Outflows, infall and evolution of a sample of embedded low-mass protostars. The William Herschel Line Legacy (WILL) survey

[Abridged] We present spectroscopic observations in H$_{2}$O, CO and related species with \textit{Herschel} HIFI and PACS, as well as ground-based follow-up with the JCMT and APEX in CO, HCO$^{+}$ and isotopologues, of a sample of 49 nearby ($d<$500\,pc) candidate protostars. These data are used to study the outflow and envelope properties of these sources. We also compile their continuum SEDs in order to constrain their physical properties. Water emission is dominated by shocks associated with the outflow, rather than the cooler, slower entrained outflowing gas probed by ground-based CO observations. These shocks become less energetic as sources evolve from Class 0 to Class I. The fraction of mass in the outflow relative to the total envelope (i.e. $M_{\mathrm{out}}/M_{\mathrm{env}}$) remains broadly constant between Class 0 and I. The median value ($\sim$1$\%$) is consistent with a core to star formation efficiency on the order of 50$\%$ and an outflow duty cycle on the order of 5$\%$. Entrainment efficiency, as probed by $F_{\mathrm{CO}}/\dot{M}_{\mathrm{acc}}$, is also invariant with source properties and evolutionary stage. The median value (6.3\kms{}) suggests an entrainment efficiency of between 30 and 60$\%$ if the wind is launched at $\sim$1AU. $L$[O\,{\sc i}] is strongly correlated with $L_{\mathrm{bol}}$ but not with $M_{\mathrm{env}}$, while low-$J$ CO is more closely correlated with the latter than the former. This suggests that [O\,{\sc i}] traces the present-day accretion activity while CO traces time-averaged accretion over the dynamical timescale of the outflow. $L$[O\,{\sc i}] does not vary from Class 0 to Class I, unlike CO and H$_{2}$O. This is likely due to the ratio of atomic to molecular gas in the wind increasing as the source evolves, balancing out the decrease in mass accretion rate. Infall signatures are detected in HCO$^{+}$ and H$_{2}$O in a few sources.
Comment: Accepted to A&A, version after language editor corrections. 47 pages, 18 figures, 15 tables. The figures have been converted to pdf due to file-size considerations, which may lead to some degradation