Reproducing $\Lambda$CDM-like Solutions in $f(Q)$ Gravity: A Comprehensive Study Across All Connection Branches

Given the remarkable success of the $\Lambda$CDM model in fitting various cosmological observations, a pertinent question in assessing the phenomenological viability of modified gravity theories is whether they can reproduce an exactly $\Lambda$CDM-like cosmic background evolution. In this paper, we address this question in the context of $f(Q)$ gravity, where $Q$ denotes the nonmetricity scalar. It is known that there are three possible symmetric teleparallel connection branches that respect the cosmological principles of spatial homogeneity, isotropy, and global spatial flatness. By enforcing a $\Lambda$CDM-like background evolution via the cosmographic condition $j(z)=1$, where $j$ is the jerk parameter, we reconstruct the $\Lambda$CDM-mimicking $f(Q)$ theory for each of the three possible connection branches. For the first connection branch, also known as the ``coincident gauge'' in cosmology, we recover the previously known result that a theory of the form $f(Q)=-2\Lambda+\alpha Q+\beta\sqrt{-Q}$ can exactly reproduce a $\Lambda$CDM-like cosmic evolution. Furthermore, we establish that the stability of the $\Lambda$CDM-like cosmic solution within this reconstructed $f(Q)$, as well as the robustness of the reconstructed $f(Q)$ form with respect to small errors in the astrophysical measurements of the jerk parameter. For the second connection branch, we analytically reconstruct the $\Lambda$CDM-mimicking $f(Q)$ to be of the form $f(Q)=-2\Lambda+\alpha Q-\beta Q^2$. For the third connection branch, we could decouple the evolution equation for the dynamical connection function, which enabled us to perform a numerical reconstruction. Our analysis proves that, at least at the background level, it is possible to obtain $\Lambda$CDM-mimicking $f(Q)$ models for all the three possible connection branches.