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Sandbox Physics
Cosmology from expansion to inference

L1 · Chapter 1

Following light through an expanding universe

Sandbox Physics ·

Expansion stretches light as it travels. Redshift tells us the scale of the universe when the light was emitted. Converting that redshift into distance requires the expansion history along the way. Separating these questions makes the distance formulas easier to understand.

Label positions that follow the expansion

Give galaxies fixed coordinates as they follow the mean expansion. Their coordinate separation is a comoving length. The physical separation is that length multiplied by the scale factor a(t)a(t). Set today’s scale factor to one. A fixed comoving separation occupied a shorter physical distance in the past.

Successive light crests stretch too. Redshift compares received wavelength λobs\lambda_{\rm obs} with emitted wavelength λem\lambda_{\rm em}. A redshift of one means the length scale doubled between emission and reception.

1+z=λobsλem=1aem,H=a˙a1+z=\frac{\lambda_{\rm obs}}{\lambda_{\rm em}}=\frac{1}{a_{\rm em}},\qquad H=\frac{\dot a}{a}
The scale factor and redshift are dimensionless. The Hubble parameter is the fractional expansion rate and has units of inverse time.

Reading: David Tong · Pantheon+ public data release

Add the small segments of a light ray

During a short time interval, light covers the speed of light times that interval in physical length. Divide by the scale factor to express it as a comoving length. The redshift relation converts the time integral into a redshift integral. Write χ\chi for radial comoving distance. In flat space it also equals the transverse comoving distance DMD_M.

For brightness, expansion lowers each photon’s energy and increases the time between arriving photons. Luminosity distance absorbs these effects into an inverse-square law. For angular size, use the physical transverse size when the light was emitted. That defines angular-diameter distance.

χ(z)=c∫0zdz′H(z′),DL=(1+z)χ,DA=χ1+z\chi(z)=c\int_0^z\frac{dz'}{H(z')},\quad D_L=(1+z)\chi,\quad D_A=\frac{\chi}{1+z}
Distances may be expressed in megaparsecs. The relation DL=(1+z)2DAD_L=(1+z)^2D_A assumes metric light propagation and conserved photon number. Curved space requires a separate transverse-distance conversion.
ObserverLight source

Light travels from right to left; the expansion history sets the comoving distance.

DL=(1+z)χ,DA=χ/(1+z)D_L=(1+z)\chi,\quad D_A=\chi/(1+z)
Flat-space schematic. Luminosity distance describes flux; angular-diameter distance describes apparent size.

Reading: David Tong · Pantheon+ public data release

Let the cosmic ingredients set the expansion

Matter dilutes as a three-dimensional volume grows. Radiation loses energy per photon as well, so its density falls by one extra power of the scale factor. A cosmological constant keeps a constant energy density. The Friedmann equation combines these contributions into the expansion history.

Write Ωi\Omega_i for a component’s present density divided by the critical density. For constant-equation-of-state dark energy, ww is pressure divided by energy density. A cosmological constant has w=−1w=-1. This equation determines a homogeneous background; dark-energy perturbations need further assumptions.

H2(z)H02=Ωr(1+z)4+Ωm(1+z)3+Ωde(1+z)3(1+w)\frac{H^2(z)}{H_0^2}=\Omega_r(1+z)^4+\Omega_m(1+z)^3+\Omega_{\rm de}(1+z)^{3(1+w)}
Flatness gives Ωr+Ωm+Ωde=1\Omega_r+\Omega_m+\Omega_{\rm de}=1. Retain radiation for the microwave-background era. The low-redshift supernova example neglects it.

Reading: David Tong · Pantheon+ public data release

Work an analytic matter-only example

Take a flat, pressureless matter-only universe with H0=70  km s−1 Mpc−1H_0=70\;\mathrm{km\,s^{-1}\,Mpc^{-1}}. Its expansion rate is proportional to redshift plus one to the power three-halves. Integrating the reciprocal gives the expression below, independently of any numerical quadrature.

At redshift one, it gives approximately 2509  Mpc2509\;\mathrm{Mpc} comoving distance, 5018  Mpc5018\;\mathrm{Mpc} luminosity distance and 1254  Mpc1254\;\mathrm{Mpc} angular-diameter distance. All three refer to the same source. Their different values answer different observational questions.

χ(z)=2cH0[1−11+z]\chi(z)=\frac{2c}{H_0}\left[1-\frac{1}{\sqrt{1+z}}\right]
Substitute the matter-only expansion rate into the distance integral. This analytic result independently checks the existing supernova distance calculation.

A horizon asks how far a signal had time to travel

A source distance stops at its emission time. The particle horizon extends back to the early universe: χhor=c∫0t0dt/a(t)\chi_{\rm hor}=c\int_0^{t_0}dt/a(t). Changing the integration limits changes the physical question.

Sound traveled more slowly through the early photon-baryon fluid. Replace light speed by the evolving sound speed and stop the integral at last scattering to obtain the sound horizon. This early ruler will let us interpret angles on the sky.

Try the prediction

Read the distance curve in public supernova data

  1. Compare low- and high-redshift points and predict which better separates expansion histories.
  2. Change the matter fraction, inspect the curve and residuals, then fit.
  3. Explain how the free brightness offset absorbs absolute calibration. The selected sample supplies relative-distance information.

Exercises and explanations

  1. At redshift two, what was the scale factor? How many times larger is luminosity distance than angular-diameter distance?

    Read the explanation

    The scale factor was one-third. The distance ratio is (1+z)2=9(1+z)^2=9, because brightness and angular size contain different redshift factors.

  2. Increase the Hubble constant by ten percent at fixed density fractions and redshift. What happens to distance?

    Read the explanation

    The whole expansion function grows by the same factor, so every distance becomes ten-elevenths as large, about nine percent shorter. A free supernova brightness offset absorbs this overall scale.

Sources and calculation scope

  1. David Tong, Cosmology, Chapter 1

    Sections 1.1.3–1.1.5 and 1.2–1.3 supply redshift, distances, horizons and fluid backgrounds. Course examples use explicitly flat models; the low-redshift supernova model omits radiation.

  2. Pantheon+ public data release

    The existing distance Lab uses forty selected Pantheon+ supernovae and their statistical-plus-systematic covariance submatrix. Its free magnitude offset removes absolute-distance calibration; this sample does not measure the Hubble constant.

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