# Constraining power of cosmological observables: blind redshift spots and optimal ranges

## TL;DR

This paper identifies the optimal and blind redshift ranges for various cosmological observables to effectively constrain parameters like matter density, dark energy equation of state, and modified gravity effects, using a new constraining power statistic.

## Contribution

It introduces a new statistic to measure the constraining power of observables across redshifts and maps the optimal and blind redshift ranges for key cosmological parameters.

## Key findings

- Blind redshift spots identified at z ≈ 1, 2, 2.7 for different parameters.
- Optimal redshift spots found at z ≈ 0.5, 0.8, 1.2 for the same parameters.
- Probing higher redshifts may be less effective than lower redshifts for constraining parameters.

## Abstract

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 cosmological observables with respect to the cosmological parameters: matter density parameter $\Omega_m$, equation of state parameter $w$ and a modified gravity parameter $g_a$ which parametrizes the evolution of an effective Newton's constant. We consider the observables: growth rate of matter density perturbations expressed through $f(z)$ and $f\sigma_8$, 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 these observables with respect to the parameters $\Omega_m$, $w$ and $g_a$. 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 be less effective than probing lower redshifts with higher accuracy.