Emulating $Lambda$CDM-like expansion on the Phantom brane

In [Schmidt PRD 80 123003 (2009)], the author suggested that dynamical dark energy (DDE) propagating on the phantom brane could mimick $Lambda$CDM. Schmidt went on to derive a phenomenological expression for $rho_{rm DE}$ which could achieve this. We demonstrate that while Schmidts central premise is correct, the expression for $rho_{rm DE}$ derived in Schmidt (2009) is flawed. We derive the correct expression for $rho_{rm DE}$ which leads to $Lambda$CDM-like expansion on the phantom brane. We also show that DDE on the brane can be associated with a Quintessence field and derive a closed form expression for its potential $V(phi)$. Interestingly the $alpha$-attractor based potential $V(phi) propto coth^2{lambdaphi}$ makes braneworld expansion resemble $Lambda$CDM. However the two models can easily be distinguished on the basis of density perturbations which grow at different rates on the braneworld and in $Lambda$CDM.