[Paper Review] Thermal fluctuations in Einstein-Cartan-Sciama-Kibble-Dirac bouncing cosmology

Nikodem J. Popławski|arXiv (Cornell University)|Dec 31, 2011

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates cosmological perturbations from thermal fluctuations in the Einstein-Cartan-Sciama-Kibble-Dirac bouncing cosmology, showing that only the Dirac spinor formulation of fermionic matter—unlike the spin fluid approximation—produces a scale-invariant spectrum of metric perturbations at the bounce. The key result is that the Dirac form yields a power spectrum with spectral index $ n_S = 1 $, matching observations, while the spin fluid model yields $ n_S = 0 $, which is inconsistent with data.

ABSTRACT

We study cosmological perturbations arising from thermal fluctuations in the big-bounce cosmology in the Einstein-Cartan-Sciama-Kibble theory of gravity. We show that such perturbations cannot have a scale-invariant spectrum if fermionic matter minimally coupled to the torsion tensor is macroscopically averaged as a spin fluid, but have a scale-invariant spectrum if the Dirac form of the spin tensor of the fermionic matter is used.

Motivation & Objective

• To determine whether thermal fluctuations at the big bounce in Einstein-Cartan-Sciama-Kibble-Dirac cosmology can generate a scale-invariant spectrum of cosmological perturbations.
• To compare the outcomes of two descriptions of fermionic matter: the spin fluid approximation and the exact Dirac spin tensor formulation.
• To assess whether the resulting power spectrum of metric perturbations matches the nearly scale-invariant spectrum observed in the cosmic microwave background.
• To evaluate the role of torsion-induced gravitational repulsion in enabling a nonsingular bounce and its impact on perturbation dynamics.
• To determine the conditions under which thermal fluctuations in the contracting phase lead to viable initial conditions for large-scale structure formation.

Proposed method

• Model the early Universe as a bouncing cosmology within the Einstein-Cartan-Sciama-Kibble (ECSK) theory, where torsion arises from fermionic spin density and avoids singularities.
• Use the longitudinal gauge for linearized perturbations around a Friedmann-Lemaître-Robertson-Walker background, with the Bardeen potential $ \Phi $ describing metric fluctuations.
• Derive the evolution equation for the Fourier-mode $ \Phi_k $, incorporating the Hubble parameter $ \mathcal{H} $, comoving wavenumber $ k $, and effective sound speed $ c_s $, under the assumption of horizon exit dominance.
• Compute the power spectrum $ P_\Phi(k) $ using the relation $ P_\Phi(k) = \frac{1}{4M_\mathrm{P}^4 H^4} \langle \delta\rho^2 \rangle $, where $ \langle \delta\rho^2 \rangle $ is derived from thermal energy-density correlations in a sphere of radius $ R(k) $.
• Apply the thermal correlation function $ \langle \delta\rho^2 \rangle \propto C_V(R) T^2 / R^6 $, with $ C_V(R) = R^3 \partial\rho / \partial T $, to compute the energy fluctuation spectrum.
• Compare the resulting power spectrum for two fermionic matter models: the spin fluid (particle approximation) and the Dirac spin tensor (exact field-theoretic formulation), focusing on the spectral index $ n_S $.

Experimental results

Research questions

• RQ1 Can thermal fluctuations at the bounce in the ECSK bouncing cosmology produce a scale-invariant spectrum of cosmological perturbations?
• RQ2 How does the choice between the spin fluid and Dirac spinor description of fermionic matter affect the resulting power spectrum of metric perturbations?
• RQ3 What is the spectral index $ n_S $ of the metric perturbation power spectrum when thermal fluctuations are used as initial conditions in the Dirac fermion model?
• RQ4 Why does the spin fluid model fail to produce a scale-invariant spectrum despite generating a nonsingular bounce?
• RQ5 How does the conformal time of horizon exit differ between the spin fluid and Dirac fermion models, and what is its impact on the final power spectrum?

Key findings

• The spin fluid approximation of fermionic matter coupled to torsion in ECSK gravity produces a power spectrum with spectral index $ n_S = 0 $, which is not scale-invariant and inconsistent with observations.
• In contrast, the Dirac spinor formulation of fermionic matter yields a power spectrum with $ n_S = 1 $, matching the nearly scale-invariant spectrum observed in the cosmic microwave background.
• The Dirac model leads to a bounce where $ \dot{a} $ jumps from $ -v $ to $ v $, causing all perturbation modes to exit the horizon simultaneously at the bounce conformal time, resulting in a $ k $-independent power spectrum.
• The spin fluid model results in horizon exit times that depend on $ k $, leading to a $ k $-dependent spectrum with $ n_S = 0 $, which fails to reproduce the observed $ n_S \approx 1 $.
• The thermal fluctuation correlation function $ \langle \delta\rho^2 \rangle \propto T^2 / R^6 $ is used to compute the energy density variance, with heat capacity $ C_V(R) = R^3 \partial\rho / \partial T $, critical for deriving the spectrum.
• The final power spectrum for the Dirac model is $ P_\Phi(k) \propto (M_\mathrm{P} a_0)^{-4} \eta_S^{-5} k^{-1} $, which corresponds to $ n_S = 1 $, confirming a scale-invariant result.