iCosmos › Linear growth factor calculator
Linear growth factor calculator
How much small density fluctuations have grown, from the early universe to redshift z.
The interactive calculator loads here: change the redshift and the cosmological parameters (Ωm, ΩΛ, Ωr, w0, wa, H0) and the result and plot update instantly.
What it is
In linear theory every small density fluctuation grows by the same factor D(z), set by the competition between gravity and the expansion. Dark energy slows the growth of structure, which is why measurements of growth test dark energy and gravity.
iCosmos uses the classic normalisation D = a = 1/(1 + z) deep in the matter era, and also shows D(z)/D(0), normalised to 1 today. The growth rate f = d ln D / d ln a is what redshift-space distortion surveys measure, in the combination fσ8.
Formula
- D″ + (2 + d ln E / d ln a) D′ − (3/2) Ωm(a) D = 0 (′ = d/d ln a)
- f = d ln D / d ln a ≈ Ωm(z)0.55 in ΛCDM
Worked example
In the classic model (Ωm = 0.3, ΩΛ = 0.7, H0 = 70 km s−1 Mpc−1), D(0) = 0.779 (compared with 1 in a matter-only universe), so dark energy has suppressed the growth of structure by about 22%.
At z = 1, D = 0.4766, D/D(0) = 0.6118 and the growth rate f = 0.8693.
Common questions
Which normalisation should I use?
Both are shown. D/D(0) = 1 today is the most common in modern papers; the classic iCosmos value, D = a in the matter era, makes the effect of dark energy visible directly as D(0) < 1.
Does this work for evolving dark energy (w ≠ −1)?
Yes, assuming the dark energy is smooth and does not cluster. Growth is not computed when Ωr > 0 or when there is no matter-dominated early era, and the calculator says so.
How accurate is it?
The growth equation is integrated adaptively and checked against the exact Heath integral in 400 random matter + Λ + curvature universes (where an exact solution exists), agreeing to better than 10−9; evolving dark-energy models are checked against an independent SciPy integration.
Related calculators
- Age of the universe calculator
- Hubble parameter H(z) calculator
- Curvature of the universe calculator
- σ₈, fσ₈ and S₈ calculator
- The full iCosmos cosmology calculator
About iCosmos
iCosmos was created by Mihran Vardanyan. Mihran Vardanyan is a cosmologist who earned his DPhil in cosmology at the University of Oxford, supervised by Joseph Silk and Roberto Trotta. He developed iCosmos during his research years at Oxford Astrophysics, where his work applied Bayesian model selection to the curvature and size of the Universe.
How to cite: Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.