Two Fluid Quantum Bouncing Cosmology I: Theoretical Model

TL;DR

This work proposes a non-singular matter-radiation bounce in a two-fluid cosmology grounded in canonical quantum gravity, where radiation in the contracting phase induces a red-tilted curvature spectrum while gravity couples the two fluids to suppress isocurvature modes. Using a coupled adiabatic vacuum for the perturbations and a two-mode decomposition ( Mode 1 with radiation sound speed $c_r$ and Mode 2 with matter sound speed $c_w$), the authors derive the background dynamics, perturbation evolution through the bounce, and predictions for scalar, tensor, and isocurvature spectra. They show that the resulting curvature spectrum is nearly scale-invariant on CMB scales with a small red tilt, whereas isocurvature perturbations remain subdominant, and tensor modes are negligible at CMB wavelengths but could be detectable at smaller scales. The model aligns with late-time observations by matching the contracting and expanding branches at $|H|=H_0$ and embedding the primordial spectrum into standard Boltzmann codes, providing a viable non-inflationary alternative with potential implications for the Hubble tension and future gravitational-wave probes.

Abstract

Bouncing cosmologies offer an alternative to inflation by resolving the initial singularity through a contracting phase followed by a bounce into expansion. In many such models, the contracting phase is dominated by a single matter component, typically pressureless dust, which leads to an almost scale-invariant spectrum of scalar cosmological perturbations with a slight blue tilt, so that generating the observed red-tilted spectrum within this framework was challenging. In this work, we consider a more realistic scenario in which the contracting phase includes both matter and radiation, as required on physical grounds. We show that the presence of radiation can naturally induce a red tilt in the spectrum of curvature perturbations seeded by quantum vacuum fluctuations in the remote past of the contraction. Since the perturbations of the two fluids are coupled via gravity, vacuum initial conditions must be carefully defined. We demonstrate that, without fine-tuning, the resulting entropy perturbations are subdominant with respect to curvature perturbations. This suggests that a minimal two-component bounce model, involving only ordinary matter and radiation, can connect to the standard expanding cosmology with observationally viable initial conditions.

Figures (19)

• Figure 1: Evolution of the gravitational weights $R_{H0}\gamma_\text{r}$ (blue) and $R_{H0}\gamma_{w}$ (orange) as functions of the logarithmic time variable $\alpha$\ref{['log_time']}. The top axis shows the corresponding scale factor $x = a_0/a$. The green curve plots the Hubble-scaled wavenumber $F_\nu$\ref{['Fnu']} for $k = 1\,\mathrm{Mpc^{-1}}$, using the fiducial cosmology of Eq. \ref{['cosmo_bg']}. Shaded regions mark the radiation-dominated (light blue) and matter-dominated (light orange) eras. Vertical lines indicate key transitions: the matter-radiation equality (dashed) and the epoch when the Hubble radius matches its present value, $x=1$ (dotted). As anticipated from Eq. \ref{['Fnu']}, $F_\nu$ and $\gamma_\text{r}$ exhibit identical time dependence up to a constant factor.
• Figure 2: WKB scales $\Delta_{Xi}$ of Eq. \ref{['eq:adiabatic-scale']} for each component (Mode 1 left panel, Mode 2 right panel), multiplied by $c_i F_\nu$, as a function of the logarithmic time variable $\alpha$ defined by Eq. \ref{['log_time']} during the contraction epoch. The upper axis shows the corresponding scale factor $x = a_0/a$. Solid lines correspond to Mode 1 and dashed lines to Mode 2. Blue denotes $\zeta$, orange denotes $Q$, and green denotes the conjugate momenta $\Pi_\zeta$ and $\Pi_Q$. Shaded regions indicate the radiation-dominated (light blue) and matter-dominated (light orange) domains. The vertical dashed line marks the matter-radiation transition, Most ratios remain close to unity, except for $Q_1$ and $\Pi_{Q1}$, which rise from near zero to unity around the matter-radiation transition. The approach to unity indicates the regime where the simpler $c_i F_\nu$ correctly approximates the error.
• Figure 3: Evolution of multiple modes, each shown in a distinct color with comoving wavelength $\lambda$ indicated in the color bar (in Mpc). In both figures, the top panels show $\lambda$ for each mode compared with the comoving Hubble radius multiplied by the relevant sound speed: $c_i F_\nu/k = c_i x R_H$ for Mode 1 (radiation SHS) on the left, and $c_w x R_H$ for Mode 2 (matter SHS) on the right. The additional factor $k^{-1}$ separates the modes visually. The bottom panels in both figures show the evolution of the corresponding perturbation: $\zeta_1$ and $\zeta_2$ in the top figure, and $\Pi_{\zeta 1}$ and $\Pi_{\zeta 2}$ in the bottom figure. Vertical lines mark the times when $\lambda = c_i x R_H$, corresponding to sound-Hubble crossing ($c_i F_\nu = 1$), and help to identify the transition between sub- and super-Hubble behavior for each mode. To emphasize oscillations inside the Hubble radius (sub-Hubble regime), only the real part of the mode functions is plotted. In addition, we also plotted the WKB approximation for each mode using the same color but dashed lines. Deep inside this regime, computing exact mode functions is computationally expensive, so the adiabatic approximation is used instead, which smooths out oscillations and explains their absence in that region. The behavior of both $\Pi_{\zeta 1}$ and $\Pi_{\zeta 2}$ illustrate the approach to the super-SHS solution: Mode 1 reaches the constant $\Pi_\zeta$ regime soon after exiting WKB, while Mode 2 shows a delayed approach due to its larger effective sound speed ($c_\zeta \simeq 1/3$) relative to $c_{w} = \sqrt{w} \ll 1$. This contrast clarifies how the crossing of SHS regimes depends on the hierarchy between propagation speeds.
• Figure 4: Evolution of the velocity isocurvature perturbation modes $\Delta\zeta_i \propto Q_i$ (see Eq. \ref{['Smode']}), analogous to Fig. \ref{['fig:mode_evolution']} for the curvature perturbations $\zeta_i$. Top panels: comoving wavelength $\lambda$ for each mode compared with the comoving Hubble radius multiplied by the relevant sound speed, with the extra factor $k^{-1}$ separating the modes visually. Bottom panels: evolution of $\Delta\zeta_1$ and $\Delta\zeta_2$ across radiation- and matter-dominated eras. Vertical lines indicate sound-Hubble crossing ($\lambda = c_i x R_H$), and only the real part of the mode functions is shown; deep inside this regime the adiabatic approximation smooths out oscillations.
• Figure 5: Evolution of the total energy density contrast $\overline{\delta}_{\rho}$, analogous to the previous figures. Top panels: comoving wavelength $\lambda$ for each mode compared with the comoving Hubble radius multiplied by the relevant sound speed, with $k^{-1}$ separating the modes visually. Bottom panels: evolution of $\overline{\delta}_{\rho,1}$ and $\overline{\delta}_{\rho,2}$ across radiation- and matter-dominated eras. Vertical lines indicate sound-Hubble crossing ($\lambda = c_i x R_H$), and only the real part of the mode functions is shown; deep inside this regime the adiabatic approximation smooths out oscillations.