Coaching Centre
Hubble and the expanding universe
How the expansion of the universe was discovered: Slipher's redshifts, Leavitt's period–luminosity law for Cepheid variables, Hubble's 1929 velocity–distance graph, Hubble's law v = H₀d, redshift, estimating the age of the universe from 1/H₀, and today's Hubble tension.
In 1912 Henrietta Swan Leavitt, paid 30 cents an hour to measure photographic plates at Harvard, noticed that in a small nearby galaxy the brighter variable stars took longer to pulse. That one graph turned a class of flickering stars into measuring sticks for the universe. Within 17 years it had shown that the Milky Way is one galaxy among millions, and that almost all of them are rushing away from us.
The universe in 1920
A century ago most astronomers thought the universe was static, and many thought the Milky Way was all of it. The fuzzy "spiral nebulae" seen through telescopes were argued about at the 1920 "Great Debate": were they clouds of gas inside our galaxy, or separate "island universes" far away? Nobody could settle it, because nobody could measure the distance to them.
Two separate lines of evidence came together to answer the question:
- Velocities from redshift. From 1912, Vesto Slipher at the Lowell Observatory measured the spectra of spiral nebulae. The absorption lines in most were shifted towards longer (redder) wavelengths, showing they were moving away, some at over 1000 km s−1, far faster than any star in our galaxy.
- Distances from Cepheid variables. Leavitt's discovery gave a way to find how far away a Cepheid star is, and so how far away the galaxy containing it is.
Cepheid variables: standard candles
A Cepheid variable is a giant star that pulses in and out, so its brightness rises and falls with a regular period, typically between 1 and 100 days. Leavitt studied Cepheids in the Small Magellanic Cloud. They are all at about the same distance from us, so the ones that look brighter really are more luminous. She found that the period is related to the luminosity: the longer the period, the more luminous the star.
Once the relation was calibrated with a few nearby Cepheids whose distances were known by parallax, any Cepheid became a standard candle: an object whose luminosity you can work out without knowing its distance. Comparing the luminosity $L$ with the apparent brightness $b$ seen from Earth then gives the distance, using the inverse square law from the data sheet:
$$b = \frac{L}{4\pi d^2} \quad\Rightarrow\quad d = \sqrt{\frac{L}{4\pi b}}$$
In 1923–24 Edwin Hubble found Cepheids in the Andromeda nebula with the new 100-inch telescope at Mount Wilson. They were so faint that Andromeda had to be far outside the Milky Way. Spiral nebulae were galaxies in their own right.
A Cepheid has a period that shows its luminosity is $L = 3.8 \times 10^{30}$ W. Its apparent brightness measured from Earth is $b = 4.0 \times 10^{-13}$ W m−2. How far away is it, in metres and in megaparsecs? (1 pc = $3.086 \times 10^{16}$ m)
$d = \sqrt{\dfrac{L}{4\pi b}} = \sqrt{\dfrac{3.8 \times 10^{30}}{4\pi \times 4.0 \times 10^{-13}}} = \sqrt{7.56 \times 10^{41}} = 8.7 \times 10^{20}$ m
$d = \dfrac{8.7 \times 10^{20}}{3.086 \times 10^{16}} = 2.8 \times 10^{4}$ pc $= 28$ kpc $= 0.028$ Mpc. That is in the outer halo of the Milky Way, beyond the edge of its disc.
Redshift and recession velocity
If a galaxy is moving away, every line in its spectrum is shifted to a longer wavelength. The fractional change in wavelength is the redshift, $z$:
$$z = \frac{\lambda_{\text{observed}} - \lambda_{\text{emitted}}}{\lambda_{\text{emitted}}} = \frac{\Delta\lambda}{\lambda}$$
For speeds much less than the speed of light, the recession velocity is $v \approx zc$. (For distant galaxies this simple formula breaks down, and the redshift is better understood as the stretching of the light's wavelength by the expansion of space while it travels.) A blueshift, where lines move to shorter wavelengths, means the object is approaching: Andromeda is one of the few galaxies with a blueshift.
The hydrogen-alpha line has a laboratory wavelength of 656.3 nm. In a galaxy's spectrum it is measured at 670.1 nm. Find the redshift and the galaxy's recession velocity.
$z = \dfrac{670.1 - 656.3}{656.3} = \dfrac{13.8}{656.3} = 0.0210$
$v \approx zc = 0.0210 \times 3.00 \times 10^{8} = 6.31 \times 10^{6}$ m s−1 $= 6310$ km s−1, moving away.
Hubble's law
In 1929 Hubble plotted the velocities of 24 galaxies (mostly Slipher's measurements) against their distances (from Cepheids and from the brightest stars in them). Despite a lot of scatter, the trend was clear: the further away a galaxy is, the faster it is moving away. Georges Lemaître had found the same relation in 1927 and explained it as the expansion of the universe. Today it is called the Hubble–Lemaître law:
$$v = H_0 d$$
$H_0$ is the Hubble constant, the gradient of the velocity–distance graph. It is usually given in km s−1 Mpc−1: for every megaparsec of distance, galaxies recede about 70 km s−1 faster.
Hubble's own value was about 500 km s−1 Mpc−1. His distances were badly underestimated: he did not know that there are two types of Cepheid with different period–luminosity relations, and some of his "brightest stars" were whole clusters of stars. When those errors were corrected through the 1950s, $H_0$ dropped to between 50 and 100.
