The perturbation series can be organized in different ways. This answer derives one way.

Recall how the Feynman-diagram series is derived from the functional integral formulation. Use the abbreviation $$ d\omega\equiv d^4x\ \sqrt{|g|} $$ and write $$ S = \int d\omega\ \left(\frac{\phi K\phi}{2}+V(\phi)\right) \tag{1} $$ with $$ K\equiv -(\nabla_\mu\nabla^\mu+m^2) \tag{2} $$ and $$ V(\phi)\equiv -\frac{\lambda}{4!}\phi^4. \tag{3} $$ Correlation functions are generated by taking derivatives of $$ Z[J]\propto \int [d\phi]\ \left(iS[\phi]+i\int d\omega\ \phi J\right) \tag{4} $$ with respect to the source $J(x)$. More explicitly, applying $$ \frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)} \tag{5} $$ to $Z[J]$ is the same as inserting a factor of $\phi(x)$ in the integrand. The small-coupling expansion comes from rewriting (4) as $$ Z[J]\propto \exp\left(i\int d\omega\ V\left(\frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)}\right)\right) Z_0[J] \tag{6} $$ and then expanding in powers of $V$, with \begin{align*} Z_0[J] &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{\phi K\phi}{2} +i\int d\omega\ \phi J\right) \\ &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{(\phi+K^{-1}J) K(\phi+K^{-1}J)}{2} -i\int d\omega\ \frac{J K^{-1}J}{2}\right) \\ &\propto \int [d\phi]\ \left(-i\int d\omega\ \frac{J K^{-1}J}{2}\right). \tag{7} \end{align*} This shows that we can account for the required factors of $\sqrt{|g|}$ by

• writing the measure of each spacetime integral as $d\omega\equiv d^4x\ \sqrt{|g|}$ instead of $d^4x$,

• using $K^{-1}=-(\nabla_\mu\nabla^\mu+m^2)^{-1}$ as the propagator, without any factor of $\sqrt{|g|}$,

• including an extra factor of $\left(\sqrt{|g|}\right)^{-4}$ for each vertex (from equation (6)).

This clearly isn't the only way to organize the series, and it may or may not be the most convenient way. The two-point correlation function is not equal to $K^{-1}$ in this approach, because applying two factors of (5) to $Z[J]$ gives $|g|^{-1/2}K^{-1}+O(\lambda)$. If we want the propagator in the perturbation series to be the same as the $\lambda=0$ version of the two-point correlation functions, then we can organize things differently.

The perturbation series can be organized in different ways. This answer derives one way.

Recall how the Feynman-diagram series is derived from the functional integral formulation. Use the abbreviation $$ d\omega\equiv d^4x\ \sqrt{|g|} $$ and write $$ S = \int d\omega\ \left(\frac{\phi K\phi}{2}+V(\phi)\right) \tag{1} $$ with $$ K\equiv -(\nabla_\mu\nabla^\mu+m^2) \tag{2} $$ and $$ V(\phi)\equiv -\frac{\lambda}{4!}\phi^4. \tag{3} $$ Correlation functions are generated by taking derivatives of $$ Z[J]\propto \int [d\phi]\ \exp \left(iS[\phi]+i\int d\omega\ \phi J\right) \tag{4} $$ with respect to the source $J(x)$. More explicitly, applying $$ \frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)} \tag{5} $$ to $Z[J]$ is the same as inserting a factor of $\phi(x)$ in the integrand. The small-coupling expansion comes from rewriting (4) as $$ Z[J]\propto \exp\left(i\int d\omega\ V\left(\frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)}\right)\right) Z_0[J] \tag{6} $$ and then expanding in powers of $V$, with \begin{align*} Z_0[J] &\propto \int [d\phi]\ \exp \left(i\int d\omega\ \frac{\phi K\phi}{2} +i\int d\omega\ \phi J\right) \\ &\propto \int [d\phi]\ \exp \left(i\int d\omega\ \frac{(\phi+K^{-1}J) K(\phi+K^{-1}J)}{2} -i\int d\omega\ \frac{J K^{-1}J}{2}\right) \\ &\propto \int [d\phi]\ \exp \left(-i\int d\omega\ \frac{J K^{-1}J}{2}\right). \tag{7} \end{align*} This shows that we can account for the required factors of $\sqrt{|g|}$ by

• writing the measure of each spacetime integral as $d\omega\equiv d^4x\ \sqrt{|g|}$ instead of $d^4x$,

• using $K^{-1}=-(\nabla_\mu\nabla^\mu+m^2)^{-1}$ as the propagator, without any factor of $\sqrt{|g|}$,

• including an extra factor of $\left(\sqrt{|g|}\right)^{-4}$ for each vertex (from equation (6)).

This clearly isn't the only way to organize the series, and it may or may not be the most convenient way. The two-point correlation function is not equal to $K^{-1}$ in this approach, because applying two factors of (5) to $Z[J]$ gives $|g|^{-1/2}K^{-1}+O(\lambda)$. If we want the propagator in the perturbation series to be the same as the $\lambda=0$ version of the two-point correlation functions, then we can organize things differently.

