Quantitative Measures

In order to make a fit-independent, quantitative prediction of the differences expected between the and pure baryon cases, I proposed (McGaugh 1999mypred) several geometry independent measures. These are the ratio of positions of observed peaks $\ell_{n+1}/\ell_n$, the absolute amplitude ratio of the peaks $(C_{\ell,n}/C_{\ell,n+1})_{abs}$,and the peak-to-trough amplitude ratio $(C_{\ell,n}/C_{\ell,n+1})_{rel}$.

Of these measures, the first is the least sensitive and the last is the most sensitive. The ratio of the positions of the first two peaks is expected to differ by only a small amount. Until this quantity is accurately measured, it does not provide a strong test. Should a second peak appear in future data, it does not necessarily favor -- a second peak is expected in either case, in roughly the same position. What does provide a clear distinction is the last measure, the peak-to-trough amplitude ratio of the first two peaks. This distinguishes between a second peak which stands well above the first trough, as expected with CDM, and one which does not, as expected without it.

These measures are readily extracted from the data. They are reported in Table 2, together with the predictions of the and pure baryon cases. The data clearly fall in the regime favored by the pure baryon case.

The result remains in the regime favored by the pure baryon case even if we adjust strategically chosen pairs of data points in the direction favorable to CDM. For example, increasing the amplitude of the point at $\ell = 500$ where the second peak should occur in by $1\sigma$ and decreasing by $1\sigma$ the amplitude of the point at $\ell = 400$ where the trough should occur does not suffice to move the result away from the range favored by the pure baryon case. This is more than a $2\sigma$ operation, as it is a coordinated move which would also impact surrounding data points. The data clearly favor the case of zero CDM.