*5.4. Differential Age and Hubble Measurements*

Another intriguing treatment, widely used in observational cosmology and also for calibrating GRB correlations, has been firstly proposed in Ref. [123]. The idea is to measure the Hubble rate by using galaxies, in a quite model-independent way. In the context of GRBs, the Hubble catalog has been widely explored. For example, in Ref. [123], the core idea is to match the observational Hubble rate data (OHD) with model independent expansion of *H* made by Bézier polynomials. At a first glance, this *differential age method* (see, e.g., Refs. [124,125]) does not require any assumption over the form of *H*, although spatial curvature can affect the overall treatment if it varies with time, instead of being fixed<sup>21</sup> .

To better introduce the method, we notice that it is well known that spectroscopic measurements of the age difference ∆*t* and redshift difference ∆*z* of couples of passively evolving galaxies lead to∆*z*/∆*t* ≡ *dz*/*dt* and so, if galaxies formed at the same time (redshift *z*), the Hubble rate can be approximated by

$$H(z) = -(1+z)^{-1} \Delta z / \Delta t \,. \tag{30}$$

Consequently, model-independent estimates may come from cosmic chronometers based on the assumption that observable Hubble rates are given by the exact formula

$$H\_{\rm obs} = -\frac{1}{(1+z)} \left(\frac{dt}{dz}\right)^{-1} \tag{31}$$

if approximated as in Equation (30). The *χ* 2 from the current 31 OHD measurements reads

$$\chi^2\_{OHD} = \sum\_{i=1}^{31} \left[ \frac{H\_{th}(\mathbf{x}\_i z\_i) - H\_{obs}(z\_i)}{\sigma\_{H,i}} \right]^2. \tag{32}$$

This procedure has the great advantage of directly considering *H* without passing through any cosmic distance.
