If there are no axions, is the universe closed

If the universe is closed, as in the standard model with k=+1, then there is no strong CP problem. The reason is that Guass's law forces the QCD vacuum angle θ to be zero when the spatial universe has no boundary. Other solutions to this problem require either axions or a massless quark. A massless quark is not indicated by current algebra. Experimental searches and astrophysical arguments seen to have ruled out axions apart for those with a decay constant in the range 10/sup 8/GeV<f/sub a/<10/sup 12/GeV. The author therefore faced with the following choices: either (1) such axions do exist or (2) the universe is closed or (3) current algebra is incomplete or (4) there is another resolution of the strong CP problem. A search for axions in the remaining range of f/sub a/ would be of great interest