  We present the first results from "Surveying the Whirlpool at Arcseconds with
NOEMA" (SWAN), an IRAM Northern Extended Millimetre Array (NOEMA)+30m large
program that maps emission from several molecular lines at 90 and 110 GHz in
the iconic nearby grand-design spiral galaxy M~51 at cloud-scale resolution
($\sim$3\arcsec=125\,pc). As part of this work, we have obtained the first
sensitive cloud-scale map of N$_2$H$^+$(1-0) of the inner $\sim5\,\times
7\,$kpc of a normal star-forming galaxy, which we compare to HCN(1-0) and
CO(1-0) emission to test their ability in tracing dense, star-forming gas. The
average N$_2$H$^+$-to-HCN line ratio of our total FoV is $0.20\pm0.09$, with
strong regional variations of a factor of $\gtrsim 2$ throughout the disk,
including the south-western spiral arm and the center. The central $\sim1\,$kpc
exhibits elevated HCN emission compared to N$_2$H$^+$, probably caused by
AGN-driven excitation effects. We find that HCN and N$_2$H$^+$ are strongly
super-linearily correlated in intensity ($\rho_\mathrm{Sp}\sim 0.8$), with an
average scatter of $\sim0.14\,$dex over a span of $\gtrsim 1.5\,$dex in
intensity. When excluding the central region, the data is best described by a
power-law of exponent $1.2$, indicating that there is more N$_2$H$^+$ per unit
HCN in brighter regions. Our observations demonstrate that the HCN-to-CO line
ratio is a sensitive tracer of gas density in agreement with findings of recent
Galactic studies which utilize N$_2$H$^+$. The peculiar line ratios present
near the AGN and the scatter of the power-law fit in the disk suggest that in
addition to a first-order correlation with gas density, second-order physics
(such as optical depth, gas temperature) or chemistry (abundance variations)
are encoded in the N$_2$H$^+$/CO, HCN/CO and N$_2$H$^+$/HCN ratios.
