The rules change every microsecond. At present the TSA does not permit any LiIon backup battery in checked baggage; they were treated as ‘raw’ lithium batteries and had to be carried in cabin baggage.
The TSA regulations are written in Wh (Watt hours) which is the stored energy and that is reasonable, Ah measures are actually the stored energy per volt and translating into energy is tricky because the voltage varies. I believe some sellers use the battery voltage, which is 3.7V, rather than the output voltage (5V to 12V typically) just to get bigger Ah figures. Here’s an example of an Amazon advert for a 100Wh battery (the checked baggage limit) which actually gives all the numbers.
Here’s the current TSA instruction set (check before every flight
I got the PDF link from this TSA url, note the words: “External chargers are also considered to be a battery.” Then there is the checked baggage restriction:
Spare lithium metal and lithium ion/polymer batteries are prohibited in checked baggage—this includes external battery packs. [* * *] “Checked baggage” includes bags checked at the gate or planeside.
The seller of that Anker 26.8Ah battery @Thomas identified is apparently quoting the capacity at 3.7V (which, guess what, multiplies out to 99.16Wh). The TalentCell item I gave a link to is $50 for exactly the same capacity.
I should add that I don’t think anyone actually believes any of these numbers; they are just made up. (For one thing, an LiIon cell starts out delivering over 4V and ends up at about 2.8V).
The other issue is that whether or not you get a fast charge out of these things depends on the match (or not) with whatever you are charging. There are a variety of conventions and at least some of these depend on having exactly the right cable connecting charger and chargee. It’s also almost impossible to work out if it is working; I have a variety of testers that I can put in the middle but these interfere with some of the protocols that the chargee uses to talk to the charger.
So far my experience is that the only way I can get my phone to charge “fast” is to use a wireless charger, even though my phone has a USB-C port and, supposedly, USB-C has standardized the whole mess. (Alas, even though it did, manufacturers don’t all seem to have read the spec)