Cosmology · Baryons, gas and cosmic neutrinos

What population constraints actually constrain

Construct distinct supplied collisionless populations with identical homogeneous energy and pressure, then identify which assumptions and observables an inference needs.

Preparation: ratios, powers of ten and ordinary arithmetic. Physical quantities and notation are introduced as they are needed.

Back to the chapter

After reading, you can

Derive number, energy and pressure scaling for supplied momentum distributions.
Explain why equal background predictions need not identify a unique population.
Separate conditional inference, prior information and additional physical observables.

A constraint belongs to a specified model

A cosmological observation responds to physical predictions through a model of signals, selection and uncertainty. Translating that response into a particle count or mass requires a relation between the population and those predictions.

A useful question is whether distinct supplied populations can produce exactly the same predictions for the observables under consideration. More precise measurements of those same predictions cannot by themselves remove such an exact degeneracy.

Construct a homogeneous example with stable, collisionless and isotropic particles. It is a mathematical family with supplied distributions, not a production model or a fit to relic-neutrino data.

The example concerns background number, energy and pressure moments. Equality of these moments does not establish equality of interactions, nucleosynthesis, laboratory signals or cosmological perturbations.

Change mass and momentum scale together

Let N be 1, 2 or 4 hypothetical identical pairs, each with g = 2 populated states. Introduce θ = N⁻¹ᐟ⁴, a positive reference temperature scale T_* and a reference mass m_* satisfying m_*c² = k_BT_*.

At scale-factor ratio q, supply T_N(q) = θT_*/q and m_N = θm_*. The temperature symbol describes the redshifting momentum shape, rather than ongoing collisions or a nonrelativistic thermodynamic temperature.

For physical momentum p, the occupation per populated state is f_N(p,q) = 1/[exp(pcq/(θk_BT_*))+1]. Chemical potential is zero. The displayed comparison uses 1 ≤ q ≤ 8, although the scaling identity holds for every positive q.

The N = 1 member defines the reference population. Other cosmological components are held identical. The construction does not explain how any of these populations was created or solve a perturbed gravitational system.

Supplied family, relative to N = 1 at the same q; equal energy and pressure are full moment identities
Pair count N	θ and m_N/m_*	n_N/n_ref	m_Nn_N/(m_*n_ref)	ε_N/ε_ref	P_N/P_ref
1	1	1	1	1	1
2	0.840896	1.189207	1	1	1
4	0.707107	1.414214	1	1	1

Follow the momentum measure and particle energy

The total proper number density is n_N = Ng/(2π²ℏ³) ∫₀∞p²f_N dp. Energy density is ε_N = Ng/(2π²ℏ³) ∫₀∞p²E_Nf_N dp, where E_N = √(p²c²+m_N²c⁴).

The isotropic pressure is P_N = Ng/(6π²ℏ³) ∫₀∞p⁴c²f_N/E_N dp. Number density has units m⁻³; energy density and pressure both have units J m⁻³. The physical-momentum measure includes ℏ rather than an unannounced dimensionless convention.

Substitute p = θu, where u still has momentum units. Then f_N(θu,q) equals the reference occupation, p²dp supplies θ³, and E_N(θu) = θE_ref(u). Hence n_N = Nθ³n_ref = N¹ᐟ⁴n_ref, while ε_N = Nθ⁴ε_ref = ε_ref.

In pressure, p⁴dp/E_N supplies θ⁴, so P_N = P_ref as well. Multiplying the number identity by m_N = θm_* gives m_Nn_N = m_*n_ref. The energy and pressure identities hold through the relativistic transition; they are not only a late-time rest-mass approximation.

