Bias loop corrections to the galaxy bispectrum

The combination of the power spectrum and bispectrum is a powerful way of breaking degeneracies between galaxy bias and cosmological parameters, enabling us to maximize the constraining power from galaxy surveys. Recent cosmological constraints treat the power spectrum and bispectrum on an uneven footing: they include one-loop bias corrections for the power spectrum but not the bispectrum. To bridge this gap, we develop the galaxy bias description up to fourth order in perturbation theory, conveniently expressed through a basis of Galilean invariants that clearly split contributions that are local and nonlocal in the second derivatives of the linear gravitational potential. In addition, we consider relevant contributions from short-range nonlocality (higher-derivative terms), stress-tensor corrections, and stochasticity. To sidestep the usual renormalization of bias parameters that complicates predictions beyond leading order, we recast the bias expansion in terms of multipoint propagators, which take a simple form in our split basis with loop corrections depending only on bias parameters corresponding to nonlocal operators. We show how to take advantage of Galilean invariance to compute the time evolution of bias and present results for the fourth-order parameters for the first time. We also discuss the possibilities of reducing the bias parameter space by using the evolution of bias and exploiting degeneracies between various bias contributions in the large-scale limit. Our baseline model allows us to verify these simplifications for any application to large-scale structure data sets.

Figure 1

The $n$th multipoint propagator is given by the sum of all reducible diagrams with $n$ legs of incoming momentum ${\mathit{k}}_1,\dots ,{\mathit{k}}_n$ (the external lines). The vertices correspond to the kernels associated with terms in the bias expansion; e.g., the vertices of the three diagrams on the right-hand side are given by ${\overline{b}}_n$, ${\overline{b}}_{n+2}$, and ${\overline{b}}_{n+4}$ for the expansion in Eq. (1). If terms nonlocal in the matter density enter the bias relation, these kernels acquire a scale dependence based on all incoming momenta. The crossed circles stand for (linear) power spectra.

Figure 2

Galaxy power spectrum, reconstructed from multipoint propagators, which are represented by the shaded circles with incoming and outgoing momentum $\mathit{k}$. The sum runs over the number of connected internal lines, each of which produces a linear power spectrum depicted by a crossed circle.

Figure 3

Galaxy bispectrum, expressed through multipoint propagators. The sum runs over the number of connected internal lines of each pair of propagators (shaded circles). At most one of the three indices can be zero, so that the overall diagram remains a connected graph.

Figure 4

Top panel: comparison of the tree-level and one-loop galaxy power spectrum for three different values of the linear bias parameter. Second- and third-order bias parameters are fixed in terms of $b_1$ by means of the peak-background split (PBS) and local Lagrangian (LL) predictions. Bottom panel: relative difference between the tree-level and one-loop models for the same three cases.

Figure 5

Relative difference between the tree-level and one-loop galaxy bispectrum as a function of $x_2=k_2/k_1$ and $x_3=k_3/k_1$, integrated over a thin shell centered on three different values of $k_1$. The linear bias parameter is given by $b_1=1.8$, while $b_2$ and $b_3$ are determined from the PBS relations, and all other parameters are fixed using the LL approximation. Note that Eq. (106) was evaluated using a grid of bispectra with fixed binning for $k_1$, $k_2$, and $k_3$, leading to an absence of very squeezed configurations.

Figure 6

Subset of diagrams that contribute to a nonzero large-scale limit ($k_1$, $k_2$, $k_3\to 0$) of the galaxy bispectrum. The first diagram appears at one-loop level, the second at two-loop, and the last two at three-loop. The bias constants $b_i$ indicate the value of the multipoint propagators (shaded circles), i.e., constants that do not vanish in the large-scale limit.