Constraining teleparallel gravity through Gaussian processes

In TG, the Levi-Civita connection ${{\mathring{{\Gamma}}}}_{\mu \nu }^{\sigma }$ (we recall that over-circles denote quantities determined by the Levi-Civita connection) is replaced with the teleparallel connection ${{\Gamma}}_{\mu \nu }^{\sigma }$ [72–74]. With this change of connection, the Riemann tensor vanishes identically. In such theories, one uses the tetrads ${e}_{\enspace \mu }^{a}$ rather than the metric tensor g_μν . These act as a soldering agent between the general manifold (greek indices) and the local Minkowski space (latin indices) [26]. Thus, tetrads can be used to raise the Minkowski metric to the general metric through the relations

$$\begin{equation}{g}_{\mu \nu }={e}_{\enspace \mu }^{a}{e}_{\enspace \nu }^{b}{\eta }_{ab}\enspace ,\quad {\eta }_{ab}={e}_{a}^{\enspace \mu }{e}_{b}^{\enspace \nu }{g}_{\mu \nu },\end{equation} \tag{ 1 }$$

where the inverse tetrads ${e}_{a}^{\enspace \mu }$ must also satisfy the orthogonality conditions

$$\begin{equation}{e}_{\enspace \mu }^{a}{e}_{b}^{\enspace \mu }={\delta }_{b}^{a}\enspace ,\quad {e}_{\enspace \mu }^{a}{e}_{a}^{\enspace \nu }={\delta }_{\mu }^{\nu },\end{equation} \tag{ 2 }$$

for consistency. The teleparallel connection can then be defined as [29]

$$\begin{equation}{{\Gamma}}_{\mu \nu }^{\sigma }{:=}{e}_{a}^{\enspace \mu }{\partial }_{\mu }{e}_{\enspace \nu }^{a}+{e}_{a}^{\enspace \sigma }{\omega }_{\enspace b\mu }^{a}{e}_{\enspace \nu }^{b},\end{equation} \tag{ 3 }$$

where ${\omega }_{\enspace b\mu }^{a}$ denotes the spin connection. In terms of a linear affine connection that is both curvatureless and satisfies metricity, this is the most general realisation [26]. The role of the spin connection is to preserve the invariance of the theory under local Lorentz transformations (LLTs) [75]. This represents the possible LLTs (three boosts and three rotations). Spin connections also appear in GR but are hidden in its internal structure except for rare circumstances such as in its spinor formalism [1, 76]. The core difference is that, in TG, the spin connection components are entirely inertial and thus there will always exist a frame where they vanish [28].

For any of the six LLTs ${{\Lambda}}_{b}^{a}$, the spin connection components can be written as ${\omega }_{\enspace b\mu }^{a}={{\Lambda}}_{c}^{a}{\partial }_{\mu }{{\Lambda}}_{b}^{c}$ [26]. For a particular metric ansatz, there will exist an infinite number of possible tetrads that satisfy equation (1) due to the active role played by LLTs through the spin connection. Thus, it is the combination of a tetrad and its associated spin connection that represent gravitational and inertial fundamental dynamical object in TG. Any action in TG will naturally lead to ten possible independent field equations as in regular GR. However, in TG, we have an additional six potential field equations which must also be satisfied. These represent the six LLTs and are determined by considering the anti-symmetric field equations [77]. The anti-symmetric field equations vanish due to the symmetries of the energy–momentum tensor and offer an avenue to relate the tetrad and spin connection components directly [28]. The gauge in which this occurs, that is where the spin connection vanished, is called the Weitzenböck connection or gauge [29].

The Riemann tensor gives a fundamental measure of curvature in GR. In the framework of TG, this is replaced by the torsion tensor [27]

$$\begin{equation}{{T}^{\sigma }}_{\mu \nu }{:=}2{{\Gamma}}_{\left[\mu \nu \right]}^{\sigma },\end{equation} \tag{ 4 }$$

where square brackets represent anti-symmetry, and where the torsion tensor is related to the gravitational field strength within the theory. As in GR, there are also two other helpful tensorial quantities in TG. First, the contorsion tensor is defined as the difference between the teleparallel connection and its Levi-Civita counterpart

$$\begin{equation}{{K}^{\sigma }}_{\mu \nu }{:=}{{\Gamma}}_{\mu \nu }^{\sigma }-{{\mathring{{\Gamma}}}}_{\mu \nu }^{\sigma }=\frac{1}{2}\left({{{T}_{\mu }}^{\sigma }}_{\nu }+{{{T}_{\nu }}^{\sigma }}_{\mu }-{{T}^{\sigma }}_{\mu \nu }\right),\end{equation} \tag{ 5 }$$

and plays a crucial role in relating TG with its standard gravity analogs. Secondly, in TG, one can define a so-called superpotential

$$\begin{equation}{{S}_{a}}^{\mu \nu }{:=}\frac{1}{2}\left({{K}^{\mu \nu }}_{a}-{{e}_{a}}^{\nu }{{T}^{\alpha \mu }}_{\alpha }+{{e}_{a}}^{\mu }{{T}^{\alpha \nu }}_{\alpha }\right),\end{equation} \tag{ 6 }$$

which has been connected to the gauge current representation of the theory [33, 34] but this questions remains largely open [78, 79].

