We observed the $A^1Sigma^+sim b^3Pito a^3Sigma^+/X^1Sigma^+$ laser-induced fluorescence (LIF) spectra of the RbCs molecule excited from the ground $X^1Sigma^+$ state by the Ti:Sapphire laser. The LIF radiation from the common perturbed levels of the singlet-triplet $Asim b$ complex was recorded by the Fourier-transform (FT) spectrometer with the instrumental resolution of 0.03~cm$^{-1}$. The relative intensity distribution in the rotationally resolved $Asim bto a^3Sigma^+(v_a)/X^1Sigma^+(v_X)$ progressions was measured, and their branching ratio was found to be about of 1$div$5$ times$10$^{-4}$ in the bound region of the $a^3Sigma^+$ and $X^1Sigma^+$ states. The experiment was complemented with the scalar and full relativistic calculations of the $A/b - X/a$ transition dipole moments (TDMs) as functions of internuclear distance. The relative systematic error in the resulting emph{ab initio} TDM functions evaluated for the strong $A - X$ transition was estimated as few percent in the energy region, where the experimental LIF intensities are relevant. The relative spectral sensitivity of the FT registration system, operated with the InGaAs diode detector and CaF beam-splitter, was calibrated in the range $[6~500,12~000]$~cm$^{-1}$ by a comparison of experimental intensities in the long $Asim bto X(v_X)$ LIF progressions of the K$_2$ and KCs molecules with their theoretical counterparts evaluated using the emph{ab initio} $A - X$ TDMs. Both experimental and theoretical transition probabilities can be employed to improve the stimulated Raman adiabatic passage process, $ato Asim b to X$, which is exploited for a laser assembling of ultracold RbCs molecules.
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