Inflation without an Inflaton II: observational predictions

TL;DR

This paper investigates inflation without an inflaton by treating the expansion as driven by a pure de Sitter background and generating scalar fluctuations at second order from tensor perturbations. By deriving and numerically integrating the full second-order Einstein kernel, the authors obtain a scale-invariant scalar power spectrum, linking its amplitude to the inflationary scale $H_{\rm inf}$ as a function of the observable e-folds $N_{\rm obs}$. Matching the observed scalar amplitude at the CMB pivot scale $k_*$ yields predictions for $H_{\rm inf}$ and the tensor-to-scalar ratio $r$, e.g., $H_{\rm inf} \sim 3\times 10^{13}$ GeV with $N_{\rm obs}=30$ and $r\sim 10^{-2}$, or $H_{\rm inf} \sim 2\times 10^{10}$ GeV with $N_{\rm obs}=60$ and $r\sim 5\times 10^{-9}$. Incorporating the quantum break-time of de Sitter space imposes an upper bound on the number of particle species, yielding $N_{\rm obs} \lesssim 126$ for typical species counts, thereby making the IWI scenario predictive and testable against standard inflaton-driven models. The work connects the observed primordial fluctuations to the quantum properties and finite lifetime of de Sitter space, offering a model-independent path to inflation with distinctive observational signatures, including a potentially small $r$ and a tilt that can be explored in future studies.

Abstract

We present a complete computation of the scalar power spectrum in the \emph{inflation without inflaton} (IWI) framework, where the inflationary expansion is driven solely by a de~Sitter (dS) background and scalar fluctuations arise as second-order effects sourced by tensor perturbations. By explicitly deriving and numerically integrating the full second-order kernel of the Einstein equations, we obtain a scale-invariant scalar spectrum without invoking a fundamental scalar field. In this framework, the amplitude of the scalar fluctuations is directly linked to the scale of inflation. More precisely, we show that matching the observed level of scalar fluctuations, $Δ_φ^2(k_\ast)\approx 10^{-9}$ at Cosmic Microwave Background (CMB) scales, fixes the inflationary energy scale $H_{\rm inf}$ as a function of the number of observed e-folds $N_{\rm obs}$. For $N_{\rm obs}\simeq 30 - 60$, we find $H_{\rm inf} \simeq 5\times 10^{13}\,\mathrm{GeV} - 2\times 10^{10}\,\mathrm{GeV}$, corresponding to a tensor-to-scalar ratio $r \simeq 0.01 - 5\times 10^{-9}$. In particular, requiring consistency with instantaneous reheating, we predict a number of e-folds of order~$\mathcal{O}(50)$ and an inflationary scale $H_{\rm inf} \simeq 10^{11}\,\mathrm{GeV}$. We also incorporate in our framework the quantum break-time of the dS state and show that it imposes an upper bound on the number of particle species. Specifically, using laboratory constraints on the number of species limits the duration of inflation to $N_{\rm obs}\lesssim 126$ e-folds. These results establish the IWI scenario as a predictive and falsifiable alternative to standard inflaton-driven models, linking the observed amplitude of primordial fluctuations directly to the quantum nature and finite lifetime of dS space.

Figures (6)

