Alleviating the Hubble tension with Torsion Condensation (TorC)

Abstract

Constraints on the cosmological parameters of Torsion Condensation (TorC) are investigated using Planck 2018 Cosmic Microwave Background data. TorC is a case of Poincaré gauge theory -- a formulation of gravity motivated by the gauge field theories underlying fundamental forces in the standard model of particle physics. Unlike general relativity, TorC incorporates intrinsic torsion degrees of freedom while maintaining second-order field equations. At specific parameter values, it reduces to the $Λ$CDM model, providing a natural extension to standard cosmology. The base model of TorC introduces two parameters beyond those in $Λ$CDM: the initial value of the torsion scalar field and its time derivative -- one can absorb the latter by allowing the dark energy density to float. To constrain these parameters, `PolyChord` nested sampling algorithm is employed, interfaced via `Cobaya` with a modified version of `CAMB`. Our results indicate that TorC allows for a larger inferred Hubble constant, offering a potential resolution to the Hubble tension. Tension analysis using the $R$-statistic shows that TorC alleviates the statistical tension between the Planck 2018 and SH0Es 2020 datasets, though this improvement is not sufficient to decisively favour TorC over $Λ$CDM in a Bayesian model comparison. This study highlights TorC as a compelling theory of gravity, demonstrating its potential to address cosmological tensions and motivating further investigations of extended theories of gravity within a cosmological context. As current and upcoming surveys -- including Euclid, Roman Space Telescope, Vera C. Rubin Observatory, LISA, and Simons Observatory -- deliver data on gravity across all scales, they will offer critical tests of gravity models like TorC, making the present a pivotal moment for exploring extended theories of gravity.

Figures (13)

• Figure 1: A schematic illustration of curvature $\tensor{\mathcal{R}}{_{\mu\nu\rho\sigma}}$ and torsion $\tensor{\mathcal{T}}{^{\alpha}_{\mu\nu}}$. The left of the diagram demonstrates curvature as taught in most introductions to GR: In the presence of curvature, the direction of vector changes when parallely transported around a loop. The right of the diagram illustrates torsion: In the presence of torsion, the infinitesimal parallelogram spanned by two vectors does not close.
• Figure 2: The effect of $\Omega_\Lambda$ on the comoving Hubble horizon on the left and the CMB power spectrum on the right. The color bar indicates the value of $\Omega_\Lambda$ in both panels. For both plots, the value for $\varpi_\mathrm{r} \sim 1$ is set as the $\Lambda\mathrm{CDM}$-equivalent case. The cosmological parameters for computing CMB power spectra are set as the $\Lambda\mathrm{CDM}$ values based on Planck results Planck:2018vyg. For comparison, the case for GR (based on $\Omega_\mathrm{r} = 9.22 \times 10^{-5}$, $\Omega_\mathrm{m} = 0.314$ and $H_0 = 67.4$ km/s/Mpc) is shown in the plots. A smaller value of $\Omega_\Lambda$ results in a smaller comoving Hubble horizon at late matter-dominated epochs, increases the amplitude of the CMB power spectrum, and shifts the peaks to lower multipoles.
• Figure 3: The effect of $\varpi_\mathrm{r}$ on the Comoving Hubble Horizon on the left and the CMB power spectrum on the right. The color bar indicates the value of $\varpi_\mathrm{r}$ in both panels. For both plots, the value for $\Omega_\Lambda \sim 0.685$ is set as the $\Lambda\mathrm{CDM}$-equivalent case Planck:2018vyg. The other cosmological parameters are set as the values of the $\Lambda\mathrm{CDM}$ model. For comparison, the case for GR (based on $\Omega_\mathrm{r} = 9.22 \times 10^{-5}$, $\Omega_\mathrm{m} = 0.314$ and $H_0 = 67.4$ km/s/Mpc) is shown in the plots. A smaller value of $\varpi_\mathrm{r}$ results in a smaller comoving Hubble horizon at early radiation-dominated epoch, increases the amplitude of the CMB power spectrum, and shifts the peaks to higher multipoles.
• Figure 4: Evolution of density parameters and effective equations of state in $\Lambda\mathrm{CDM}$ and TorC cosmologies. The top panel shows the standard $\Lambda\mathrm{CDM}$ components: radiation (yellow), matter (red), and dark energy (purple). The second panel shows the corresponding quantities in TorC cosmology. The additional TorC effect appears as an extra component (blue), representing the torsion contribution to the total energy budget of the Universe. A grey dotted line marks the combined contribution of the standard components (radiation, matter, and dark energy). The third panel displays the evolution of the effective dark energy density parameter in TorC, combining the bare dark energy and torsion contributions. The fourth and fifth panels show the equations of state for the additional extra torsional component and effective dark energy component, respectively. Both exhibit radiation-like behaviour ($w=1/3$) at early times, while the effective dark energy transitions to a cosmological-constant-like behaviour ($w=-1$) at late times. Since we focus on the effect of $\varpi_\mathrm{r}$, no rescaling of the scale factor $a$ is applied between models. TorC parameters are set to $\Omega_\Lambda = 0.685$ and $\varpi_\mathrm{r} = 0.8$, and $\Lambda\mathrm{CDM}$ parameters follow the Planck 2018 results Planck:2018vyg.
• Figure 5: The figure shows the full posterior distributions of the cosmological parameters for $\Lambda\mathrm{CDM}$ (Planck 2018 data (purple)) and TorC (Planck 2018 data (yellow) and joint Planck+SH0ES data (red)). Nested sampling was performed with PolyChord using 1000 live points, and the corner plot was generated with anesthetic.