Cosmology · Galaxies as tracers of cosmic structure

Why galaxies do not trace all matter equally

Compare matter and two tracer contrasts in four supplied cells, separating a cross-correlation slope from an amplitude ratio.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Define galaxy-number and matter-mass contrasts using their respective means.
Explain what a simple bias relation does and does not say about individual objects.
Calculate how a prescribed residual changes variance without changing a cross slope.

Counting galaxies does not directly weigh all matter

Galaxies provide positions and observable properties, but their number density is a different field from total matter mass density. Gas, stars and dark matter do not all contribute one counted galaxy per fixed amount of mass.

Define matter contrast δ_m = ρ_m/ρ̄_m − 1 and tracer contrast δ_g = n_g/n̄_g − 1 at a specified time and smoothing scale. Each field uses its own mean; mass density and number density also have different units.

A biased tracer can have stronger fractional spatial variation than matter. A relation such as δ_g = bδ_m compares fluctuations about the respective means. It does not assign a mass b times larger to every individual galaxy.

Which galaxies are selected matters. Formation, merging, star formation and feedback affect the luminous population, and selection can favour different environments. A relation for one sample, scale and epoch need not hold for another.

A finite construction makes the contrast arithmetic explicit

Choose four equal physical-volume cells at one reference time. Their supplied matter masses are 900, 900, 1100 and 1100 in an arbitrary positive mass unit M_*. The mean is 1000M_* per cell.

Assign tracer A counts 90, 90, 110 and 110, with mean 100. Its contrasts are −0.1, −0.1, 0.1 and 0.1, exactly matching the matter contrasts in these four cells.

Assign tracer B counts 78, 82, 118 and 122, also with mean 100. Its contrasts are −0.22, −0.18, 0.18 and 0.22. They can be written δ_B = 2δ_m + ε with residuals −0.02, 0.02, −0.02 and 0.02.

These are declared inventories, not observed galaxies or random Poisson draws. The known matter mass is supplied for the exercise; a real catalogue does not reveal that mass merely by counting its objects.

Four supplied equal-volume cells at one epoch; mass is in M_* and each tracer has its own mean count 100
Cell	Matter mass/M_*	Tracer A count	Tracer B count	δ_m	δ_B	Residual ε
1	900	90	78	−0.1	−0.22	−0.02
2	900	90	82	−0.1	−0.18	0.02
3	1100	110	118	0.1	0.18	−0.02
4	1100	110	122	0.1	0.22	0.02

The residual leaves one relation unchanged and changes another

For optional finite-cell statistics, average with denominator four throughout. All contrasts and ε have zero mean. The residual variance is 0.0004, and its covariance with δ_m is zero in this supplied construction.

The matter variance is 0.01. The covariance of matter with B is 0.02, so the cross slope b_cross = Cov(δ_m,δ_B)/Var(δ_m) equals 2.

The B variance is 4Var(δ_m) + Var(ε) = 0.0404. The root-mean-square amplitude ratio is therefore b_rms = √[Var(δ_B)/Var(δ_m)] = √101/5 ≈ 2.009975, slightly larger than 2.

Their finite-cell correlation coefficient is 10/√101 ≈ 0.995037. It is close to one but does not erase the paired residuals. Equal matter masses in cells 1 and 2 coexist with different supplied B counts.

