Total and differential cross sections of the $\boldsymbol{dp\to {}^3}\textrm{He}\,\boldsymbol{\eta}$ reaction at excess energies between 1 and 15 MeV

New high precision total and differential cross sections are reported for the $dp\to {}^3\textrm{He}\,\eta$ reaction close to threshold. The measurements were performed using the magnetic spectrometer ANKE, which is an internal fixed target facility at the COSY cooler synchrotron. The data were taken for deuteron beam momenta between $3.14641~\textrm{GeV}/c$ and $3.20416~\textrm{GeV}/c$, which corresponds to the range in excess energy $Q$ for this reaction between $1.14~\textrm{MeV}$ and $15.01~\textrm{MeV}$. The normalization was established through the measurement in parallel of deuteron-proton elastic scattering and this was checked through the study of the $dp\to {}^3\textrm{He}\,\pi^0$ reaction. The previously indicated possible change of sign of the slope of the differential cross sections near the production threshold, which could be explained by a rapid variation of the $s$- and $p$-wave interference term, is not confirmed by the new data. The energy dependence of the total cross section and the $90^{\circ}$ slope parameter are well explained by describing the final state interaction in terms of a complex Jost function and the results are significant in the discussion of $\eta$-mesic nuclei. In combination with recently published WASA-at-COSY data [P. Adlarson $et\, al.$, Phys. Lett. B 782, 297 (2018)], a smooth variation of the slope parameter is achieved up to an excess energy of $80.9~\textrm{MeV}$.