FR08-A · The sky is one draw

One Sky and Cosmic Variance

Why does a perfect map still leave uncertainty about the Universe? A sky map samples a finite number of patterns at each angular scale. Even a perfect telescope sees only this one realization, so its measured power fluctuates around the underlying spectrum. Large patterns have fewer independent modes and fluctuate more.

Rotate a synthetic microwave sky and compare its measured angular power with the spectrum that generated it. Draw another sky, or blur the same one with a wider telescope beam.

Filtered sky RMS $45.3\,\mu\mathrm K$

Selected power / theory $0.417$

Intrinsic fractional scatter $53.5\%$

Independent modes at this degree $7$

A finite view of the Universe

Keep the theory, change the sky

Finest angular pattern

\ell_{\max}

18

Beam width

\theta_{\rm FWHM}/{}^\circ

0

Input fluctuation amplitude

\sqrt{D_\ell}/(\mu\mathrm K)

30

Drag the sphere to look around. A wider beam smooths the same sky; drawing again changes its random patterns.

Select a scale and a seed

Selected angular degree

\ell

3

Show only the selected degree on the sphereRandom seed

Model, normalization and uncertainty

$D_\ell=\ell(\ell+1)C_\ell/(2\pi)$ is the scaled angular power. It is constant in this teaching sky. Monopole and dipole are absent, and no acoustic peaks are included.

$B_\ell=\exp[-\ell(\ell+1)\sigma_b^2/2]$ filters each saved coefficient. The spectrum compares the original realization, the blurred realization and the input theory. All maps and records are synthetic, with no sky mask, foreground or detector noise.

The shaded intervals describe repeated skies under a fixed theory. They are central sampling intervals, not posterior limits on cosmological parameters. The ensemble is regenerated deterministically from a separate fixed seed.

Model assumptions

The synthetic full sky contains Gaussian, statistically isotropic spherical harmonics from degree two to at most twenty-four. Its input angular power is prescribed and has a constant scaled spectrum. A Gaussian beam filters the saved coefficients. The calculation includes no mask, detector noise, foregrounds, acoustic peaks or Planck map. Cosmic-variance intervals assume this full-sky Gaussian model.

Synthetic full-sky record

One realization wanders around the theory

Angular power

Repeated skies at the selected scale

At each degree, the independent harmonic modes set the scatter. The blue interval contains ninety-five percent of repeated skies under this input spectrum.

Exact sampling relation

\frac{(2\ell+1)\widehat C_\ell}{C_\ell}\sim\chi^2_{2\ell+1},\qquad \frac{\sigma(\widehat C_\ell)}{C_\ell}=\sqrt{\frac{2}{2\ell+1}}

0.241<\widehat C_3/C_3<2.29

The record saves the seed, parameters, complex coefficients for nonnegative orders, filtered coefficients, spectra and interval convention. Negative orders follow the conjugacy relation required by a real temperature field.

Knox (1995), full-sky mode-counting limit

Continue from angular patterns to cosmological inference

Start with the essentials

Focus question: Why does a perfect map still leave uncertainty about the Universe?
One-sentence intuition: Large angular patterns have fewer independent modes, so one sky gives a less precise estimate of their average power.

Core mathematical model

Temperature patterns on the sphere

\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}^*

Orthonormal complex spherical harmonics describe angular patterns. The conjugacy relation makes the temperature real. The monopole and dipole are excluded, and all temperatures are in microkelvin.

Estimating power from one sky

\widehat C_\ell=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|a_{\ell m}|^2

At each degree, sum the squared coefficient amplitudes and average over the independent real modes. The map and this estimate use exactly the same saved coefficients.

The full-sky Gaussian sampling law

(2\ell+1)\widehat C_\ell/C_\ell\sim\chi^2_{2\ell+1},\qquad \sigma(\widehat C_\ell)/C_\ell=\sqrt{2/(2\ell+1)}

The squared independent Gaussian coordinates add to a chi-square variable. Its central sampling interval is asymmetric, especially at low angular degree. This is scatter among possible skies at fixed theory.

Common difficulties

Uncertainty survives a perfect measurement

Typical misconceptionA noiseless map must reveal the theoretical spectrum exactly.

Better mental modelA noiseless map gives the exact coefficients of our realization. The theoretical spectrum describes an ensemble of realizations, and the finite number of modes leaves sampling scatter.

Run the experiment

01
Keep the theory fixed
Before pressing Draw another sky, predict whether the white theoretical line or the gold points will move. Draw several skies.
What to observe: The theory stays fixed and the gold points wander. A low or high point can arise from the finite random sample alone.
02
Compare large and small patterns
Choose A few large patterns, then Small patterns appear. Under Select a scale and a seed, move Selected angular degree from the lowest to a higher degree.
What to observe: The repeated-sky distribution narrows at higher degree. More independent modes give a more stable estimate of the input power.
03
Blur the same sky
Predict which patches disappear first, then widen Beam width without drawing another sky. Save the sky and harmonics.
What to observe: Small patches fade fastest. The cyan power curve drops at high degree, and the original gold points stay fixed. The saved coefficients specify both views.

One Sky and Cosmic Variance

One realization wanders around the theory

Angular power

Repeated skies at the selected scale