Stockholm University · Department of Astronomy

AS5003 (Cosmology) HT26

Lecture 2 (5 Nov 2026)

Timeline of the Universe from the Big Bang to today

Hubble's 1929 diagram: velocity against distance for nearby galaxies

Practical information

Homework assignment: Problem set 1 is online in Canvas, under Homework Assignments in Modules, or under Assignments.
Due Tuesday 10 November, 10:00.
- Upload your answers in Canvas before the deadline, as a single file (preferably PDF).
- Scanned, handwritten work is acceptable, but needs to be clearly legible (I can only grade what I can read).
Literature review: the topic choice is posted as an assignment under Literature Assignment, titled Literature Topic Selection.
First topic choice due Thursday 12 November.

Relevant units

Distance: 1 megaparsec (Mpc) $\approx 3\times10^{22}$ m
distance to Andromeda $\approx 0.75$ Mpc
Time: 1 gigayear (Gyr) $= 10^9$ yr $\approx 3\times10^{16}$ s
age of the Sun $\approx 4.57$ Gyr
Mass: 1 solar mass ($M_\odot$) $\approx 2\times10^{30}$ kg
mass of the Milky Way $\approx 10^{12}\,M_\odot$

Recap: fundamental observations

The night sky is dark
The Universe has a finite age, so we only see stars out to $r \sim c t_0 \ll \lambda$.
The universe is isotropic and homogeneous on large scales
Cosmic Microwave Background
Proportions of light elements (H, D, He, Li)
Galaxies are receding from each other with a velocity proportional to their distance

Recap

Try it: is the real Universe homogeneous?

Data: 2dFGRS (Colless et al. 2001), both strips, projected

Far away the survey only sees the brightest galaxies, so we only compare circles at the same redshift.

Redshift of the circles Circle diameter

Answer to the L1 question "why did near patches look lumpy and far ones smooth?": far away the survey sees only bright galaxies, so fewer dots; structure is there at all distances. Pick a redshift, grow the circles: small circles differ a lot, ~100–200 Mpc circles agree to ~10%. Real data (2dF, z ≈ 0.08–0.1): ~±25% at 100 Mpc, ~±10% at 200 Mpc, ~±7% at 300 Mpc. Same idea at every redshift; far away the counts are smaller, so counting noise (√N) adds to the spread. The redshift slider stops at z = 0.12: beyond that the spread grows again because of the survey itself (patchy completeness, small changes of the magnitude limit across the sky matter a lot when we only see the brightest galaxies), not because the Universe is lumpier.
Homogeneous ≠ smooth: each big circle still has filaments and voids inside, just the same amount. From one place we measure isotropy (two opposite strips alike); homogeneity needs the Copernican principle, tested indirectly (distant clusters see an isotropic CMB).

Recap: fundamental observations

The night sky is dark
The universe is isotropic and homogeneous on large scales
Cosmic Microwave Background
Proportions of light elements (H, D, He, Li)
Galaxies are receding from each other with a velocity proportional to their distance ← Today's topic!

How do we measure velocity? How do we measure distance?

Measuring velocities

The Doppler effect

A moving light source: the waves are bunched up in front of it (blueshift) and stretched behind it (redshift)

A source moving away from (or towards) us at speed $v \ll c$ shifts the wavelength:

$$\lambda_\text{obs} = \lambda_\text{em}\left(1 + \frac{v}{c}\right)$$

$$\frac{\lambda_\text{obs} - \lambda_\text{em}}{\lambda_\text{em}} = \frac{v}{c}$$

Moving away: longer wavelength, redshift
Moving towards us: shorter wavelength, blueshift

Warm-up exercise: Doppler effect

You are pulled over by the police for not stopping at a red light. Having just learned about the Doppler effect, you try this defence: "At the speed I was going, the red light appeared green!"

How fast would you have to be driving for red light to appear green?

(Take green light as 500 nm and red light as 600 nm.)

$$\frac{v}{c} = \frac{\lambda_\text{obs} - \lambda_\text{em}}{\lambda_\text{em}} = \frac{500 - 600}{600} = -\frac{1}{6} \;\Rightarrow\; |v| \approx 5\times10^4\ \text{km/s}$$ About $1.8\times10^8$ km/h: you get a speeding ticket instead.

Try it: drive through the red light

The three lamps of a traffic light (630, 590 and 505 nm), seen by a driver moving at speed $v$. Negative $v$: driving towards the light.

