Tensor-to-Scalar Ratio Calculator
Calculate the tensor-to-scalar ratio r — the key observable that probes primordial gravitational waves from cosmic inflation.
About the Tensor-to-Scalar Ratio Calculator
The Tensor-to-Scalar Ratio Calculator is a precision scientific tool designed for cosmologists, astrophysicists, and students to compute the tensor-to-scalar ratio r — one of the most important parameters in modern cosmology. This ratio quantifies the relative amplitude of primordial gravitational waves (tensor modes) to density perturbations (scalar modes) generated during cosmic inflation. The current observational upper limit from Planck + BICEP/Keck (2021) is r < 0.036 at 95% confidence, with future experiments like CMB-S4, LiteBIRD, and the Simons Observatory aiming to reach σ(r) ≈ 0.001.
Scientific Foundation & Formulas
In single-field slow-roll inflation, the tensor-to-scalar ratio is related to the first slow-roll parameter by the Lyth relation:
where ε = −Ḣ/H² is the slow-roll parameter. Alternatively, from primordial power spectra:
This calculator supports both methods and automatically selects the most accurate one based on available input.
Importance of Measuring r
Detecting a non-zero tensor-to-scalar ratio would be revolutionary: it would confirm the quantum origin of cosmic structure, directly probe physics at energy scales ~10¹⁶ GeV (near Grand Unification), and distinguish between hundreds of inflationary models. A value of r ≈ 0.01 would favor large-field models (e.g., chaotic inflation), while r < 10⁻³ would support small-field or hybrid models.
When & Why You Should Use This Calculator
- Testing theoretical inflation models against current observational constraints
- Interpreting results from CMB polarization experiments (BICEP, POLARBEAR, SPT, ACT)
- Preparing research papers or grant proposals in cosmology
- Teaching advanced cosmology and general relativity courses
- Comparing predictions of Starobinsky, Higgs inflation, α-attractors, fiber inflation, etc.
User Guidelines
Enter any one or more of the following:
- Slow-roll ε → directly computes r = 16ε
- nₛ + model type → estimates r using consistency relations
- P_T and Δ²ₛ → computes r = P_T / P_S
The calculator prioritizes the most precise method available and clearly states its assumptions.
Current Observational Status (2025)
The latest combined analysis from Planck 2018 + BICEP/Keck 2021 gives:
r < 0.036 (95% CL) at k = 0.05 Mpc⁻¹
This already rules out simple monomial potentials like m²φ² (r ≈ 0.13) and strongly constrains natural inflation unless the axion decay constant is very large.
Future Prospects
Next-generation experiments will dramatically improve sensitivity:
- CMB-S4: expected σ(r) ≈ 0.0005
- LiteBIRD: σ(r) ≈ 0.001
- Simons Observatory: σ(r) ≈ 0.003
A detection at r > 0.01 would be a landmark discovery in fundamental physics.
Learn More
For a comprehensive technical guide, read the ultimate resource on Tensor-to-Scalar Ratio. Also explore agricultural applications of data science and modeling at Agri Care Hub.
References & Peer-Reviewed Sources
- BICEP/Keck Collaboration (2021), Phys. Rev. Lett. 127, 151301
- Planck Collaboration (2020), Astron. Astrophys. 641, A6
- Lyth & Riotto (1999), Phys. Rep. 314, 1
- Baumann & Zaldarriaga, Inflation lectures (arXiv:0907.5424)
This Tensor-to-Scalar Ratio Calculator is built with scientific rigor and designed for the cosmology community worldwide.
