Cosmology · Cosmic maps, uncertainty and fair comparisons

Cosmic variance: learning from one observable sky

Separate detector noise from the finite number of independent large-scale patterns available in one sky, using a conditional power-estimation model.

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

Distinguish repeated instrument measurements from new realizations of a cosmic field.
Explain why independent pattern counts matter more than display pixels.
Interpret a finite-mode standard deviation without turning it into a universal cosmological limit.

A clearer view can still be one realization

Suppose two instruments observe the same large-scale sky pattern. Better sensitivity can reduce instrumental noise, and improved calibration can improve the measurement. Yet both instruments still observe the same particular arrangement of the cosmic field in those directions.

Many cosmological models describe statistical properties: how strong patterns of different scales are expected to be over possible realizations of a field. Our observed sky supplies one realization. The measured pattern strength can differ from that ensemble expectation even if the instrument has no noise.

This finite-realization uncertainty is called cosmic variance in the relevant measurement. It is different from an instrument’s inability to read a fixed signal precisely. Repeating the same sky observation can improve the reading without producing a fresh independent sky.

The idea does not mean that cosmology cannot learn more. New observables, wider usable coverage and improved treatment of other uncertainties can greatly improve many inferences. The limit has to be stated for a defined observable and model.

Independent coefficients are different from pixels

A sky field can be decomposed into patterns of different angular sizes, rather as a sound can be decomposed into different frequencies. Each pattern coefficient tells how strongly a specified pattern is present. The full mathematical construction is a later return path; the counting idea is enough here.

Broad angular patterns leave relatively few independent coefficients to estimate their statistical strength. Finer angular patterns can supply more. Increasing the number of pixels used to draw the same smooth pattern does not create additional independent coefficients.

In the ideal full-sky analysis of a statistically isotropic field, the angular order is labelled ℓ, pronounced ell. At a given ℓ there are 2ℓ + 1 real degrees of freedom. Isotropic here means that the model’s statistics do not select a preferred direction; it does not require each realized sky to look identical in every direction.

The result below also assumes a Gaussian field and an estimator of its power at one angular order. Masks, noise, correlations and departures from those assumptions require a fuller treatment. A bare number of map pixels cannot stand in for the relevant mode count.

A model with more independent modes estimates power more precisely

Consider n independent real Gaussian (normal) coefficients, each with mean zero and common positive variance C > 0. The variance C describes their expected squared amplitude and is the power quantity being estimated. The average of their squared amplitudes estimates C.

For this supplied model, the estimator’s standard deviation divided by C is √(2/n). A standard deviation describes spread across realizations of the estimator. The ratio is dimensionless, so it can be compared for different powers without choosing an amplitude unit.

For n = 5, the relative standard deviation is about 0.632. For n = 21 it is about 0.309, and for n = 201 it is about 0.0998. These nonzero spreads remain even when detector noise is set to zero. They arise from the finite count of independently realized coefficients.

In the ideal full-sky model, these counts correspond to ℓ = 2, 10 and 100 through n = 2ℓ + 1. The table illustrates the estimator’s precision under the stated assumptions; it contains no measured CMB powers or error bars from an actual experiment.

Conditional analytic power-estimator spread with independent Gaussian modes and no detector noise
Independent real modes n	Optional full-sky order ℓ	Relative standard deviation √(2/n)
5	2	0.632 approximately
21	10	0.309 approximately
201	100	0.0998 approximately

For n independent real zero-mean Gaussian coefficients with common positive variance C>0, the power estimator Ĉ is their average squared amplitude. The analytic relative standard deviation is SD(Ĉ)/C=√(2/n), even without detector noise. A smooth formula line uses a logarithmic horizontal axis from n=5 to 201 and a linear vertical axis from 0 to 0.7; direct labels and circle, diamond and square model markers pick the admissible integer counts 5,21,201. Their relative standard deviations are approximately 0.632,0.309,0.0998. The continuous connecting formula illustrates the relationship, not fractional measured mode counts or empirical observations. In the ideal full-sky isotropic single-field setting n=2ℓ+1 gives ℓ=2,10,100. C is squared-amplitude units; the plotted ratio is dimensionless. A standard deviation does not automatically specify a symmetric 68% interval, and rereading the same modes does not create independent realizations. Masks, noise and multiple observables require fuller analysis. Original Ark teaching model.

A standard deviation is not automatically a symmetric interval

With few modes, the average of squared amplitudes has a skewed distribution: it cannot be negative, and occasional large amplitudes produce a long upper tail. Its standard deviation therefore should not automatically be drawn as a symmetric 68 percent probability interval.

This connects with the preceding lesson on uncertainty conventions. The formula √(2/n) specifies a relative standard deviation for one estimator. It does not by itself specify every quantile, a confidence procedure for an unknown C, or a posterior distribution for a cosmological parameter.

A model parameter can affect several angular orders and several observables together. Its uncertainty is not obtained by applying the table’s percentage universally to that parameter. Correlations and the information carried by the other measurements must be considered.

Real sky cuts also mix information between patterns, and instrument or foreground effects can limit which modes are usable. Replacing the full-sky count by an arbitrary pixel count, or simply assuming all modes remain independent after a mask, would change the statistical problem.

Name the uncertainty before claiming a limit

If a measurement is dominated by instrument noise, better sensitivity can help. If its useful sky coverage is restricted by foregrounds, improved foreground information can help. If an ideal single-field power estimate is already limited by a small number of independent modes, repeated measurement of those same modes does not erase that finite-realization spread.

These statements concern different bottlenecks. They explain why cosmic variance is useful as a conditional benchmark without declaring that every future observation is futile or every cosmological error is irreducible.

For a first reading, remember the difference between reading the same five coefficients more accurately and acquiring 201 independent coefficients in a specified model. The first reduces reading noise; the second changes the finite-sample estimate of statistical power.

Worked reasoning

Calculate a conditional finite-mode spread

For n = 21 independent Gaussian modes in the supplied power-estimation model, what is the relative standard deviation? Would zero detector noise make it vanish?

Use the stated relation √(2/n) = √(2/21).
The result is approximately 0.3086, or a standard deviation about 30.9 percent of C.
The expression already assumes no detector noise. Its nonzero value comes from finite independent mode count.

About 0.309, or 30.9 percent of C; it does not vanish when detector noise is zero.

Try the reasoning

Does redrawing a fixed smooth sky pattern with four times as many screen pixels automatically supply four times as many independent modes?
Show the answer and reasoning
No.
Display sampling is different from independently realized physical coefficients.
Can repeated noiseless readings of the same five modes turn them into 201 independent modes?
Show the answer and reasoning
No.
They can improve knowledge of those same amplitudes, but cannot generate new independent realizations.
Does √(2/n) set the same fractional limit on every cosmological parameter?
Show the answer and reasoning
No.
It describes the stated single-field power estimator; parameter inference can combine multiple scales and observables under a fuller model.

What to carry forward

Cosmic variance concerns finite realization sampling for a defined statistical measurement.
Independent modes are different from image pixels or repeated readings of the same sky.
The finite-mode formula has explicit assumptions and does not impose a universal limit on every cosmological inference.

References

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

Cosmic Microwave Background Anisotropies — CMB Temperature FieldWayne Hu; Scott DodelsonTeaching resourcebackground.uchicago.eduRead source