Speed $v/c$

Such a line shift is visible in galaxy spectra

Absorption lines in a spectrum: unshifted, shifted to the red, and shifted to the blue

Wavelength increases to the right (violet → red): redshifted lines move right, blueshifted lines left

By measuring the observed wavelengths of identifiable spectral features (e.g. the Balmer lines of hydrogen), and comparing them with the rest wavelengths, we can calculate the relative velocities.

This is described by the redshift $z$:

$$z \equiv \frac{\lambda_\text{obs} - \lambda_\text{em}}{\lambda_\text{em}} \quad \left(= \frac{v}{c}\right)$$

Try it: redshift a spectrum

Slide the redshift and watch the emission lines move to longer wavelengths.

Redshift $z$

Schematic spectrum built from real rest wavelengths, not a real object. The rainbow bar marks visible light.

Example: observed quasar spectra

Which of these is furthest away? Which is closest?

Spectra of three quasars, each with a strong Lyman-alpha emission line at a different observed wavelength

z = ? z = ? z = ?

The circled emission line is the Lyα (2→1) transition of hydrogen in all three spectra, i.e. at the same rest wavelength (1216 Å).

The top one is furthest: its Lyα is shifted the most ($z = 6.3$). Then $z = 4.4$ and $z = 2.8$.

All but the very closest galaxies show a positive redshift

Vesto Slipher, director of Lowell Observatory in Arizona

Slipher's 1917 table of the radial velocities of 25 spiral nebulae, most of them positive

Velocities of 25 spiral nebulae (Slipher 1917)

These are the spiral nebulae: inside or outside our Galaxy?

Most are redshifted, some by more than 1000 km/s. A few, such as Andromeda (NGC 224), are blueshifted.

Slipher had the velocities. What was missing was the distances.

Measuring distances

The Shapley–Curtis Great Debate (1920)

Portraits of Harlow Shapley and Heber Curtis

Harlow Shapley (left) and Heber Curtis (right)

The spiral nebulae, seen since 1845, were still a mystery. In 1920 the question came to a head in the Great Debate.

Central question: how big is our Universe? Is the Milky Way all there is?

Shapley argued that the Milky Way Galaxy was the entire universe, and that "spiral nebulae" were inside the Milky Way.

Curtis argued that these "nebulae" were galaxies like our own: Slipher's speeds were too fast for objects bound to the Milky Way.

It could not be settled in 1920: nobody could measure the distance to a nebula.

Slipher's spirals raise the question: are they part of the Milky Way? In 1920 the open question was how big the Universe is. To settle it we need distances.
Slipher's speeds (1912–17) were the new evidence.
Nebulae: "nebula" meant any fuzzy patch on the sky (Messier 1781, Herschel). Huggins (1864) found emission lines in the Orion nebula: a glowing gas cloud. But the spiral nebulae (Lord Rosse saw the spiral in M51 in 1845) had a continuous, star-like spectrum. Clouds forming stars inside the Milky Way, or "island universes" like our own (Kant 1755)?
Great debates drive a field: two clear, opposite positions force sharp tests.

Newton vs Huygens (late 1600s): light as particles (corpuscles) or as waves. Young's double slit (1801) favoured waves; Einstein's photons (1905) brought particles back: wave–particle duality.
Bohr vs Einstein (from the Solvay conference 1927): is quantum mechanics complete? Einstein: "God does not play dice"; EPR paradox (1935). Settled by Bell's inequality (1964) and experiments (Aspect 1982; Nobel Prize 2022): quantum mechanics wins.
In cosmology itself, later: steady state (Hoyle) vs Big Bang (Gamow, Lemaître), settled by the CMB (1965). Coming up in this course.

Today: Shapley–Curtis, settled by measuring distances.

Cepheid variable stars: the first "standard candle"

Henrietta Leavitt (1868–1921)

Leavitt discovered that the luminosity of Cepheids is related to their pulsation period: bright ones pulsate slowly, faint ones fast.
Leavitt's law: $L = K\,P^{\alpha}$ (slope $\alpha$ from her diagram)

Calibrate $K$ with a Cepheid of known distance (e.g. parallax). Then:

$$f = \frac{L}{4\pi r^2} \;\Rightarrow\; r = \sqrt{\frac{L}{4\pi f}}$$

$L$: from the period $P$ · $f$: the flux we measure

The "dimness" of Cepheids in Andromeda and other "spiral nebulae" (Hubble 1925) showed that they are far outside our Galaxy. Curtis was right.

Leavitt's 1912 figure: magnitude against the logarithm of the period for Cepheids in the Small Magellanic Cloud

Leavitt (1912): apparent magnitude (brighter is up) against the logarithm of the period. Each star twice: at its brightest and faintest.

