an improved version of the code [GADGET-2]{} [@2005MNRAS.364.1105S], created for GR simulations. Dark matter halo catalogues were produced using the [SUBFIND]{} algorithm [@2001MNRAS.328..726S]. Each model has been simulated for one random realization.

For more detailed discussions on the N-body implementations, we refer interested readers to the corresponding publications.

Convolution Lagrangian Perturbation Theory for biased tracers in MG {#sec:LPTDM}
-------------------------------------------------------------------

### LPT for dark matter {#LPTdarkmatter}

The Lagrangian Perturbation Theory approach to structure formation has been extensively studied in the context of $\Lambda$CDM scenarios. Opposite to the Eulerian picture, in which one monitors the evolution of the desired quantities at a given, fixed, position, in LPT one instead tracks down the evolution of a given fluid element over time. Starting from an initial, Eulerian, comoving position $\bold{q}$ at a desired early time $t_0$, each mass element is mapped to its comoving Lagrangian position $\bold{x}(\bold{q},t)$ at time $t$, through the relationship $$\label{Lagpos}
\bold{x}(\bold{q},t) = \bold{q} + \bold{\Psi}(\bold{q},t).$$ The Lagrangian displacement $\bold{\Psi}(\bold{q},t)$, taken to be 0 at the initial time $t_0$, is the fundamental quantity of interest in LPT. Furthermore, enforcing mass conservation, through the continuity equation, between the initial and final infinitesimal volume elements centered around $\bold{q}$ and $\bold{x}$, respectively, gives $\rho_{m}(\bold{x},t)d^3x=\rho_{m}(\bold{q},t_0)d^3q$. 