Untangling PBH overproduction in $w$-SIGWs generated by Pulsar Timing Arrays for MST-EFT of single field inflation

TL;DR

The paper addresses PBH overproduction in scalar-induced gravitational waves (SIGWs) inferred from Pulsar Timing Array data within a single-field MST-EFT inflation framework. It develops a rigorous renormalization- and DRG-resummation-based treatment of the tree-level and one-loop scalar power spectrum across six sharp transitions, ensuring perturbativity and yielding a broad PBH mass range. It finds that a radiation-like equation of state $w=1/3$ with $1 \leq c_s \leq 1.17$ best fits NANOGrav-15 data while avoiding PBH overproduction, and it demonstrates that large PBHs with ${M_{\rm PBH} \sim {\cal O}(10^{-6}-10^{-3}) M_{\odot}}$ can arise without conflicting with microlensing constraints. The work also connects DRG resummation with the $\delta N$ formalism, compares sharp and smooth transition pictures, and provides predictions for the SIGW spectrum across PTA-relevant frequencies, with implications for dark matter candidates and future GW observations.

Abstract

Our work highlights the crucial role played by the equation of state (EoS) parameter $w$ within the context of single field inflation with Multiple Sharp Transitions (MSTs) to untangle the current state of the PBH overproduction issue. We examine the situation for a broad interval of EoS parameter that remains most favourable to explain the recent data released by the pulsar timing array (PTA) collaboration. Our analysis yields the interval, $0.2 \leq w \leq 1/3$, to be the most acceptable window from the SIGW interpretation of the PTA signal and where sizeable PBHs abundance, $f_{\rm PBH} \in (10^{-3},1)$, is observed. We also obtain $w=1/3$, radiation-dominated era, to be the best scenario to explain the early stages of the Universe and address the overproduction problem. Within the range of $1 \leq c_{s} \leq 1.17$, we construct a regularized-renormalized-resummed scalar power spectrum whose amplitude obeys the perturbativity criterion while being substantial enough to generate EoS dependent scalar induced gravitational waves ($w$-SIGWs) consistent with NANOGrav-15 data. Working for both $c_{s} = 1\;{\rm and}\;1.17$, we find the $c_{s}=1.17$ case more favourable for generating large mass PBHs, $M_{\rm PBH}\sim {\cal O}(10^{-6}-10^{-3})M_{\odot}$, as potential dark matter candidates with substantial abundance after constraints coming from microlensing experiments.

Figures (13)

• Figure 1: This branching diagram illustrates the crucial issues addressed in our research, showcasing the interconnected relationships and various dimensions of the set-up of Multiple Sharp Transitions. Each branch delves into specific aspects, contributing to a holistic understanding while discussing these prime subjects in detail.
• Figure 2: Representative diagram highlighting the equivalence between the DRG resummation approach and the $\delta{\cal N}$ formalism.
• Figure 3: The one-loop corrected renormalized and DRG-resummed scalar power spectrum with $c_s =1$ represented by red line and $c_s = 1.17$ by blue line. The ash shaded region displays the allowed parameter space from the theory satisfying $1 \leqslant c_{s} \leqslant 1.17$.
• Figure 4: Figure depicts the variation of amplitude $A$ of the one-loop renormalized and DRG-resummed scalar power spectrum with the transition wavenumber, which avoids PBH overproduction in the MST setup. The effective sound speed parameter is fixed to take $c_{s}=1$. Red and blue lines enclose the region of sizeable abundance $f_{\rm PBH} \in (1,10^{-3})$. Orange, Magenta, Cyan, and Brown colours denote the specific regions related to the EoS parameter $w \in \{1/3,0.2,0,-0.01\}$. The posteriors for the NANOGrav 15 and EPTA data, represented in the green and light-blue colors, respectively, are taken from Franciolini:2023pbf.
• Figure 5: Figure depicts the variation of amplitude $A$ of the one-loop renormalized and DRG-resummed scalar power spectrum with the transition wavenumber, which avoids PBH overproduction in the MST setup. The effective sound speed parameter is fixed to take $c_{s}=1.17$. Red and blue lines enclose the region of sizeable abundance $f_{\rm PBH} \in (1,10^{-3})$. Orange, Magenta, Cyan, and Brown colours denote the specific regions related to the EoS parameter $w \in \{1/3,0.2,0,-0.01\}$. The posteriors for the NANOGrav 15 and EPTA data, represented in the green and light-blue colors, respectively, are taken from Franciolini:2023pbf.