Improved measurements of the absolute branching fractions of the inclusive decays $D^{+(0)}tophi X$

By analyzing 2.93 fb$^{-1}$ of $e^+e^-$ annihilation data taken at the center-of-mass energy $sqrt s=$ 3.773 GeV with the BESIII detector, we determine the branching fractions of the inclusive decays $D^+tophi X$ and $D^0tophi X$ to be $(1.135pm0.034pm0.031)%$ and $(1.091pm0.027pm0.035)%$, respectively, where $X$ denotes any possible particle combination. The first uncertainties are statistical and the second systematic. We also determine the branching fractions of the decays $Dtophi X$ and their charge conjugate modes $bar{D}tophi bar{X}$ separately for the first time, and no significant CP asymmetry is observed.