For all intents and purposes..NO, a loose rear differential should not cause a vehicle to fail an emissions or "smog" test.
The two systems are entirely: different, separate and have no common elements, or components.
The Emissions system, which controls the quantity of un-burned hydrocarbons (gasoline) in modern vehicles employes a host of mechanical, thermo-electric, and voltage sensing, components to send information signals back to the: ECM, ECU,..or most plainly put...the COMPUTER. The computer then interprets the Data, and adjusts various components based on info received from the sensors thru the computer. This keeps the vehicle operating at Federal Emission Standards.
A loose differential will cause: noise, vibration, wheel shudder, and possible steering problems, and if bad enough...eve
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