Approximate Lookback Time From z
Bounded 2048-panel Simpson quadrature in flat matter+Lambda Fried mann model, radiation/curvature omitted; angular-distance and lookback integrals are not interchangeable. Model only, not an ephemeris, measured stellar identity or observational guarantee.
Description
Lookback time measures elapsed model time between emission and observation.
Lookback time measures elapsed model time between emission and observation.1
It differs from light-path distance and depends on the chosen cosmological parameters. Use Approximate Lookback Time From z to evaluate the supplied inputs under the model explained below.
When to use Approximate Lookback Time From z
Use Approximate Lookback Time From z to evaluate the supplied inputs under the model explained below. Establish the units, measurement scope and model assumptions before entering values.
How to use Approximate Lookback Time From z and understand its results
- Establish the units, measurement scope and model assumptions before entering values.
- Enter Redshift, Hubble km per second per mpc, Matter density fraction using the field table and validation rules below, then run the tool.
- Read Flat model lookback gyr; interpret the units, signs, ordering and unknown-value placeholders described below.
- Correct reported input errors and rerun. Apply successful results according to their specific meanings and check the assumptions before using them in a decision.
| Input field | Meaning | Value type | Requirement |
|---|---|---|---|
| redshift | Redshift | number | Required; constraints below |
| hubbleKmPerSecondPerMpc | Hubble km per second per mpc | number | Required; constraints below |
| matterDensityFraction | Matter density fraction | number | Required; constraints below |
Enter redshift 0–20, positive Hubble parameter in km/s/Mpc and matter density fraction 0–1. Set E(z)=sqrt(Ωm(1+z)³+1−Ωm) and numerically integrate with composite Simpson quadrature over 2048 equal subintervals. Integrate 1/((1+z)E(z)), then convert 1/H0 using 3.0856775814913673e19 km/Mpc and 365.25 days/year for Gyr. Zero redshift returns zero. This is a flat matter-plus-Lambda model with radiation omitted, no error estimate or inferred cosmological parameters.
lookbackGyr=convertedHubbleTime×∫0..z 1/((1+x)E(x)) dx Use the field order, units and branch conditions described above.
| Output field | Meaning | Value type |
|---|---|---|
| flatModelLookbackGyr | Flat model lookback gyr | number |
{
"input": {
"redshift": 1,
"hubbleKmPerSecondPerMpc": 70,
"matterDensityFraction": 0
},
"output": {
"flatModelLookbackGyr": 9.682198880449796
}
}Limitations and assumptions
It differs from light-path distance and depends on the chosen cosmological parameters. Apply the exact scope and assumptions explained above.
All listed fields require explicit values. Unexpected fields, incorrect types, nonfinite numbers and sparse arrays fail validation. Domain-specific bounds, alignments and supported tokens are explained above; extremely large inputs or outputs can also be rejected. Input errors differ from successful false results, empty match lists and unknown-evidence placeholders.
Alternative or Complementary methods
Use Approximate Luminosity Distance From z for a related task: Luminosity distance relates source luminosity to observed flux in a cosmological model.
Use Approximate Comoving Distance From z for a related task: Comoving distance removes cosmic expansion from a model coordinate separation.
References
-
Approximate Lookback Time From z — tool.io
Similar or alternative tools
- Approximate Luminosity Distance From z
Luminosity distance relates source luminosity to observed flux in a cosmological model.
- Approximate Comoving Distance From z
Comoving distance removes cosmic expansion from a model coordinate separation.