*5.5. The* R *Parameter*

The CMB represents a cosmic recombination epoch remnant and contains abundant early universe information. Consequently, the acoustic peak positions [120,126] can be used to characterize a given cosmological model by means of the shift parameter, defined as [127]

$$\mathcal{R} = \frac{\sqrt{\Omega\_M}}{\sqrt{|\Omega\_k|}} \text{sinc}\left(\sqrt{|\Omega\_k|} \int\_0^{z\_k} \frac{dz}{E(\mathbf{x}, z)}\right) = 1.70 \pm 0.03\,\text{.}\tag{33}$$

The last scattering redshift, namely *z*ls, is fixed to

$$z\_{\rm ls} = 1048[1 + 0.00124(\Omega\_b h^2)^{-0.738}][1 + g\_1(\Omega\_M h^2)^{\otimes 2}].\tag{34}$$

where *g*<sup>1</sup> = 0.078(Ω*bh* 2 ) <sup>−</sup>0.238[1 + 39.5(Ω*bh* 2 ) 0.763] <sup>−</sup><sup>1</sup> and *g*<sup>2</sup> = 0.56[1 + 21.1(Ω*bh* 2 ) 1.81] −1 [128] and the *χ* 2 reads

$$
\chi^2\_{\rm CMB} = \frac{(\mathcal{R} - 1.70)^2}{0.03^2}.\tag{35}
$$

In analogy with BAO measures, the shift parameter is not fully model-independent.