Contracting the torsion tensor with its superpotential directly leads to the torsion scalar

$$\begin{equation}T{:=}{{S}_{a}}^{\mu \nu }{{T}^{a}}_{\mu \nu },\end{equation} \tag{ 7 }$$

which is dependent on the teleparallel connection in an analogous way to the dependence of the Ricci scalar on the Levi-Civita connection. Given the intrinsic use of the contorsion tensor means that the Ricci and torsion scalars can be related to each other, and turn out to differ by a total divergence term [45, 54]

$$\begin{equation}R={\mathring{R}}+T-\frac{2}{e}{\partial }_{\mu }\left(e{T}_{\enspace \sigma }^{\sigma \mu }\right)=0,\end{equation} \tag{ 8 }$$

where $e=\mathrm{det}\left({e}_{\enspace \mu }^{a}\right)=\sqrt{-g}$ represents the tetrad determinant, and R is the Ricci scalar as calculated with the teleparallel connection, which vanishes, while ${\mathring{R}}$ is the regular Ricci scalar in standard gravity. Straightforwardly, this means that the Ricci and torsion scalars are equal up to a boundary term

$$\begin{equation}{\mathring{R}}=-T+\frac{2}{e}{\partial }_{\mu }\left(e{T}_{\enspace \sigma }^{\sigma \mu }\right){:=}-T+B.\end{equation} \tag{ 9 }$$

This point alone guarantees that the Ricci and torsion scalars will produce the same dynamical equations. Thus, TEGR can be defined as the theory in which the Lagrangian is simply the torsion scalar T, or where the action is represented by

$$\begin{equation}{\mathcal{S}}_{\text{TEGR}}=-\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\enspace eT+\int {\mathrm{d}}^{4}x\enspace e{L}_{\mathrm{m}},\end{equation} \tag{ 10 }$$

where κ² = 8πG is the gravitational coupling and L_m is the matter Lagrangian.

Following the same reasoning as $f\left({\mathring{R}}\right)$ gravity [24, 25], the TEGR Lagrangian can straightforwardly be elevated to a generalised f(T) gravity framework [37–41]. In this context, the action will then be given as

$$\begin{equation}{\mathcal{S}}_{\tilde {f}\left(T\right)}=\frac{1}{2{\kappa }^{2}}\int {\mathrm{d}}^{4}x\enspace e\tilde {f}\left(T\right)+\int {\mathrm{d}}^{4}x\enspace e{L}_{\mathrm{m}},\end{equation} \tag{ 11 }$$

which produces second order equations of motion. This is only possible to a weakened Lovelock theorem in TG [30–32]. This can recover ΛCDM in the limit where $\tilde {f}\left(T\right)=-T+{\Lambda}$. On another note, f(T) gravity also shares a number of interesting properties with GR such as having the same number of associated polarisation modes of its gravitational wave signature [57, 80–83], and being Gauss–Ostrogradsky ghost free (since it remains second-order) [28, 74]. Performing a variation with respect to the tetrad finally gives the field equations as

$$\begin{align}\hfill {e}^{-1}& {\partial }_{\nu }\left(e{e}_{a}^{\enspace \rho }{S}_{\rho }^{\enspace \mu \nu }\right){\tilde {f}}_{T}-{e}_{a}^{\enspace \lambda }{T}_{\enspace \nu \lambda }^{\rho }{S}_{\rho }^{\enspace \nu \mu }{\tilde {f}}_{T}+\frac{1}{4}{e}_{a}^{\enspace \mu }\tilde {f}\left(T\right)\hfill \\ \hfill & +{e}_{a}^{\enspace \rho }{S}_{\rho }^{\enspace \mu \nu }{\partial }_{\nu }\left(T\right){\tilde {f}}_{TT}+{e}_{b}^{\enspace \lambda }{\omega }_{\enspace a\nu }^{b}{S}_{\lambda }^{\enspace \nu \mu }{\tilde {f}}_{T}={\kappa }^{2}{e}_{a}^{\enspace \rho }{{\Theta}}_{\rho }^{\enspace \mu },\hfill \end{align} \tag{ 12 }$$

where subscripts denote derivatives, and ${{\Theta}}_{\rho }^{\enspace \nu }$ is the regular energy–momentum tensor. In the ensuing work, we consider the flat Friedmann–Lemaître–Robertson–Walker (FLRW) cosmology which has been shown to be compatible with the Weitzenböck gauge [55, 58, 62] which we shall be adopting. Moreover, we also choose to consider f(T) as an extension through the transformation

$$\begin{equation}\tilde {f}\left(T\right)\to -T+f\left(T\right),\end{equation} \tag{ 13 }$$

where f(T) will appear as an extension to TEGR.

The flat homogeneous and isotropic FLRW metric represented by

$$\begin{equation}\mathrm{d}{s}^{2}=-\mathrm{d}{t}^{2}+{a}^{2}\left(t\right)\left(\mathrm{d}{x}^{2}+\mathrm{d}{y}^{2}+\mathrm{d}{z}^{2}\right),\end{equation} \tag{ 14 }$$

can be produced by the tetrad choice

$$\begin{equation}{e}_{\enspace \mu }^{a}=\mathrm{diag}\left(1,a\left(t\right),a\left(t\right),a\left(t\right)\right),\end{equation} \tag{ 15 }$$

where a(t) is the scale factor which is compatible with the Weitzenböck gauge, i.e. ${\omega }_{\enspace b\mu }^{a}=0$ in the f(T) context [75, 84]. Using the torsion scalar definition in equation (7) immediately gives

$$\begin{equation}T=6{H}^{2},\end{equation} \tag{ 16 }$$

where the boundary term will be $B=6\left(3{H}^{2}+\dot {H}\right)$, which straightforwardly gives the expected standard Ricci scalar for the flat FLRW setting, i.e. ${\mathring{R}}=-T+B=6\left(\dot {H}+2{H}^{2}\right)$ (note the use of the standard metric convention [85], rather than the one used in references [27, 39, 40] which differs by a sign difference in T). Evaluating the field equations in equation (12) results in the Friedmann equations

$$\begin{equation}3{H}^{2}={\kappa }^{2}\left({\rho }_{\mathrm{m}}+{\rho }_{\text{eff}}\right),\end{equation} \tag{ 17 }$$

