Will a cultivator get through chalky soil?
Generally no. I hired a cultivator to make my vegetable beds. It was one of those monster petrol ones. They tell you that you need to get the turf up for it to ‘bite’ into the earth. I ended up fighting like mad to keep the cultivator in one place until the blades hadd a chance to dig in. I wouldn’t have thought you would have to dig up soil for drainage – after all the rain must go somewhere now. You could either leave the work to the worms (perhaps put a layer of leaves/grass or other organic matter down before the topsoil). Alternatively try using apick-axee or mattock to break the surface. Wetting the surface first or working after rain will probably make things a lot easier.