4.3.1. Primary Distance Indicators

The above first typology mainly includes *Variable stars*, i.e., among which, Cepheids, RR Lyrae, and Mira. Here, the variable star type is based on the possible correlation between their period of variation, steadily measured, and their luminosity. Even though this set of stars mainly constitutes the primary indicators, further typologies are main sequence and red clump stars. Here, using the luminosity-temperature relations from the standard Hertzsprung–Russell diagram, one deduces stellar luminosity within a fairly narrow range. Last but not least, eclipsing binaries are also primary distance indicators, since their luminosity is computed by the Stefan–Boltzmann law through a direct estimate of their radius, by means of a Doppler measurement of orbital velocities combined with the light–curve data, together with the temperature, deduced from the spectrum.

#### 4.3.2. Secondary Distance Indicators

On the other hand, the second class of standard candles is essentially based on very different indicators with respect to the first case. For instance, the prototypes of such indicators are the *properties of galaxies*, among all, the Tully–Fisher relation. This law matches spiral galaxy rotation speed and stellar luminosity. In particular, to argue the spiral galaxy rotation speed, one can consider, for example, the spectral line width. Another relation, widely adopted as an underlying second type of indicator, is the Faber– Jackson relation. Here, it is possible to infer elliptical galaxy random stellar velocities using the total luminosity. Again, the way to get these velocities consists of the use of spectral line widths. Another quite relevant relation is the fundamental plane law, i.e., a treatment that extends the Faber–Jackson one by including surface brightness as an additional observable parameter.

Besides galaxy properties, another second typology of standard candles is represented by *SNe Ia*, i.e., probably the most used cosmological standard candles to accredit the late time cosmic speed up. The scenario in which they form is due to thermonuclear explosions of WDs that exceed the Chandrasekhar's limit, namely ∼ 1.4*M*. For such objects, we see a correlation between the time scale of the explosion and the peak luminosity. The corresponding light curves follow given shapes, in agreement with the so-called Phillips curve [118]. As stated, SNe Ia are the most fruitful standard candles. For each event, even if the luminosity is clearly different for every SN, the Phillips curve relates the B magnitude peak to the luminous decay after 15 days with an overall set of SNe distributed in the range *z* = 0–2.5. These redshifts span between decelerating and accelerating phases of universe's evolution, corresponding to the matter and DE dominated epochs19. Last but not least, these indicators are present in all galaxies, except in the arms of spiral galaxies, but their physical internal processes are still the object of investigations as they are not fully-interpreted.

#### **5. Going Ahead with Standard Indicators: The** *χ* <sup>2</sup> **Analysis**

Using standard candles, it is possible to establish data catalogs that can be used and matched with GRB data. Hence, to experimentally fit a given model with a given set of free parameters, one requires the definition of a merit function that quantifies the overall agreement between the working model with the aforementioned cosmic data. Equivalently, it is of utmost importance to get best fit parameters and corresponding estimates of error bars, together with a method to possibly measure the goodness of fit. The parameter fitting treatment commonly makes use of least-squares analyses, based on the combination among

data points, say *D<sup>i</sup>* , a model for these data, namely the *y*(*x*,~*θ*), function of *θ*. Naively, the simplest approach to least squares for uncorrelated data becomes

$$\chi^2 = \sum\_{i} w\_i [D\_i - y(\mathbf{x}\_i|\vec{\theta})]^2 \tag{22}$$

where the weights *w<sup>i</sup>* reach the maximum variance in case *w<sup>i</sup>* = 1/*σ* 2 *i* , with *σ<sup>i</sup>* the data point errors. For correlated data, we have

$$\chi^2 = \sum\_{ij} (D\_i - y(\mathbf{x}\_i|\theta)) Q\_{ij}^{-1} (D\_j - y(\mathbf{x}\_j|\theta)) \tag{23}$$

in which the inverse of covariance matrix, *Q*, has been introduced describing the degree of correlations among data. Minimizing the *χ* 2 is equivalent to getting suitable sets of findings that represent the best fit for our procedure. Different *χ* <sup>2</sup> values lead to probability distribution around the minimum.
