Complementary signatures of $α-$attractor inflation in CMB and cosmic string Gravitational Waves

TL;DR

This work links α-attractor inflationary models to observable signatures from cosmic strings formed during inflation, showing that the SGWB from string loops encodes the strings' inflationary history via a turning-point frequency $f_{\Delta}$ and the number of e-folds they experience, $N_{CS}$. Using the Velocity-dependent One-Scale (VOS) framework and turning-point analysis, the authors connect CMB observables $n_s$ and $r$ to GW signatures, focusing on T-Model and Polynomial α-attractors. They derive explicit complementarity relations for local and global cosmic strings, demonstrating that GW detectors such as LISA, SKA, and ET can probe $N_{CS}$ ranges that map to distinctive $n_s$–$r$ windows, thereby breaking degeneracies present in CMB data alone. The results outline concrete, testable predictions for joint CMB-GW analyses and offer guidance for upcoming CMB experiments like LiteBIRD and CMB-S4 in concert with future GW observations.

Abstract

When cosmic strings are formed during inflation, they regrow to reach a scaling regime, leaving distinct imprints on the stochastic gravitational wave background (SGWB). Such signatures, associated with specific primordial features, can be detected by upcoming gravitational wave observatories, such as the LISA and Einstein Telescope (ET). Our analysis explores scenarios in which cosmic strings form either before or during inflation. We examine how the number of e-folds experienced by cosmic strings during inflation correlates with the predictions of inflationary models observable in cosmic microwave background (CMB) measurements. This correlation provides a testable link between inflationary physics and the associated gravitational wave signals in a complementary manner. Focusing on $α$-attractor models of inflation, with the Polynomial $α$-attractor serving as an illustrative example, we find constraints, for instance, on the spectral index $n_s$ to $0.962 \lesssim n_s \lesssim 0.972$ for polynomial exponent $n=1$, $0.956 \lesssim n_s \lesssim 0.968$ for $n=2$, $0.954 \lesssim n_s \lesssim 0.965$ for $n=3$, and $0.963 \lesssim n_s \lesssim 0.964$ for $n=4$, which along with the GW signals from LISA, are capable of detecting local cosmic strings that have experienced $\sim 34 - 47$ e-folds of inflation consistent with current Planck data and are also testable in upcoming CMB experiments such as LiteBIRD and CMB-S4.

Figures (14)

• Figure 1: Potential profiles of the E-model $\alpha$-attractors given by Eq. (\ref{['E-Model eq1']}), illustrating the dependence on the parameters $\alpha$ and $n$. (a) The potential $V(\phi)/V_0$ is plotted as a function of $\phi/M_{Pl}$ for fixed $n=1$ and varying $\alpha$. (b) The potential is shown for fixed $\alpha=1$ and varying $n$. These plots highlight the sensitivity of the inflationary potential's shape to changes in $\alpha$ and $n$ - the key parameters in the $\alpha$-attractor framework.
• Figure 2: Potential profiles of the T-model of $\alpha$-attractors, defined in Eq. (\ref{['T-Model eq1']}). (a) The potential $V(\phi)/V_0$ is plotted as a function of $\phi/M_{Pl}$ for fixed $n=1$ and varying $\alpha$. (b) The potential is shown for fixed $\alpha=1$ and varying $n$.
• Figure 3: Potential profiles of the simplest Polynomial $\alpha$-attractor, defined in Eq. (\ref{['Poly Eq1']}). (a) The potential $V(\phi)/V_0$ is plotted as a function of $\phi/M_{Pl}$ for fixed $n=1$ and varying $\mu$. (b) The potential is shown for fixed $\mu=1$ and varying $n$.
• Figure 4: The $n_{s}-r$ prediction for T-Model of $\alpha$-attractor with $n=1$. The dots along each line represent a specific value of $N_{e}$. Going from right to left, $N_{e}=65, 60, 55, 50, 45.$ Planck I correspond to Planck TT, TE, EE+lowE+lensing+BK18+BAO at $1-\sigma$ and Planck II implies Planck TT, TE, EE+lowE+lensing+BK18+BAO at $2-\sigma$.
• Figure 8: Variation with different $G\mu$ / $\eta^{local}$ keeping $r$ constant.