How do I invite an expert to contribute to a review article I'm writing? I am in the process of writing my PhD thesis, and I would like to extend the state-of-the-art part into a review article. Although I am confident that I have investigated and understood the existing literature thoroughly, I think the overall quality and visibility of the review article would increase if I find an expert to collaborate with on the paper. My PhD supervisor is not an expert in the field and I expect his willingness to actively contribute to be near zero.

I have already identified a rising expert (junior PI) that I would like to contact. A strong plus is also that he seems to be an expert on the area of the article where I have no published results, whereas I have some published research in the other of the two subareas of the review article. However, I am not sure how and when to contact this person (I have not met him in person).

One approach would be to (nearly) finish the paper and then send him a direct email, asking whether he would like to contribute, and send him the paper after he agreed to handle the draft confidentially so he can decide if he wants to contribute. I would also mention that I am willing to do the ground work but expect him to comment on the draft and help improving the overall quality of the paper.

Another approach would be to upload a finished (to the best of my abilities) paper on arXiv and than send him the mail asking for his willingness to contribute.

Is there a better approach?
I suggest that you contact the other person and ask for their advice on your ideas, but not for collaboration, initially. See where that goes. It allows the person to make a small commitment without feeling any pressure for a larger one. 

Your idea for a review or summary article might be a bit unusual, but in many fields these are occasionally needed as the field splits but the different pursuits have some common but unnoticed features. Partly for this reason, I don't think that the other party would think about going into competition with you on the idea, and might want to participate. The advice would be valuable in any case. 

If the collaboration starts naturally then you can write a joint paper, perhaps after you complete your degree so that your advisor has less sway. Write and explain your ideas, expanded from what you write here, send a draft, and ask for any feedback that he/she might give.

The other person might also be able to suggest others with whom you might profitably share your ideas. 



I mention the competition issue only because you don't really know the person. Most everyone behaves ethically, but not all do. So in some situations it is necessary to be a bit cagey when you are a newcomer to academia. I don't really see it as an issue in this case, however, as mentioned above.