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Kay Lehnert edited this page Feb 24, 2026 · 11 revisions

This Wiki presents notable changes to the CLASS source code and their physical and mathematical motivation.

What makes our model different from $\Lambda$CDM is that the DM models and DE (in the form of quintessence) are coupled to each other. Everything else remains equal to the concordance model. Therefore, we can focus on these terms in the Lagrangian:

$$L\supset-\frac{\nabla^\mu\phi\nabla_\mu\phi}{2}-V\left(\phi\right)-\rho_\textnormal{DM}.$$

We derive all equations in an abstract form in the Mathematica notebook xPert.nb. We use these equations in xPand.nb to get all necessary equations up to first-order perturbations in the Newtonian gauge. Those equations are then implemented in CLASS. In this Wiki, we elaborate on the technical implementation of those equations.

We considered three different models of dark matter:

  1. Standard Cold Dark Matter with $m$ constant
  2. Hubbelian Dark Matter, where $m=\sqrt{3}Hm_{0}$ with $H$ the Hubble parameter.
  3. Interacting Dark Matter, with $m\left(\phi\right)=\frac{m_0}{2}\left(1-\tanh\left(c\phi\right)\right)$, with $\phi$ the scalar dark energy field.

In addition, we consider a generalised coupling between dark matter and dark energy of the form

$$\Xi=\frac{3H}{\left(\rho_\textnormal{DM}+\rho_\textnormal{DE}\right)^{l-1}} \left(\mathfrak{a}\rho_\textnormal{DE}^{l-k}\cdot\rho_\textnormal{DM}^k+\mathfrak{b}\rho_\textnormal{DM}^{l-k}\cdot\rho_\textnormal{DE}^k\right) +\dot{\phi}\left(\frac{\mathfrak{c}\rho_\textnormal{DM}}{M}+\mathfrak{d}\rho_\textnormal{DM}\prime\right).$$

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