tokens: similar to those presented in this paper to fully validate such methods. APPENDIX A: Implementing <italic toggle="yes">f</italic>(<italic toggle="yes">R</italic>) model in <sc>mgenzo</sc> We present here a brief overview of the modified gravity (Hu–Sawicki) model used in this paper which has been implemented via our mgenzo software. This software is a variant of the well-established enzo code and is fully described in Zhao et al. (in preparation). mgenzo has been extensively studied using several independent N -body codes including mgmlapm (Zhao, Li Koyama 2011 ) and ecosmog (Li et al. 2012 ). The mgenzo code uses the same algorithm to solve for the non-linear scalar field equations as the mgmlapm and ecosmog code. Previous results from mgenzo have been validated against other N -body and hydrodynamical simulations of the Hu–Sawicki model, including the code comparison work of Winther et al. ( 2015 ) and Hammami et al. ( 2015 ). These papers show that all these independent codes give consistent solutions for the scalar field, as well as the power spectrum and the mass function of dark matter. In detail, we can write the action of the f ( R ) model as (Capozziello 2002 ) where with . Under the quasi-static approximation, the equation of motion of the scalar field δ f R can be obtained as (Noller, von Braun-Bates Ferreira 2014 ) where , and . One can invert equation ( A2 ) to relate R to f R using where the approximation holds if the background cosmology is close to a ΛCDM+GR model and, in this case, one can approximate as At redshift z = 0, where a flat universe is assumed. Combining equations ( A4 ) and ( A6 ), one can rewrite equation ( A4 ) in terms of f R0 , which is the background value of f R at redshift z = 0, as and δ R is given explicitly as The scalar field f R can then be solved numerically by combining equation ( A3 ) with equation ( A8 ), given the model parameters f R0 and n with background cosmological parameters. The modified Poisson equation for the gravitational potential Φ can be obtained by summing the 00 and ii component of the modified Einstein equation in the Hu–Sawicki f(R) model (Hu Sawicki 2007 ), namely The dynamics of the system are determined by equations ( A3 ) and ( A9 ). Equation ( A3 ) is a non-linear Poisson equation, and it has to be solved numerically on a regular, or self-adaptive, grid using iteration methods (Oyaizu 2008 ; Zhao, Li Koyama 2011 ; Li et al. 2012 ; Puchwein, Baldi Springel 2013 ). We chose Hu–Sawicki model as it has been implemented to several simulations and, as shown in Terukina et al. ( 2014 ) and Wilcox et al. ( 2015 ), is insensitive to the form of the potential of the chameleon field. Thus our constraints are applicable to a wide class of chameleon models, and contains GR as a limiting case. Thus our test also serves as a consistency check of the LCDM model. The weak equivalence principle is not violated in this model as all matter is coupled to the scalar field universally. However,
ner_tags: O O O O O O O O O O O O O O O B-Tag I-Tag B-Tag O O B-Tag O O O O O O O O O O B-Model O O O O O O O O O O O B-Software O O O O O O O O O B-Software O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation B-Software O O O O O O O B-Model I-Model O O B-Software B-Citation I-Citation I-Citation I-Citation O O B-Software B-Citation I-Citation I-Citation I-Citation O O B-Software O O O O O O O O O O O O O O O B-Software O B-Software O O O O O O O O O O B-Model I-Model O B-Model I-Model O O B-Model I-Model O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model I-Model B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O O B-Model I-Model O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O