Null tests of the standard model using the linear model formalism

We test both the Friedmann-Lemaître-Robertson-Walker geometry and $\mathrm{\Lambda }\mathrm{CDM}$ cosmology in a model-independent way by reconstructing the Hubble function $H(z)$, the comoving distance $D(z)$, and the growth of structure $f{\sigma }_8(z)$ using the most recent data available. We use the linear model formalism in order to optimally reconstruct the above cosmological functions, together with their derivatives and integrals. We then evaluate four of the null tests available in the literature that probe both background and perturbation assumptions. For all the four tests, we find agreement, within the errors, with the standard cosmological model.

Figure 1

The 31 cosmic chronometer data points and the corresponding linear model best fit (in light blue) that we use in this work. See Table 1 for the numerical values.

Figure 2

The compressed JLA compilation (Ref. [3], Appendix E) and the corresponding linear model best fit (in light blue) that we use in this work. For this plot, we have adopted $M=0$.

Figure 3

The 18 $f{\sigma }_8$ data points [40] and the corresponding linear model best fit (in light blue) that we use in this work.

Figure 4

The four null tests considered in this work with $1\sigma$ bands. The flat $\mathrm{\Lambda }\mathrm{CDM}$ model is falsified if the $Om$ diagnostics of Sec. 3a do not give a constant value; a reference value of ${\mathrm{\Omega }}_{m_0}=0.3$ is shown (solid line) to guide the eye. All the FLRW models are ruled out if the $Ok$ diagnostic of Sec. 3b is not compatible with a constant value (the solid line shows the reference value ${\mathrm{\Omega }}_{k_0}=0$). Finally, the (possibly curved) $\mathrm{\Lambda }\mathrm{CDM}$ model is falsified if the $ns$ diagnostic of Sec. 3c is incompatible with a constant value of unity. As these plots show, the latest cosmological data pass all the standard-model null tests. For the $O{m}_1$ and $Ok$ tests, we adopted the following values of the Hubble constant: the $H_{0,\mathrm{cmb}}$ value of Eq. (21) (solid line and gray bands), $H_0=70\mathrm{Km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ (dashed line), and the $H_{0,\mathrm{loc}}$ value of Eq. (20) (dot-dashed line). For the $ns$ test, we adopt the values given in Eq. (19): ${\sigma }_{8,0}=0.804$ (black solid line), ${\sigma }_{8,0}=0.832$ (black dashed line), and ${\sigma }_{8,0}=0.747$ (black dotted line). See Sec. 6 for more details.