Choose hypothetical identical pair counts N = 1, 2, 4 with g =two populated states per pair, zero chemical potential and frozen collisionless isotropic Fermi–Dirac momentum occupation. Set θ =N^(−1/4), mass m_N =θm_* and momentum-temperature T_N =θT_*/q, with m_*c² =k_BT_*. Occupation per state is f_N(p, q) =1/[exp(pcq/(θk_BT_*))+1] using physical momentum p, not an equilibrium energy distribution for a massive gas. Other components remain identical. The top table gives N, θ =m_N/m_* and n_N/n_ref: rows(1, 1, 1), (2, 0.840896, 1.189207), (4, 0.707107, 1.414214), each relative to N =one at the same positive q. Panel A uses linear q from 1 to 8 and proper-number ratios from 1 to 1.5, with baseline one. Solid blue circles, dashed orange squares and dotted violet diamonds are the constant number ratios for N = 1, 2, 4. These ratios are constant while each absolute proper number density dilutes with expansion. Panel B is separate with a zero-based ratio scale from zero to 1.5. A single explicitly shared solid green line at one with circle samples represents ALL N and ALL three ratios: full energy ε_N/ε_ref, full pressure P_N/P_ref and rest-mass density m_Nn_N/(m_*n_ref). No artificial separation or observational scatter is introduced. Proper number n =Ng/(2π²ℏ³)∫p²f dp, energy ε =Ng/(2π²ℏ³)∫p²E f dp and pressure P =Ng/(6π²ℏ³)∫p⁴c²f/E dp, where E =√(p²c²+m²c⁴). Substitution p =θu leaves the reference occupation, gives p²dp factor θ³ and E_N(θu) =θE_ref(u). Thus n ratio is Nθ³ =N^(1/4), energy ratio Nθ⁴ =one. In pressure p⁴dp/E supplies θ⁴, so pressure ratio is also one; multiplying n by m_N supplies another θ and rest-mass density ratio one. These full energy/pressure identities hold through the relativistic transition for every q>zero, not merely a late nonrelativistic approximation. Only finite 1≤q≤8 is drawn. Number has units m⁻³, energy and pressure J m⁻³, and rest-mass density kg m⁻³; the respective ratios are dimensionless. A common homogeneous prediction model and same joint likelihood/covariance/nuisance inputs cannot identify N from only these identical moments. No data vector, fitted covariance, confidence contour, perturbation system, production history or laboratory/nucleosynthesis equivalence is supplied. BN 06 concerns a different nonthermal population family and conditional approximate degeneracies, not a proof of this separately derived exact identity. Additional observables or temperature/production priors require their own assumptions. Original Ark teaching model.

Identical background inputs give identical background likelihood inputs

Within a common homogeneous expansion model using only these total energy and pressure histories, the family supplies the same background inputs. A background prediction computed from identical other components and boundary conditions therefore receives the same population moments.

For any fixed data vector, the same joint likelihood using only those identical predictions, the same covariance and the same nuisance model receives identical inputs. No data vector, uncertainty, likelihood centre or confidence interval is supplied here.

Multiplying likelihoods for separate probes requires the relevant conditional-independence assumptions. Correlated probes instead need an appropriate joint description. Either treatment retains an exact degeneracy when all of its population-dependent predictions remain identical.

A prior relation fixing a common temperature or a production mechanism can restrict the family. That restriction should be reported as an assumption or additional information, rather than attributed to background measurements that cannot distinguish the supplied members.

Additional physics can supply additional information

An effective relativistic energy normalization is not automatically an integer count of particle pairs. Here Nθ⁴ stays one even though N changes; a temperature relation is needed to convert an energy normalization into a count.

The example does not establish identical perturbations, interaction rates, nucleosynthesis outcomes or laboratory observables. Those require their own predictions and can introduce information absent from the supplied homogeneous moments.

Studies of nonstandard relic momentum distributions illustrate the dependence of cosmological mass inference on population assumptions. Their particular distribution families and approximate degeneracies are distinct from the exact rescaling identity derived here.

A useful population constraint names the observables, distribution and mass assumptions, nuisance treatment and statistical model. It then states what those ingredients identify, without turning a conditional background result into an unconditional particle census.

Worked reasoning

Check the N = 2 construction

How can two supplied pairs have a larger number density while retaining the reference energy and pressure histories?

Set θ = 2⁻¹ᐟ⁴ ≈ 0.840896, scaling both the mass and the momentum distribution.
The number measure gives Nθ³ = 2¹ᐟ⁴ ≈ 1.189207, while the extra particle-energy or pressure factor gives Nθ⁴ = 1.
Thus n increases, m decreases, and the full energy and pressure moments remain equal at every common q.

The family changes the particle inventory and individual scale together; homogeneous energy and pressure alone do not uniquely identify N.

Try the reasoning

Does a measured relativistic energy normalization always directly count integer particle pairs?
Show the answer and reasoning
No.
The conversion requires population assumptions, including the momentum or temperature relation; Nθ⁴ is fixed across this supplied family.
Do equal homogeneous ε and P establish identical laboratory signals and cosmological perturbations?
Show the answer and reasoning
No.
Those predictions require additional physical models and were not derived by the homogeneous moment identities.
Can arbitrarily precise additional probes that use only the same identical background predictions distinguish these family members?
Show the answer and reasoning
Not within that common likelihood and prediction model.
Their inputs remain identical. Additional assumptions or observables with genuinely different predictions are needed to identify a member.

What to carry forward

Number, energy and pressure weight a momentum distribution differently.
Distinct supplied populations can share exact homogeneous energy and pressure histories while differing in count and mass.
Population inference is conditional on distribution assumptions, observables and the statistical model.

References

Sources supporting the explanations, physical distinctions and examples in this lesson.