$$\begin{equation}3{H}^{2}+2\dot {H}=-{\kappa }^{2}\left({p}_{\mathrm{m}}+{p}_{\text{eff}}\right),\end{equation} \tag{ 18 }$$

where ρ_m and p_m represent the energy density and pressure of the matter content respectively, and where f(T) gravity can be interpreted as an effective fluid with components

$$\begin{equation}{\rho }_{\text{eff}}{:=}\frac{1}{2{\kappa }^{2}}\left(2T{f}_{T}-f\right),\end{equation} \tag{ 19 }$$

$$\begin{equation}{p}_{\text{eff}}{:=}-\frac{1}{{\kappa }^{2}}\left[2\dot {H}\left({f}_{T}+2T{f}_{TT}\right)\right]-{\rho }_{\text{eff}}.\end{equation} \tag{ 20 }$$

The effective fluid coincidentally also satisfies the standard conservation equation

$$\begin{equation}{\dot {\rho }}_{\mathrm{e}\mathrm{ff}}+3H\left({\rho }_{\mathrm{e}\mathrm{ff}}+{p}_{\mathrm{e}\mathrm{ff}}\right)=0,\end{equation} \tag{ 21 }$$

and can be used to define an effective equation of state (EoS) giving [58, 62]

$$\begin{equation}{\omega }_{\text{eff}}{:=}\frac{{p}_{\text{eff}}}{{\rho }_{\text{eff}}}=-1+\left(1+{\omega }_{\mathrm{m}}\right)\frac{\left(T+f-2T{f}_{T}\right)\left({f}_{T}+2T{f}_{TT}\right)}{\left(-1+{f}_{T}+2T{f}_{TT}\right)\left(-f+2T{f}_{T}\right)}.\end{equation} \tag{ 22 }$$

An interesting point to highlight is that the ΛCDM scenario is recovered when f(T) = Λ.

As previously mentioned, GP extend the idea of a Gaussian distribution by defining a mean function μ(z) together with a two-point covariance function $\mathcal{C}\left(z,{z}^{\prime }\right)$ to form a GP as a continuous curve

$$\begin{equation}\xi \left(z\right)\sim \mathcal{GP}\left(\mu \left(z\right),\mathcal{C}\left(z,{z}^{\prime }\right)\right),\end{equation} \tag{ 23 }$$

together with its associated error regions Δξ(z), thus resulting in ξ(z) ± Δξ(z). Without losing generality, the mean can be set to zero for all points in the reconstruction, since each point is not very sensitive to this. For the redshifts z* of the reconstruction at which we do not have data points, we can define a kernel function for the covariance such that $\mathcal{C}\left({z}^{{\ast}},{z}^{{{\ast}}^{\prime }}\right)=\mathcal{K}\left({z}^{{\ast}},{z}^{{{\ast}}^{\prime }}\right)$, which will form the majority of points. The kernel will embody all the information about the strength of the correlations between these reconstructed values as well as the amplitude of the deviations from the mean [94]. The only generic property about this is that it must be a symmetric function. On the other hand, for observational data points $\tilde {z}$ we have available the associated errors and covariance matrix $\mathcal{D}\left(\tilde {z},{\tilde {z}}^{\prime }\right)$ between the points, so that the covariance can be written as $\mathcal{C}\left(\tilde {z},{\tilde {z}}^{\prime }\right)=\mathcal{K}\left(\tilde {z},{\tilde {z}}^{\prime }\right)+\mathcal{D}\left(\tilde {z},{\tilde {z}}^{\prime }\right)$ which will give information about the kernel. Lastly, observational points and reconstructed points will be correlated by the kernel alone through $\mathcal{C}\left({z}^{{\ast}},{\tilde {z}}^{\prime }\right)=\mathcal{K}\left({z}^{{\ast}},{\tilde {z}}^{\prime }\right)$ [100].

There exists a number of kernel function choices [68] where some behave slightly better in certain situations which remains an open discussion in the literature [87]. In this work, we consider the effect of choosing a number of kernel functions, namely the general purpose square exponential

$$\begin{equation}\mathcal{K}\left(z,\tilde {z}\right)={\sigma }_{f}^{2}\enspace \mathrm{exp}\left[-\frac{{\left(z-\tilde {z}\right)}^{2}}{2{l}_{f}^{2}}\right],\end{equation} \tag{ 24 }$$

together with the Cauchy, Matérn and rational quadratic kernels which are respectively given by

$$\begin{equation}\mathcal{K}\left(z,\tilde {z}\right)={\sigma }_{f}^{2}\left[\frac{{l}_{f}}{{\left(z-\tilde {z}\right)}^{2}+{l}_{f}^{2}}\right],\end{equation} \tag{ 25 }$$

$$\begin{equation}\mathcal{K}\left(z,\tilde {z}\right)={\sigma }_{f}^{2}\left(1+\frac{\sqrt{3}\vert z-\tilde {z}\vert }{{l}_{f}}\right)\mathrm{exp}\left[-\frac{\sqrt{3}\vert z-\tilde {z}\vert }{{l}_{f}}\right],\end{equation} \tag{ 26 }$$

$$\begin{equation}\mathcal{K}\left(z,\tilde {z}\right)={\sigma }_{f}^{2}{\left[1+\frac{{\left(z-\tilde {z}\right)}^{2}}{2\alpha {l}_{f}^{2}}\right]}^{-\alpha },\end{equation} \tag{ 27 }$$

where σ_f , l_f and α are the kernel hyperparameters which relate the strength and scope of the correlations between the reconstructed data points respectively [101]. Here, the hyperparameters appear as constants and are called hyperparameters since their values point to the behaviour of the underlying function rather than a model that mimics this behaviour. The observational data being input into a GP then appears as a subset of Gaussian points that can be extended through the GP approach. This is achieved by maximizing the likelihood of the GP producing the observational data by estimating the underlying functional behaviour for different values of the hyperparameters for particular kernel instances.

