Quantum creation of an open inflationary universe

We discuss the dramatic difference between the description of the quantum creation of an open universe using the Hartle-Hawking wave function and the tunneling wave function. Recently Hawking and Turok have found that the Hartle-Hawking wave function leads to a universe with $\Omega =0.01,$ which is much smaller than the observed value of $\Omega .$ Galaxies in such a universe would be ${10}^{{10}^8}$ light years away from each other, and so the universe would be practically structureless. We argue that the Hartle-Hawking wave function does not describe the probability of the creation of the universe. If one uses the tunneling wave function for the description of the creation of the universe, then in most inflationary models the universe should have $\Omega =1,$ which agrees with the standard expectation that inflation makes the universe flat. The same result can be obtained in the theory of a self-reproducing inflationary universe, independently of the issue of initial conditions. However, there exist some models where $\Omega$ may take any value, from $\Omega >1\mathrm{}$ to $\Omega \ll 1.$