Computational Cosmology Calculator

Formula used

The calculator evaluates standard ΛCDM background relations with curvature and an optional radiation term.

• E(z) = \sqrt{Ωr(1+z)⁴ + Ωm(1+z)³ + Ωk(1+z)² + ΩΛ}
• H(z) = H0 · E(z)
• D_H = c / H0 (Hubble distance)
• χ(z) = ∫₀^z dz′ / E(z′) (dimensionless comoving coordinate)
• D_c(z) = D_H · χ(z)
• Transverse comoving distance D_M using curvature:
  If Ωk ≈ 0: D_M = D_c
  If Ωk > 0: D_M = (D_H/\sqrt{Ωk}) sinh(\sqrt{Ωk} χ)
  If Ωk < 0: D_M = (D_H/\sqrt{|Ωk|}) sin(\sqrt{|Ωk|} χ)
• D_L = (1+z) D_M, D_A = D_M/(1+z)
• t_L(z) = (1/H0) ∫₀^z dz′ / ((1+z′)E(z′)) (lookback time)
• t₀ ≈ (1/H0) ∫₀^zmax dz′ / ((1+z′)E(z′)) (age; zmax approximates ∞)
• μ = 5 log₁₀(D_L/\text{Mpc}) + 25 (distance modulus)
• ρ_c,0 = 3H0²/(8πG), ρ_m(z) = Ωm ρ_c,0(1+z)³
• q(z) = [0.5Ωm(1+z)³ + Ωr(1+z)⁴ − ΩΛ]/E(z)²
• dV/(dz dΩ) = D_H · D_M² / E(z)

Numerical integration uses Simpson’s rule with the chosen step count.

How to use this calculator

1. Enter your redshift z and H0.
2. Set density parameters Ωm, ΩΛ, and optionally Ωr.
3. Leave curvature on auto for Ωk = 1 − Ωm − ΩΛ − Ωr, or enable manual Ωk.
4. Choose an integration step count; increase it for higher precision.
5. Set zmax for the age integral; larger values better approximate infinity.
6. Click Submit to view results below the header.
7. Use Download CSV or Download PDF to export the current results.

Computational cosmology notes

1) Background expansion in ΛCDM

This calculator evaluates E(z) and H(z) from matter, radiation, curvature, and dark energy terms. For a flat late-time model, Ωm + ΩΛ ≈ 1, and the radiation contribution is usually negligible for z below a few hundred.

2) Realistic parameter sets

To mirror common literature, try a reference set such as H0 = 67.4 km/s/Mpc, Ωm = 0.315, ΩΛ = 0.685, and Ωr ≈ 0. With these values, the tool typically returns an age near 13.8 Gyr and a Hubble distance around 4.45 Gpc.

3) Numerical integration and stability

The distance and time integrals are computed with Simpson’s rule, which converges quickly for smooth integrands. Use at least 1200 steps for routine work reliably, and raise the value for very high redshift or tight tolerances. If you observe small shifts when increasing steps, treat the difference as an estimate of numerical uncertainty.

4) Distances used in mock catalogs

Comoving distance Dc sets the radial mapping between redshift and coordinate position in light-cone outputs. DM controls transverse separations, while DL and DA connect theory to fluxes and angular sizes.

5) Lookback time and cosmic age

Time integrals help translate redshift snapshots into physical epochs. The lookback time is the difference between the current age and the age at z. For many concordance-like models, z = 1 corresponds to a lookback of roughly 7–8 Gyr, while z = 3 is commonly near 11–12 Gyr.

6) Curvature and geometric distances

When Ωk differs from zero, transverse distances use trigonometric or hyperbolic functions of the comoving coordinate. Even small curvature can shift DM, altering inferred volumes and angular scales. If your model is constrained to be flat, keep auto-curvature enabled and ensure Ωm + ΩΛ + Ωr remains close to one.

7) Volume element for number counts

The differential comoving volume dV/(dz dΩ) combines geometry and expansion, making it essential for survey forecasting. Multiply by a solid angle and a redshift bin width to estimate the comoving volume probed. In practice, this supports rapid checks on expected halo counts, galaxy densities, and selection-function impacts.

8) Growth diagnostics for structure

Beyond background evolution, structure formation depends on how density perturbations grow. The optional output estimates a normalized linear growth factor D(a) and a growth rate f. These are useful for sanity checks against approximate relations like f ≈ Ωm(z)^{0.55} in dark-energy dominated epochs.

FAQs

1) What does E(z) represent?

E(z) is the dimensionless expansion rate, defined by H(z) = H0·E(z). It collects the redshift scaling of matter, radiation, curvature, and dark energy into one function used in distance and time integrals.

2) Why is the age integral using a large zmax?

The age integral formally extends to infinite redshift. A large zmax approximates that limit while keeping computation finite. Increase it if you include radiation or require higher precision at very early times.

3) How should I choose between DL and DA?

DL converts luminosity to observed flux, while DA converts physical sizes to angular sizes. They are related by DL = (1+z)^2 DA, so the correct choice depends on whether your observable is flux-based or angle-based.

4) When should I set Ωk manually?

Use manual Ωk when exploring non-flat cosmologies or matching a published parameter table that specifies curvature explicitly. Otherwise, auto-curvature ensures internal consistency via Ωk = 1 − Ωm − ΩΛ − Ωr.

5) What is distance modulus used for?

The distance modulus μ converts a luminosity distance to magnitudes, which is convenient for supernova analyses and brightness comparisons. It is computed from μ = 5 log10(DL/Mpc) + 25.

6) How accurate is the numerical integration?

With smooth ΛCDM integrands, Simpson’s rule converges rapidly. Accuracy typically improves as you increase the step count. A practical approach is to run two step values and treat the change in results as a numerical error estimate.

7) What do the growth outputs mean?

D(a) is the normalized linear growth factor and f is the logarithmic growth rate. They help connect background parameters to structure formation, but they remain an approximation when radiation or exotic components become important.