Finally, the GP approach is model-independent in the context of a physical model and instead assumes a particular statistical kernel which dictates the correlation between the reconstructed points. In our analysis, we show that for Hubble data, this dependence is very weak and is almost totally model-independent for most intents and purposes, i.e. differing kernels produce the same results to a reasonable extent. In this sense, the GP approach is non-parametric in terms of physical models. In the following, we apply this approach to the case of Hubble data for several data sets.

We now apply the GP approach with the kernels specified in equations (25)–(27) to a number of different H(z) data sources, from which we reconstruct H₀. We do this using three principal sources of H(z) data, namely cosmic chronometers (CC), supernovae of type Ia (SN) and baryonic acoustic oscillation (BAO). CC are very efficient in obtaining H(z) data at redshifts of z ≲ 2 and do not rely on any cosmological models. They also avoid using Cepheids as distance scale indicators and are instead built on spectroscopic dating techniques. In this work, we only adopt those points in reference [102] that are independent of BAO observations in order to retain independence of cosmological models.

For the SN data, we use a combination of the compressed Pantheon compilation [103] together with the CANDELS and CLASH multi-cycle treasury data [104]. We use the Hubble rate parameter measurements of E(z) = H(z)/H₀ along with the corresponding correlation matrix, where only five of the reported six data points are adopted since the z = 1.5 data point is not Gaussian–distributed (similar to reference [94]). In order to incorporate the SN data set in our GP analyses, we make use of an iterative numerical procedure [94] to determine an H₀ value. We first infer an H₀ value by applying GP to the CC data set only, and then we promote the SN E(z) data points to the corresponding H(z) = H₀ E(z) values via a Monte Carlo routine. A number of successive GP reconstructions are applied on the combined CC + SN data set, until the resulting value of H₀ and its uncertainty converge to ≲10⁻⁴.

We also include eight BAO data points from the sloan digital sky survey (data release 12 and 14) as reported in references [105–107], along with the corresponding correlation matrices. We follow a similar iterative technique for the inclusion of the BAO data set as the one adopted for the SN measurements. While BAO measurements are not entirely independent from ΛCDM, particularly due to the assumption of a fiducial radius of the sound horizon r_d = 147.78  Mpc (adopted for all BAO data points), within this context, they add perspective to the growing tension in the value of H₀. We should remark that other well-constrained values of r_d have been considered, however the final results were not altered. A final important point is that whenever a cosmological model is used, it is always assumed to be flat which at these redshifts would have a very small impact in any case. However, the latest Planck 2018 (P18) results report a spatial curvature density which is very small at Ω_k (z = 0) = 0.001 ± 0.002 [14].

Recently there has been an increase in the reported value of H₀ due to the growing tension in its value against the predicted value using the ΛCDM model with the latest cosmic microwave background (CMB) data from the Planck mission [14]. The Planck collaboration report a low value of 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ while the dark energy survey collaboration give a similar value of $67.{4}_{-1.2}^{+1.1}\enspace {\mathrm{k}\mathrm{m}\enspace \mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ [108]. These predicted values of H₀ contrast with the cosmology independent estimates from late-time observations. The highest of these values is the Riess prior which is ${H}_{0}^{\mathrm{R}}=74.22{\pm}1.82\enspace {\mathrm{k}\mathrm{m}\enspace \mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ [12]. This value comes from long period observations of Cepheids in the Large Magellanic Cloud using the Hubble space telescope which has significantly reduced the uncertainty in this measurement. Another important recent announcement of H₀ is that from the H0LiCOW Collaboration [13] which uses strong lensing from quasars and gives a value of ${H}_{0}^{\text{HW}}=73.{3}_{-1.8}^{+1.7}\enspace {\mathrm{k}\mathrm{m}\enspace \mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$. We also consider measurements using the tip of the red giant branch as a standard candle where ${H}_{0}^{\text{TRGB}}=69.8{\pm}1.9\enspace {\mathrm{k}\mathrm{m}\enspace \mathrm{s}}^{-1}\enspace {\mathrm{M}\mathrm{p}\mathrm{c}}^{-1}$ [109]. While other measurements exist such as the novel approach of gravitational wave standard sirens [23], the aforementioned measurements are the most representative model-independent values that exist to reasonable uncertainty in the literature.

In the present study, we perform a number of GP analyses using a combination of two choices, the first being the prior which is selected from having no prior, or one of the ${H}_{0}^{\mathrm{R}},\enspace {H}_{0}^{\text{HW}},\enspace {H}_{0}^{\text{TRGB}}$ values, while the second involves the choice of GP kernel. GP are model-independent but they do depend on the kernel hyperparameters. Given the level of precision in the discrepancy in H₀, we consider the kernels in equations (25)–(27) in order to reduce any fine differences between these covariance functions on the estimation of H₀. To do this, we use a modified version of the public code GaPP (Gaussian processes in Python) ⁸ [101] which implements the GP approach using these kernels.

We now apply the GP approach explained in subsection 3.1 to the various sources of Hubble data together with the priors described here. The results pertaining to the value of H₀ are presented in tables 1–4 which contain the principal results for the square exponential, Cauchy, Matérn and rational quadratic kernels respectively. In each table, we present the GP inferred value of H₀ for the case of taking no prior, and the ${H}_{0}^{\mathrm{R}}$, ${H}_{0}^{\text{TRGB}}$ and ${H}_{0}^{\text{HW}}$ priors. In every case of this analysis, we determine the distance (in units of σ) between the GP determined value against the literature priors discussed above, so that this distance will be defined as

$$\begin{equation}d\left({H}_{0,i},{H}_{0,j}\right)=\frac{{H}_{0,i}-{H}_{0,j}}{\sqrt{{\sigma }_{i}^{2}+{\sigma }_{j}^{2}}},\end{equation} \tag{ 28 }$$

where H_0,i and H_0,j are two respective values of the present value of the Hubble parameter together with their respective 1σ uncertainties σ_i and σ_j .

