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Sandbox Physics
Cosmology from expansion to inference

L4 · Chapter 8

What sky patterns imply about dark components

Sandbox Physics ·

The microwave sky contains peak positions, heights and lensing signals. A physical model converts them into cosmic ingredients. Different observations cut through different parameter compensations; combining them makes those ingredients easier to distinguish.

Different ingredients leave overlapping signatures

Baryons add inertia to the acoustic fluid and change relative compression and rarefaction heights. Cold dark matter affects gravitational potentials, matter-radiation equality and subsequent structure growth. Dark energy changes late expansion, distance to last scattering, evolving potentials and lensing. These paths overlap enough for some parameter changes to compensate.

Physical density ωb=Ωbh2\omega_b=\Omega_bh^2 differs from the density fraction Ωb\Omega_b. The former determines baryon density per physical volume at a fixed redshift; the latter divides by today’s critical density. Their conversion changes with the Hubble constant. Holding physical density fixed and holding the density fraction fixed define different comparisons. Here h=H0/(100  km s−1 Mpc−1)h=H_0/(100\;\mathrm{km\,s^{-1}\,Mpc^{-1}}) is the dimensionless Hubble constant.

Reading: Planck 2018 results. VI. Cosmological parameters

Convert a precise angle into an expansion rate

Planck 2018 section 3.1 discusses its acoustic-angle measurement. The abstract reports 100θ∗=1.0411±0.0003100\theta_*=1.0411\pm0.0003. Combining temperature, polarization and lensing reconstruction in spatially flat six-parameter cosmological-constant plus cold-dark-matter cosmology, it infers H0=(67.4±0.5)  km s−1 Mpc−1H_0=(67.4\pm0.5)\;\mathrm{km\,s^{-1}\,Mpc^{-1}}. These are sixty-eight-percent intervals. The first measures an angle; the second uses a model to connect the early ruler to late distance.

A stable angular measurement need not imply an equally tight Hubble constant in every extension. Freeing curvature or the dark-energy equation of state adds distance compensations. Baryon acoustic oscillations, or BAO, leave a sound ruler in the galaxy distribution. They measure radial and transverse distance combinations at other redshifts; supernovae give relative luminosity distances. Their information restricts different directions.

The BAO sound horizon rdr_d ends at the baryon drag epoch. The CMB acoustic angle instead uses rs(z∗)r_s(z_*) at last scattering. These nearby epochs give different ruler lengths. Both depend on early physics and must be calculated separately in a joint analysis.

The course’s selected supernova sample lacks absolute luminosity calibration and cannot independently measure the Hubble constant. Comparing its distance curve with acoustic models teaches these information directions. A full joint analysis would need consistent parameters, selection assumptions and correlated uncertainties.

Reading: Planck 2018 results. VI. Cosmological parameters · Pantheon+ public data release

Extra early radiation changes several rulers

Varying the effective relativistic-species number NeffN_{\rm eff} changes early expansion. Faster expansion usually shortens the time sound has to travel and reduces the sound horizon. Diffusion damping, phase shifts from free-streaming particles and nucleosynthesis also change. Simply shifting a peak sequence horizontally misses those tests.

Planck Figure 35 plots the effective relativistic-species number against the Hubble constant. The colored sample band shows how more early radiation can accompany faster late expansion while preserving the acoustic angle. The solid black contours also add lensing reconstruction and BAO. Track that data change before attributing a narrower contour to a physical effect.

When reading Planck section 7.5.2, first record the temperature, polarization, lensing and BAO combination. Check whether helium abundance and neutrino mass are fixed. Intervals from different assumptions cannot be attributed solely to extra data. This is a reading method: the acoustic Lab varies only its three declared grid parameters and fixes radiation content.

Reading: Planck 2018 results. VI. Cosmological parameters

Use lensing amplitudes to ask a limited question

The preceding chapter’s AκA_\kappa rescales one fixed convergence-power template. It checks released bandpowers relative to that template and shows how prior width affects evidence. A real change of matter density also changes distances, growth, shape and reconstruction response. One overall amplitude cannot represent all those effects.

