Hubble's law

Edwin Hubble, from the Mount Wilson Observatory in California, systematically observed the spectra of galaxies between 1924 and 1929 and noticed two surprising facts.

The first: all galaxies outside our own Milky Way show spectra shifted towards the red. The known absorption lines — those of hydrogen, of calcium — appear at a longer wavelength than measured in the laboratory. The second: the ratio $z = \Delta\lambda/\lambda$ grows linearly with the galaxy’s distance. The farther away a galaxy is, the more its spectrum is shifted towards the red.

Hubble–Lemaître law

The apparent recession velocity of a galaxy at distance $d$ is proportional to $d$: $\ev{v_\text{rec} = H_0\,d}$ with $H_0$ the Hubble constant, estimated today at $H_0 \approx 70$ km/(s·Mpc).

From this relation one also extracts a fundamental order of magnitude: the age of the universe, as the characteristic time of the expansion, is about $1/H_0 \approx 1.4\cdot 10^{10}$ years.

How is the spectral shift $z$ related to the velocity? By the relativistic Doppler effect, in the limit $v \ll c$, $z = \frac{\Delta\lambda}{\lambda} \approx \frac{v}{c} \quad\Longleftrightarrow\quad v = c\,z.$ Hubble’s law is therefore measured by comparing $z$, obtained from spectral lines, with the distance $d$, obtained from standard indicators. The two most important are Cepheids, stars that pulsate with a period precisely linked to their luminosity, and type Ia supernovae, “standard candles” of nearly constant luminosity: knowing the true luminosity and measuring the apparent one, the distance can be deduced.

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Topics: Relatività ristretta Concepts: Effetto Doppler

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