Cosmological Perturbation Theory, 26.4.2011 version

17 FLUID COMPONENTS 4117.2 Gauge TransformationsPerhaps I’ll write more here someday ...Note that gauge conditions that refer to fluid perturbations, refer usually to the total fluid.Thus, e.g., in the comoving gauge the total velocity perturbation v vanishes, but the componentvelocities v i do not vanish (unless they happen to be all equal). Thus, the velocity v thatappears in the gauge transformation equations to comoving gauge refers to the total densityperturbation. For example the component gauge tranformation equations that correspond toEq. (15.14) readδρ C i = δρ N i − ¯ρ ′ i vN (17.9)= δp N i − ¯p ′ iv Nδp C iδ C i = δ N i − ¯ρ′ i¯ρ vN .Since the gauge transformations are the same for all components, we find some gauge invariances.The relative velocity perturbation between two components i and j,v i − v j is gauge invariant. (17.10)Like the total anisotropic stress, also the component anisotropic stressesΠ i are gauge invariant. (17.11)We can also define a kind of entropy perturbation( )δρ iS ij ≡ −3H¯ρ ′ − δρ ji¯ρ ′ j(17.12)between two fluid components i and j which turns out to be gauge invariant due to the waythe density perturbations δρ i transform. The above is a special case of a generalized entropyperturbation( δxS xy ≡ H¯x ′ − δy )ȳ ′ (17.13)between any two 4-scalar quantities x and y, which is gauge invariant due to the way 4-scalarperturbations transform. Beware of the many different quantities called “entropy perturbation”!Some of them can be interpreted as perturbations in some entropy/particle ratio. What iscommon to all of them, is that they all vanish in the case of adiabatic perturbations.17.3 EquationsThe Einstein equations (both background and perturbation) involve the total fluid quantities.The metric perturbations are not divided into components due to different fluid components!There is a single gravity, due to the total fluid, which each fluid component obeys.If there is no energy transfer between the fluid components in the background universe, thebackground energy continuity equation is satisfied separately by such independent components,and in that case we can write Eq. (17.3) as¯ρ ′ i = −3H(¯ρ i + ¯p i ), (17.14)c 2 s = ∑ ¯ρ i + ¯p i¯ρ + ¯p c2 i , (17.15)