The realistic models of neutron stars are considered for simple $R+\alpha R^2$ gravity and equivalent Brance-Dicke theory with dilaton field in Einsein frame. For negative values of $\alpha$ we have no acceptable results from astrophysical viewpoint: the resulting solution for spherical stars doesn't coincide with Schwarzschild solution on spatial infinity... The mass of star from viewpoint of distant observer tends to very large values. For $\alpha>0$ it is possible to obtain solutions with required asymptotics and well-defined star mass. The mass confined by stellar surface decreases with increasing of $\alpha$ but we have some contribution to mass from gravitational sphere appearing outside the star. The resulting effect is increasing of gravitational mass from viewpoint of distant observer. But another interpretation take place in a case of equivalent Brance-Dicke theory with massless dilaton field in Einstein frame. The mass of star increases due to contribution of dilaton field inside the star. We also considered the possible constraints on $R^{2}$ gravity from GW 170817 data. According to results of Bauswein et al. the lower limit on threshold mass is $2.74^{+0.04}_{-0.01}$ $M_{\odot}$. This allows to exclude some equations of state for dense matter. But in $R^2$ gravity the threshold mass increases for given equation of state with increasing of $\alpha$. In principle it can helps in future discriminate between General Relativity and square gravity (of course one need to know equation of state with more accuracy rather than now. read more

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General Relativity and Quantum Cosmology