  This paper presents new deep and wide narrow-band surveys undertaken with
UKIRT, Subaru and the VLT; a combined effort to select large, robust samples of
H-alpha (Ha) emitters at z=0.40, 0.84, 1.47 and 2.23 (corresponding to
look-back times of 4.2, 7.0, 9.2 and 10.6 Gyrs) in a uniform manner over ~2
deg^2 in the COSMOS and UDS fields. The deep Ha surveys reach ~3M_sun/yr out to
z=2.2 for the first time, while the wide area and the coverage over two
independent fields allow to greatly overcome cosmic variance. A total of 1742,
637, 515 and 807 Ha emitters are homogeneously selected at z=0.40, 0.84, 1.47
and 2.23, respectively, and used to determine the Ha luminosity function and
its evolution. The faint-end slope is found to be -1.60+-0.08 over z=0-2.23,
showing no evolution. The characteristic luminosity of SF galaxies, L*, evolves
significantly as log[L*(z)]=0.45z+log[L*(z=0)]. This is the first time Ha has
been used to trace SF activity with a single homogeneous survey at z=0.4-2.23.
Overall, the evolution seen in Ha is in good agreement with the evolution seen
using inhomogeneous compilations of other tracers of star formation, such as
FIR and UV, jointly pointing towards the bulk of the evolution in the last 11
Gyrs being driven by a strong luminosity increase from z~0 to z~2.2. Our
uniform analysis reveals an Ha star formation history of the Universe which can
be simply parameterised by log(SFRD)=-2.1/(1+z) for z<2.23. Both the shape and
normalisation of the Ha star formation history are consistent with the
measurements of the stellar mass density growth, confirming that our Ha
analysis traces the bulk of the formation of stars in the Universe up to z~2.2.
Star formation activity since z~2.2 is responsible for ~95% of the total
stellar mass density observed locally today (with half being assembled between
z=1.2-2.2, and the rest since z<1.2).