Table 1. Different GP reconstructions of H₀ with the square exponential kernel function of equation (24). The reconstructed values of H₀ are complemented by their distance (in units of σ) from literature priors.

Data set(s)	H0	$d\left({H}_{0},{H}_{0}^{\mathrm{R}}\right)$	$d\left({H}_{0},{H}_{0}^{\text{TRGB}}\right)\hspace{2pt}$	$\hspace{2pt}d\left({H}_{0},{H}_{0}^{\text{HW}}\right)$	$\hspace{2pt}d\left({H}_{0},{H}_{0}^{\text{P}18}\right)$
CC	67.539 ± 4.7720	−1.3037	−0.4408	−1.1334	0.0290
CC + SN	67.001 ± 1.6531	−3.2253	−1.1183	−2.6165	-0.2309
CC + SN + BAO	66.197 ± 1.4639	−3.8407	−1.5127	−3.1132	-0.7776
CC + ${H}_{0}^{\mathrm{R}}$	73.782 ± 1.3743	−0.12556	1.7106	0.2166	4.3640
CC + SN + ${H}_{0}^{\mathrm{R}}$	72.022 ± 1.0756	−1.1271	1.0265	−0.6220	3.8969
CC + SN + BAO+${H}_{0}^{\mathrm{R}}$	71.180 ± 1.0245	−1.6279	0.6447	−1.0457	3.3155
CC + ${H}_{0}^{\text{TRGB}}$	69.604 ± 1.7557	−1.9599	−0.0760	−1.4908	1.2076
CC + SN + ${H}_{0}^{\text{TRGB}}$	68.468 ± 1.2212	−2.9695	−0.5942	−2.2641	0.8096
CC + SN + BAO + ${H}_{0}^{\text{TRGB}}$	67.811 ± 1.1470	−3.4070	−0.9036	−2.6233	0.3284
CC + ${H}_{0}^{\text{HW}}$	72.966 ± 1.6636	−0.4863	1.2617	−0.1382	3.2043
CC + SN + ${H}_{0}^{\text{HW}}$	70.850 ± 1.1991	−1.7111	0.4710	−1.1550	2.6555
CC + SN + BAO + ${H}_{0}^{\text{HW}}$	69.911 ± 1.1276	−2.2717	0.0506	−1.6280	2.0355

Table 2. Different GP reconstructions of H₀ with the Cauchy kernel function of equation (25). The reconstructed values of H₀ are complemented by their distance (in units of σ) from literature priors.

Data set(s)	H0	$d\left({H}_{0},{H}_{0}^{\mathrm{R}}\right)$	$d\left({H}_{0},{H}_{0}^{\text{TRGB}}\right)$	$\hspace{4pt}d\left({H}_{0},{H}_{0}^{\text{HW}}\right)$	$\hspace{2pt}d\left({H}_{0},{H}_{0}^{\text{P}18}\right)$
CC	69.396 ± 5.1862	−0.8618	−0.0732	−0.7132	0.3831
CC + SN	67.082 ± 1.6819	−3.1566	−1.0780	−2.5619	−0.1814
CC + SN + BAO	66.179 ± 1.4717	−3.8392	−1.5173	−3.1144	−0.7858
CC + ${H}_{0}^{\mathrm{R}}$	73.802 ± 1.3757	−0.1152	1.7187	0.2256	4.3738
CC + SN + ${H}_{0}^{\mathrm{R}}$	72.056 ± 1.0826	−1.1055	1.0404	−0.6045	3.9046
CC + SN + BAO + ${H}_{0}^{\mathrm{R}}$	71.166 ± 1.0279	−1.6340	0.6377	−1.0516	3.2943
CC + ${H}_{0}^{\text{TRGB}}$	69.695 ± 1.7603	−1.9168	−0.0408	−1.4524	1.2541
CC + SN + ${H}_{0}^{\text{TRGB}}$	68.508 ± 1.2327	−2.9366	−0.5749	−2.2386	0.8330
CC + SN + BAO + ${H}_{0}^{\text{TRGB}}$	67.796 ± 1.1512	−3.4101	−0.9094	−2.6275	0.3156
CC + ${H}_{0}^{\text{HW}}$	73.003 ± 1.6665	−0.4690	1.2755	−0.1228	3.2205
CC + SN + ${H}_{0}^{\text{HW}}$	70.892 ± 1.2087	−1.6830	0.4887	−1.1323	2.6695
CC + SN + BAO + ${H}_{0}^{\text{HW}}$	69.895 ± 1.1323	−2.2767	0.0434	−1.6335	2.0160

Table 3. Different GP reconstructions of H₀ with the Matérn kernel function of equation (26). The reconstructed values of H₀ are complemented by their distance (in units of σ) from literature priors.

