important idea is calibration. Each rung works because a lower rung teaches us how to interpret a higher rung.

Standard candle

An object whose luminosity can be inferred — a Cepheid from its period, a Type Ia supernova from its light curve — allowing distance to be estimated from the observed flux via the inverse-square law.

Quick check

Why would redshifts alone not have been enough to discover Hubble’s law?

Part 2: Hubble’s Law Is an Expansion Pattern

At low redshift, the observed pattern is summarized by Hubble’s law:

Unpack it: $v$ is the recession speed inferred from redshift (usually km/s), $d$ is distance (often megaparsecs for galaxies), and $H_0$ is the Hubble constant, the present-day expansion rate, about $70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$. On large enough scales, recession speed increases linearly with distance.

Hubble's law

The large-scale relation $v = H_0 d$, connecting galaxy recession speed and distance at low redshift. It is the empirical signature of cosmic expansion.

Hubble constant

The present-day expansion rate $H_0$ in Hubble’s law, usually written in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$. Independent measurement chains currently disagree — early-universe methods give roughly $67$, local distance-ladder methods roughly $73$ — a gap called the Hubble tension, an active research frontier.

This is not a rule for every object near us. The Moon is not expanding away from Earth by Hubble’s law. The Solar System is bound. The Milky Way is bound. The Local Group is bound. Hubble’s law describes the large-scale expansion pattern after local gravitational motions average out.

The Hubble constant can be measured in more than one way. That is powerful because independent methods test the model. It is also where modern cosmology becomes alive: when different measurement chains disagree, astronomers have to decide whether the issue is unrecognized systematic error, incomplete modeling, or new physics. The expansion rate is not just a number; it is a constraint on cosmic history — including its age.

Worked Example 1A Recession Speed and the Age of the Universe

Problem

Take $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}}$. (a) A galaxy lies at $d = 100~\mathrm{Mpc}$; find its recession speed. (b) Invert $H_0$ to estimate the age of the universe. Use $1~\mathrm{Mpc} = 3.086\times10^{19}~\mathrm{km}$ and $1~\mathrm{yr} = 3.16\times10^{7}~\mathrm{s}$.

Step·Recession speed from v = H₀ d

$v = H_0 d = (70~\mathrm{km\,s^{-1}\,Mpc^{-1}})(100~\mathrm{Mpc}) = 7{,}000~\mathrm{km\,s^{-1}}$ — about $2\%$ of the speed of light.

Step·The Hubble time, 1/H₀

Convert $H_0$ to inverse seconds: $H_0 = 70~\mathrm{km\,s^{-1}\,Mpc^{-1}} / (3.086\times10^{19}~\mathrm{km\,Mpc^{-1}}) = 2.27\times10^{-18}~\mathrm{s^{-1}}$. Then $1/H_0 = 4.4\times10^{17}~\mathrm{s} \approx 14~\mathrm{Gyr}$.

Dimensional check

$[H_0 d] = (\mathrm{km\,s^{-1}\,Mpc^{-1}})(\mathrm{Mpc}) = \mathrm{km\,s^{-1}}$, a speed; and $[1/H_0] = 1/(\mathrm{s^{-1}}) = \mathrm{s}$, a time. Both units land where they should.

Result

The Hubble time $1/H_0 \approx 14~\mathrm{Gyr}$ is a first estimate of the age — what you would get if the universe had always expanded at the same speed (neither slowing nor speeding up). The true age (about $13.8~\mathrm{Gyr}$) is close, but getting it exactly requires the full expansion history — how the rate sped up and slowed down — which is the next reading’s job. Keep the number; we will earn it properly in L4.

Now push the law where it surprises us.

Quick check

A student computes $v = H_0 d$ for a very distant galaxy and gets $v = 1.3c$, then concludes the calculation must be wrong because nothing travels faster than light. What is the student missing?

Quick check

Why does Hubble’s law work better for large-scale galaxy samples than for nearby bound systems such as the Local Group?

Part 3: Redshift as a Scale-Factor Record

In Module 1, we learned the Doppler shift: motion along the line of sight stretches or compresses wavelengths. That model still works for many nearby objects. But on cosmological scales, there is a deeper interpretation. Light from a distant galaxy travels while the universe expands. As space expands, the wavelength of that light stretches too.

This equation has three connected ways to read the same redshift, where $z$ is the fractional wavelength increase, $1+z$ is the total stretch factor (the ratio of observed to emitted wavelength), and that stretch equals the scale-factor ratio $a_0/a_{\text{emit}}$.

The quantity $z$ is defined as

$$z = \frac{\lambda_{\text{obs}} - \lambda_{\text{emit}}}{\lambda_{\text{emit}}}.$$

If we set today’s scale factor to $a_0 = 1$, then light emitted when the universe had scale factor $a_{\text{emit}} = 1/2$ is observed with $1+z = 2$, or $z = 1$. The universe doubled in scale while the light was traveling.

