Starobinsky's f(R) gravity in mgcamb

[Chudaikin Anton]

I want to apply Starobisky's gravity to mgcamb code. This f(R) gravity looks as

f(R) = R + labmda * Rs * [ (1+(R/Rs)^2)^(-nn)-1 ]

<nn, lambda, Rs> - parametres of this model and only (nn, lambda) are free parametres.

R=6*(adotoa*adotoa+Hdot*Hdot)
nn=2
lambda=2
Rs=70**2*4.38/lambda

To fit these two parametres in CosmoMC I decided to write down expression "mu" and "gamma" in mgcamb code explicitly. Let me remember view of these functions:

mu = [ 1 + 4 * (k/a)^2 * (f''/f') ] / [1 + 3 * (k/a)^2 * (f''/f') ]
gamma = [ 1 + 2 * (k/a)^2 * (f''/f') ] / [1 + 4 * (k/a)^2 * (f''/f') ]

I rewrited it by following way in high curvature regime R>>Rs:

MG_mu = (1.d0 + 4*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)/(1.d0 + 3*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)

MG_mudot = ((B1 - 1.d0) * 3*( adotoa-2*nn*(2*nn+1)*lambda*a**(4*nn+4)*(R/Rs)**(-2*nn-2)/Rs )*(4*nn+2)*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs) / ((1.d0 + 3*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)**2.d0)

MG_gamma = (1.d0 + 2*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)/(1.d0 +4*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)

MG_gammadot = ((B2 - 1.d0) * 4*( adotoa-2*nn*(2*nn+1)*lambda*a**(4*nn+4)*(R/Rs)**(-2*nn-2)/Rs )*(4*nn+2)*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs) / ((1.d0 + 4*k2*a**(4*nn+2)*2*nn*(2*nn+1)*lambda*(R/Rs)**(-2*nn-2)/Rs)**2.d0)

And when I run mgcamb I get that sigma8=10^(-11)...

The problem maybe consists in that when I derived last fourh equations I considered that R'=0. But in my opinion it isn't so. I have done it because I didn't know the explicit value of a'''/a, which is neccessary to write down R'. If I know R' I know Hdotdot = a'''/a - (adotdota) * (adotoa) - 2 * adotoa * dotH.

PS Folloing by the camb code there was used Newtonian metric where ' - conformed time-derivation and

adotoa = a'/a = H
adotdota = a''/a
Hdot = H' = a''/a - (adotoa)^2

I want to ask, am I on the right path to do like this or maybe exists another solution of such problem here. And how can I get (a'''/a) or H'' in the mgcamb code?

Anton

[Brent Follin]

In many cases, f(R) gravity is observationally equivalent to coupled dark energy models, see e.g. http://arxiv.org/abs/1006.0877.

Since these models correspond only to changes in the dark sector densities and pressures, and not changes in the background gravity theory, you might find them easier to incorporate into CAMB.

[Chudaikin Anton]

You are absolutely right. In Starobinsky's model I I don't differ background evolution from background evolution in LCDM model. But in all parametrization in CAMB I need "MG_mu" and "MG_gamma" and therefore I need R' in GR. But I don't know how to express it from variables mentioned above according to mgcamb code. How can I do it?