Data set(s)	H0	$d\left({H}_{0},{H}_{0}^{\mathrm{R}}\right)$	$d\left({H}_{0},{H}_{0}^{\text{TRGB}}\right)$	$\hspace{2pt}d\left({H}_{0},{H}_{0}^{\text{HW}}\right)$	$\hspace{2pt}d\left({H}_{0},{H}_{0}^{\text{P}18}\right)$
CC	68.434 ± 5.0295	−1.0709	−0.2545	−0.9139	0.2045
CC + SN	66.981 ± 1.6798	−3.2049	−1.1187	−2.6052	−0.2393
CC + SN + BAO	66.139 ± 1.4729	−3.8572	−1.5337	−3.1310	−0.8110
CC + ${H}_{0}^{\mathrm{R}}$	73.777 ± 1.3760	−0.1279	1.7078	0.2143	4.3560
CC + SN + ${H}_{0}^{\mathrm{R}}$	72.016 ± 1.0821	−1.1279	1.0223	−0.6239	3.8728
CC + SN + BAO + ${H}_{0}^{\mathrm{R}}$	71.148 ± 1.0286	−1.6438	0.6292	−1.0603	3.2767
CC + ${H}_{0}^{\text{TRGB}}$	69.629 ± 1.7597	−1.9462	−0.0663	−1.4791	1.2186
CC + SN + ${H}_{0}^{\text{TRGB}}$	68.457 ± 1.2322	−2.964	−0.5975	−2.2626	0.7952
CC + SN + BAO + ${H}_{0}^{\text{TRGB}}$	67.772 ± 1.1521	−3.4226	−0.9203	−2.6386	0.2959
CC + ${H}_{0}^{\text{HW}}$	72.963 ± 1.6668	−0.4875	1.2592	−0.1396	3.1966
CC + SN + ${H}_{0}^{\text{HW}}$	70.840 ± 1.2081	−1.7108	0.4657	−1.1567	2.6312
CC + SN + BAO + ${H}_{0}^{\text{HW}}$	69.872 ± 1.1328	−2.2888	0.0330	−1.6442	1.9967

Table 4. Different GP reconstructions of H₀ with the rational quadratic kernel function of equation (27). The reconstructed values of H₀ are complemented by their distance (in units of σ) from literature priors.

Data set(s)	H0	$d\left({H}_{0},{H}_{0}^{\mathrm{R}}\right)$	$d\left({H}_{0},{H}_{0}^{\text{TRGB}}\right)$	$\hspace{3pt}d\left({H}_{0},{H}_{0}^{\text{HW}}\right)$	$\hspace{3pt}d\left({H}_{0},{H}_{0}^{\text{P}18}\right)$
CC	70.672 ± 5.4918	−0.5920	0.1502	−0.4560	0.5933
CC + SN	67.099 ± 1.6865	−3.1438	−1.0699	−2.5515	−0.1712
CC + SN + BAO	66.195 ± 1.4652	−3.8398	−1.5129	−3.1129	−0.7782
CC + ${H}_{0}^{\mathrm{R}}$	73.851 ± 1.3767	−0.0907	1.7391	0.2473	4.4042
CC + SN + ${H}_{0}^{\mathrm{R}}$	72.081 ± 1.0852	−1.0904	1.0514	−0.5918	3.9179
CC + SN + BAO + ${H}_{0}^{\mathrm{R}}$	71.620 ± 1.0851	−1.3486	0.8388	−0.8160	3.5320
CC + ${H}_{0}^{\text{TRGB}}$	69.782 ± 1.7636	−1.8761	−0.0070	−1.4159	1.2994
CC + SN + ${H}_{0}^{\text{TRGB}}$	68.523 ± 1.2349	−2.9263	−0.5679	−2.2303	0.8430
CC + SN + BAO + ${H}_{0}^{\text{TRGB}}$	67.810 ± 1.1469	−3.4074	−0.9039	−2.6237	0.3280
CC + ${H}_{0}^{\text{HW}}$	73.077 ± 1.6685	−0.4349	1.3042	−0.0921	3.2593
CC + SN+${H}_{0}^{\text{HW}}$	70.917 ± 1.2118	−1.6674	0.4997	−1.1194	2.6831
CC + SN + BAO + ${H}_{0}^{\text{HW}}$	69.911 ± 1.1280	−2.2716	0.0504	−1.6280	2.0348

The full results for each of the square exponential, Cauchy, Matérn and rational quadratic kernels are respectively presented in figures 1–4. These figures depict the wider GP reconstruction for the range within which the data appears. The GP approach is used for each set of priors for H₀ as shown in the sub-figures. Moreover, for every GP reconstruction, the 1σ and 2σ regions are shown. As a reference point, we present the ΛCDM behaviour in all instances. In addition to the reconstructions, each kernel GP is complemented by consistency tests for the ΛCDM model.

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There also exist diagnostic tools to assess preferences in the reconstructions towards deviations from ΛCDM [110, 111]. Considering the GR Friedmann equation for a cosmos filled with a dark fluid EoS w(z)

$$\begin{equation}\frac{{H}^{2}\left(z\right)}{{H}_{0}^{2}}={{\Omega}}_{m}^{0}{\left(1+z\right)}^{3}+{{\Omega}}_{k}^{0}{\left(1+z\right)}^{2}+{{\Omega}}_{{\Lambda}}^{0}\enspace \mathrm{exp}\left[3{\int }_{0}^{z}\frac{1+w\left({z}^{\prime }\right)}{1+{z}^{\prime }}\enspace \mathrm{d}{z}^{\prime }\right],\end{equation} \tag{ 29 }$$

which can be rearranged to reconstruct the EoS of the dark fluid via

$$\begin{equation}w\left(z\right)=\frac{2\left(1+z\right)E\left(z\right){E}^{\prime }\left(z\right)-3{E}^{2}\left(z\right)}{3\left[{E}^{2}\left(z\right)-{{\Omega}}_{\mathrm{m}}^{0}{\left(1+z\right)}^{3}\right]},\end{equation} \tag{ 30 }$$

where we have assumed spatial flatness. We report the GP reconstructions of w(z) in A, where it is clear that w = −1 is not excluded by the currently available data. However, we should point out that this reconstruction is dependent on the matter density parameter, which restricts us from constructing physical models. On the other hand, one could test the flat ΛCDM model by considering the following diagnostic redshift function

