How the light-year number is calculated from the scale factor and Hubble expansion/ The integrals that give distances in light-years/ Why an integral?

 


The “light-year number” for distant objects in cosmology is a distance (in light-years) obtained from the expansion history via the scale factor (a(t)) and the Hubble parameter \(H = \dot{a}/a\). There is no single distance; cosmologists use several related measures that all start from the same integral along a light ray.

Key relations

  • Scale factor (a(t)) is conventionally normalized so \(a(\text{today}) = 1\).
  • Redshift of light emitted when the scale factor was \(a_e\): \[ 1 + z = \frac{1}{a_e}. \]
  • Hubble parameter: \(H(a) = \dot{a}/a\), or equivalently (H(z)). In the standard flat \(\Lambda\)CDM model \[ H(z) = H_0\, E(z),\qquad E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_\Lambda + \Omega_k(1+z)^2}. \](\(H_0\) is the present Hubble constant; the \(\Omega\)’s are density parameters.)


Light travels on null geodesics (\(ds=0\)). In an expanding FLRW universe the radial comoving coordinate distance accumulated by a photon is obtained by integrating
\[ c\,\frac{dt}{a(t)}. \] Changing variable with \(dt = da/(a H(a))\) (or the equivalent redshift form) yields the fundamental integral cosmologists use.

The integrals that give distances in light-years

1. Line-of-sight comoving distance (the most fundamental quantity; often the number people convert to light-years when they quote “how far away a galaxy is today”):
\[ d_C(z) = \frac{c}{H_0}\int_0^z\frac{dz'}{E(z')} = c\int_{a_e}^{1}\frac{da}{a^2 H(a)}. \] (This is also written \(D_H\int_0^z dz'/E(z')\) where \(D_H = c/H_0\) is the Hubble distance \(\approx 3000\,h^{-1}\) Mpc.)

  • Today’s proper (physical) distance to the same object is simply \(d_P(\text{today}) = d_C\) (because \(a=1\)).
  • At emission the proper distance was \(a_e\,d_C = d_C/(1+z)\).

2. Light-travel (look-back) distance (the distance light would have covered if space were static; equals (c) times the look-back time):
\[ d_T(z) = \frac{c}{H_0}\int_0^z\frac{dz'}{(1+z')E(z')} = c\int_{t_e}^{t_0}dt. \] This is the number that is closest to a naïve “light-years of travel,” but it is not the present-day separation.

3. Other common distances derived from the same integral

  • Angular-diameter distance: \(d_A = d_M/(1+z)\)
  • Luminosity distance: \(d_L = (1+z)\,d_M\) where \(d_M\) is the transverse comoving distance (identical to \(d_C\) in a flat universe).


Converting to light-years

Once any of the above distances is computed in Mpc (or metres), multiply by the conversion factor
\[ 1\,\text{Mpc} \approx 3.26156\times10^6\,\text{light-years}. \] Typical numerical results for the particle horizon (comoving distance to \(z\to\infty\)) in \(\Lambda\)CDM are \(\sim 46.5\) billion light-years today; the corresponding light-travel distance is only \(\sim 13.8\) billion light-years (the age of the Universe times (c)).

Why an integral is needed

Because (H) (and therefore the expansion rate) changes with time/redshift, the distance is not simply \(cz/H_0\). For small (z) the linear Hubble law is recovered, but for high-redshift objects the full integral must be evaluated numerically (or with analytic approximations valid for particular cosmologies).

In short: start from the null geodesic condition, change variables using (H(a)) or (H(z)), integrate, and convert units. The comoving-distance integral is the one most often meant when cosmologists quote a “distance in light-years.”

Text: Grok
Images: Blogger

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