Input parameters
Example data table
Illustrative outputs using H0=70, Ωm=0.3, ΩΛ=0.7, Ωr=0, w0=−1, wa=0.
| z | DL (Mpc) | DA (Mpc) | Lookback time (Gyr) |
|---|---|---|---|
| 0.50 | 2832.938 | 1259.084 | 5.041 |
| 1.00 | 6607.658 | 1651.914 | 7.715 |
| 3.00 | 25422.742 | 1588.921 | 11.354 |
Formula used
The dimensionless expansion rate is:
E(z) = \sqrt{\Omega_m(1+z)^3 + \Omega_r(1+z)^4 + \Omega_k(1+z)^2 + \Omega_\Lambda f_{DE}(z)}
With curvature:
\Omega_k = 1 - \Omega_m - \Omega_r - \Omega_\Lambda
CPL dark-energy scaling:
f_{DE}(z) = (1+z)^{3(1+w0+wa)}\,\exp\!\left(-\frac{3\,wa\,z}{1+z}\right)
Comoving radial distance:
D_C = \frac{c}{H0} \int_0^z \frac{dz'}{E(z')}
Transverse comoving distance:
D_M = \begin{cases} \frac{c}{H0\sqrt{\Omega_k}}\sinh\!\left(\sqrt{\Omega_k}\int_0^z \frac{dz'}{E(z')}\right), & \Omega_k>0\\ D_C, & \Omega_k=0\\ \frac{c}{H0\sqrt{|\Omega_k|}}\sin\!\left(\sqrt{|\Omega_k|}\int_0^z \frac{dz'}{E(z')}\right), & \Omega_k<0 \end{cases}
Angular diameter and luminosity distances:
D_A = \frac{D_M}{1+z}, \quad D_L = (1+z)D_M
Lookback time:
t_L = \frac{1}{H0}\int_0^z \frac{dz'}{(1+z')E(z')}
Integrals are evaluated with a composite Simpson rule.
How to use this calculator
- Enter a redshift z for your source or epoch.
- Set cosmology: H0, Ωm, ΩΛ, and optional Ωr.
- Optionally change w0 and wa for evolving dark energy.
- Choose the output unit for distance values.
- Increase integration steps for higher-precision results.
- Press Compute Distances to display outputs above.
- Use the download buttons to export CSV or PDF.
Professional notes on cosmological distances
1) Why multiple distance definitions exist
Observations do not measure a single "distance". Telescopes record angles, fluxes, redshifts, and number counts. Each maps to a different distance concept in an expanding universe. This calculator reports comoving distances for geometry, angular diameter distance for size, and luminosity distance for brightness.
2) Redshift as the practical input
Redshift z is measured from spectral features and is robust across instruments. For nearby sources, z roughly tracks recessional velocity. At higher z, it encodes the integrated expansion history. Using z keeps the workflow consistent across supernovae, galaxies, and the cosmic microwave background.
3) Expansion history through E(z)
The key quantity is E(z)=H(z)/H0. Matter scales as (1+z)3, radiation as (1+z)4, and curvature as (1+z)2. Dark energy is modeled with the CPL form using w0 and wa, allowing mild evolution while recovering the constant-w limit when wa=0.
4) Comoving radial and transverse distances
The comoving radial distance DC integrates 1/E(z) from 0 to z and sets line-of-sight separations. Curvature changes transverse paths, so the calculator converts DC to DM using sin or sinh forms when 03A9k 2260 0. These distinctions matter for wide-field surveys.
5) Angular diameter versus luminosity distance
Angular diameter distance DA=DM/(1+z) links a physical size to an observed angle. Luminosity distance DL=(1+z)DM links intrinsic luminosity to observed flux. Their ratio is fixed by Etherington2019s reciprocity, so consistent results provide a useful sanity check.
6) Time and horizon intuition
The lookback time integrates 1/[(1+z)E(z)], returning how long light has traveled in gigayears. Combined with DH=c/H0, it helps interpret which epochs contribute most to an integral. Increasing steps improves stability for very large z or rapidly varying models.
7) Survey-friendly outputs and the BAO scale
The calculator also reports DV, a volume-averaged distance used in baryon acoustic oscillation analyses: DV=[(cz/H(z))DM2]1/3. For supernova work, the distance modulus 03BC relates directly to magnitude fits through 03BC=5log10(DL/10 pc).
8) Practical parameter checks
Flat 039BCDM typically satisfies 03A9m+03A9039B+03A9r22481. If you explore curvature, verify 03A9k remains modest and E(z) stays positive for all z in your range. For z2248020133 applications, 03A9r can be near zero; for early-universe ranges, include it explicitly.
