---
title: "One Sky and Cosmic Variance Interactive Lab | Sandbox Physics"
description: "Why does a perfect map still leave uncertainty about the Universe? A sky map samples a finite number of patterns at each angular scale."
canonical_url: "https://sandboxphysics.com/labs/one-sky-cosmic-variance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/one-sky-cosmic-variance/"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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](https://arxiv.org/abs/astro-ph/9504054v1)

[Continue from angular patterns to cosmological inference](https://sandboxphysics.com/stories/cosmology/)

Physics tutorial

## One Sky and Cosmic Variance

**Background** A theory predicts the average strength of temperature patterns across many possible skies. We observe one sky, with one particular set of hot and cold patches.

**Why it matters** Even if a telescope measured every patch exactly, its angular-power estimate would still fluctuate because there are only finitely many independent patterns at each scale.

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 misconception** A noiseless map must reveal the theoretical spectrum exactly.

**Better mental model** A 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Read one sky as a finite random sample, and distinguish cosmic variance from telescope smoothing.

## Related experiments and articles

Cosmology from expansion to inference · Reading path

### Expansion, sound, sky statistics and inference

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How much can one sky tell us about its power?**](https://sandboxphysics.com/stories/cosmology/one-sky-statistics/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)