Models, assumptions and useful simplifications

Why describe a clumpy sky with a smooth model?

Look at a city from a window and you see individual buildings, streets and gaps. A map of average population density describes something different. It can answer a regional question without claiming that every square metre holds the same number of people. Whether that average is useful depends on the scale of the question and the variation it hides.

Cosmology uses a related distinction between a large-scale background and departures from it. Galaxies, clusters and voids are real. A smooth background model describes average expansion and contents; a description of perturbations adds variations that can grow into structure. The analogy with a city concerns averaging only. It does not supply equations for gravity or demonstrate that the Universe is actually homogeneous.

A model is a deliberately limited representation with variables, relationships and conditions. It is useful when its predictions answer the intended question accurately enough. An approximation should be judged against the scale, precision and observations being considered, rather than defended as a literal picture of every part of reality.

Position and direction are different assumptions

Homogeneity means statistical sameness from place to place on the scales under discussion. Isotropy means statistical sameness with direction. Neither means every small region is identical. A model can treat the background as homogeneous and isotropic while adding the fluctuations responsible for the patterns we actually study.

The two ideas are logically different. An observer at the centre of a spherically layered distribution could see the same pattern in every direction, even if density changed with distance from that observer. Directional symmetry about one location would not establish sameness from every location. This is a thought experiment about geometry, not evidence that we occupy such a special centre.

The cosmological principle uses large-scale homogeneity and isotropy as an organizing assumption. It can be confronted with observations of the CMB and large-scale structure, and with models that allow particular departures. A successful test constrains the departures it was designed to detect; it does not prove perfect symmetry throughout unobservable space.

A scale factor records a relative change

Choose two ideal markers following the average cosmic flow. In a homogeneous expansion model their proper separation on a surface of equal cosmic time can be written D(t) = a(t)χ. Here t is cosmic time, D is the separation, a is a dimensionless scale factor, and χ is the constant comoving separation in a convention where χ has units of length. These markers are idealized; a bound pair of stars need not follow this rule with constant χ.

Set a = 1 at the reference time. If χ = 100 megaparsecs, then D = 100 megaparsecs there. At a = 0.5 the same ideal pair has D = 50 megaparsecs. A megaparsec, abbreviated Mpc, is one million parsecs, approximately 3.086 × 10²² metres. The arithmetic describes a spatial separation assigned at each cosmic time; it is not a direct reading of light-travel distance.

The normalization of a is arbitrary. If we instead multiply every a by two and divide every χ by two, D stays the same. A physical prediction cannot depend on that relabelling. Ratios such as a₂/a₁ have physical meaning within the model. Saying 'the Universe doubled in size' without this qualification suggests an edge we have not measured and can confuse a length ratio with a volume ratio.

For a comoving three-dimensional region in this model, volume scales as a³. Doubling the scale factor multiplies its volume by 2³ = 8. If it contains a conserved number of nonrelativistic particles, their number density falls to one eighth. Radiation has additional energy redshift, so its energy density does not follow that matter-density rule. Later thermal-history lessons derive the distinction.

Keep the average and its departures together

Smoothing can discard exactly the effect a question needs. A perfectly smooth background cannot itself explain why one small region collapses into a galaxy and another becomes a void. To study that growth, we describe departures from the average, their evolution under gravity and eventually processes such as gas cooling and stellar feedback.

Conversely, resolving every star is unnecessary for many background-distance questions. The appropriate model separates scales while checking whether neglected effects matter at the required precision. 'More detailed' is not automatically 'more useful': each added ingredient needs physical motivation, constraints and a check that it improves the prediction.

This distinction also protects against a common misconception about expansion. D = aχ for comoving markers does not require your body, the Solar System or a gravitationally bound galaxy to grow in proportion to a. Local dynamics determine those systems. A background relationship is not a universal rule for every pair of objects.

A fitted parameter depends on the chosen model

A parameter is a quantity specifying one member of a family of models. The expansion rate today is one example. Data constrain parameters through predicted observables. Changing a model can change the relation between a parameter and the observed pattern, and therefore change the inferred value even when the underlying data remain the same.

The commonly used base Lambda cold dark matter model, written base ΛCDM, is a particular framework. Λ denotes a cosmological constant; cold dark matter is matter whose random motions are small enough to permit the relevant structure growth. Base ΛCDM fixes spatial flatness and makes specific choices about primordial fluctuations and other ingredients. Those are part of the model definition. They are not all separately measured facts supplied by a telescope.

For example, Planck's final full-mission analysis reported H₀ = 67.4 ± 0.5 km/s/Mpc under its base-ΛCDM assumptions, quoting a 68% region. H₀ means the present expansion rate: a value in kilometres per second per megaparsec relates a recession-rate scale to distance in that model. This is a dated Planck 2018 result, published in 2020, used here to illustrate conditional inference; it is not a claim to summarize every current estimate.

A broader model can free an assumption and test whether data support the extension. Extra flexibility may improve a fit, but it can also fit noise or weaken constraints. A convincing comparison asks what new prediction is gained, how the extra parameters are constrained, and whether the improvement survives independent data and systematic checks.

An assumption can become the next test

An assumption should be visible enough to challenge. If a model assumes direction-independent expansion, ask what pattern a direction-dependent alternative would create. Saadeh and colleagues used Planck temperature and polarization to constrain anisotropic expansion within a specified class of Bianchi models. That is a concrete model test, not a proof about every imaginable asymmetric Universe.

When an approximation fails, the result can be informative. It may identify a scale where more physics is needed, a faulty measurement model, or a promising extension. Keep its successes and its limits together. The aim is a model whose relationships can be used and tested, not a picture protected from every possible observation.

Worked reasoning

Try the reasoning

1. Does a sky that looks statistically similar in every direction from our position alone logically prove that every place in the Universe is alike?

   Show the answer and reasoning

   No. Isotropy about one position alone does not establish homogeneity everywhere.

   A spherically layered distribution is a counterexample to that logical implication. Cosmological inference uses additional assumptions and observations.

2. A conserved population in a comoving region expands from a = 0.5 to a = 1. What happens to number density?

   Show the answer and reasoning

   It falls by a factor of eight.

   Linear dimensions double, volume grows eightfold, and the same particle number occupies that larger volume. This reasoning assumes conservation and comoving boundaries.

3. Why is it misleading to say a parameter fit 'measured zero curvature' if its base model fixed curvature to zero before fitting?

   Show the answer and reasoning

   A fixed assumption was not a freely estimated parameter in that fit.

   Testing curvature requires a model allowing it to vary and a comparison with the data, including degeneracies with other parameters.

What to carry forward

• Homogeneity concerns position and isotropy concerns direction, on stated scales.
• A background model and its fluctuations serve different, connected questions.
• Scale-factor normalization is conventional; physical relationships use invariant ratios and defined separations.
• Quote parameter constraints with their dataset, model and uncertainty, and make assumptions available for testing.