Modulus of continuity of averages of SRB measures for a transversal family of piecewise expanding unimodal maps

Let $f_t:[0,1]\to [0,1]$ be a family of piecewise expanding unimodal maps with a common critical point that is dense for almost all $t\in [a,b]$. If ${\mu }_t$ is the corresponding SRB measure for $f_t$, we study the regularity of $\mathrm{\Gamma }(t)=\int \varphi d{\mu }_t$ when assuming that the family is transversal to the topological classes of these maps, more precisely, we prove that if $J_t(c)={\sum }_{k=0}^{\mathrm{\infty }}\frac{v_t(f_t^k(c))}{D{f}_t^k(f_t(c))}\mathrm{\ne }0$ for all $t$, where $v_t(x)={\mathrm{\partial }}_s{f}_s(x){\mathrm{\mid }}_{s=t}$, then $\mathrm{\Gamma }(t)$ is not Lipschitz for almost all $t\in [a,b]$. Furthermore, we give the exact modulus of continuity of $\mathrm{\Gamma }(t)$.