Planck section 6.2 uses another phenomenological quantity, ALA_{\rm L}, to scale lensing when calculating the smoothing of temperature and polarization peaks. Lensing reconstruction instead estimates the lensing field through higher-order correlations. These measurements can test each other, but their definitions and information differ.

A bounded reproduction is useful: freeze the release, template, bins and covariance, and check the Gaussian amplitude slice. Then identify the cosmological response and additional observations needed for a larger inference. Carrying the limited question into a paper reading preserves what the calculation actually established.

Reading: Qu et al. · NASA LAMBDA · Planck 2018 results. VI. Cosmological parameters

Work through a disagreement between estimates

Take two independent Gaussian teaching measurements of the same quantity: 67±167\pm1 and 73±173\pm1 in matching units. Their difference has variance equal to the sum of variances. A difference of six corresponds to about 4.244.24 standard deviations of that difference. These illustrative numbers are not a pair of named cosmological results.

Shared calibration introduces covariance. If one estimate assumes a cosmological model, changing that model can change its central value and uncertainty. Truncated priors and non-Gaussian tails can also invalidate a simple standard-deviation comparison.

An extension can lower the best-fit residual because it has more freedom. Assessing support for that freedom also requires independent predictions, identifying which data improve, and checking prior sensitivity. A proposed new component should also make independently testable predictions beyond the measurements whose disagreement it was introduced to explain.

Z=∣q^1−q^2∣σ12+σ22−2Cov⁡(q^1,q^2)Z=\frac{|\hat q_1-\hat q_2|}{\sqrt{\sigma_1^2+\sigma_2^2-2\operatorname{Cov}(\hat q_1,\hat q_2)}}
The comparison requires estimates of the same parameter and a suitable joint sampling approximation. Independent Gaussian estimates have zero covariance.

Try the prediction

Separate estimation from model preference

  1. Keep the observations and template fixed and compare default and broad priors.
  2. Record the mean, interval and evidence ratio, then explain the effect of prior volume.
  3. Add the optional response uncertainty and describe its effect on amplitude information without converting it directly into dark energy.

Turn geometric compensation into a testable question

  1. Choose two nearly aligned spectra and state which parameters remain fixed.
  2. Compare the acoustic scale and full-spectrum differences against the grid’s interpolation accuracy.
  3. Choose an independent observable, such as a distance or lensing measurement, and explain why it could distinguish the models.

Exercises and explanations

  1. Does a lensing amplitude slightly above one establish a larger dark-matter density?

    Read the explanation

    It establishes a relative excess against the chosen template. Matter density, initial fluctuation amplitude, growth, distances and reconstruction response all affect physical lensing. A density interpretation needs their consistent joint model.

  2. Why retain the likelihood, prior and data combination when comparing extensions?

    Read the explanation

    The likelihood specifies the data constraint; the prior specifies allowed parameter weights and model averaging; the data combination specifies the information used. A best-fit point alone cannot reproduce posterior intervals or evidence, or identify which observation breaks a degeneracy.

Sources and calculation scope

  1. 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.

  2. Qu et al., The Atacama Cosmology Telescope: A Measurement of the DR6 CMB Lensing Power Spectrum and Its Implications for Structure Growth, arXiv:2304.05202v3

    Section VIII.2 and Figure 25 discuss a fixed-template lensing amplitude; equation (45) in section IX.1 gives a Gaussian bandpower likelihood. Our fixed-template amplitude slice uses the released bandpowers and covariance, with the exact template and omissions documented beside the Lab.

  3. NASA LAMBDA, ACT DR6 lensing likelihood data

    Frozen public ACT DR6 lensing product. The data manifest records the release, original filenames, binning, template, covariance, checksums and attribution. Additional calibration uncertainty in the Lab is an explicitly chosen teaching extension.

  4. Pantheon+ public data release

    The existing distance Lab uses forty selected Pantheon+ supernovae and their statistical-plus-systematic covariance submatrix. Its free magnitude offset removes absolute-distance calibration; this sample does not measure the Hubble constant.

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