The perturbation series can be organized in different ways. This answer suggests a natural way.

Recall how the Feynman-diagram series is derived from the functional integral formulation. Use the abbreviation $$ d\omega\equiv d^4x\ \sqrt{|g|} $$ and write $$ S = \int d\omega\ \left(\frac{\phi K\phi}{2}+V(\phi)\right) \tag{1} $$ with $$ K\equiv -(\nabla_\mu\nabla^\mu+m^2) \tag{2} $$ and $$ V(\phi)\equiv -\frac{\lambda}{4!}\phi^4. \tag{3} $$ Correlation functions are generated by taking derivatives of $$ Z[J]\propto \int [d\phi]\ \left(iS[\phi]+i\int d\omega\ \phi J\right) \tag{4} $$ with respect to the source $J(x)$. More explicitly, applying $$ \frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)} \tag{5} $$ to $Z[J]$ is the same as inserting a factor of $\phi(x)$ in the integrand. The small-coupling expansion comes from rewriting (4) as $$ Z[J]\propto \exp\left(i\int d\omega\ V\left(\frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)}\right)\right) Z_0[J] \tag{6} $$ and then expanding in powers of $V$, with \begin{align*} Z_0[J] &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{\phi K\phi}{2} +i\int d\omega\ \phi J\right) \\ &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{(\phi+K^{-1}J) K(\phi+K^{-1}J)}{2} -i\int d\omega\ \frac{J K^{-1}J}{2}\right) \\ &\propto \int [d\phi]\ \left(-i\int d\omega\ \frac{J K^{-1}J}{2}\right). \tag{7} \end{align*} This shows that we can account for the required factors of $\sqrt{|g|}$ by

• writing the measure of each spacetime integral as $d\omega\equiv d^4x\ \sqrt{|g|}$ instead of $d^4x$,

• using $K^{-1}=-(\nabla_\mu\nabla^\mu+m^2)^{-1}$ as the propagator, without any factor of $\sqrt{|g|}$,

• including an extra factor of $\left(\sqrt{|g|}\right)^{-4}$ for each vertex (from equation (6)).

This clearly isn't the only way to organize the series, but it's a natural way.

The perturbation series can be organized in different ways. This answer derives one way.

Recall how the Feynman-diagram series is derived from the functional integral formulation. Use the abbreviation $$ d\omega\equiv d^4x\ \sqrt{|g|} $$ and write $$ S = \int d\omega\ \left(\frac{\phi K\phi}{2}+V(\phi)\right) \tag{1} $$ with $$ K\equiv -(\nabla_\mu\nabla^\mu+m^2) \tag{2} $$ and $$ V(\phi)\equiv -\frac{\lambda}{4!}\phi^4. \tag{3} $$ Correlation functions are generated by taking derivatives of $$ Z[J]\propto \int [d\phi]\ \left(iS[\phi]+i\int d\omega\ \phi J\right) \tag{4} $$ with respect to the source $J(x)$. More explicitly, applying $$ \frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)} \tag{5} $$ to $Z[J]$ is the same as inserting a factor of $\phi(x)$ in the integrand. The small-coupling expansion comes from rewriting (4) as $$ Z[J]\propto \exp\left(i\int d\omega\ V\left(\frac{-i}{\sqrt{|g|}} \frac{\delta}{\delta J(x)}\right)\right) Z_0[J] \tag{6} $$ and then expanding in powers of $V$, with \begin{align*} Z_0[J] &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{\phi K\phi}{2} +i\int d\omega\ \phi J\right) \\ &\propto \int [d\phi]\ \left(i\int d\omega\ \frac{(\phi+K^{-1}J) K(\phi+K^{-1}J)}{2} -i\int d\omega\ \frac{J K^{-1}J}{2}\right) \\ &\propto \int [d\phi]\ \left(-i\int d\omega\ \frac{J K^{-1}J}{2}\right). \tag{7} \end{align*} This shows that we can account for the required factors of $\sqrt{|g|}$ by

• writing the measure of each spacetime integral as $d\omega\equiv d^4x\ \sqrt{|g|}$ instead of $d^4x$,

• using $K^{-1}=-(\nabla_\mu\nabla^\mu+m^2)^{-1}$ as the propagator, without any factor of $\sqrt{|g|}$,

• including an extra factor of $\left(\sqrt{|g|}\right)^{-4}$ for each vertex (from equation (6)).

This clearly isn't the only way to organize the series, and it may or may not be the most convenient way. The two-point correlation function is not equal to $K^{-1}$ in this approach, because applying two factors of (5) to $Z[J]$ gives $|g|^{-1/2}K^{-1}+O(\lambda)$. If we want the propagator in the perturbation series to be the same as the $\lambda=0$ version of the two-point correlation functions, then we can organize things differently.