  To explore the doping dependence of the recently discovered charge density
wave (CDW) order in YBa2Cu3Oy, we present a bulk-sensitive high-energy x-ray
study for several oxygen concentrations, including strongly underdoped
YBa2Cu3O6.44. Combined with previous data around the so-called 1/8 doping, we
show that bulk CDW order exists at least for hole concentrations (p) in the
CuO2 planes of 0.078 <~ p <~ 0.132. This implies that CDW order exists in close
vicinity to the quantum critical point for spin density wave (SDW) order. In
contrast to the pseudogap temperature T*, the onset temperature of CDW order
decreases with underdoping to T_CDW ~ 90K in YBa2Cu3O6.44. Together with a
weakened order parameter this suggests a competition between CDW and SDW
orders. In addition, the CDW order in YBa2Cu3O6.44 shows the same type of
competition with superconductivity as a function of temperature and magnetic
field as samples closer to p = 1/8. At low p the CDW incommensurability
continues the previously reported linear increasing trend with underdoping. In
the entire doping range the in-plane correlation length of the CDW order in
b-axis direction depends only very weakly on the hole concentration, and
appears independent of the type and correlation length of the oxygen-chain
order. The onset temperature of the CDW order is remarkably close to a
temperature T^\dagger that marks the maximum of 1/(T_1T) in planar 63^Cu
NQR/NMR experiments, potentially indicating a response of the spin dynamics to
the formation of the CDW. Our discussion of these findings includes a detailed
comparison to the charge stripe order in La2-xBaxCuO4.