This original finite construction contains four equal physical-volume cells at one reference time. Matter masses are 900, 900, 1100, 1100 M_* with mean 1000 M_*. Tracer A counts are 90, 90, 110, 110 and B counts are 78, 82, 118, 122; each tracer has its own mean count 100. Matter mass contrast and tracer number contrast use their respective means and are dimensionless. The linear graph has matter contrast on the horizontal axis and tracer contrast on the vertical axis, with explicit zero guides. Solid blue slope-one guide and circle symbols show A = delta_m. The two pairs of A values coincide exactly and are labelled A cells 1,2 or 3,4 rather than being jittered. Orange squares give the four B contrasts -0.22,-0.18,+0.18,+0.22 at matter contrasts -0.1,-0.1,+0.1,+0.1. The dashed orange guide is 2 delta_m only over the supplied matter range -0.1 to +0.1. Green dotted vertical segments represent prescribed signed residuals -0.02,+0.02,-0.02,+0.02 from that guide; they are not uncertainty intervals or random draws. Both contrast fields and residual have zero finite mean. Every variance and covariance below uses denominator four. Matter variance is 0.01, residual variance 0.0004 and matter-residual covariance zero. Matter-B covariance is 0.02, so cross slope equals 2. B variance is 0.0404, giving RMS amplitude ratio sqrt(101)/5, approximately 2.009975, and correlation 10/sqrt(101), approximately 0.995037. Zero covariance here does not prove independence in a probability law. The known matter inventory is supplied, not weighed by counting galaxies. These four cells are not a cosmological power spectrum, calibrated stochastic-bias field or Poisson realization. No extrapolation to deeper voids, halos, other samples, scales or epochs is asserted. A bias of two relates fractional spatial fluctuations and assigns no fixed mass to each galaxy. Original Ark teaching model.

A squared bias factor requires stated assumptions

If a deterministic proportional relation holds for specified fluctuations with no residual terms, its two-point amplitude is multiplied by b². Additional residual correlations or scale-dependent terms change that statement.

The four-cell average above illustrates the distinction between a cross slope and a variance ratio. It is not itself a cosmological power spectrum, an ensemble calibration of stochastic bias or a demonstration that real residuals obey a particular noise law.

Zero covariance in these four cells is a supplied algebraic property. It does not prove independence in a probability distribution, nor does it guarantee the residual remains uncorrelated at other separations, scales or epochs.

The displayed matter contrasts are only ±0.1. Extrapolating the same linear rule into a deep void or a dense halo can fail physically, including by predicting invalid number densities. A validity range belongs with any approximation.

Matter inference needs more than a galaxy clustering amplitude

Galaxy clustering can constrain the spatial distribution of a selected tracer. Relating it to total matter requires a bias model and the other observational and cosmological assumptions entering the calculation.

Additional tracers or gravitational information can supply different combinations of matter and tracer properties. Their usefulness still depends on selection, calibration, common uncertainties and a joint statistical treatment.

Galaxy creation and merging also mean that a chosen galaxy sample is not automatically a conserved pressureless matter fluid. Importing a matter continuity equation without accounting for the tracer definition would add an untested assumption.

A useful result states which tracer is counted, how contrasts are defined, the range of scales and epochs, and which deterministic and residual terms are allowed. The next lesson asks how pair counts become a clustering statistic only after comparison with a reference.

Worked reasoning

Compare two finite-cell bias summaries

For the supplied cells, why is the cross slope 2 while the RMS ratio is slightly larger?

Var(δ_m) = 0.01 and Cov(δ_m,ε) = 0, so Cov(δ_m,δ_B) = 2 × 0.01 = 0.02.
The cross slope is 0.02/0.01 = 2. The B variance includes the residual variance: 4 × 0.01 + 0.0004 = 0.0404.
The RMS ratio is √(0.0404/0.01) = √4.04 ≈ 2.009975.

The prescribed residual adds B variance while leaving the matter–B covariance unchanged in this construction.

Try the reasoning

Should tracer contrast be computed by dividing galaxy counts by the mean matter mass?
Show the answer and reasoning
No.
Number contrast uses the tracer’s own mean number density; mass contrast uses mean mass density.
Does b_cross = 2 mean each B galaxy contains twice the total matter of an A galaxy?
Show the answer and reasoning
No.
The coefficient relates fractional spatial fluctuations rather than assigning an object-by-object mass.
Does the supplied zero matter–residual covariance establish a universal independent noise law?
Show the answer and reasoning
No.
It is an exact finite-cell property under the declared arithmetic, not a calibrated probability law for all scales.

What to carry forward

Matter mass and galaxy number define different fields and use their own mean densities.
A bias relation concerns selected spatial fluctuations, with sample, scale and epoch limits.
Residual terms can separate cross slopes from amplitude ratios and must accompany two-point inference.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.