Science was in the state of gathering momentum = is taking off
Science was in the state of embryo = is taking off
With what word can I describe better the state of very beginning of improving?
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"Nascent" is often used to describe something that has just been "born" and is expected to grow.
In the 1990s Internet search was a nascent technology that few would have predicted would grow into the dominant entity that is Google.
Depending on what you want to say, you could similarly describe a particular field of scientific study as nascent. Another possibility is embryonic, although this implies the subject has "yet to be born" and is still "gestating".
Despite the hype, some consider self-driving cars to be an embryonic technology which has yet to prove its economic viability.
If you want to use idiom, any variation on either "moving fast" or "exploding" will work:
The technology is about to take off
The technology is about to blow up / go nuclear.