$$\begin{equation}{\mathcal{O}}_{\mathrm{m}}^{\left(1\right)}\left(z\right){:=}\frac{{E}^{2}\left(z\right)-1}{z\left(3+3z+{z}^{2}\right)},\end{equation} \tag{ 31 }$$

which reduces to ${\mathcal{O}}_{\mathrm{m}}^{\left(1\right)}\left(z\right)={{\Omega}}_{\mathrm{m}}^{0}$ in the ΛCDM scenario. We also report the GP reconstructions of ${\mathcal{O}}_{\mathrm{m}}^{\left(1\right)}\left(z\right)$ in appendix A, where one could clearly notice that the considered data sets are in a very good agreement with the ΛCDM predictions, although the H₀ prior has a significant effect on the reconstruction and hence the viability of the concordance model of cosmology. A more effective diagnostic is the vanishing of the derivative of ${\mathcal{O}}_{\mathrm{m}}^{\left(1\right)}\left(z\right)$, denoted by

$$\begin{equation}{\mathcal{L}}^{\left(1\right)}\left(z\right)=3\left(1-{E}^{2}\left(z\right)\right){\left(1+z\right)}^{2}+2z\left(3+3z+{z}^{2}\right)E\left(z\right){E}^{\prime }\left(z\right).\end{equation} \tag{ 32 }$$

Any deviation from ${\mathcal{L}}^{\left(1\right)}\left(z\right)=0$ represents a deviation from ΛCDM, which makes ${\mathcal{L}}^{\left(1\right)}\left(z\right)$ a good diagnostic over which to assess the behaviour of the concordance model. For the square exponential, Cauchy, Matérn and rational quadratic kernels used in this study, the ${\mathcal{L}}^{\left(1\right)}\left(z\right)$ diagnostic is presented for each reconstruction in figures 5–8. In the scenario where no prior is assumed, to a fairly consistent degree, the square exponential kernel produces lower values of H₀ while the CC data always produce higher reconstructed H₀ values which occur due to the data being placed at lower values of redshift. This is brought out in tables 1–4. However, in all cases, the results remain within the 1σ confidence levels across all the kernels for their respective reconstructions. Throughout, the ${H}_{0}^{\mathrm{R}}$ prior produces the highest values of H₀ with the rational quadratic giving the highest H₀ within that subset which is achieved when the CC data are taken alone. The impact of BAO data in all instances is to reduce the reconstructed value of H₀ which holds through for each of the priors and kernels. This property follows from the fact that BAO data appears at higher redshifts and acts to favour smaller uncertainties at those redshifts. The ${H}_{0}^{\text{HW}}$ prior produces similar but lower values for H₀ when compared with the ${H}_{0}^{\mathrm{R}}$ prior since their values are fairly close. The lowest inferred values of H₀ with a prior are found with the ${H}_{0}^{\text{TRGB}}$ prior, since this is the lowest of the three. In fact, it is the ${H}_{0}^{\text{TRGB}}$ prior that gives the lowest tension with the recent results by the Planck collaboration. This feature is brought out in the last column of tables 1–4 which illustrates the distance between the reconstructed H₀ and ${H}_{0}^{\text{P}18}$, which produces the lowest discrepancy for the TRGB setting.

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In figure 1, the square exponential kernel GP reconstructions are shown for the redshift range of the full data set. In all cases, the BAO data reduce the 1σ and 2σ uncertainties at higher redshifts since the other data sets do not feature points in that regime. In fact, in the cases of CC and CC + SN, the ΛCDM theoretical prediction only deviates into the 2σ uncertainty region for these high values of redshift, and only outside of both when the BAO data are included. The BAO data are dependent on the concordance model of cosmology, and so one would expect it to produce issues of this kind since the other two data sets are independent of cosmological models. The remainder of the reconstruction regions remain in close range of the ΛCDM prediction. This is further exposed by the diagnostic consistency test shown in figure 5 where a number of regions mark slight deviations from ΛCDM. However, it is the BAO data set that exposes this deviation at high redshifts. Another interesting feature of these diagnostic tests is that as with the H₀ reconstructions in table 1, the ${H}_{0}^{\mathrm{R}}$ prior brings about the largest deviation of the reconstructions.

Concerning the 1σ confidence regions, a similar picture unfolds for the GP Hubble reconstructions and diagnostic tests for the Cauchy, Matérn and rational quadratic kernels which are respectively shown in figures 2 and 6, figures 3 and 7 and figures 4 and 8. The resonance between these GP reconstructions over the kernel choices is a crucial property to the model independence of the analysis which shows that while dependent on a covariance model, the resulting Hubble parameter evolution is independent of a cosmological model.

In the above analyses, the GP approach was used to reconstruct the Hubble parameter history of the evolution of the Universe with a focus on the inferred value of H₀. However, another value of growing importance is that of the redshift value at which the Universe transitioned from a decelerating cosmos to its present state of acceleration, z_t [5]. This second probe of dark energy provides further details on its properties and possible information on its eventual interpretation which may include new physics beyond ΛCDM. Given that the transition occurs at a different point in the evolution of the Universe, this may also reveal any potential time dependence of dark energy, as well as the evolution of the ratio of matter to dark energy along the cosmic timeline [112]. In the ΛCDM model, for a flat FLRW cosmology, the transition redshift turns out as [113]

$$\begin{equation}{z}_{t}={\left[\frac{2\left(1-{{\Omega}}_{m}^{0}\right)}{{{\Omega}}_{m}^{0}}\right]}^{1/3}-1,\end{equation} \tag{ 33 }$$

which for P18 values gives z_t ≃ 0.63.

