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Sandbox Physics
Cosmology from expansion to inference

L3 · Chapter 5

How much can one sky tell us about its power?

Sandbox Physics ·

A perfect telescope still leaves uncertainty about the average strength of large sky patterns. Each angular scale has finitely many independent modes, and we observe one random realization. Cosmic variance describes this sampling scatter.

Use wave patterns adapted to a sphere

A plane has sinusoidal waves; a sphere has spherical harmonics YℓmY_{\ell m}. Higher degree ℓ\ell gives finer patterns. Different mm values describe directional structure at that degree. The coefficients aℓma_{\ell m} specify amplitudes and phases. After removing the mean and dipole we begin at degree two.

Complex harmonics are convenient, but temperature is real. Negative-index coefficients are fixed by conjugacy and cannot be independent complex draws. Each degree retains exactly twice the degree plus one independent real coordinates. An orthonormal real-harmonic basis gives the same count and calculation.

ΔT(n^)=∑ℓ=2ℓmax⁡∑m=−ℓℓaℓmYℓm(n^),aℓ,−m=(−1)maℓm∗\Delta T(\hat n)=\sum_{\ell=2}^{\ell_{\max}}\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell m}(\hat n),\quad a_{\ell,-m}=(-1)^m a_{\ell m}^{*}
The harmonics are orthonormal over the whole sphere. Temperature and coefficients use microkelvin; angular power uses square microkelvin.

Estimate a theoretical mean from one realization

Statistical isotropy assigns the same theoretical variance CℓC_\ell to every independent real coordinate at a degree. In a Gaussian field, divide each coordinate by its standard deviation to obtain independent standard normal numbers. Their squared sum has a chi-square distribution with one degree of freedom per coordinate.

Power therefore scatters around its theoretical mean. At low degree the distribution is visibly skewed. Better instrumentation reduces measurement noise. The finite number of modes in this one sky still sets the remaining cosmic variance.

C^ℓ=12ℓ+1∑m∣aℓm∣2,(2ℓ+1)C^ℓCℓ∼χ2ℓ+12\widehat C_\ell=\frac{1}{2\ell+1}\sum_m|a_{\ell m}|^2,\quad\frac{(2\ell+1)\widehat C_\ell}{C_\ell}\sim\chi^2_{2\ell+1}
The denominator is fixed theoretical power; the numerator varies across generated skies. This sampling law assumes a full-sky Gaussian field.
E[C^ℓ]=Cℓ,Var⁡(C^ℓ)Cℓ=22ℓ+1\mathbb E[\widehat C_\ell]=C_\ell,\qquad\frac{\sqrt{\operatorname{Var}(\widehat C_\ell)}}{C_\ell}=\sqrt{\frac{2}{2\ell+1}}
A standard chi-square variable has mean equal to its degrees of freedom and variance twice that value. Substitution gives the unbiased estimator and its fractional scatter.

Reading: Lloyd Knox · Planck 2018 results. VI. Cosmological parameters

Calculate the scatter for large and small patterns

Degree two has five independent coordinates, giving fractional power scatter 2/5=0.632\sqrt{2/5}=0.632. Degree twenty has forty-one coordinates, giving 2/41=0.221\sqrt{2/41}=0.221. Higher-degree power still fluctuates, but concentrates more tightly across draws. These percentages describe power, not temperature amplitude.

The central ninety-five-percent sampling band at degree two is 0.166<C^2/C2<2.570.166<\widehat C_2/C_2<2.57; at degree twenty it is approximately 0.615<C^20/C20<1.480.615<\widehat C_{20}/C_{20}<1.48. The low-degree interval is strongly asymmetric. A symmetric mean-plus-or-minus-standard-deviation band misses that shape.

Power also discards some map information. Changing positive-order coefficient phases while preserving real-field conjugacy rearranges hot and cold patches. If the squared sum at each degree stays fixed, so does angular power. Many different maps can share the same spectrum.

Two interval questions need different endpoints

With known theoretical power, ask where repeated estimates fall. Let qν,pq_{\nu,p} be a chi-square quantile. Multiplying theoretical power by the lower and upper quantiles and dividing by the mode count gives a central ninety-five-percent sampling range.

With an observed estimate and unknown theoretical power, solve the same probability statement for the parameter instead. This produces a frequentist confidence interval: across repeated experiments, the constructed intervals cover the true power at the stated rate. Its endpoints differ from the sampling band. A Bayesian posterior interval depends jointly on the likelihood and the chosen prior.

Cℓqν,0.025ν≤C^ℓ≤Cℓqν,0.975ν,ν=2ℓ+1\frac{C_\ell q_{\nu,0.025}}{\nu}\leq\widehat C_\ell\leq\frac{C_\ell q_{\nu,0.975}}{\nu},\quad\nu=2\ell+1
Fix the theory and locate random estimates. The experiment’s spectrum displays this sampling range.
νC^ℓqν,0.975≤Cℓ≤νC^ℓqν,0.025\frac{\nu\widehat C_\ell}{q_{\nu,0.975}}\leq C_\ell\leq\frac{\nu\widehat C_\ell}{q_{\nu,0.025}}
Fix the observed estimate and invert for unknown theoretical power. The quantiles now divide the estimate and exchange order.

Reading: Lloyd Knox · Planck 2018 results. VI. Cosmological parameters

Smoothing cannot create another independent sky

An ideal circular Gaussian beam multiplies each degree’s coefficients by a fixed window. Power falls by its square, but the number of independent coordinates stays the same. If the theory is smoothed consistently, fractional cosmic variance remains set by mode count. Real observations also add noise and masks, treated next.

This generator stops at degree twenty-four and uses a prescribed teaching spectrum. It illustrates mode counting and sampling scatter, with no acoustic peaks or observed Planck map. The separate acoustic-grid experiment studies the high-degree peak structure.

Try the prediction

Keep the theory and draw more skies

  1. Choose the large-pattern preset and compare low-degree measured power with theory.
  2. Draw more skies, then increase maximum degree and compare the finer patterns.
  3. Widen the beam while retaining the coefficients, then export the record to inspect the same filtered sky.

Exercises and explanations

  1. Can rotating the same full-sky map reduce its cosmic variance?

    Read the explanation

    Rotation changes how the harmonic coefficients are represented, but preserves their squared sum at each degree. Full-sky angular power is unchanged, and no new independent sky has been observed.

  2. Does one low-degree point outside a central ninety-five-percent sampling band reject the model?

    Read the explanation

    A preselected point falls outside that band about five percent of the time. A model test also needs a joint statistic, a stated selection procedure and the relevant noise and foreground assumptions.

Sources and calculation scope

  1. Lloyd Knox, Determination of Inflationary Observables by CMB Anisotropy Experiments, astro-ph/9504054v1 (1995)

    Full-sky Gaussian mode counting, cosmic variance and Gaussian beam suppression. The course derives the chi-square sampling law from independent real harmonic coordinates; no inflationary forecast is reproduced.

  2. Planck 2018 results. VI. Cosmological parameters, arXiv:1807.06209v4 (2021)

    Sections 2.2–2.3 define the data likelihoods, section 3.1 discusses the acoustic scale, and sections 6.2 and 7.3–7.5 separate lensing smoothing, curvature, dark energy and relativistic species. Published in A&A 641, A6 (2020), with the v4 corrigendum. This course does not reproduce the full Planck likelihood or chains.

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