Hellings-Downs Curve Calculator for Pulsar Pairs

Two sky positions, one correlation

Turn separation into expectation

Each millisecond pulsar is a clock. For two distinct pulsars, an isotropic tensor background predicts a correlation that depends only on their angle on the sky.

x = (1 - cos ζ) / 2 Γ(ζ) = 1/2 + (3/2)x ln x - x/4

Choose two positions Name-decoded example pair

Pulsar A right ascension, degrees Pulsar A declination, degrees

Pulsar B right ascension, degrees Pulsar B declination, degrees

Computing the pair...

The live sky map will describe the current pair here.

The live curve will describe the current point here.

The curve is a spatial fingerprint, not a time-series waveform. A clock error is common to every pulsar and produces a monopole. A solar-system ephemeris error produces a dipole. The tensor-background expectation has the folded shape drawn above.

Shape, covariance, spectrum

Ask what the curve can actually support

The points in a pulsar-pair plot are correlated with one another. This layer keeps the full covariance matrix in the calculation and keeps published results separate from a constructed exercise.

Data gate: the official Figure 1 repository links the full pair products and covariance bundle through an external Google Drive download that was unavailable in this build environment. The official covariance-aware binned chi-square notebook cell also currently has a dimension mismatch. No collaboration fit, Bayes factor, p-value, or significance is reconstructed below.

Constructed covariance exercise Shape used to construct the live vector Off-diagonal covariance strength Computing... Deterministic perturbation size Computing...

This is not NANOGrav data. Fifteen angles, a disclosed covariance kernel, and a deterministic sine perturbation are free choices used to expose the generalized least-squares machinery.

Full-covariance generalized least squares

â = (mᵀC⁻¹y) / (mᵀC⁻¹m) χ² = (y - âm)ᵀC⁻¹(y - âm)

Shape m	Best scale â	χ²	χ² / dof
Computing...

Computing all three fits...

The live exercise plot will be described here.

Characteristic-strain spectrum

Amplitude A at one cycle per year, in 10⁻¹⁵ Strain spectral index α Frequency, nanohertz

Computing the spectrum...

One spectrum, two slope conventions

h_c(f) = A(f / f_yr)^α γ = 3 - 2α

The live spectrum will be described here.

Published checkpoints, kept in their own lane

These values are rendered from the cited source record. They are not outputs of the constructed exercise.

Analysis	Pulsars	Fixed-slope A at 1/yr	Spatial evidence summary	Status
Loading the checked source record...

The visible check

Recomputing the page state...

Conventions

Angles are degrees; right ascension and declination use a spherical coordinate convention.
The distinct-pulsar normalization gives Γ(0°) = 0.5. The same-pulsar autocorrelation delta term is excluded.
The limit x ln x = 0 is used at x = 0.
One year is the Julian year of 365.25 days.

Uncertainties and free choices

The named coordinates are rounded values decoded from pulsar J-names, not catalogue astrometry.
The exercise uses fifteen evenly spaced angles, a unit scale, a sine perturbation, and C_ij = 0.08²[δ_ij + ρ exp(-|i-j|/3)].
Real inference depends on pulsar noise, chromatic-noise treatment, sky coverage, covariance estimation, spectral model, and whether the slope is fixed.

Approximations and exclusions

No released pair product or collaboration covariance value is copied into this page.
The sky picture is an equirectangular drawing; the separation itself uses the spherical cosine rule.
Finite-source effects and cosmic variance are not included in the ideal Hellings-Downs expectation.

Still open

The pulsar-timing-array evidence supports a gravitational-wave background interpretation, but does not by itself establish a particular source population.
The 2026 NANOGrav result is a preprint reanalysis of the same 15-year data, not a statistically independent data set.
This page does not claim a broad-band five-sigma detection.

Sources checked

Peer reviewed Hellings and Downs, Upper limits on the isotropic gravitational radiation background from pulsar timing analysis, Astrophysical Journal Letters 265 (1983). The original angular-correlation derivation.
Peer reviewed Romano and Allen, Answers to frequently asked questions about the pulsar timing array Hellings and Downs correlation curve, Classical and Quantum Gravity 41 (2024). Normalization, conventions, and interpretation.
Peer reviewed Agazie et al., The NANOGrav 15-year Data Set: Evidence for a Gravitational-wave Background, Astrophysical Journal Letters 951 L8 (2023). The 67-pulsar evidence and fixed-slope amplitude.
Preprint Agarwal et al., The NANOGrav 15 yr Data Set: Impacts of Customized Chromatic Noise Models on Gravitational Wave Analyses (2026). Same timing data with a custom chromatic-noise framework.
Official repository NANOGrav, 15-year stochastic-analysis Figure 1 release. Code and links to the pair-product bundle; repository material is not peer reviewed.
Peer reviewed Antoniadis et al., EPTA and InPTA results, Astronomy and Astrophysics 678 A50 (2023).
Peer reviewed Reardon et al., Parkes Pulsar Timing Array third data release, Astrophysical Journal Letters 951 L6 (2023).
Peer reviewed Miles et al., MeerKAT Pulsar Timing Array first data release, Monthly Notices of the Royal Astronomical Society 536 (2025). Its spatial-significance result is explicitly noise-model sensitive.