How long does it take to cook eggs safely as a function of temperature?

You can find out more than you'd ever want to know about cooking eggs safely in the International Egg Pasteurization Manual. The standard USDA guidelines for pasteurization of plain whole eggs require 3.5 minutes minimum holding time at 140F (60C).

(Note: Holding temp here refers to egg temperature itself in liquid form: see edit for clarification below. These times/temps are NOT sous vide water bath temperatures for eggs in shells.)

As you can read in that document (particularly Table 6 on page 12 and Table 29 on page 25), adding in small amounts of salt or sugar increases the requirement for pasteurization by various amounts, up to 6 degrees F (3.3 degrees C). Yolks or whites alone also have different numbers.

So, it's really hard to generalize here. Particularly if you start to include any dish that may contain eggs, the required pasteurization temperature may vary a lot. If you mix in eggs with a good growth medium and then slow-cook that mixture for a couple hours before it hits ~130F, there will be a lot more bacteria to kill than in an intact whole egg, hence requiring a longer or higher temperature process to make sure the food is safe. And different textures of food, different moisture levels, etc. will also affect pasteurization. Yes, in general if you hold eggs long enough above 130F, you should eventually kill off enough bacteria to make the food safe to eat. But there's no easy formula for it, particularly if you include mixing other things with the eggs.

If you want to be safe, the easiest thing is often to buy pasteurized eggs or follow the 60C/140F recommendation for 3.5 minutes with whole eggs to pasteurize them whole yourself. Then you can mix or combine them with other foods with less worry. The only major egg property impaired by pasteurization at 140F is the foaming of egg whites. (Egg white alone can be pasteurized at 136F (57.7C) for 6.3 minutes, which will retain much of the foaming properties. With whole eggs, going above 140F, even by a degree or two, begins to inhibit other baking properties; see the linked document for details.)

EDIT: After a comment pointed out some ambiguity here, let me note that the linked document is only about liquid egg mixtures. That is, if you break your whole eggs and bring that mixture to 140F, they will be pasteurized in 3.5 minutes. If you are cooking eggs in shells in a water bath, that's going to be a lot more unpredictable. Larger eggs will take longer to come to temperature, eggs that start colder will take longer to come to temperature, and water baths that have a very active circulator will heat eggs a lot more quickly than a static water bath. If you want to ensure safety for eggs cooked in shells, you'll have to hold quite a bit longer to ensure they reach adequate internal temp, but that time is unpredictable because of the above factors. There are sous vide guides out there, as noted in the question, that give much longer times to try to ensure safety for eggs in shells. Sous vide guides can't give minimum temp/time with as much precision as the pasteurization guide in the link here, because most people aren't monitoring the internal temp of their food in a water bath.

I was originally trying to address the question about cooking eggs in general (not just "boiled eggs in shell") and dishes containing them safely, which would mostly involve liquid eggs and various mixtures including them. And the only way to be precise about minimum time/temp is to specify final internal holding temp, as you'd measure a turkey or whatever. Minimum overall cook times will always be estimates.

You can refer tho the FDA food charts. If you sous-vide egg in a pouch, or water bath, if placed in cold water and then gradually heated, the core of the egg will reach the temperature of the bath uniformly with bath temperature rise.

This document might help you: it is from http://www.foodprotect.org http://www.foodprotect.org/issues/packets/2012packet/attachments/iii_018__all.pdf