Why don't they use nitrogen to fill footballs, I believe nitrogen isn't affected by temprature
That is NOT true.
Every gas is the same. The reason some put Nitrogen in tires rather than the mixture of gases in air, is to delay the decay and ageing of rubber by exposure to Oxygen which is the second most common gas in air after Nitrogen.
We now have a possible entirely natural explanation for what happened,
IF the Colts balls were stored in a truck/bus outside, in the cold, while the Patriot balls were storred inside in the warmth. No wrongdoing by anyone.
If anything, the Colts might have very temporarily and inadvertantly, played with over-inflated footballs early in the game, as the cold balls after checking/certification, warmed up, and over-pressured, and had not cooled back down to outside temperatures. early in the game.
As for ascertaining the "weight difference", I defy anyone blindfolded, to select between two identical footballs with one having a torn-in-half, half of a dollar bill resting on it. That is the difference in weight between the amount of air inside a football to yeild a 2 PSI difference in the small volume of a football.
Dequell Jackson rumored to be the one who noted under-pressure said it was not him, nor could he tell. He kept it merely because he wanted the souvenir of his INT in the AFCCG.
This natural explanation explains how both sets of balls could be tested and be certified prior to the game.
This natural explanation explains how the warm Patriot footballs could lower pressure when cooled on the sidelines, and fall out ofcompliance when measured at halftime.
This natural explanation explains why the cold Colt footballs were cooled back to conpliance, if they overinflated, after certification testing, in the 2 hours prior to the game.
The one suposedly Patriot football in compliance, the "12th ball", was probably missing, and never tested, as La Garette Blount had thrown it into the stands after scoring his TD.
This natural explanation explains why re-inflated Patriot footballs during the short 12 minute halftime in the cold, were unchanged after the game. It even proves that the Patriot footballs were not damaged or leaking, since they held compliance for the second half.
By Occam's Razor this is the likely truth. Without resorting to Conspiracy Theories and wrongdoing, or mal or mis-feasance on anyone's part, either Patriots or Refs or NFL ballboys assistants.
BTW, genuine scientific calibration certifications generally list pressure at some temperature. e.g. 13 PSI Gauge at "Standard Temperature and (atmospheric)Pressure", aka "STP" which the ambiguous NFL rules do NOT do. (I,m certain they will when revised, next year.)