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.

Inferred CMB temperature $2.7255\,\mathrm K$

Fitted dust brightness $24\,\mathrm{MJy}\,\mathrm{sr}^{-1}$

Residual RMS $0.35\,\mathrm{MJy}\,\mathrm{sr}^{-1}$

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 · Planck 2013 XI · NASA FIRAS

Read the discoveries behind these models →

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 misconceptionA smooth-looking blackbody fit must have found the correct cosmic temperature.

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

Run the experiment

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.
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.
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.
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.

What remains after the fit

Core mathematical model

The cosmic background spectrum

A background plus a foreground

Common difficulties

A measurement to interpret carefully

Run the experiment

Start with a clear sky

Leave out a real source of light

Fit the foreground

Limit the frequency coverage