[Paper Review] Impact of Lorentz Violation on Cosmology

Jiro Soda, Sugumi Kanno|ArXiv.org|Dec 12, 2006

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper proposes Lorentz-violating gravitational models to address cosmological acceleration in both early (inflation) and late-time (current) universe scenarios. By introducing a preferred frame via a dynamical vector field and modifying gravity with curvature and derivative terms, the model enables stable, ghost-free late-time acceleration in DGP-type braneworld models, offering a potential resolution to the ghost problem while predicting flat primordial spectra in Lorentz-violating inflation.

ABSTRACT

We discuss the impact of Lorentz violation on the cosmology. Firstly, we show that the Lorentz violation affects the dynamics of the chaotic inflationary model and gives rise to an interesting feature. Secondly, we propose the Lorentz violating DGP brane models where the Lorentz violating terms on the brane accelerate the current universe. We conjecture that the ghost disappears in the Lorentz violating DGP models.

Motivation & Objective

• To investigate the impact of Lorentz symmetry violation on cosmological dynamics, particularly in inflation and current cosmic acceleration.
• To address the instability of the DGP braneworld model, which suffers from a ghost mode, by introducing Lorentz-violating terms on the brane.
• To explore whether Lorentz violation can stabilize the DGP model and eliminate the ghost while still producing late-time acceleration.
• To examine how Lorentz violation modifies chaotic inflation, especially the dynamics and power spectrum of primordial perturbations.
• To assess the potential observational signatures of Lorentz violation in the CMB, such as anomalies in polarization or tensor-scalar ratios.

Proposed method

• Introduce a spontaneously broken Lorentz symmetry via a timelike vector field $u^\mu$ with $\langle u^\mu u_\mu \rangle = -1$, defining a preferred frame aligned with the CMB.
• Construct a generalized gravitational action including curvature terms $K^{ij}K_{ij}$, $K^2$, and derivative terms of $u^\mu$ with field-dependent coupling functions $\beta_i(\phi)$, $\gamma_i(\phi)$, and a Lagrange multiplier $\lambda(u^\mu u_\mu + 1)$.
• Apply the model to the chaotic inflation scenario by setting $\beta = \xi\phi^2$, $V = \frac{1}{2}m^2\phi^2$, and derive modified Friedmann equations in a homogeneous, isotropic spacetime.
• Analyze the DGP braneworld model by adding Lorentz-violating terms $\beta_1 K^{ij}K_{ij} - \beta_2 K^2$ and derivative terms in $u^\mu$, leading to modified junction conditions and effective Friedmann equations.
• Use the junction condition $K_{\mu\nu} - g_{\mu\nu}K = \kappa_5^2 T_{\mu\nu}$ to derive the effective 4D Friedmann equation, showing that $\beta_1$ and $\beta_2$ can induce late-time de Sitter expansion.
• Study linear perturbations in the Lorentz-violating inflation regime, deriving the quadratic action for tensor modes and showing a flat power spectrum despite rolling inflaton.

Experimental results

Research questions

• RQ1 How does Lorentz violation modify the dynamics of chaotic inflation, particularly in terms of the Hubble parameter and scalar field evolution?
• RQ2 Can Lorentz-violating terms on the brane generate late-time acceleration in a DGP-type braneworld model without introducing a ghost?
• RQ3 What is the critical condition for Lorentz violation to become relevant during inflation, and how does it affect the e-fold number and power spectrum?
• RQ4 Does the inclusion of Lorentz-violating derivative terms in the brane action alter the ghost structure of the DGP model, potentially stabilizing it?
• RQ5 What observational imprints could Lorentz violation leave on the CMB, such as in the tensor-scalar ratio or polarization spectrum?

Key findings

• In the Lorentz-violating inflation model with $\beta = \xi\phi^2$, the condition $\xi > 1/(72\pi) \sim 1/226$ ensures Lorentz violation is relevant when $\phi_i \sim 3M_{\text{pl}}$, modifying the standard inflationary dynamics.
• During the Lorentz-violating inflationary phase, the Hubble parameter remains approximately constant, leading to a flat primordial power spectrum despite the inflaton rolling down the potential.
• The tensor perturbation spectrum is completely flat in the Lorentz-violating regime, a direct prediction of the model independent of the potential shape.
• In the proposed Lorentz-violating DGP model, the effective Friedmann equation $H = \frac{2}{\kappa_5^2 \beta}$ yields a de Sitter solution at late times, indicating stable late-time acceleration.
• The model suggests that the ghost may be absent in Lorentz-violating DGP models due to the modified tensor structure, offering a potential resolution to the ghost problem.
• The vector modes of perturbations may survive to the last scattering surface, potentially imprinting detectable signatures on the CMB polarization spectrum.