  Upon embedding the axion-inflation in the minimal left-right symmetric gauge
extension of the SM with gauge group $SU(2)_L\times SU(2)_R \times U(1)_{B-L}$,
[arXiv:2012.11516] proposed a new particle physics model for inflation. In this
work, we present a more detailed analysis. As a compelling consequence, this
setup provides a new mechanism for simultaneous baryogenesis and right-handed
neutrino creation by the chiral anomaly of $W_R$ in inflation. The lightest
right-handed neutrino is the dark matter candidate. This setup has two unknown
fundamental scales, i.e., the scale of inflation and left-right symmetry
breaking $SU(2)_R\times U(1)_{B-L}\rightarrow U(1)_{Y}$. Sufficient matter
creation demands the left-right symmetry breaking scale happens shortly after
the end of inflation. Interestingly, it prefers left-right symmetry breaking
scales above $10^{10}~GeV$, which is in the range suggested by the
non-supersymmetric SO(10) Grand Unified Theory with an intermediate left-right
symmetry scale. Although $W_R$ gauge field generates equal amounts of
right-handed baryons and leptons in inflation, i.e. $B-L=0$, in the Standard
Model sub-sector $B-L_{SM}\neq 0$. A key aspect of this setup is that $SU(2)_R$
sphalerons are never in equilibrium, and the primordial $B-L_{SM}$ is conserved
by the Standard Model interactions. This setup yields a deep connection between
CP violation in physics of inflation and matter creation (visible and dark);
hence it can naturally explain the observed coincidences among cosmological
parameters, i.e., $\eta_{B}\simeq 0.3 P_{\zeta}$ and $\Omega_{DM}\simeq
5\Omega_{B}$. The $SU(2)_R$-axion inflation comes with a cosmological smoking
gun; chiral, non-Gaussian, and blue-tilted gravitational wave background, which
can be probed by future CMB missions and laser interferometer detectors.
