On density of modular points in pseudo-deformation rings

Given a continuous, odd, reducible and semi-simple $2$-dimensional representation $\bar\rho_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd characteristic $p$, we study the relation between the universal deformation ring of the pseudo-representation corresponding to $\bar\rho_0$ (pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the maximal reduced quotient of the pseudo-deformation ring is isomorphic to the local component of the big $p$-adic Hecke algebra corresponding to $\bar\rho_0$ if a certain global Galois cohomology group has dimension $1$. This partially extends the results of B\"{o}ckle to the case of residually reducible representations. We give an application of our main theorem to the structure of Hecke algebras modulo $p$. As another application of our methods and results, we prove a result about non-optimal levels of newforms lifting $\bar\rho_0$ in the spirit of Diamond-Taylor. This also gives a partial answer to a conjecture of Billerey-Menares.