I'm more interrested in how to improve research in a wide civilization. At this point he game is very inviting to a playstyle with 4 cities and tradition, because tradition gives quite good bonusses that are important early on, happyness is also limited early on, settlers are costly to build. Tradition is very good for varied reasons, growth, wonders and happyness.
Yet, consider though that at some point in the game, youre probably sitting there with 4 well developed city's and actually plenty of happyness, and while it would cost you indeed 5% more techcost for an added city, add actually 4 city's that grow to roughly the same size and the others and youll probably have almost doubled youre science for a far smaller penalty, even if youre capital is still doing more than other city's. One can have plenty of excess hapyness by the indusrial age, at which point it would actually be better to have something like 10 city's, but it's not like youre going to have so many beautifull settling options by then than possibly at game start. You could try to conquer the city's, but that scenario has its own implications. Having virgin land to settle in by the industrial age would be very optimal though, especially if it comes with still extra luxury's if asside for hapyness also for the resource diversity, but i guess by the industrial age most empire's can support 6 to 8 city's in happyness depending how much hapyness buildings they can gather, and i'm pretty sure that before 10 city's those added city's can still add to technological speed rather than impair it provided good growth, so actually you probably should try to grow to something like 8 city's.
I too have been looking at things that can aid me with an early wide empire. There are logical things youd probably need, like enough variation of luxury's, lots of workers to turn the land productive fast. A first thing that sounds to me as a very handy bonus is the pantheon belief messenger of the gods, which gives +2 additional science for each city connected to youre capital iirc. If we consider that one for ex grows to 8 city's early on rather than 4, then 7 city's connected to the capital give an additional 14 science i guess which can be modified by buildings. It would though take some time to build the roads, the city's must have grown to a point to make those roads profitable somewhat too, though one probably would have more gold trough a more varied resource diversity.
An bigger empire might actually also catch more trade routes created by other civs, there are more city's to send to and you get more scinece depending on how backward you are, and there are ways to make it more interresting for another civilization to send a trade route to you, so that seems worth some considerations, as in maybe adding caravanseray's might actually add science to youre civ. Afcourse the larger empire should really have the maximum amount of caravans running around too, every precious science will count. Science buildings in all city's is obvious, the national college even though it offers a very nice bonus for a 4 city nation with a powerfull capitali might actually impede expansion plans for the broad building empire and be of a lesser inpact if one didn't take tradition (no growth and hapiness bonus in capital) but rather Liberty for the bonus to building settlers and the speed at which workers can work.
Civilizations of interrest here might be Babylonians, The maya and Koreans. the Babylonians get an early free great scientist and double great scientist generation, not bad when you have an empire which will have more scientists slots possiblu due to more science buildings, whereas the korean specialists all produce an added 2 science whatever their type, that might be interresting depending how fast you can get a specialist building up, marketplaces cost no maintainance and are a good investment for money in my book, spend money to increase money, in the case of korea you could try to use youre larger empire to gain more gold, aim towards currency as a tech not to far away and then buy a market in every city and assign a specialist, then thats another 16 science for Korea (asside of a lot more gold) and currency is not that far in the tech tree. Consider then 8 city's, +14 sciency from messenger of gods with connected city's, + 16 science from specialists in markets by currency, thats +30 base science, not that bad as an addition to what youre making, might actually help quite a bit in making the gap much smaller, with the thought that later on youll have a far more advantagous position with all youre added poppulation. When as korea youve build all those marketplaces, you might as well build the east indian company too in some city with a caravanseray bordering many city's of other civilizations, as a magnet pole for science bringing caravans. The maya get an even better early building in their shrine which also produces 2 science, thats an quite early boost.
One question i have to ask, i guess the national college adds the science across the whole civilization? in that case she is as important as ever, although you need library's in all city's afcourse to build it so that won't be an easy one. That is what youre competing wioth though, civilizations who take tradition and build the national college very early in a fast growing capital can go up to quite a lot of science early, but time should eventually be on the side of the larger empire.