Sensitivity and Error Budget

A sensitivity analysis asks a sharp question: if one input moves, how much does the output move, and which input matters most? For $P(t)$ the answer has an unusually clean structure, because the master equation is multiplicative. This chapter separates the two things a sensitivity analysis conflates — the response to a change and the contribution to uncertainty — and shows that for $P(t)$ they behave very differently. It builds directly on the uncertainty analysis and shares its machinery with the reproducibility notebook.

Every factor has unit elasticity

Write the aggregated output schematically as a product of its factors, $P = \prod_i x_i$ with $x_i \in \{\Phi, \sigma, \Delta p, v_{\text{ph}}, \eta_{\text{conv}}\}$. The elasticity of $P$ with respect to any factor — the fractional change in output per fractional change in input — is then exactly one:

$$\frac{\partial \ln P}{\partial \ln x_i} = 1 \qquad \text{for every } i. \tag{1}$$

A one-percent change in any single factor changes the output by one percent, no more and no less. First-order response is therefore uniform: no factor is intrinsically more “leveraged” than another. This is a genuine property of the multiplicative form, and it means the usual local-derivative ranking is uninformative here — it would rank all five factors equally.

What actually differs is the uncertainty

If the factors are not equally responsive, they are certainly not equally known. The useful sensitivity question for $P(t)$ is therefore an error budget: how much of the uncertainty in the output does each factor contribute? Treating the factors as independent and log-normally distributed — the natural choice for quantities uncertain across an order of magnitude — the log-variances add:

$$\operatorname{Var}(\ln P) = \sum_i \operatorname{Var}(\ln x_i) = \sum_i \sigma_i^{2}, \tag{2}$$

so the fractional contribution of factor $i$ to the output variance is simply

$$c_i = \frac{\sigma_i^{2}}{\sum_j \sigma_j^{2}}. \tag{3}$$

The error budget is set entirely by the relative uncertainties $\sigma_i$, not by the values of the factors. This is the quantity worth ranking.

Error budget (illustrative)

The table uses illustrative log-scale uncertainties — flagged as such, chosen to reflect that the neutrino cross-section is the least constrained term and the phonon velocity among the best constrained. Real values would come from measurement (measurement and falsifiability).

Factor	Illustrative $\sigma_i$ (ln units)	Variance share $c_i$
$\sigma(E)$ — scattering cross-section	0.6	40.0%
$\Phi$ — radiation flux density	0.5	27.8%
$\Delta p$ — momentum transfer	0.4	17.8%
$\eta_{\text{conv}}$ — conversion efficiency	0.3	10.0%
$v_{\text{ph}}$ — phonon velocity	0.2	4.4%

What the budget tells you

Two practical conclusions follow, and they are the reason the analysis is worth doing. First, measurement priority: because the cross-section term carries the largest share of the variance, an experimental programme reduces output uncertainty fastest by constraining $\sigma(E)$ first; halving its $\sigma_i$ removes more of the total variance than halving any other. Second, where precision is wasted: tightening the phonon velocity, already the best-constrained term, barely moves the total. Combined with the unit-elasticity result, the picture is complete — every factor matters equally for the value of $P(t)$, but they matter very unequally for its uncertainty, and the error budget says exactly where to look. As with all numbers on this site, the $\sigma_i$ here are illustrative; the method — decompose the log-variance, rank the shares, act on the largest — is the transferable content, and it is reproducible from the notebook.