On Cosmological Correlators at One Loop

TL;DR

This paper presents a systematic framework for one-loop cosmological correlators of massless scalars in flat space, revealing why correlator loops are simpler than wavefunction corrections via a cosmological Baikov parametrisation and time-ordered decompositions. Through detailedBubble and Triangle analyses, it derives reduced loop integrals, classifies kinematic singularities with Landau techniques, and demonstrates a clean dilogarithmic structure for the triangle. A partial-energy factorisation theorem shows how singular kinematics relate to flat-space amplitudes and lower-point correlators, enabling a tree-level-like interpretation even at one loop. Overall, the work provides explicit, tractable analytic results and a roadmap toward a broader one-loop basis for flat-space cosmology with potential phenomenological applications in primordial non-Gaussianity.

Abstract

We study equal-time in-in correlators of massless scalar fields in flat space at one loop. Using the time-ordered decomposition of correlators together with a cosmological analogue of the Baikov representation, we systematically construct relatively simple loop integrals and make manifest why, in this setting, loop corrections to correlators are simpler than those of wavefunction coefficients. As benchmark examples, we analyse the bubble and triangle diagrams. The bubble exhibits a UV divergence that can be removed by a local counterterm, while the triangle yields a finite result, which we evaluate explicitly in terms of dilogarithms using an integral transform for the Laplacian Green's function. We classify the kinematic singularities of these diagrams using Landau analysis, identifying novel types of singular behaviour, and validate this analysis against the explicit results. Finally, we derive a factorisation property of one-loop cosmological correlators at singular kinematics, relating them to flat-space loop amplitudes and lower-point tree-level correlators.

Figures (32)

• Figure 1: Real (solid line) and imaginary (dashed line) parts of the triangle integral $I_\triangle$ with $X_1=3.55$, $X_2=2.76$, $k_1=1.15$, $k_2=0.93$, and $k_3=1.00$. Two distinct $i\varepsilon$ prescriptions have been implemented, as discussed in detail in Sec. \ref{['sec:ExplicitResults']}.
• Figure 2: The nine different time-orderings for a three-site graph grouped into the three different sectors.
• Figure 3: Four examples of consistent tubings for the time-ordered three-site graphs.
• Figure 4: The nature of the in-in time contour is such that not all graph topologies appear in the correlator. In the example of the bubble, there is no diagram involving one time-ordered and one disconnected propagator, they always come in pairs.
• Figure 5: The first two diagrams contribute to the wavefunction but not to the in-in correlator. The rightmost diagram is excluded from both due to an ill-defined time ordering.