Riccati equations for holographic 2-point functions

Understand

Any second order homogeneous linear ordinary differential equation can be transformed into a first order non-linear Riccati equation.

• We argue that the Riccati form of the linearized fluctuation equations that determine the holographic 2-point functions simplifies considerably the numerical computation of such 2-point functions and of the corresponding transport coefficients by computing directly the response functions, eliminating the arbitrary source from the start.
• Moreover, it provides a neat criterion for the infrared regularity of the fluctuations.
• In particular, it is shown that the infrared regularity conditions for scalar and tensor fluctuations coincide, and hence they are either both regular or both singular.