As soon as, No sooner than, The instant that, No sooner had
Structure
Verb[る/た] + が早いか、Phrase[た]
(A) が早いか (B) is a formal written expression that is not often used in speech, meaning that as soon as (A) happens, (B). In other words, (B) happens directly after (A).
(B) is often surprising, spontaneous, or habitual and usually expresses someone’s actions (行く、駆け出す）. Due to this, it is uncommon with expressions that are beyond human control, like 雨が降った.
The following phrase is always in the past tense, so it cannot be used to express future intentions with volitional form or expressions like つもり.
It is also not used when talking about speaker/writer, however, the subjects of (A) and (B) might differ.