The commonly used isotropic baryon acoustic oscillation ( BAO ) equation is an approximation derived from empirical and geometric arguments , and the equivalent anisotropic BAO equations were written down by analogy .
Using fit-lines to CMB compatible solutions , \Omega _ { m } and H _ { 0 } values have been derived for BAO studies without recourse to those equations , and have been applied for their appraisal .
The isotropic expression becomes problematic at precision levels of \sim 1 percent or better , and at high redshift values of z \gtrsim 1 .
Most revealing , the anisotropic equations , D _ { M } ( z ) / D _ { M, \text { fid } } ( z ) = \alpha _ { \perp } r _ { d } / r _ { d, \text { fid } } , and D _ { H } ( z ) / D _ { H, \text { fid } } ( z ) = \alpha _ { \| } r _ { d } / r _ { d, \text { fid } } , are invalid when \alpha \sim 1 , since under that condition , D _ { M } ( z ) / D _ { M, \text { fid } } ( z ) = D _ { H } ( z ) / D _ { H, \text { fid } } ( z ) = r _ { d } / r _ { d, \text { fid% } } \sim 1 , and neither equation is satisfied with anisotropic data .
( The ratios are respectively , the angular distance , the inverse Hubble parameter , and the comoving acoustic horizon , each divided by its fiducial value ) . \alpha can be driven towards unity for any BAO study , e.g. , by applying the derived \Omega _ { m } , H _ { 0 } pair as the core of a second iteration fiducial parameter-set .
Thus , the anisotropic equations are untenable .
Dissociated from the BAO equations , we have extracted weighted mean values of \bar { \Omega } _ { mw } = 0.299 \pm 0.011 and \bar { H } _ { 0 w } = 68.6 \pm 0.7 km \text { s } ^ { -1 } \text { Mpc } ^ { -1 } for eight uncorrelated BAO data sets .