Conformal Time Calculator
Calculate Conformal Time η(z)
About the Conformal Time Calculator
The Conformal Time Calculator is a precision cosmology tool that computes the conformal time η from the Big Bang to a given redshift z in a flat ΛCDM universe. Conformal time is a fundamental coordinate in modern cosmology because the Friedmann-Lemaître-Robertson-Walker (FLRW) metric becomes conformally flat when expressed in (η, χ) coordinates — making light propagation extremely simple: photons travel along straight lines at 45° on a η–χ diagram.
This calculator uses the exact analytic integral derived by Liddle & Lyth (1993), Baumann (2009), and implemented in major codes like CAMB and CLASS:
η(z) = ∫0z dz' / [H₀ (1+z') √(Ωₘ(1+z')³ + Ωₗ)]
Why Conformal Time is Crucial in Cosmology
In physical time t, the scale factor evolves as a(t) ∝ t2/3 during matter domination, but in conformal time, the comoving horizon grows linearly. This property dramatically simplifies calculations of the particle horizon, sound horizon at recombination, and angular diameter distance to the last scattering surface — all essential for interpreting the Cosmic Microwave Background (CMB) power spectrum.
Purpose of This Calculator
Researchers, lecturers, and students use the Conformal Time Calculator to:
- Determine the size of the particle horizon at any epoch
- Compute the comoving distance to CMB (z ≈ 1100)
- Verify results against Boltzmann solvers
- Teach the geometric meaning of conformal coordinates
When & Why You Should Use It
Use this tool whenever you need:
- The exact conformal age of the universe at recombination (ηrec)
- Comparison between physical time t(z) and conformal time η(z)
- Input values for theoretical CMB or BAO analysis
- Educational demonstrations of light-cone structure
User Guidelines
Recommended Planck 2018 baseline values are pre-filled:
- H₀ = 67.4 km/s/Mpc
- Ωₘ = 0.315 (includes baryons + cold dark matter)
- Ωₗ = 0.685 (cosmological constant)
- z = 1100 → η ≈ 14,100 Mpc (comoving)
You may change any parameter to explore alternative cosmologies (e.g., EdS, open/closed models).
Scientific Foundation
The conformal time integral for a flat ΛCDM universe has no elementary antiderivative, but it can be expressed in terms of hypergeometric functions or evaluated numerically to arbitrary precision. This implementation uses adaptive Gauss–Kronrod quadrature (accuracy > 10-12), identical to methods in CAMB and CLASS.
Key References
- Dodelson, *Modern Cosmology* (2003)
- Weinberg, *Cosmology* (2008)
- Liddle & Lyth, *Cosmological Inflation and Large-Scale Structure* (1993)
- Baumann, TASI Lectures (2009)
Applications in Modern Cosmology
Conformal time appears in:
- CMB anisotropy calculations (CAMB, CLASS)
- Primordial non-Gaussianity templates
- Inflationary perturbation evolution
- Particle horizon and causality arguments
Conclusion
The Conformal Time Calculator brings professional-grade cosmological computation to everyone. Whether you are writing a research paper, preparing a lecture, or simply curious about the fabric of spacetime, this tool delivers accurate, publication-quality results instantly.
For more science tools, visit Agri Care Hub and learn more about Conformal Time on Wikipedia.
