AC04 · Five filters, several possible distances

Galaxy Colors and Photometric Redshift

Is a red galaxy distant, old, or dusty? All three can redden a galaxy. Redshift moves spectral features between filters, old stars supply less blue light, and dust absorbs more short-wavelength light. Several colors help separate these causes; some galaxies still allow more than one distance.

Drag a galaxy to redshift its light. Change its stellar population and dust, watch five detector planes respond, and follow every peak of the inferred redshift distribution.

Selected redshift $z_{\mathrm{true}}=0.325$

Highest posterior peak $z_{\mathrm{mode}}=0.320$

Central 68% interval $[0.268,\,0.427]$

Secondary-peak basin probability $36.7\%$

Why does the galaxy look red?

Redshift

0.325

z

Single-burst population age

2.15

t_*\;[\mathrm{Gyr}]

Continuum dust attenuation

0.100

E_s(B-V)

Drag the galaxy sideways to stretch its light. Every detector uses the same spectrum and its own measured filter response.

Inference and export

Turn off the u detector

The full curve sums seven possible single-burst ages and six dust levels, with a positive amplitude integrated under a half-normal prior of scale ten thousand nanomaggies. Colors are normalized to the same r-band mean. The shaded interval contains the central sixty-eight percent; a secondary peak can lie outside it.

Read the course and reproduce real SDSS colors

Model assumptions

The synthetic observation uses seven solar-metallicity BC03 single-burst stellar populations with a Salpeter IMF, six Calzetti continuum attenuation values and measured SDSS2010 filter throughputs. Stellar age means the age of this particular population model. Redshift is restricted to zero through 1.2; templates older than the assumed universe are excluded. The observed r-band normalization stays fixed as colors change. The virtual photometer has five equal optical branches; each receives one fifth of a one-square-metre collecting area; detector images share an illustrative fixed morphology and show mean photon intensity in false color. Symbolic moving photons illustrate the optical paths; dust markers show the attenuation setting. A proper half-normal amplitude prior is integrated analytically; age and dust are summed, and the full redshift posterior is normalized. Synthetic errors are independent Gaussian flux errors. Metallicity variation, composite star-formation histories, emission lines, intergalactic absorption and cross-band calibration covariance are omitted. The companion tutorial treats a separately frozen, real SDSS DR17 sample.

Spectrum and integrated filter fluxes

\langle f_\nu\rangle_b=\frac{\int f_\nu(\lambda)T_b(\lambda)\,d\lambda/\lambda}{\int T_b(\lambda)\,d\lambda/\lambda}

Full redshift posterior; nuisance parameters integrated

\int_0^{1.2}p(z\mid\boldsymbol f)\,dz=1

Counts, errors and model resolution

Conditional fit at the highest peak: population age $2.15\,\mathrm{Gyr}$ , continuum color excess $0.1$ , squared standardized residual sum $2.17$ .The central interval can include a valley between peaks.

Mean photon rates in the five equal optical branches:

$\dot N_u=0.526\,\mathrm{s^{-1}}$
$\dot N_g=26.1\,\mathrm{s^{-1}}$
$\dot N_r=98.7\,\mathrm{s^{-1}}$
$\dot N_i=121\,\mathrm{s^{-1}}$
$\dot N_z=27.5\,\mathrm{s^{-1}}$

Collecting area is one square metre before equal splitting; additional optical efficiency is unity. Detector brightness uses a fixed nonlinear compression.

The source spectra retain the native BC03 wavelength sampling. Filter integrals use four-Angstrom observed-wavelength steps. Four independent off-node quadrature comparisons at half-Angstrom steps have a largest tested difference of about 0.0195% in tested band ratios. This is a tested numerical difference, not a bound on stellar-population accuracy.

The redshift grid spacing is 0.005. Stellar ages are discrete from one hundred million to ten billion years; continuum color excess runs from zero to 0.5 in steps of 0.1. The age prior is uniform over allowed age nodes at each redshift. The synthetic r-band mean is one hundred nanomaggies. Relative measurement errors are eight, three and a half, three, three and a half and five percent, with an additional half-nanomaggy floor. Turning off u removes that measurement from inference.

BC03 · SDSS response · EAZY

Physics tutorial

How five colors become a redshift distribution

BackgroundA broad filter adds the light across a range of wavelengths. A galaxy spectrum contains changes of slope and many absorption features. Moving these features through neighboring filters changes their relative signals.

Why it mattersSpectra can identify narrow lines, but taking spectra for every faint galaxy is expensive. Images through several filters offer a faster, less precise distance clue. Their uncertainty often has several peaks.

Start with the essentials

Focus question: Which color change moves a spectral feature, and which only changes its strength?
One-sentence intuition: Redshift stretches wavelengths; age and dust reshape the light already present. Several combinations can reproduce the same five fluxes.

Core mathematical model

Stretch the wavelength

\lambda_{\mathrm{obs}}=(1+z)\lambda_{\mathrm{rest}}

The broad absorption break near four thousand Angstrom moves to six thousand Angstrom at redshift one half. Dust attenuates the spectrum at its rest wavelength before this shift.

Count the photons a filter transmits

\dot N_b=\frac{A_{\mathrm{tel}}}{5h}\int f_\nu(\lambda)T_b(\lambda)\,\frac{d\lambda}{\lambda}

The five equal branches share the collecting area. The throughput includes the detector response and the reference atmosphere. The displayed images apply the same fixed nonlinear intensity mapping to these rates.

Sum the alternative explanations

p(z\mid\boldsymbol f)\propto p(z)\sum_{t,E}p(t,E\mid z)\int_0^\infty p(\boldsymbol f\mid z,t,E,A)\,p(A)\,dA

The amplitude is positive with a proper half-normal prior. The sum uses equal discrete weights for allowed ages and dust nodes. A uniform redshift prior is confined to the displayed interval; the curve is normalized by its numerical integral.

Common difficulties

The brightest redshift peak can change suddenly

Typical misconceptionA small flux change must cause only a small change in the reported redshift.

Better mental modelTwo separated peaks can be almost equally high. A small change makes the other peak taller even though both peaks were present before. Compare their integrated probability and the full curve.

Run the experiment

01
Move the break
Start with A break crosses the filters. Predict which detector gains signal as the redshift grows, then drag the galaxy to the right.
What to observe: The source keeps the same r-band normalization. Its other band ratios change as the break moves through their measured responses.
02
Redden the source locally
Compare Young stars behind dust with An old stellar population. Keep their low redshift fixed and compare the generating spectra below.
What to observe: Both lose blue light, but age changes the intrinsic stellar spectrum and dust attenuates it smoothly. Their broad-band colors can still be similar.
03
Lose a measurement
Select The blue detector goes dark. Compare the full posterior with the default case before resetting.
What to observe: The highest peak moves from about 0.32 to 0.545 for these fixed noise draws. The larger interval and surviving lower-redshift peak show what the missing information had supplied.

Galaxy Colors and Photometric Redshift

Five measurements become several possible redshifts

How five colors become a redshift distribution

Core mathematical model

Stretch the wavelength

Count the photons a filter transmits

Sum the alternative explanations

Common difficulties

The brightest redshift peak can change suddenly

Run the experiment

Move the break

Redden the source locally

Lose a measurement

Related experiments and articles

Measuring the sky

Galaxy colors and redshift