---
title: "CMB Spectrum Thermometer Interactive Lab | Sandbox Physics"
description: "How can a small foreground make a precise cosmic thermometer read too warm? Warm Galactic dust adds extra intensity, especially at high microwave frequencies. A fit that assigns all of that light to one blackbody raises the inferred temperature. Fitting the dust spectrum separately recovers the injected cosmic temperature in this model."
canonical_url: "https://sandboxphysics.com/labs/cmb-spectrum-thermometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/cmb-spectrum-thermometer/"
last_modified: "2026-10-24"
---

F-X1a · Find the temperature beneath Galactic dust

# CMB Spectrum Thermometer

**How can a small foreground make a precise cosmic thermometer read too warm?** Warm Galactic dust adds extra intensity, especially at high microwave frequencies. A fit that assigns all of that light to one blackbody raises the inferred temperature. Fitting the dust spectrum separately recovers the injected cosmic temperature in this model.

Sweep a microwave receiver across a synthetic sky spectrum. Compare a blackbody fit with a fit that also includes warm Galactic dust, and watch the temperature and residuals change.

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

The record is synthetic: a blackbody at 2.7255 kelvin plus one modified-blackbody dust component and fixed-seed Gaussian noise. Dust temperature is fixed at 19.6 kelvin and its emissivity index at 1.6; its amplitude is defined at 353 gigahertz. The receiver uses forty-three monochromatic channels between thirty and six hundred gigahertz, with a known additive noise level. Temperature is fitted by least squares and dust amplitude is fitted linearly when enabled. The hardware drawing and visible ray colors are schematic. No FIRAS observations, bandpasses, absolute calibration uncertainties, correlated noise, other foregrounds or spectral distortions are included. The displayed residual RMS describes this synthetic fit, not a historical measurement uncertainty.

Synthetic model data

## The fitted sky spectrum

Amber points: synthetic channels with one-noise-standard-deviation error bars. Teal: fitted total. Dashed blue: fitted cosmic background alone.

$\nu/\mathrm{GHz},\qquad I_\nu/(\mathrm{MJy}\,\mathrm{sr}^{-1})$

### What remains after the fit

Every residual is a synthetic channel intensity minus the fitted total. A broad curve points to a missing spectral component.

$\nu/\mathrm{GHz},\qquad (I_\nu-I_{\nu,\rm fit})/(\mathrm{MJy}\,\mathrm{sr}^{-1})$

Sources and definitions

The reference temperature follows Fixsen (2009). Dust uses a fixed modified blackbody, with the form discussed in Planck 2013 XI. The forty-three teaching channels are generated here. [Fixsen 2009](https://arxiv.org/abs/0911.1955) · [Planck 2013 XI](https://doi.org/10.1051/0004-6361/201323195) · [NASA FIRAS](https://lambda.gsfc.nasa.gov/product/cobe/firas_monopole_get.html)

[Read the discoveries behind these models →](https://sandboxphysics.com/stories/astronomy-gravity/)

Physics tutorial

## CMB Spectrum Thermometer

**Background** The cosmic microwave background fills the sky with a spectrum close to that of a cold blackbody. Warm dust in our own Galaxy adds radiation to the same receiver. The two contributions have different frequency shapes.

**Why it matters** A very precise fit can still use the wrong model. The shape of the leftover signal helps us discover what the fit missed.

Start with the essentials

- **Focus question**: How can a small foreground make a precise cosmic thermometer read too warm?
- **One-sentence intuition**: A cosmic temperature measurement needs a model for the foreground light that shares the receiver.

### Core mathematical model

#### The cosmic background spectrum

$$
B_\nu(T)=\frac{2h\nu^3}{c^2}\left[\exp\!\left(\frac{h\nu}{k_BT}\right)-1\right]^{-1}
$$

This is radiance per unit frequency. Frequency is measured in hertz, temperature in kelvin, and the constants are Planck’s constant, the speed of light and Boltzmann’s constant.

#### A background plus a foreground

$$
I_\nu=B_\nu(T)+A_d\left(\frac{\nu}{\nu_0}\right)^{1.6}\frac{B_\nu(19.6\,\mathrm K)}{B_{\nu_0}(19.6\,\mathrm K)}+\epsilon_\nu
$$

The reference frequency is 353 gigahertz. Dust amplitude is its brightness there, and the final term is the channel noise. Dust temperature and spectral index stay fixed in this teaching model.

### Common difficulties

#### A measurement to interpret carefully

**Typical misconception** A smooth-looking blackbody fit must have found the correct cosmic temperature.

**Better mental model** Foreground light can move the best-fit temperature. Compare the residual with the prescribed channel noise and include the missing spectral component.

### Run the experiment

1. 01

   #### Start with a clear sky

   Choose Clear sky and compare the amber samples with the fitted curve.

   **What to observe:** The fitted dust amplitude is near zero, and the temperature is close to the injected 2.7255 kelvin.

2. 02

   #### Leave out a real source of light

   Choose Dust ignored and inspect both the temperature and the residual below the apparatus.

   **What to observe:** The fitted temperature rises. A broad leftover curve remains because a warmer cosmic blackbody cannot reproduce the dust shape.

3. 03

   #### Fit the foreground

   Choose Dust fitted, then change Dust brightness at the reference channel.

   **What to observe:** The inferred dust amplitude grows with the injected brightness. The cosmic temperature stays close to the injected value.

4. 04

   #### Limit the frequency coverage

   Choose Narrow band and compare it with Dust fitted.

   **What to observe:** Low frequencies contain less of the dust component. Broad frequency coverage gives more leverage to separate the two spectral shapes.

Physical conclusion

A cosmic temperature measurement needs a model for the foreground light that shares the receiver.

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)