FAQs
1) What values should I use for a standard model?
Try H0=70 km/s/Mpc, 03A9m=0.3, 03A9039B=0.7, 03A9r=0, w0=22121, wa=0. This is a common flat 039BCDM reference for comparisons and sanity checks.
2) Why do DA and DL differ?
Expansion stretches photon wavelengths and arrival rates. That adds (1+z) factors between geometric size distances and brightness distances. The calculator enforces DL=(1+z)2DA, so the pair remains self-consistent.
3) When should I include radiation 03A9r?
For low redshift, radiation contributes negligibly to E(z). For higher redshift, especially z2265100, it can matter. If you want a simple inclusion, use a small 03A9r such as 0.00005 as a placeholder.
4) What do w0 and wa control?
w0 sets the present-day dark-energy equation of state. wa adjusts its evolution with scale factor using the CPL form. Setting wa=0 gives a constant-w model, and w0=22121 matches a cosmological constant.
5) How many integration steps are enough?
For z22643, 100020132000 steps typically gives stable results. For larger z or extreme parameters, increase steps gradually and confirm outputs converge. Very high values may slow the page, so balance precision and speed.
6) What is DV used for?
DV combines radial and transverse distances into one scale for BAO constraints. It is common in survey summaries where the observed clustering feature is averaged over angle, especially when detailed anisotropic fits are not used.
7) Why does curvature change DM but not DC?
DC is a line-of-sight integral of 1/E(z). Curvature affects transverse geodesics, altering how radial separation maps into angular separation. That is why the sin/sinh transformation appears only when computing DM.
Professional notes on cosmological distances
1) Why multiple distance definitions exist
Cosmology separates distance measures because light travels through an expanding spacetime. A single observed redshift can map to different operational distances depending on whether you are predicting flux, angles, volumes, or time intervals.
2) Redshift as the primary observable
For galaxies and supernovae, redshift z is measured from spectral features. The calculator treats z as the integration upper limit that sets the scale factor a = 1/(1+z), linking observations to the expansion history.
3) Expansion history and E(z)
The dimensionless expansion rate E(z) combines matter, radiation, curvature, and dark energy. Matter scales as (1+z)3, radiation as (1+z)4, and curvature as (1+z)2. With H0, the tool returns H(z)=H0E(z) for model checking.
4) Comoving distances and curvature response
The comoving radial distance DC follows c/H0 times an integral of 1/E(z). When curvature Ωk ≠ 0, the transverse distance DM uses sin or sinh, changing how areas and volumes grow with redshift.
5) Angular and luminosity distances
Angular diameter distance DA sets physical size per observed angle, while luminosity distance DL sets flux dimming. They are related by DL=(1+z)2DA, so a consistent model must satisfy both simultaneously.
6) Lookback time in gigayears
Lookback time integrates 1/[(1+z)E(z)] and converts through H0 to seconds and then Gyr. This helps translate a redshift into “how long ago” the light was emitted, which is often more intuitive than distance alone.
7) Survey-ready summary distances
The tool also reports DV, a volume-averaged distance used in baryon acoustic oscillation analyses: DV ≈ [(cz/H(z))DM2] 1/3. It compresses radial and transverse information into one scale for quick comparisons.
8) Accuracy and practical modeling tips
Numerical precision depends on integration steps and parameter stiffness. For z > 5 or rapidly varying dark energy (large |wa|), increase steps to reduce Simpson-rule error. Use Ωr≈0.00005 for early-universe sensitivity, and verify Ωk=1−Ωm−Ωr−ΩΛ stays physical.
FAQs
1) What distance should I use for brightness predictions?
Use luminosity distance DL. It connects intrinsic luminosity to observed flux through the inverse-square law modified by expansion, and it is the distance used in distance modulus calculations.
2) What distance should I use for angular sizes?
Use angular diameter distance DA. Multiply DA by an observed angular size (in radians) to estimate the transverse physical size at the source redshift.
3) Why can DA stop increasing at high redshift?
Because of expansion geometry. DA=DM/(1+z) can peak and then decrease as the (1+z) factor overtakes the growth of DM, making very distant objects appear larger again.
4) How do I represent a flat universe in the inputs?
Choose Ωk=0 implicitly by setting ΩΛ ≈ 1 − Ωm − Ωr. The calculator reports Ωk so you can confirm near-zero curvature.
5) When should I include radiation Ωr?
