Cosmological Calculator

Calculate cosmological distances, times, and scale factors for any redshift using Planck 2018 ΛCDM parameters.

Input Parameters

Redshift (z):

01.0010

Or enter exact value:

Cosmological Parameters (Planck 2018):

H₀: 67.4 km/s/Mpc

Ωₘ: 0.315

Ωₗ: 0.685

Ωₖ: 0.000

Equations Used

Friedmann Equation

$$H(z) = H_0 E(z) = H_0\sqrt{\Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda}$$

Comoving Distance

$$d_C(z) = \frac{c}{H_0}\int_0^z \frac{dz'}{E(z')}$$

Luminosity Distance

$$d_L(z) = (1+z)d_C(z)$$

Angular Diameter Distance

$$d_A(z) = \frac{d_C(z)}{1+z}$$

Lookback Time

$$t_{\text{lookback}}(z) = t_H\int_0^z \frac{dz'}{(1+z')E(z')}$$

where $t_H = 1/H_0$ is the Hubble time.

Scale Factor

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

Physical Interpretation

Comoving Distance

The distance to an object measured along a spatial hypersurface of constant cosmic time. This is the proper distance that would be measured "now" if we could freeze the expansion.

Luminosity Distance

The distance inferred from the observed flux and intrinsic luminosity: $F = L/(4\pi d_L^2)$. This is larger than the comoving distance due to cosmological dimming from expansion.

Angular Diameter Distance

The distance inferred from an object's angular size and physical size: $\theta = D/d_A$. At high redshifts, this distance actually decreases, making distant objects appear larger!

Lookback Time

The difference between the current age of the universe and the age when the light we observe was emitted. This tells us how far back in time we're looking.