Cosmic Inflation Calculator
The cosmic inflation calculator computes the exponential growth of the scale factor during inflation from the number of e-folds and the inflationary Hubble parameter. Cosmic inflation, first proposed by Alan Guth in 1981, provides the most successful explanation for the large-scale uniformity of the universe and the near-perfect flatness of spacetime. During inflation, the universe underwent exponential de Sitter expansion: a(t) = a_i * exp(H_inf * t). The number of e-folds N = H_inf * (t_end - t_start) determines the total expansion factor. CMB observations from Planck constrain the scale of inflation and the spectrum of primordial perturbations.
Inflation formulas
Scale factor: a(t) = a_i * exp(H_inf * t)
e-folds: N = H_inf * delta_t = ln(a_f/a_i)
Scale factor ratio: a_f/a_i = exp(N)
Duration: delta_t = N / H_inf
H_inf in s^-1 = H_inf(GeV) * 1.52e24 s^-1/GeV
Inflationary observables (Planck 2018)
- Spectral index of scalar perturbations: ns = 0.9649 +/- 0.0042 (Planck 2018).
- Upper limit on tensor-to-scalar ratio: r less than 0.056 (BICEP/Keck 2021).
- Constraint on scale of inflation: H_inf less than 2.5e-5 * M_Pl.
- Minimum e-folds to solve horizon problem: N greater than 60.
Cosmic inflation: frequently asked questions
What is cosmic inflation?
Cosmic inflation is a period of extremely rapid exponential expansion of the universe that is theorized to have occurred approximately 10^-36 to 10^-32 seconds after the Big Bang. The scale factor grew by a factor of at least e^60 (about 10^26) in a tiny fraction of a second. Inflation explains the observed flatness, isotropy, and homogeneity of the universe, and the origin of density perturbations that seeded cosmic structure.
What is an e-fold?
An e-fold is a factor of e (Euler's number, approximately 2.718) in the expansion of the universe. After N e-folds, the scale factor has grown by a factor of e^N. The standard inflationary models require at least N = 60 e-folds to solve the flatness and horizon problems. After 60 e-folds: scale factor increase = e^60 approximately 10^26. After 70 e-folds: e^70 approximately 2.5e30.
What is the inflationary Hubble parameter?
During inflation, the Hubble parameter H_inf is approximately constant (de Sitter expansion): H_inf = V(phi)/(3*M_Pl^2), where V(phi) is the inflaton potential and M_Pl = 2.435e18 GeV is the reduced Planck mass. Observational constraints from CMB measurements give H_inf less than 2.5e-5 * M_Pl approximately 6e13 GeV = 6e22 MeV. The inflationary energy scale is (V(phi))^(1/4) approximately 10^16 GeV.
What observational evidence supports inflation?
Cosmic microwave background (CMB) observations from Planck and BICEP2/Keck support inflation. Key signatures: (1) flat universe (Omega_total = 1.0000 to within 0.5%); (2) scale-invariant spectrum of density perturbations (spectral index ns = 0.965); (3) Gaussian statistics of CMB fluctuations; (4) superhorizon correlations. A definitive detection of primordial gravitational waves (tensor modes) in B-mode CMB polarization would provide the strongest confirmation.
What ended inflation?
Inflation ended when the inflaton field (the hypothetical scalar field driving inflation) reached the minimum of its potential. The inflaton then oscillated and decayed into Standard Model particles through a process called reheating. The temperature after reheating was likely above 10^9 GeV, allowing electroweak symmetry breaking, baryogenesis, and eventually Big Bang nucleosynthesis to proceed. The exact mechanism of reheating depends on the specific inflationary model.
Official sources
- ESA Planck: Planck 2018 Results X: Inflation (ESA).
- NASA: NASA LAMBDA CMB Data Products.
Reviewed by the CalculatorHub team, edited by James Graham, 14 June 2026. See our methodology.