Talk Polywell has some calculations of what might be achieved with the Harold White space warping work given plausible power generation and propulsion systems. This information was provided by Paul March who is working on the NASA project to try to create a detectable warping of space.
1. First have a successful creation and detection of a one part in ten million space warp
2. Develop and increase the level of space warping to a full warping of space
3. Develop advanced space propulsion to achieve about 10% of light speed (nuclear fusion propulsion, nuclear fission propulsion, power beamed propulsion)
4. Apply the advanced design of warping technology to the sublight space vehicle
Assuming we use a 100,000 kg vehicle with an initial velocity of 0.1 times the speed of light (c), with green light (6.0x10^14Hz) lasers for our warp field oscillators, a toroidal warp field cavity that has a superconductive Q-Factor of 10^8, with an input power of 1,000 GWe or 1.0 terawatt (T), we could expect a net light speed boost factor of only 4.02c. With 10 TWe input power we get a net c boost factor of 12.72c. Of course I could have used even higher frequencies for the warp field oscillators of say 1,000x the green light frequency, which would reduce these power levels by a factor of ~30 for the same boost factor, but we really don't know how to build X-Ray lasers yet.
Paul March did some more calculations with the warp-field analysis tool and found the following -
by decreasing the resonant cavity dielectric density down to a lunar like vacuum level of 5x10^-12 Torr and increasing the warp-core torodial resonant cavity size up to 20 meter OD by 15 meter ID by 20 meter long while still using green light laser frequency for the RF source and using "just" 1.0 gigawatt of electrical input power, that one might be able to obtain a c boost factor of 88,000 times the speed of light. If one pulled back to using an infrared 1x10^12 Hz (THz) RF source using the same 1.0 GWe of input power, then the c boost factor lowers down to ~3,600c.
As you can tell there from these comments, there are many design parameters that go into this simulation, so your obtainable c boost factor will depend on just how clever we are in the actual design and buildup of the starship.
Here is information from a presentation by Harold White that explains the test setup and physics around the concept.
H. G. White and E. W. Davis- The AlcubierreWarp Drive in Higher Dimensional Spacetime
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