Explicitly, our mode functions are $$f_{\mu }^{+}(k\tau )=\frac{\sqrt{\pi }}{2i}{e}^{+\pi \mu /2}{H}_{i\mu }^{(2)}(-k\tau ),$$(A22)$$f_{\mu }^{-}(k\tau )=\frac{i\sqrt{\pi }}{2}{e}^{-\pi \mu /2}{H}_{i\mu }^{(1)}(-k\tau ).$$(A23)and obey $f_{\mu }^{+}=(f_{\mu }^{-}{)}^{*}$.

Concretely, consider a diagrammatic expansion of the in-in path integral in which each vertex is labeled by $\pm$ according to its branch of the integration contour. At leading order, this gives $iG={\mathrm{\Pi }}^{++}+{\sum }_{\alpha ,\beta }{\mathrm{\Pi }}^{+\alpha }{\mathrm{\Sigma }}^{\alpha \beta }{\mathrm{\Pi }}^{\beta +}+\dots$, where ${\mathrm{\Pi }}^{\alpha \beta }$ are the free theory propagators for $\varphi$ with different operator orderings. Comparing with the leading order expansion of (9) leads to (15). Some combinatorics then shows that (9) will match the in-in sum over vertices at every order in $\mathrm{\Sigma }$.

Here we refer to the causality condition that the analytic structure of $G(\nu )$, which encodes the time-ordering, remains the same in the interacting theory: namely an even combination of $e^{\pm \pi \nu }$ terms which have singularities in the upper/lower half planes respectively.

Notice that $g_2=0$ would require $\rho =0$ for all energies above $\mathrm{\Lambda }$, i.e., that the theory is free. These bounds are strict inequalities when applied to theories with nontrivial interactions.

Alternatively, notice that the analytic structure of $\mathrm{\Sigma }$ implied by causality means we can express it as $\frac{1}{2\pi i}{\oint }_{\nu }d{\mu }^{\prime }2{\mu }^{\prime }\mathrm{\Sigma }({\mu }^{\prime })/({\mu }^{\prime 2}-{\nu }^2-i\epsilon )$ around the pole in the lower half of the complex plane, and then deform this contour so that it runs over ${\int }_{-\infty }^{+\infty }d{\mu }^{\prime }$ and then change variables to ${\mu }^{\prime 2}$ to produce (29).

Note that a low-energy observer can only determine $g_n(\mathrm{\Lambda })$ from the EFT coefficients when $\mathrm{\Lambda }$ is below threshold, in this case $\mathrm{\Lambda }<2\mu$. In Figure 2 we have plotted all values of $\mathrm{\Lambda }$ to illustrate that our bounds are always satisfied, regardless of the EFT series (33) converging.

This is analoguous to scattering with spacelike momenta on Minkowski, for which the $i\epsilon$ becomes unimportant because there is no operator ordering ambiguity.

(58) makes it clear that ${\rho }_J$ and $\mathrm{\Sigma }$ are related to ${\rho }_{\varphi }$ and $⟨\varphi \varphi ⟩$ by an amputation of two propagators that closely resembles LSZ reduction: $\mathrm{\Sigma }$ is therefore closer in spirit to an $S$-matrix, and is the natural object for EFT matching.

For the equal-mass case considered above, these singularities are canceled by zeroes of $\sinh (\pi \nu )$ and the cutoff is instead set by singularities at $|\nu {|}^2=4{\mu }^2+d^2/4$).

See for instance [129] for a description of how to “square” an in-out $S$-matrix on de Sitter into an in-in Bunch-Davies correlator.

Explicitly, they are given by $$F_{\nu }^j(1/z)=\frac{1}{\sqrt{2}}\mathrm{\Gamma }(j\pm i\nu )P_{i\nu -\frac{1}{2}}^{\frac{1}{2}-j}(z){\left(\sqrt{z^2-1}\right)}^{\frac{1}{2}-j}$$(A24)and are specified by a half-integer index $j=\frac{d-2}{2}(n-1)$ that depends on the number of spatial dimensions and the number of particles at each vertex. $j=1/2$ for $⟨{\sigma }^4{⟩}_s$ in $d=3$.