Baryon density, nuclear rates and abundance predictions

A nuclear network connects laboratory physics and cosmic conditions

A nucleosynthesis prediction requires more than a list of nuclear binding energies. It uses reaction coefficients, weak processes and decay inputs together with baryon density, temperature and expansion history.

For a two-body process, the thermal coefficient κ = ⟨σv⟩ averages cross section times relative speed over the stated particle distributions. In a length-based unit system, κ has units of length³/time.

Multiplying κ by the partner number density gives a per-target rate λ = n_partner κ with units of inverse time. Confusing κ with λ would omit how often targets encounter partners. Published conventions containing an Avogadro factor also require the appropriate unit conversion.

Laboratory cross-section measurements, their normalizations and correlations influence the thermal coefficients. A full network carries those input choices into several abundances, with reverse reactions and common cosmological inputs as well.

Isolate one destruction mechanism with a prescribed bath

Consider a deliberately isolated trace-deuterium destruction interval in a nearly unchanged prescribed proton bath. A capture can convert deuterium plus a proton into helium-3 plus a photon. Hold the bath density, thermal coefficient and duration constant in this local account.

Ignore inverse and other reactions and common expansion dilution during the supplied interval. Holding the proton bath fixed is a trace-species approximation or external prescription; it is not exact closure of a finite reacting parcel, since each capture also uses a proton.

Choose reference inputs n_ref, κ_ref and Δt so that n_ref κ_ref Δt = ln 2. Let b = n_p/n_ref and r = κ/κ_ref be positive dimensionless density and coefficient factors. Here r denotes a rate-coefficient factor, not the earlier neutron/proton ratio.

The trace count obeys dN_D/dt = −n_pκN_D. Its survival fraction is S = N_D,out/N_D,in = exp(−br ln 2) = 2^(−br). This is exact for the declared one-path constant-input model, not a fitted total primordial D/H response.

One survival fraction does not separate both inputs

At b = r = 1, the relative exposure br is 1 and S = 1/2. Raising either b alone or r alone to 1.2 gives the same exposure 1.2 and survival about 0.435275.

A density increase can also be offset in this account: b = 1.2 and r = 5/6 have br = 1 and recover S = 1/2. One isolated survival fraction therefore cannot distinguish these input combinations.

A full nucleosynthesis network can respond differently because its other channels and time-dependent conditions add information. The algebraic ambiguity here should not be promoted to a universal exact degeneracy of actual light-element observations.

Increasing the capture coefficient leaves less of the trace deuterium in this chosen destruction model. The sign and magnitude of a full-network response must still be obtained from that network rather than copied from this exponential.

Covariance changes a local standard uncertainty

For an optional local uncertainty calculation about b = r = 1, use natural-log inputs ln b and ln r. Since ln S = −br ln 2, its derivatives with respect to both log inputs are −ln 2 at that reference point.

Choose input standard deviations u(ln b) = 0.05 and u(ln r) = 0.10, solely as teaching assumptions. First-order propagation gives u(ln S) = ln 2 √[0.05² + 0.10² + 2ρ(0.05)(0.10)], where ρ is the supplied correlation.

For ρ = 0 the result is about 0.0774962. For ρ = 0.5 the covariance is 0.0025 and the result is about 0.0916948. Positive dependence increases the local output uncertainty here because the two sensitivities have the same sign.

These are standard uncertainties for the linearized log output under supplied inputs. They are not measured nuclear uncertainties, exact nonlinear output distributions or automatic 68 percent intervals. A coverage probability would require additional distributional conditions.

A useful abundance comparison must preserve the whole input model

Real reaction coefficients vary with temperature and are derived from energy-dependent cross sections, with experiment-specific normalization and uncertainty issues. One constant κ over a supplied interval is a mechanism illustration, not a replacement for those inputs.

Baryon density changes partner encounters across many channels, while reaction-rate changes can have distinct effects on connected nuclei. Independent cosmological information or laboratory measurements can help distinguish explanations that one isolated output cannot.

A proposed adjustment intended to improve one abundance must be checked against the rest of the network and the other observations. Shared inputs and correlations prevent those comparisons from being treated as unrelated votes.

When reporting a prediction, state the abundance convention, cosmological history, rate datasets, numerical treatment and uncertainty model. This makes a disagreement a specific test of a physical account instead of an unexplained difference between numbers.

Worked reasoning

Try the reasoning

1. If n_p is in m⁻³ and κ in m³/s, what are the units of λ = n_pκ?

   Show the answer and reasoning

   s⁻¹.

   The volume units cancel, leaving a per-target inverse-time rate.

2. Does increasing the capture coefficient leave more of the trace deuterium in this model?

   Show the answer and reasoning

   No; it leaves less.

   The destruction exposure increases and S = exp(−exposure) decreases.

3. Do the chosen log-input standard deviations describe actual measured BBN errors or imply a 68 percent interval?

   Show the answer and reasoning

   No.

   They are supplied local uncertainty inputs, with no observed dataset or output coverage distribution attached.

What to carry forward

• A thermal coefficient needs a partner density to become a per-target rate.
• Density and coefficient changes share an exposure in the chosen isolated account.
• Input covariance and approximation scope must be preserved when an uncertainty is propagated.