Cosmological redshift is therefore a time-and-scale clue. It tells us that the light we receive has crossed an evolving universe. To turn redshift into a precise distance or age, we need a cosmological model, including the effects of matter, radiation, curvature if present, and dark energy. But the core idea is already enough here: redshift records expansion. We call this stretch the cosmological redshift.

Cosmological redshift

Wavelength stretching caused by the expansion of the universe while light travels — distinct from an ordinary Doppler shift due to motion through space. It records the ratio of the present scale factor to the scale factor at emission.

Multiple choice

If today’s scale factor is $a_0=1$ and light was emitted when $a_\text{emit}=1/4$, what redshift do we observe?

$z = 3$$z = 1/4$$z = 1$

Numeric answer

A galaxy has redshift $z = 3$. What is the wavelength stretch factor $1+z$ (the ratio of observed to emitted wavelength)?

Your answer

Part 4: Looking Far Away Means Looking Back

Light travels at a finite speed. That simple fact turns distance into time. When we observe a nearby object, we see it as it was a short time ago. When we observe a distant galaxy, we see it as it was millions or billions of years ago.

Deep fields make this idea visible. They contain galaxies at many distances, which means they contain galaxies at many lookback times. A single image becomes a layered history of galaxy formation and evolution.

Lookback time

The time light has spent traveling from an object to us, so distant objects are seen as they were in the past. A deep field mixes many lookback times in one frame.

This is one of the most important habits in cosmology: do not read a deep field as a flat photograph. Read it as a time archive. A small, faint, red galaxy in the background may not be a small faint version of a nearby galaxy. It may be a distant galaxy observed when the universe was much younger, with its light stretched and dimmed by cosmic expansion.

JWST pushes this logic closer to the first generations of galaxies. Very high-redshift objects test how quickly stars, gas, dust, and black holes assembled after the Big Bang.

The frontier is exciting because it is a stress test. If galaxies appear earlier, brighter, or more massive than expected, then models of early galaxy formation must explain how gas cooled, formed stars, and assembled structure quickly enough. As always, the observation is not the final answer. The observation is the constraint that the model must survive.

Quick check

Why should we read a deep field as a time archive rather than a flat collection of galaxies?

Summary

We started from Module 2’s question — how far away is that object? — and finished at the expansion of the cosmos. The distance ladder climbs from parallax to standard candles to the Hubble flow, each rung calibrating the next. Hubble’s law, $v = H_0 d$, turns distance into recession speed, and inverting the expansion rate gives a first age of the universe, $1/H_0 \approx 14~\mathrm{Gyr}$. Cosmological redshift records the scale factor directly through $1+z = a_0/a$, and — pushed far enough — the same expansion carries distant galaxies apart faster than light, because space itself is stretching. Deep fields turn distance into lookback time, so the farthest galaxies show us the youngest universe.

We have measured how fast the universe expands, how far we can see, and roughly how old it is. The next question is what governs that expansion: what the universe is made of, and how its contents set its history. That is where we turn next.

Retrieval — equation: cosmological-redshift▸

Glossary

Cosmological redshift

Wavelength stretching caused by the expansion of the universe while light travels — distinct from an ordinary Doppler shift due to motion through space. It records the ratio of the present scale factor to the scale factor at emission.

Distance ladder

A chain of distance-measurement methods in which nearby calibrated methods support more distant methods — parallax calibrates Cepheids, Cepheids calibrate supernovae, and so on out to cosmological distances.

Hubble constant

The present-day expansion rate $H_0$ in Hubble’s law, usually written in $\mathrm{km\,s^{-1}\,Mpc^{-1}}$. Independent measurement chains currently disagree — early-universe methods give roughly $67$, local distance-ladder methods roughly $73$ — a gap called the Hubble tension, an active research frontier.

Hubble's law

The large-scale relation $v = H_0 d$, connecting galaxy recession speed and distance at low redshift. It is the empirical signature of cosmic expansion.

Lookback time

The time light has spent traveling from an object to us, so distant objects are seen as they were in the past. A deep field mixes many lookback times in one frame.

Scale factor

A quantity, written $a(t)$, that describes the relative size of the universe at a given cosmic time. Normalizing today to $a_0 = 1$, a past epoch with $a = 1/2$ is when the universe was half its present scale.

Spacetime

The combined description of space and time. In cosmology, expanding spacetime means the scale factor changes with time, stretching large-scale distances and the wavelengths of traveling light.

Standard candle

An object whose luminosity can be inferred — a Cepheid from its period, a Type Ia supernova from its light curve — allowing distance to be estimated from the observed flux via the inverse-square law.