What the expansion means
Every galaxy sees every other galaxy receding, with speed proportional to distance. That is exactly what happens if space itself is stretching uniformly: imagine dots on a balloon being inflated, or raisins in a rising loaf. No galaxy is at the centre.
If the galaxies are moving apart now, they were closer together in the past. Running the expansion backwards (and assuming constant speed), all the galaxies were together at a time
$$t = \frac{d}{v} = \frac{1}{H_0}$$
ago. This gives a rough age of the universe, and was some of the first evidence for the Big Bang. (Galaxies within a few megaparsecs, held by gravity in our Local Group, don't follow the law: that is why Andromeda can be approaching.)
Taking $H_0 = 70$ km s−1 Mpc−1, estimate the age of the universe in years.
1 Mpc $= 3.086 \times 10^{22}$ m and 70 km s−1 $= 7.0 \times 10^{4}$ m s−1, so $H_0 = \dfrac{7.0 \times 10^{4}}{3.086 \times 10^{22}} = 2.27 \times 10^{-18}$ s−1
$t = \dfrac{1}{H_0} = 4.41 \times 10^{17}$ s $= \dfrac{4.41 \times 10^{17}}{3.156 \times 10^{7}} = 1.4 \times 10^{10}$ years, about 14 billion years. The careful modern value, allowing for the expansion slowing and then speeding up, is 13.8 billion years.
- Choose Galaxy 1. Move the period slider until the orange predicted peaks sit on the measured light curve. The best match is at 12 days. Read the distance: it should be 8 Mpc.
- Try a period of 6 days instead. Does the galaxy come out nearer or further? Explain using the period–luminosity relation.
- Find the periods for Galaxies 2, 3 and 4 (24, 36 and 48 days) and add each to the Hubble graph. Check you get 15, 26 and 34 Mpc.
- With all four galaxies plotted, read the gradient: it should be about 70 km s⁻¹ Mpc⁻¹, giving 1/H₀ ≈ 13.9 billion years.
- Deliberately use the wrong periods for every galaxy. How does a systematic error in distance change $H_0$ and the age? This is the kind of error Hubble made.
The Hubble tension
Modern measurements of $H_0$ fall into two groups. Methods using the "distance ladder" of Cepheids and supernovae in nearby galaxies give about 73 km s−1 Mpc−1. Methods using the cosmic microwave background and the standard cosmological model give about 67–68. Both are now precise to about 1–2%, so the gap is real and is called the Hubble tension. Either there are hidden errors in one of the methods, or our model of the universe is missing something. For calculations, the HSC usually accepts a value of about 70.
Since 1998, measurements of distant supernovae have also shown that the expansion is accelerating. The Australian astronomer Brian Schmidt (now at the ANU) led one of the two teams that discovered this, sharing the 2011 Nobel Prize in Physics. The cause is given the name "dark energy".
Test yourself
Why could Leavitt compare the luminosities of Cepheids in the Small Magellanic Cloud directly from how bright they looked?
With the same distance, differences in apparent brightness reflect differences in luminosity.
A galaxy is 120 Mpc away. With $H_0 = 70$ km s⁻¹ Mpc⁻¹, its recession velocity is
$v = H_0 d = 70 \times 120 = 8400$ km s⁻¹.
If $H_0$ were measured to be larger, the estimated age of the universe would be
Age ≈ 1/H₀: faster expansion means the galaxies took less time to reach their present separation.
Hubble's 1929 value of about 500 km s⁻¹ Mpc⁻¹ was too large mainly because
Gradient = v/d. Underestimated distances make the gradient too large. His Cepheid calibration and "brightest star" assumptions were both wrong.
A line of wavelength 486.1 nm is observed at 491.0 nm. The recession velocity is closest to
$z = 4.9/486.1 = 0.0101$; $v = zc = 0.0101 \times 3.00 \times 10^{5} = 3.0 \times 10^{3}$ km s⁻¹.
Label the Cepheid diagrams.
Put the steps for finding a galaxy's distance with a Cepheid in order.
A Cepheid with a longer period is more . Because its luminosity can be found independently, it is called a standard . Spectral lines of distant galaxies are shifted to longer wavelengths, called a . Hubble's law states that recession velocity is proportional to . The inverse of the Hubble constant estimates the of the universe.
Analyse how the work of Leavitt, Slipher and Hubble led to the conclusion that the universe is expanding. Include a calculation of the age of the universe using $H_0 = 68$ km s⁻¹ Mpc⁻¹.
Leavitt found that a Cepheid's period of variation is related to its luminosity. Once calibrated, this made Cepheids standard candles: measuring a Cepheid's period gives its luminosity, and comparing that with its apparent brightness ($d = \sqrt{L/4\pi b}$) gives its distance. Hubble used Cepheids to show that the spiral nebulae were distant galaxies, and estimated distances for many of them. Slipher had independently measured the spectra of these galaxies and found most were redshifted, so they were moving away ($v \approx zc$). Hubble combined Slipher's velocities with his own distances and found that recession velocity is proportional to distance, $v = H_0d$. Neither set of data alone would have shown this. A velocity proportional to distance in every direction is what is expected if space itself is expanding uniformly, so the universe is expanding and was smaller and denser in the past.
Age: $H_0 = \dfrac{68 \times 10^{3}}{3.086 \times 10^{22}} = 2.20 \times 10^{-18}$ s⁻¹, so $t = 1/H_0 = 4.54 \times 10^{17}$ s $= \dfrac{4.54 \times 10^{17}}{3.156 \times 10^{7}} = 1.44 \times 10^{10}$ years.