This is what Leavitt discovered (about 110 years ago, Cepheids in the Small Magellanic Cloud, all at about the same distance). Cepheids are stars that pulsate: they swell and shrink, getting brighter and fainter with a regular period. The rate depends on their brightness: the bright one pulsates slowly, so its period is longer; the faint one pulsates fast, so its period is shorter. Leavitt's diagram shows exactly that: brighter (up) at longer period (right).
All her stars are at the same distance, so f ∝ L: the diagram gives the shape of the law, L = K P^α (the slope α), but not the constant K.
[click] Calibration: take a Cepheid whose distance r is known another way, measure its f and P: K = 4πr²f / P^α. Then for any Cepheid: P gives L, we measure f, so the formula gives r.
Parallax in one line: as the Earth orbits the Sun, a nearby star shifts against the background; a shift of p arcseconds means d = 1/p parsec (that is where the parsec comes from). (In 1913 no Cepheid was close enough for a direct parallax; Hertzsprung used statistical parallaxes. Direct Cepheid parallaxes came with Hipparcos (1997) and Gaia.)
The big plot: y = apparent photographic magnitude (smaller = brighter, so brighter is up), x = log10 of the period in days. Two lines: each Cepheid at maximum and at minimum light. All in the SMC, so apparent brightness ∝ luminosity.

Combining distance and velocity: Hubble's law

E. Hubble (1929), PNAS 15, 168

Edwin Hubble

$$v = H_0\, D$$

$H_0$: the Hubble constant, in km/s/Mpc
$v = cz$: the velocity
$D$: the distance

Hubble's estimate: $H_0 \approx 500$ km/s/Mpc.

A modern Hubble diagram

Velocity against distance for galaxies with Cepheid distances out to 22 Mpc, with lines for H0 = 67, 75 and 83 km/s/Mpc

Freedman et al. (2001), ApJ 553, 47 (Ryden Fig. 2.5)

Distances from Cepheids (Hubble Space Telescope)
Lines: $H_0 = 67$, $75$, $83$ km/s/Mpc
Hubble (1929): $H_0 \approx 500$ km/s/Mpc

All of Hubble's distances were too small. For example, galaxies with $v \approx 1000$ km/s:

Hubble: $D \approx 2$ Mpc
Today: $D \approx 15$ Mpc

Distances about 7× too small → $H_0$ about 7× too large

Measurement of $H_0$ over time

Measurements of the Hubble constant since 1920: from about 500 down to about 70 km/s/Mpc

From about 500 (Hubble 1929) down to 67–73 km/s/Mpc today.

But the two best methods disagree:

Early Universe (CMB, Planck):
$H_0 = 67.4 \pm 0.5$
Local distance ladder (Cepheids + supernovae, SH0ES):
$H_0 = 73.0 \pm 1.0$

The Hubble tension (about 5σ): an open question, and a good literature-project topic.

Does Hubble's law make our location special?

Almost every galaxy moves away from us, faster the further away it is.

Are we at the centre of the expansion?

Think of raisins in a rising bread dough: from every raisin, all the others move away, faster when they are further.

To check that this works for galaxies, describe the expansion with one function: the scale factor $a(t)$.

Click a raisin to make it our galaxy: the dough then rises.

Hubble's law arises naturally in a universe undergoing homogeneous & isotropic expansion

$$r_{12}(t) = a(t)\,r_{12}(t_0), \quad r_{23}(t) = a(t)\,r_{23}(t_0), \quad r_{31}(t) = a(t)\,r_{31}(t_0)$$

$a(t)$: the scale factor, normalised to $a(t_0) = 1$ today
homogeneous: $a$ is a function of $t$, but not of $\vec r$
isotropic: $a$ is a scalar, the same in every direction

$$v_{12}(t) = \frac{dr_{12}}{dt} = \frac{da}{dt}\,r_{12}(t_0) = \frac{\dot a}{a}\,r_{12}(t)$$

$$v = H(t)\,r, \qquad H(t) \equiv \frac{\dot a}{a}$$ $H(t)$: the "Hubble parameter"; $H_0 = H(t_0)$: the "Hubble constant"

Hubble time

$$H_0 \approx 70\ \text{km/s/Mpc}$$

$H_0$ is today's rate of expansion, with units 1/time:

$$H_0 \approx 70\,\frac{\text{km}}{\text{s Mpc}} \times \frac{10^3\,\text{m}}{1\,\text{km}} \times \frac{1\,\text{Mpc}}{3.1\times10^{22}\,\text{m}} \approx 2.2\times10^{-18}\,\text{s}^{-1}$$

Every second, all distances between galaxies grow by a fraction $2.2\times10^{-18}$.

$$\frac{1}{H_0} \approx 4.4\times10^{17}\,\text{s} \approx 1.4\times10^{10}\ \text{years} \approx 14\ \text{Gyr}$$

So if the Universe has been expanding at a constant rate $H_0$, it is about 14 billion years old.

