Adding the second metric to the GR dust solution introduces a system of coupled ODE governing two conformal factors of two metrics.

In such a case, the ODE for the initial data becomes:

---

---

Here we demanded that the matter distribution profile is the same as in GR case. This is a generalized Lane-Emden equation. In GR, the Lane-Emden equation occurs in the case of a polytropic fluid (which is a special case in the above equation for a fixed beta model). Hence, adding a second metric makes a pressureless fluid in one sector to appear as the influence of a nontrivial polytropic fluid on both metrics.

An example of the initial conditions is given in Figure 4.

---

---

In Figure 4, the testbed GR initial conditions are shown in magenta for comparison. The bimetric initial conditions are depicted in blue/red and appear as a wavy departure from GR. The evolution is under development :)

---

---