• Figure 1: Energy scale of inflation $H_{\rm inf}(\rm GeV)$ and the tensor-to-scalar ratio $r(k_\ast)$ as a function of $N_{\rm obs}$ used in the numerical integration of Eq. \ref{['eq: dimensionless power spectrum to compute']}, when implementing the super-horizon requirement and stopping the integration at $x_{\rm max}= e^{N_{\rm obs}}$ as explained in the main text. We show as a gray shaded area the region excluded since it gives a tensor-to-scalar ratio inconsistent with current upper bounds at the CMB pivot scale $k_\ast$, i.e. $r(k_\ast)\gtrsim0.07$Planck:2018jri.
• Figure 2: Dimensionless scalar and tensor power spectra as a function of the external wavenumber $k$. $\Delta^2_\phi(k)$ is shown as a solid black line and it matches the observed level of scalar fluctuations at CMB scales Planck:2018jriPlanck:2018vyg. As a dashed blue line, we plot $\Delta^2_h(k)$ for $N_{\rm obs}=30$. In this case, we fix the energy scale to $H_{\rm inf}\simeq~3\times~10^{13}\,\rm{GeV}$ to match Planck constraints at $k_\ast$Planck:2018jriPlanck:2018vyg, as explained in the main text. Thus, the tensor-to-scalar ratio results $r(k_\ast)\simeq0.01$. The dotted red line shows $\Delta^2_h(k)$ for the case $N_{\rm obs}=60$, in which $H_{\rm inf}\simeq2\times10^{10}\,\rm{GeV}$ and $r(k_\ast)\simeq5\times10^{-9}$.
• Figure 3: Energy scale of inflation $H_{\rm inf}(\rm GeV)$ as a function of the minimum number of e-folds $N_{\rm obs}$. The solid black line represents our result from the numerical integration of Eq. \ref{['eq: dimensionless power spectrum to compute']} (same as Figure \ref{['fig: Hinf VS Nmin']}). We plot the relation between $H_{\rm inf}$ and $N_{\rm obs}$ given by Eq. \ref{['eq: upper bound particle species']} for different numbers of particle species $\mathcal{N}^{\rm max}_{\rm sp}\simeq5\times10^{9},9\times10^{15}$ with a dashed blue line and a dotted red line, respectively. By construction, the lines cross at $[N_{\rm obs},H_{\rm inf}(\mathrm{GeV})]\simeq[30, 3\times10^{13}\;\rm GeV]$ if $\mathcal{N}^{\rm max}_{\rm sp}\simeq~5\times10^{9}$, and at $[N_{\rm obs},H_{\rm inf}(\mathrm{GeV})]\simeq[60, 2\times10^{10}\;\rm GeV]$ if $\mathcal{N}^{\rm max}_{\rm sp}\simeq~9\times~10^{15}$.
• Figure 4: Dimensionless scalar and tensor power spectra as a function of the energy scale $H_{\rm inf}$. We fix $N_{\rm obs}$ and we obtain the scalar power spectrum $\Delta_\phi^2(k_\ast)$ from Eq. \ref{['eq: dimensionless power spectrum to compute']} for different values of $H_{\rm inf}$. As a dashed blue line, we show $\Delta_\phi^2(k_\ast)$ for $N_{\rm obs}=30$, as a dotted blue line, we plot $\Delta_\phi^2(k_\ast)$ for $N_{\rm obs}=60$, while the solid red line is the result for the tensor power spectrum $\Delta_h^2(k_\ast)$. We set the pivot scale to $k_\ast=~0.01\;\rm Mpc^{-1}$. The blue points correspond to the scalar amplitude $\Delta^2_\phi(k_{\rm CMB}=k_\ast)\simeq2.1\times10^{-9}$ constrained by PlanckPlanck:2018jriPlanck:2018vyg. For the case $N_{\rm obs}=30$, $\Delta^2_\phi(k_{\rm CMB})$ corresponds to $H_{\rm inf}\simeq3\times10^{13}\;\rm GeV$, therefore the tensor power spectrum and the tensor-to-scalar ratio are $\Delta^2_h(k_\ast)\simeq~3\times~10^{-11}$ and $r(k_\ast)\simeq0.01$, respectively. For the case $N_{\rm obs}=60$, $\Delta^2_\phi(k_{\rm CMB})$ corresponds to $H_{\rm inf}\simeq2\times10^{10}\;\rm GeV$, therefore $\Delta^2_h(k_\ast)\simeq 10^{-17}$, and $r(k_\ast)\simeq5\times10^{-9}$.
• Figure 5: Energy scale of inflation $H_{\rm inf}(\rm GeV)$ as a function of the minimum number of e-folds $N_{\rm obs}$. The solid black line represents our result from the numerical integration of Eq. \ref{['eq: dimensionless power spectrum to compute']}. We plot as a dotted red line the number of e-foldings for the case of instantaneous reheating, i.e., Eq. \ref{['eq: efolds instantaneous reheating']}. This crosses our result at $[N_{\rm obs},H_{\rm inf}(\mathrm{GeV})]\simeq[53, 10^{11}\;\rm GeV]$ and corresponds to a maximum number of particle species of $\mathcal{N}^{\rm max}_{\rm sp}\simeq3\times10^{14}$ using Eq. \ref{['eq: upper bound particle species']} (dashed blue line).