Single field inflation in the light of Pulsar Timing Array Data: Quintessential interpretation of blue tilted tensor spectrum through Non-Bunch Davies initial condition

In this work, we present a quintessential interpretation of having a blue-tilted tensor power spectrum for canonical single-field slow-roll inflation to explain the recently observed Pulsar Timing Array (NANOGrav 15-year and EPTA) signal of Gravitational Waves (GW). We formulate the complete semi-classical description of cosmological perturbation theory in terms of scalar and tensor modes using the Non-Bunch Davies initial condition. We found that the existence of the blue tilt ($n_t$) within the favoured range $1.2<n_t<2.5$ can be explained in terms of a newly derived consistency relation. Further, we compute a new field excursion formula using the Non-Bunch Davies initial condition, that validates the requirement of Effective Field Theory in the sub-Planckian regime, $|\Delta\phi|\ll M_{\rm pl}$ for the allowed value of the tensor-to-scalar ratio, $r<0.06$ from CMB observations. In our study, we refer to this result as Anti Lyth bound as it violates the well-known Lyth bound originally derived for Bunch Davies initial condition. Further, we study the behaviour of the spectral density of GW and the associated abundance with the frequency, which shows that within the frequency domain $10^{-9}{\rm Hz}<f<10^{-7}{\rm Hz}$ the outcome obtained from our analysis is completely consistent with the Pulsar Timing Array (NANOGrav 15-year and EPTA) signal. Also, we found that the behaviour of GW spectra satisfies the CMB constraints at the low frequency, $f_*\sim 7.7\times 10^{-17}{\rm Hz}$ corresponding to the pivots scale wave number, $k_*\sim 0.05{\rm Mpc}^{-1}$. Finally, the sharp falling behaviour of the GW spectra within the frequency domain $10^{-7}{\rm Hz}<f<1{\rm Hz}$ validates our theory in the comparatively high-frequency regime as well.