The Big Bang — Question 10

Question 10

Independent abundance measurements test a shared cosmic baryon density

In an illustrative nucleosynthesis model, deuterium abundance is $D/H=2.5\times10^{-5}(6/\eta_{10})^{1.6}$, where $\eta_{10}=10^{10}\eta$. An observation gives $D/H=3.0\times10^{-5}$. An independent background fit gives $\eta_{10}=6.0$.

1. Infer eta10 from the deuterium model.

2. Find the abundance predicted by the background value and the observed fractional excess; state what uncertainties are needed before judging a discrepancy.

Question 10 - Solution

Independent abundance measurements test a shared cosmic baryon density

(a) Method and calculation

$$\frac{3.0}{2.5}=(6/\eta_{10})^{1.6}\Rightarrow\eta_{10}=6(2.5/3.0)^{1/1.6}.$$$$\boxed{\eta_{10}\simeq5.354}.$$

(b) Interpretation and further calculation

$$\boxed{(D/H)_{\mathrm{background}}=2.5\times10^{-5},\quad\frac{D_{\mathrm{obs}}-D_{\mathrm{pred}}}{D_{\mathrm{pred}}}=.20}.$$$$z=\frac{D_{\mathrm{obs}}-D_{\mathrm{pred}}}{\sqrt{\sigma_{\mathrm{obs}}^2+\sigma_{\mathrm{pred}}^2}}\quad\text{for independent errors}.$$

Physical check

A difference of central values alone is not a statistical rejection. Nuclear rates, measurement errors and the cosmological model enter the comparison.