Δχ² landscape
Hover parameter space to inspect the fit.
-0.9592
Best wₚ
-0.1946
Best wₐ
15.12
Minimum χ²
0.94
Reduced χ²
These constraints are produced from simplified mock datapoints and demonstrate the inference procedure only. They are not measurements of the real Universe.
Baryon Acoustic Oscillations
A fixed physical scale can become a cosmological measuring tool.
Baryon acoustic oscillations preserve a characteristic scale in the large-scale distribution of matter. Unlike a standard candle, whose known luminosity is compared with its observed brightness, BAO acts as a standard ruler: a known physical scale whose apparent size depends on cosmological distance.
If the expansion history changes, the inferred distance to a given redshift changes. The same 150 Mpc ruler then subtends a different angle, allowing cosmological parameters to be tested against the observed BAO signal.
Core relation
θBAO = lBAOd(z)
lBAO = 150 Mpc
d(z) is the cosmology-dependent comoving distance.
θBAO is the corresponding angular scale.
Standard Ruler
Connect redshift to distance, then distance to angle.
For each redshift, the expansion history determines the comoving distance. Dividing the fixed BAO scale by that distance gives the predicted angular scale.
Move through the mock redshift sample to see how the same physical ruler is viewed at different cosmological distances.
Standard ruler
ΛCDM reference
Redshift
1.2
z
The physical BAO ruler remains fixed at 150 Mpc. Increasing redshift moves the observer farther from that ruler in this illustrative geometry, reducing the angle it subtends.
Diagram geometry is illustrative; the distance and angular values are calculated from the cosmological model.
Mock Measurements
Start with the measurements the fit is actually asked to explain.
The analysis uses 18 simple mock BAO measurements between z = 0.3 and z = 2.0.
Important
These are simplified mock datapoints used to demonstrate the cosmological inference method. They are not measurements of the real Universe and should not be interpreted as a real Euclid cosmological constraint.
What is measured?
Mock BAO angular residual relative to the Lambda-CDM reference cosmology.
z
Redshift
Position of the mock observation in cosmic history.
Δθ
Angular residual
Difference from the reference ΛCDM BAO prediction.
σθ
Uncertainty
A fixed 5.0e-5 uncertainty is used in this demonstration.
Complete input data
All values used by the demonstration.
| Point | Redshift z | Mock Δθ | σθ |
|---|---|---|---|
| 01 | 0.3 | +3.420e-4 | 5.00e-5 |
| 02 | 0.4 | +2.490e-4 | 5.00e-5 |
| 03 | 0.5 | +2.280e-4 | 5.00e-5 |
| 04 | 0.6 | +1.410e-4 | 5.00e-5 |
| 05 | 0.7 | +1.270e-4 | 5.00e-5 |
| 06 | 0.8 | +6.200e-5 | 5.00e-5 |
| 07 | 0.9 | +1.180e-4 | 5.00e-5 |
| 08 | 1.0 | +9.900e-5 | 5.00e-5 |
| 09 | 1.1 | +2.800e-5 | 5.00e-5 |
| 10 | 1.2 | +2.000e-5 | 5.00e-5 |
| 11 | 1.3 | +1.500e-5 | 5.00e-5 |
| 12 | 1.4 | +4.700e-5 | 5.00e-5 |
| 13 | 1.5 | +4.900e-5 | 5.00e-5 |
| 14 | 1.6 | -1.800e-5 | 5.00e-5 |
| 15 | 1.7 | +2.300e-5 | 5.00e-5 |
| 16 | 1.8 | +1.000e-6 | 5.00e-5 |
| 17 | 1.9 | -4.000e-5 | 5.00e-5 |
| 18 | 2.0 | -1.770e-4 | 5.00e-5 |
Infer Cosmology
Change the expansion history and the BAO prediction moves.
The model allows dark energy to evolve with scale factor using
w(a) = wp + wa(1 − a)
Each pair (wₚ, wₐ) produces a different distance-redshift relation and therefore a different set of BAO angular residuals.
01
Choose cosmology
Select a trial pair of dark-energy parameters within wₚ = -1.00 to -0.90 and wₐ = -0.35 to 0.35.
02
Infer distance
Numerically integrate the expansion history to obtain d(z) for every mock redshift.
03
Predict BAO
Convert each cosmological distance into an angular BAO prediction and subtract the ΛCDM reference.
04
Score the model
Compare the predicted residuals with the mock observations using χ².
From model to prediction
The best-fit cosmology traces the mock BAO residuals across redshift.
Points show the simplified mock observations. The line shows the angular residual predicted by the optimised cosmological model.
Parameter Search
Find the cosmology that minimises the discrepancy.
For every trial cosmology, the predicted BAO residuals are compared with the mock measurements. Their uncertainty-weighted differences are summed to obtain χ².
χ² = Σ (Δθobs − Δθmodel)²σ²
Computational method
01
Original
Nested brute-force parameter loops with repeated numerical integrations.
02
Optimisation
Vectorised cumulative integration and numerical minimisation locate the χ² minimum efficiently.
03
Surface
151 × 151 parameter points are then evaluated to map the confidence landscape.
Preliminary Results
The minimum is only one point. The surrounding landscape tells us what is constrained.
The optimisation identifies the lowest χ², while a parameter grid maps how rapidly the fit deteriorates away from that solution.
Joint confidence regions for two fitted parameters are defined by Δχ² ≤ 2.30 at 1σ and Δχ² ≤ 6.18 at 2σ.
Δχ² landscape
Hover parameter space to inspect the fit.
Inferred Cosmology
Translate the χ² minimum into an inferred dark-energy model.
For this mock dataset, the numerical optimiser identifies the parameter pair that most closely reproduces the supplied BAO residuals.
Interpretation limit
These values are a demonstration of parameter inference from simplified mock measurements. They must not be interpreted as a measurement of the actual dark-energy equation of state.
Best-fit wₚ
-0.9592
Best-fit wₐ
-0.1946
Minimum χ²
15.116
16 degrees of freedom
Reduced χ²
0.945
Projected parameter ranges
ΛCDM reference
The ΛCDM point (wₚ = −1, wₐ = 0) has Δχ² ≈ 118.62 on the sampled grid and lies outside the plotted 2σ region.
Implementation
Small enough to expose the complete numerical workflow.
The analysis is intentionally compact. The published Python implementation performs the cosmological distance integration, χ² minimisation, confidence mapping and website-data export.
A compact NumPy archive preserves the numerical output, while the JSON export supplies the values displayed on this page.
This study is a methodological demonstration using simplified mock BAO measurements. It demonstrates the path from standard-ruler observations to cosmological parameter inference, not a contemporary Euclid cosmological result.