In <flat gauge in cosmology>, the supplied expressions imply
$$
K_i{}^j-\delta_i{}^jK
=-2H\delta_i{}^j+2H\delta N\,\delta_i{}^j
-\partial_i\partial^j\psi+\delta_i{}^j\partial^2\psi
$$
to first order. The last two terms cancel after taking $\partial_j$, so the geometric part of the momentum constraint is $2H\partial_i\delta N$. Its matter part is
$$
-\frac{P_{,X}\dot{\bar\phi}}{M_{\rm Pl}^2}\partial_i\varphi.
$$
Therefore
$$
2H\partial_i\delta N
-\frac{P_{,X}\dot{\bar\phi}}{M_{\rm Pl}^2}\partial_i\varphi=0,
$$
and, after discarding a spatially homogeneous lapse mode,
$$
\boxed{\delta N=C\varphi,
\qquad
C=\frac{P_{,X}\dot{\bar\phi}}{2HM_{\rm Pl}^2}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 312 / 2 / iii / Solution

Flat gauge sets the scalar perturbation of the spatial metric to zero. Scalar fluctuations then reside in the lapse, shift, and matter fields, making the field fluctuation directly visible while the lapse and shift remain constrained variables.


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