There are several synonyms for the phrase "according to careful estimates" that one can use when they want to convey the same meaning. Some of these synonyms include "based on informed calculations," "as per expert projections," "according to precise approximations," and "based on meticulous evaluations." All these synonyms are useful when one wants to emphasize that their information is based on a rigorous and thorough process of analysis. By using these synonyms, one can add authority to their writing and provide the reader with the confidence that the information they are presenting is accurate and reliable. Ultimately, picking the best synonym depends on the context and the style of writing, but all these synonyms can help elevate one's writing.