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I'm reading Kinney's lectures on inflation: http://arxiv.org/abs/0902.1529

and got stuck trying to show that for some comoving length scale [itex]\lambda[/itex], the quantity

[tex]\left(\frac{\lambda}{d_h}\right)^2 |\Omega -1| [/tex]

is conserved, if [itex]w[/itex] is constant in the equation of state. Here [itex]d_h[/itex] is the horizon size; it's not clear if he means the comoving or physical scale. I've been assuming he means the comoving scale; then, differentiating, it looks to me as if this is constant if

[tex]\ddot{a}d_h +H=0[/tex]

where a dot denotes a derivative with respect to time and H is the hubble constant, but I don't see why this should be true, or how to get rid of [itex]d_h[/itex] so that I could use the Einstein equations for the scale factor.

Thanks in advance.

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# Conserved charge in FRW expansion

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