[Paper Review] Modified Einstein-Cartan Gravity and its Implications for Cosmology

Wei Lu|arXiv (Cornell University)|Jun 29, 2014

Cosmology and Gravitation Theories12 references3 citations

TL;DR

This paper proposes a modified Einstein-Cartan gravity theory that breaks local Lorentz symmetry via a new torsion coupling, introducing a characteristic Hubble scale $h_0$ linked to the cosmological constant. The modified Friedmann equations predict late-time cosmic acceleration without a cosmological constant and eliminate the need for dark matter by enhancing baryonic density parameters at low Hubble rates.

ABSTRACT

We propose a modification of Einstein-Cartan gravity equations. The modified cosmology departs from the standard model of cosmology for small Hubble parameter. A characteristic Hubble scale h0, which is intrinsically related to cosmological constant, marks the boundary between the validity domains of the standard model of cosmology and modified cosmology. For large Hubble parameter, the standard model of cosmology is restored. In the opposite limit of small Hubble parameter, which is the case for present epoch, Lorentz-violating effects would manifest themselves. One of the implications is that there may be no need to invoke dark matter to account for cosmological mass discrepancies.

Motivation & Objective

• To develop a relativistic extension of MOND that preserves diffeomorphism invariance while breaking local Lorentz symmetry.
• To address cosmological mass discrepancies without invoking cold dark matter (CDM) by modifying the Friedmann equations.
• To explore whether late-time cosmic acceleration can emerge naturally from modified gravity without a cosmological constant.
• To link the MOND acceleration scale $a_0$ to the vacuum expectation value of the gravity gauge field and the cosmological constant.
• To derive a modified Hubble parameter $\tilde{H} \simeq (H/h_0)H$ that interpolates between standard and modified cosmology depending on $H \lesssim h_0$.

Proposed method

• Formulate gravity as a de Sitter gauge theory using Clifford-valued 1-forms for vierbein $e$ and spin connection $\omega$, with curvature $F = R + \frac{1}{l}T + \frac{1}{l^2}e^2$.
• Introduce a modified torsion coupling to spin current via a dimensionless parameter $\delta$, breaking local Lorentz symmetry while preserving diffeomorphism invariance.
• Derive modified Einstein-Cartan equations by varying the action $S = S_G + S_M$ with respect to $e$ and $\omega$, leading to $\tilde{H} \simeq (H/h_0)H$ in the Friedmann equations.
• Define a modified Hubble parameter $\tilde{H}$ such that $\tilde{H}^2 \simeq (H/h_0)^2 H^2$, which interpolates between standard and modified dynamics at low $H$.
• Update the Friedmann equations by replacing $H^2$ with $\tilde{H}^2$, resulting in modified energy density and acceleration terms dependent on $h_0$ and curvature $\kappa$.
• Analyze the dynamics in the limit $H \ll h_0$, showing that $\dot{a} \sim t^{1/3}$ implies late-time acceleration without $\Lambda$.

Experimental results

Research questions

• RQ1 Can a relativistic modification of Einstein-Cartan gravity explain galactic rotation curves without dark matter?
• RQ2 Does the introduction of a characteristic Hubble scale $h_0$ linked to the cosmological constant lead to a viable alternative to the standard model of cosmology?
• RQ3 Can late-time cosmic acceleration emerge in this model without a cosmological constant?
• RQ4 How does the modified Hubble parameter $\tilde{H} \simeq (H/h_0)H$ alter the Friedmann equations and the evolution of the scale factor $a(t)$?
• RQ5 To what extent can the modified baryonic density parameter $\tilde{\Omega}_b = (H^2 / \tilde{H}^2) \Omega_b$ account for observed mass discrepancies without invoking CDM?

Key findings

• The modified Friedmann equations predict a transition from decelerating to accelerating expansion when $H \ll h_0$, enabling late-time cosmic acceleration without a cosmological constant.
• For $H \gg h_0$, the standard model of cosmology is recovered, confirming consistency with early-universe dynamics.
• The present-day value of the factor $H^2 / \tilde{H}^2$ is estimated at 1.6 or 6.9 for $\alpha_T / \alpha_S = 1$ or $18$, respectively, enhancing baryonic density parameters.
• The characteristic Hubble scale $h_0 \simeq \frac{1}{18} \frac{\alpha_T}{\alpha_S} H_0$ links the MOND acceleration scale $a_0$ to the cosmological constant via the gravity gauge field VEV.
• Numerical simulations show that with $\kappa > 0$ and $\Lambda > 0$, the universe can undergo two cycles of deceleration and acceleration, one driven by matter and the other by curvature and $\Lambda$.
• The model explains galactic rotation curves via modified torsion without requiring cold dark matter, suggesting that cosmological mass discrepancies may be explained by modified gravity alone.