With FTL, the rate of expansion would rise exponentially, though. This could create balance problems, especially if FTL rushing was a viable option (I know nothing about the research system in this hypothetical game). Note that colony rush is a very frequent problem in 4X games.I don't think that this is a huge problem. At the beginning you should be busy colonizing the solar system. And to the next stars (alpha centaury 4 light years) can be colonized by subsace/impulse egnines. With fusion reactors and an ion-engine spaceships can make it. 1 lightyear = 40.000.000.000.000 km. Lightspeed is 1.080.000.000 km/h. 8.544,6. With 1/10 lightspeed you will make it in 40 years. Some scientists think that you can travel with 4/10 lightspeed so 10 Years would be possible. That is not much more as you need to get an army without fuel and supplies trough russia^^. So you don't even need generationships for close stars like Alpha Centauri, Sirius, etc.
Also, Google says that 1 light year is 9.46 trillion kilometres. The value you gave is the distance to Alpha Centauri (4.37 light years).