Include Ωr for high-redshift work or early-universe comparisons. For low redshift (z ≲ 2), Ωr contributes very little, so setting it to zero is often acceptable.
6) What do w0 and wa change in the results?
They modify the dark-energy density evolution using the CPL form. Deviations from w0=−1 or nonzero wa alter E(z), changing all integrated distances and the lookback time, especially at intermediate to high redshift.
7) Why do I see small changes when I increase integration steps?
Because the distances come from numerical integrals. More steps reduce integration error, which matters most at large z or sharp parameter combinations. If results stabilize, you have likely reached sufficient accuracy.
Professional notes on cosmological distances
1) Why multiple distance definitions exist
In an expanding universe, "distance" depends on the observable you care about. Photon energies redshift, arrival rates dilate, and angles project through a changing spacetime. Using one redshift, these measures can differ by factors of (1+z). This calculator reports standard distances for consistent comparisons.
2) Redshift as the primary input
Redshift z relates to the scale factor a through a = 1/(1+z). For small z, distance roughly follows Hubble\'s law v ≈ cz and D ≈ cz/H0, but beyond z ~ 0.1 the full expansion history matters. Enter z from observations, then choose H0 to set the overall scale.
3) Expansion history and model parameters
The expansion history is encoded in E(z) = H(z)/H0. Matter scales as (1+z)^3, radiation as (1+z)^4, and curvature as (1+z)^2. Dark energy is modeled with a constant density (Lambda) or with the CPL form using w0 and wa, which alters H(z) across cosmic time.
4) Comoving radial distance DC
The comoving radial distance integrates 1/E(z) from 0 to z and is returned in Mpc, Gpc, or Gly. DC tracks the present-day separation to the source\'s comoving coordinate and is the backbone for turning redshift catalogs into three-dimensional comoving maps.
5) Transverse distance and curvature effects
If curvature is nonzero, transverse comoving distance DM differs from DC through sine or hyperbolic sine functions. Positive Ωk indicates open geometry and increases DM, while negative Ωk indicates closed geometry and reduces it. The calculator computes Ωk from your density inputs.
6) Angular diameter and luminosity distances
Angular diameter distance DA = DM/(1+z) links physical size to observed angle via size = θDA. Luminosity distance DL = DM(1+z) links intrinsic luminosity to observed flux. They satisfy reciprocity: DL = (1+z)2DA, so model changes shift both predictions coherently.
7) Lookback time as an epoch indicator
Lookback time uses an integral of 1/((1+z)E(z)) and is reported in gigayears. It estimates how long light traveled before reaching us. Combining time and distance helps connect redshift to physical epochs such as reionization, peak star formation, and the onset of accelerated expansion.
8) Survey outputs: μ and DV
Two outputs target common analyses. The distance modulus μ converts DL to magnitudes used in Type Ia supernova Hubble diagrams. The volume-averaged distance DV = [(cz/H(z))DM2]^{1/3} is widely used in BAO constraints because it blends radial and transverse information. For high z or extreme parameters, increase steps until results stabilize.
FAQs
1) Which distance should I use for supernovae?
Use luminosity distance and distance modulus. Supernova flux scales as 1/DL2, and μ = 5 log10(DL/10 pc). Compare μ(z) from your model to calibrated supernova magnitudes.
2) Which distance should I use for galaxy sizes or rulers?
Use angular diameter distance. Convert an observed angle to physical size with size = θDA. Standard ruler analyses may also use DM depending on whether you work with transverse comoving coordinates.
3) What is a typical value for Ωr?
For low-redshift work, Ωr is often negligible. A commonly used order of magnitude is about 5×10−5 for photons plus relativistic neutrinos, but the best value depends on the assumed radiation content.
4) Why do distances change when I change H0?
Most distances scale roughly as 1/H0 because the Hubble distance DH = c/H0 sets the overall length scale. Changing Ω parameters reshapes the redshift dependence, but H0 largely rescales distances together.
5) What do w0 and wa represent?
They parameterize dark-energy pressure via the CPL form w(a)=w0+wa(1−a). w0 is today\'s value and wa controls evolution with scale factor. A cosmological constant corresponds to w0 = −1 and wa = 0.
6) How accurate is the numerical integration?
Composite Simpson integration is high-order for smooth functions like 1/E(z). Accuracy improves as you increase even steps. For very large z or unusual parameters, double the steps and confirm outputs change only slightly.
7) Why can DA decrease at high redshift?
DA grows with z at first, reaches a maximum, then decreases because emission occurred when the universe was smaller, while light had more time to travel. The peak location depends on Ω values and dark-energy behavior.