Clarification on gr.fac. D and Rescaling P(k,0) in CLASS

[zyuan007]

Dear CLASS Developers,

I'm working with the public version of CLASS (with standard ΛCDM settings) and had a question about interpreting the background output quantity gr.fac. D in background.dat.

I understand that gr.fac. D is intended to represent the linear growth factor D(z), normalized to 1 at z=0. Naively, I expected that I could recover the matter power spectrum at any redshift using P(k,z) = P(k,0)*D(z)^2. However, when I compare this to the linear matter power spectrum output by CLASS at redshift z, I observe discrepancies at both ends of the spectrum: at k<0.01 h/Mpc and k>10 h/Mpc.

Initially, I suspected the discrepancy at high k could be due to non-linear effects, but since I have nonlinear turned off, the power spectrum should remain linear and the growth should be scale-independent, even at those scales. This makes the disagreement at high k as puzzling as that at low k.

Looking into the code (around line 2629 in background.c), I see that D(τ) is computed by solving a differential equation, but the physical meaning of that equation isn't immediately clear to me. Could you please clarify:

• What exactly does gr.fac. D measure and how is it computed?
• Is it tied to the growth of CDM+baryon perturbations, total matter, or another component?
• Is it valid to use D(z)^2 to rescale P(k,0) and if not, what are the limitations or caveats?
• Why might this approach fail at large and small k even in linear theory and with standard cosmology?

I’d really appreciate any clarification on these issues. Thank you for your time!

Best regards,
Linda