OmegaM split: Implement differential equation for the growth G(z), rescale the power spectrum

[FelicitasKeil]

📝 Issue Summary

This issue is related to the growth-geometry split for $\Omega_m$ (Issue #323 ), focused at the moment on photometric observables (3x2pt). The goal is to implement the differential equation to calculate the growth G(z).

$G''+\left(4+\frac{H'}{H}\right)G'+\left(3+\frac{H'}{H}\frac{3}{2}\Omega_m(z)\right)G=0$

This will then be used to rescale the linear matter power spectrum.

$P_{\rm split}^{\rm lin}(k,z)= P_{\rm geo}^{\rm lin,EBS}(k,z)\left(\frac{G_{\rm growth}^{\rm ODE}(z)}{G_{\rm geo}^{\rm ODE}(z)}\right)^2$

For this, we need to call the function that calculates the ODE twice: once with $\Omega_{\rm m}^{\rm geo}$ and once with $\Omega_{\rm m}^{\rm growth}$. To do this, we also have to introduce a new cosmological parameter $\Omega_{\rm m}^{\rm growth}$.

🎯 Goals / Acceptance Criteria

• Goal 1: New sampling parameter $\Omega_{\rm m}^{\rm growth}$
• Goal 2: Differential equation, callable with different $\Omega_{\rm m}$
• Goal 3: Rescaled linear matter power spectrum with the growth from the ODE

🔄 Steps to Reproduce

1. Add a $\Omega_{\rm m}^{\rm growth}$ parameter
2. Implement the ODE growth_function_ODE(...) in derived_cosmology.py
3. In derived_cosmology, call the linear perturbations from an EBS (we are planning to use CLASS), call the growth ODE twice with both $\Omega_{\rm m}$ and rescale the linear power spectrum.
4. Save the non-linear boost before the rescaling and multiply it afterwards to get the non-linear power spectrum.

✅ Expected Outcome

Produce growth-geometry split linear and non-linear matter power spectra, so that the matter power spectrum can be computed with a different $\Omega_{\rm m}^{\rm geo}$ and $\Omega_{\rm m}^{\rm growth}$.

📌 Additional Notes

Here is our ongoing overleaf project with more information on the equations we are using.

👥 Assignees

• Myself
• Tagging also: @zsirap , @itutusaus , @matmartinelli