Description
Return a distance-duality residual.
Cosmology uses several distance measures and often expresses the expansion rate in mixed astronomical units. Scale-factor relations, inverse-Hubble scales and relativistic velocity composition answer different questions; units and the distinction between local algebra and an integrated expansion history are essential.
Return a distance-duality residual.
When to use this tool
Use Luminosity Angular Distance Duality Residual when checking one explicit cosmological distance, redshift-factor or unit convention.
How to use the tool and how it works
- Enter luminosityDistance, angularDistance, redshift. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read predictedLuminosityDistance, residual, ratio. Use the output convention below.
Return a distance-duality residual. Require luminosityDistance >= 0, angularDistance > 0 and z > -1; predicted luminosity distance = angularDistance*(1+z)², residual = supplied-predicted and ratio divides them.1
Cosmological distance, volume and lookback-time relations require the appropriate expansion-rate and distance conventions; local factors do not replace integration.2
| Input field | Meaning and units | Accepted input |
|---|---|---|
| luminosityDistance | Luminosity Distance | Required. Finite number; minimum 0. |
| angularDistance | Angular Distance | Required. Finite number; strictly above 0. |
| redshift | Dimensionless redshift z; the stated scaling formulas require z > -1. | Required. Finite number; strictly above -1. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| predictedLuminosityDistance | Predicted Luminosity Distance | Required. Finite number. |
| residual | Signed difference in the direction stated above, with matching units. | Required. Finite number. |
| ratio | luminosityDistance/[angularDistance*(1+redshift)^2]; unity matches the supplied duality relation. | Required. Finite number. |
{
"input": {
"luminosityDistance": 400,
"angularDistance": 100,
"redshift": 1
},
"output": {
"predictedLuminosityDistance": 400,
"residual": 0,
"ratio": 1
}
}Limitations and assumptions
- Distances share units; the reciprocity assumptions are not verified and a residual alone does not identify a physical violation.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Use a complete cosmological distance/time calculation and the appropriate observation model when an actual source distance, age or velocity is required.
References
Similar or alternative tools
- Flux Luminosity Reference Ratio
Return a luminosity ratio from two fluxes and distances.
- Distance Modulus Hubble Rescale Term
Rescale a distance modulus under inverse-Hubble distance scaling.