A cosmological observable measured in a range of redshifts can be used as a probe of a set of cosmological parameters .
Given the cosmological observable and the cosmological parameter , there is an optimum range of redshifts where the observable can constrain the parameter in the most effective manner .
For other redshift ranges the observable values may be degenerate with respect to the cosmological parameter values and thus inefficient in constraining the given parameter .
These are blind redshift ranges .
We determine the optimum and the blind redshift ranges of basic cosmological observables with respect to three cosmological parameters : the matter density parameter \Omega _ { m } , the equation of state parameter w ( assumed constant ) , and a modified gravity parameter g _ { a } which parametrizes a possible evolution of the effective Newton ’ s constant as G _ { eff } ( z ) = G _ { N } ( 1 + g _ { a } ( 1 - a ) ^ { 2 } - g _ { a } ( 1 - a ) ^ { 4 } ) ( where a = \frac { 1 } { 1 + z } is the scale factor and G _ { N } is Newton ’ s constant of General Relativity ) .
We consider the following observables : the growth rate of matter density perturbations expressed through f ( z ) and f \sigma _ { 8 } ( z ) , the distance modulus \mu ( z ) , baryon acoustic oscillation observables D _ { V } ( z ) \times \frac { r _ { s } ^ { fid } } { r _ { s } } , H \times \frac { r _ { s } } { r _ { s } ^ { fid } } and D _ { A } \times \frac { r _ { s } ^ { fid } } { r _ { s } } , H ( z ) measurements and the gravitational wave luminosity distance .
We introduce a new statistic S _ { P } ^ { O } ( z ) \equiv \frac { \Delta O } { \Delta P } ( z ) \cdot V _ { eff } ^ { 1 / 2 } , including the effective survey volume V _ { eff } , as a measure of the constraining power of a given observable O with respect to a cosmological parameter P as a function of redshift z .
We find blind redshift spots z _ { b } ( S _ { P } ^ { O } ( z _ { b } ) \simeq 0 ) and optimal redshift spots z _ { s } ( S _ { P } ^ { O } ( z _ { s } ) \simeq max ) for the above observables with respect to the parameters \Omega _ { m } , w and g _ { a } .
For example for O = f \sigma _ { 8 } and P = ( \Omega _ { m } ,w,g _ { a } ) we find blind spots at z _ { b } \simeq ( 1 , 2 , 2.7 ) , respectively , and optimal ( sweet ) spots at z _ { s } = ( 0.5 , 0.8 , 1.2 ) .
Thus probing higher redshifts may in some cases be less effective than probing lower redshifts with higher accuracy .
These results may be helpful in the proper design of upcoming missions aimed at measuring cosmological obsrevables in specific redshift ranges .