Given that we use the GP approach to reconstruct the Hubble parameter beyond the indicative ΛCDM value of the transition redshift, we can also make a determination of this value for each of the reconstructions that form part of the study up to this point. These results are reported in table 5 where, respectively, we give the reconstructed transition redshifts for the square exponential, Cauchy, Matérn and rational quadratic kernels for the various data sets and prior combinations. These values are inferred from the GP reconstructed deceleration parameter (which are illustrated in appendix A)

$$\begin{equation}q\left(z\right)=\left(1+z\right)\frac{{H}^{\prime }\left(z\right)}{H\left(z\right)}-1,\end{equation} \tag{ 34 }$$

from which the transition time is straightforwardly inferred.

Table 5. Values of z_t along with their corresponding 1σ uncertainties inferred via the GP reconstructions of q(z) with different data sets and kernel functions as specified in equations (24)–(27).

	zt
Data set(s)	Square exponential	Cauchy	Matérn	Rational quadratic
CC	$0.61{6}_{-0.137}^{+\mathrm{n}\mathrm{a}\mathrm{n}}$	$0.57{2}_{-0.118}^{+0.267}$	$0.59{1}_{-0.129}^{+\mathrm{n}\mathrm{a}\mathrm{n}}$	$0.55{3}_{-0.110}^{+0.211}$
CC + SN	$0.60{7}_{-0.084}^{+0.132}$	$0.60{0}_{-0.087}^{+0.132}$	$0.60{5}_{-0.089}^{+0.131}$	$0.59{8}_{-0.088}^{+0.133}$
CC + SN + BAO	$0.66{7}_{-0.075}^{+0.095}$	$0.65{8}_{-0.081}^{+0.105}$	$0.65{9}_{-0.082}^{+0.108}$	$0.66{4}_{-0.079}^{+0.101}$
CC + ${H}_{0}^{\mathrm{R}}$	$0.55{1}_{-0.079}^{+0.112}$	$0.54{0}_{-0.081}^{+0.114}$	$0.54{6}_{-0.082}^{+0.115}$	$0.52{8}_{-0.082}^{+0.114}$
CC + SN + ${H}_{0}^{\mathrm{R}}$	$0.70{2}_{-0.112}^{+0.246}$	$0.68{7}_{-0.114}^{+0.268}$	$0.69{4}_{-0.116}^{+0.263}$	$0.67{9}_{-0.114}^{+\mathrm{n}\mathrm{a}\mathrm{n}}$
CC + SN + BAO + ${H}_{0}^{\mathrm{R}}$	$0.78{3}_{-0.088}^{+0.107}$	$0.77{0}_{-0.095}^{+0.119}$	$0.77{2}_{-0.097}^{+0.123}$	$0.78{3}_{-0.095}^{+0.118}$
CC+${H}_{0}^{\text{TRGB}}$	$0.57{3}_{-0.096}^{+0.165}$	$0.55{2}_{-0.099}^{+0.161}$	$0.56{4}_{-0.102}^{+0.167}$	$0.53{7}_{-0.098}^{+0.159}$
CC + SN + ${H}_{0}^{\text{TRGB}}$	$0.63{3}_{-0.091}^{+0.149}$	$0.62{4}_{-0.094}^{+0.148}$	$0.62{9}_{-0.094}^{+0.148}$	$0.62{2}_{-0.094}^{+0.148}$
CC + SN + BAO + ${H}_{0}^{\text{TRGB}}$	$0.70{3}_{-0.079}^{+0.099}$	$0.69{3}_{-0.085}^{+0.109}$	$0.69{5}_{-0.087}^{+0.112}$	$0.69{9}_{-0.084}^{+0.106}$
CC + ${H}_{0}^{\text{HW}}$	$0.55{6}_{-0.083}^{+0.120}$	$0.54{4}_{-0.085}^{+0.122}$	$0.55{1}_{-0.087}^{+0.124}$	$0.53{1}_{-0.085}^{+0.121}$
CC + SN + ${H}_{0}^{\text{HW}}$	$0.67{9}_{-0.104}^{+0.196}$	$0.66{6}_{-0.107}^{+0.197}$	$0.67{1}_{-0.108}^{+0.200}$	$0.66{1}_{-0.107}^{+0.201}$
CC + SN + BAO + ${H}_{0}^{\text{HW}}$	$0.75{2}_{-0.085}^{+0.106}$	$0.74{0}_{-0.091}^{+0.116}$	$0.74{3}_{-0.094}^{+0.119}$	$0.74{5}_{-0.091}^{+0.113}$

Similar to the inferred values of H₀ reported in tables 1–4, the ${H}_{0}^{\mathrm{R}}$ prior produces the most extreme reconstructed parameter values of the transition redshift. In this case, the Riess prior produces the lowest values of z_t which results from the fact that a higher prior would imply a lower redshift turning point for the acceleration of the Universe. While this occurs consistently for all three respective data sets, the lowest transition redshift occurs for the CC data set since this produces the highest reconstructed H₀ value. To a lesser extent, the ${H}_{0}^{\text{HW}}$ prior produces the next lowest inferred z_t reconstructions followed by the ${H}_{0}^{\text{TRGB}}$ prior which is expected since ${H}_{0}^{\text{TRGB}}{< }{H}_{0}^{\text{HW}}$. Another important similarity with the H₀ reconstructions is that since the CC, CC + SN and CC + SN + BAO produce the leading values of H₀ in that order, then they will produce, by and large, the leading z_t in ascending order since the transition will occur closest to the present time. Finally, on the issue of uncertainties, since the ΛCDM-dependent BAO data sets are characterised by values of the Hubble parameter at the highest redshifts, they produce the lowest uncertainties, particularly with respect to the CC inferred errors.