  The genus of the iso-density contours is a robust measure of the topology of
large scale structure, and it is relatively insensitive to nonlinear
gravitational evolution, galaxy bias and redshift-space distortion. We show
that the growth of density fluctuations is scale-dependent even in the linear
regime in some modified gravity theories, which opens a new possibility of
testing the theories observationally. We propose to use the genus of the
iso-density contours, an intrinsic measure of the topology of large scale
structure, as a statistic to be used in such tests. In Einstein's general
theory of relativity, density fluctuations are growing at the same rate on all
scales in the linear regime, and the genus per comoving volume is almost
conserved as structures are growing homologously, so we expect that the
genus-smoothing scale relation is basically time-independent. However, in some
modified gravity models where structures grow with different rates on different
scales, the genus-smoothing scale relation should change over time. This can be
used to test the gravity models with large scale structure observations. We
studied the case of the f(R) theory, DGP braneworld theory as well as the
parameterized post-Friedmann (PPF) models. We also forecast how the modified
gravity models can be constrained with optical/IR or redshifted 21cm radio
surveys in the near future.
