Evading the BBN bound with a soft stiff period

TL;DR

Post-inflationary stiff periods can boost primordial gravitational waves but risk violating BBN bounds on the GW energy density. The authors propose softening the stiff phase via a waterfall-field–driven dynamics in a modified hybrid inflation model, producing a gradually varying EOS $w(t)$ that yields a characteristic rounded peak in the GW spectrum while satisfying BBN and $ ext{Δ}N_{ ext{eff}}$ constraints. They solve the background evolution with efolds $N$ and the GW mode equations, computing $ ext{Ω}_{GW}(f)$ today including relativistic-degree-of-freedom corrections and BBN checks. A representative parameter choice yields a GW spectrum overlapping with the sensitivities of upcoming detectors like the Einstein Telescope and Cosmic Explorer, indicating a viable observational window into the post‑inflationary EOS. The work demonstrates a theoretically motivated, testable path to probe the early Universe's EOS without spoiling standard cosmology.

Abstract

Cosmic inflation is the leading theory to explain early Universe history and structure formation. Non-oscillatory inflation is a class of models which can naturally introduce a post-inflationary stiff period of the Universe's evolution which boosts the signal of primordial gravitational waves (GWs), making it possible to observe them in forthcoming GW experiments. However, this pushes the GW energy density high enough to destabilise the process of Big Bang Nucleosynthesis (BBN). This problem can be overcome by "softening" the stiff period, so that the field is gradually tending towards freefall from a frozen start. Here, we consider a modified hybrid inflation model where the stiff period is driven by the waterfall field, allowing the barotropic parameter of the Universe to vary, so that it does not violate the BBN constraint but produces a characteristic gravitational wave spectrum soon to be observable.

Figures (9)

• Figure 1: Conceptual sketch of how the $\Omega_{GW}$ spectrum behaves when subject to a scalar-field-driven period directly after inflation, with different barotropic parameters. Kination (blue) has the steepest proportionality and violates the BBN constraint if it continues for too long, so even if kination does occur, it is unlikely to be observable. A stiff period of barotropic parameter $w=\frac{1}{2}$ (orange) does not violate BBN, but is difficult to observe at lower frequencies. A period of slowly varying barotropic parameter (green) is expected to form a curved peak. This makes the spectrum more accessible to future observations. Here the variation in $w$ is modelled as linear in $\log(f)$ for illustrative purposes. The purple dashed line shows an example of how an instrumental observation curve can overlap with this slowly varying stiff period but not a stiff period of constant barotropic parameter. At the low-f end of the spectrum, the spectrum rises again due to the matter-dominated period at late times, and will eventually turn downward once more during the $\Lambda$-dominated era, forming a second peak.
• Figure 2: Schematic representation of the $V(\phi)$ potential given in Eq. \ref{['V']}. The units appearing in the graph are fiducial.
• Figure 3: Schematic representation of the $W(\phi,\sigma)$ potential. For $\sigma>0$, the system is driven inside a deep valley with $\phi=0$ as dictated by the potential $\Delta V$. The valley is characterised by a gentle slope given by $V(\sigma)$, which brings the system towards $\sigma=0$. There, the valley opens up and the system slides along the $\phi$-direction, following the potential $V(\phi)$ given in Eq. \ref{['V']}. The units appearing in the graph are fiducial.
• Figure 5: The barotropic parameter of the background (radiation, matter and $\Lambda$) of the Universe (blue), the Universe overall (orange), and the field $\phi$ (green), for the point with model parameters $\lambda = 72$, $\kappa=0.0226$, and $\log_{10}(\rho_{r}^{\textnormal{ini}}/m_{\text{P}}^{4})=-15.25$, as a function of elapsing efolds.
• Figure 6: The density parameters of the field $\phi$ (blue), the cosmological constant $\Lambda$ (orange), matter (green) and radiation (red) for the point model parameters $\lambda = 72$, $\kappa=0.0226$, and $\log_{10}(\rho_{r}^{\textnormal{ini}}/m_{\text{P}}^{4})=-15.25$, as a function of elapsing efolds.