Radiative Symmetry breaking, Cosmic Strings and Observable Gravity Waves in $U(1)_R$ symmetric $SU(5) \times U(1)_{\chi}$

We implement shifted hybrid inflation in the framework of supersymmetric $SU(5) \times U(1)_{\chi}$ GUT model which provides a natural solution to the monopole problem appearing in the spontaneous symmetry breaking of $SU(5)$. The $U(1)_{\chi}$ symmetry is radiatevely broken after the end of inflation at an intermediate scale, yielding topologically stable cosmic strings. The Planck's bound on the gravitational interaction strength of these strings, characterized by $G_N \mu_s$ are easily satisfied with the $U(1)_{\chi}$ symmetry breaking scale which depends on the initial boundary conditions at the GUT scale. The dimension-5 proton lifetime for the decay $p \rightarrow K^+ \bar{\nu}$, mediated by color-triplet Higgsinos is found to satisfy current Super-Kamiokande bounds for SUSY breaking scale $M_{\text{SUSY}} \gtrsim 12.5$ TeV. We show that with minimal K\"ahler potential, the soft supersymmetry breaking terms play a vital role in bringing the scalar spectral index $n_s$ within the Planck's latest bounds, although with small tensor modes $r \lesssim 2.5 \times 10^{-6}$ and $SU(5)$ gauge symmetry breaking scale in the range ($2 \times 10^{15} \lesssim M_{\alpha} \lesssim 2 \times 10^{16}$) GeV. By employing non-minimal terms in the K\"ahler potential, the tensor-to-scalar ratio approaches observable values ($r \lesssim 10^{-3}$) with the $SU(5)$ symmetry breaking scale $M_{\alpha} \simeq 2 \times 10^{16}$ GeV.