Acoustics of early universe. II. Lifshitz vs. gauge-invariant theories
  Appealing to classical methods of order reduction, we reduce the Lifshitz system to a second order differential equation. We demonstrate its equivalence to well known gauge-invariant results. For a radiation dominated universe we express the metric and density corrections in their exact forms and discuss their acoustic character. 
Zdzislaw A. Golda, Andrzej Woszczyna