Try it: Hubble time

$H_0$ (km s⁻¹ Mpc⁻¹)

Try Hubble's 500 km/s/Mpc.

Relation between redshift and scale factor

Scale factor when the light was emitted, $a(t_\text{em})$

Think of the redshift of galaxies as the wavelength being "stretched" as space expands. The ratio of observed and emitted wavelengths is the ratio of scale factors:

$$\frac{\lambda_\text{obs}}{\lambda_\text{em}} = \frac{a(t_\text{obs})}{a(t_\text{em})} = \frac{1}{a(t_\text{em})}$$

From the definition of redshift:

$$1 + z = \frac{1}{a(t_\text{em})}$$

(We derive this properly from the Robertson–Walker metric next week; see also Ryden §3.5.)

Example: back to our quasars

Spectra of three quasars at redshifts 6.3, 4.4 and 2.8

How small was our Universe when the light from the middle quasar was emitted?

$$a = \frac{1}{1+z} = \frac{1}{1 + 4.4} \approx 0.19$$

The Universe was about 5.4× smaller: all distances between galaxies were shorter by this factor.

Common confusion: relativistic Doppler effect and redshift

For large enough $D$, Hubble's law gives velocities greater than $c$. Should we use the special-relativistic Doppler formula instead?

Doppler ($v \ll c$)

$$\frac{\lambda_\text{obs}}{\lambda_\text{em}} = 1 + \frac{v}{c}$$

Special relativity

$$\frac{\lambda_\text{obs}}{\lambda_\text{em}} = \sqrt{\frac{1 + v/c}{1 - v/c}} \quad ?$$

No. The redshifts of galaxies come from the expansion of space, not from motion through space. We define

$$\frac{\lambda_\text{obs}}{\lambda_\text{em}} = 1 + z$$

… and $v = cz$ can be larger than $c$ when $z > 1$.

Expansion and peculiar motion

Hubble: the Universe is expanding, so galaxies farther away recede from us faster.

But gravity also moves things: the Earth orbits the Sun, the Sun orbits the centre of the Milky Way.

This motion through space is called peculiar motion.

Peculiar velocities and redshifts

The velocity we measure from the Doppler shift has two parts:

cosmological redshift, due to the expansion of the universe
an extra Doppler shift, due to the peculiar velocity: real motion through space, so never faster than $c$

$$z \approx z_\text{cosmological} + \frac{v_\text{peculiar}}{c}$$

Peculiar velocities are $\lesssim 3000$ km/s, so the expansion dominates beyond $z \gtrsim 0.01$.

● real positions, arrows: velocities. ● where a survey puts them

CfA redshift survey slice: clusters look stretched along the line of sight because of their galaxies' peculiar velocities

z ≈ 0.017, ≈ 70 Mpc z ≈ 0.033, ≈ 140 Mpc Coma cluster: a "finger"
pointing at us

CfA redshift slice (de Lapparent, Geller & Huchra 1986). We are at the bottom tip; radial axis cz in km/s, out to z ≈ 0.05 (≈ 210 Mpc, ≈ 0.7 Gyr ago)

The clearest sign of the cosmological principle: the CMB

Galaxy surveys are messy: clustering, peculiar velocities, survey edges.
The CMB has the same temperature in every direction, $T_0 = 2.7255$ K, to about 1 part in 100,000 (the colours).
More in the CMB lecture.

Planck's all-sky map of the cosmic microwave background: tiny hot (red) and cold (blue) spots on an otherwise uniform sky

The CMB over the whole sky, from Planck. Image: ESA and the Planck Collaboration (CC BY 4.0)

Back to Slipher's spirals

Velocities of 25 spiral nebulae (Slipher 1917)

Can you now guess why a few of these spirals, such as Andromeda (NGC 224), are blueshifted?

$z \approx z_\text{cosmological} + v_\text{peculiar}/c$

They are very close. For Andromeda ($D \approx 0.8$ Mpc) the expansion gives only $H_0 D \approx 55$ km/s.

Its peculiar velocity towards us is a few hundred km/s, so close by the peculiar velocity wins.

Exercise session

Exercises on redshifts: available on Canvas under the Lecture 2 module.

Before next time

Homework 1 is due Tuesday 10 November, 10:00, in Canvas
Read Ryden chapter 3: curvature and the Robertson–Walker metric
Literature project: first topic choice due Thursday 12 November