Effective field theory of large-scale structure (EFTofLSS)

The effective field theory of large-scale structure represents unresolved short-scale dynamics by an effective stress tensor. Its leading deterministic correction is proportional to $k^2{\delta }^{(1)}$ and renormalizes ultraviolet-sensitive loop contributions.

For the one-dimensional B411 contribution to the one-loop matter bispectrum, contracting two hard inputs in the fourth-order field gives $-(I_{\Lambda }/2)k^2{\delta }^{(2)}$, where $I_{\Lambda }={\int }_{q_h<|q|<\Lambda }dq{P}_L(q)/(2\pi q^2)$. A counterterm $C(\Lambda ){\partial }_x^2{\delta }^{(2)}$ cancels this cutoff-dependent part when $C(\Lambda )=-I_{\Lambda }/2+C_{\mathrm{finite}}$. This sign follows from ${\partial }_x^2\leftrightarrow -k^2$; the finite coefficient describes unresolved short-scale dynamics.

The leading deterministic power-spectrum counterterm in the effective field theory of large-scale structure is proportional to $-2c_{\mathrm{eff}}^2{k}^{2}P(k)$. Its coefficient depends on the coarse-graining convention and runs so that it cancels the cutoff dependence of the one-loop $P_{13}$ contribution.

For internal momentum $q\gg k$, the P22 contribution to the one-loop matter power spectrum starts at $k^4$, while twice the P13 contribution to the one-loop matter power spectrum starts at $k^{2}P(k)$. The latter has the same momentum dependence as the effective sound-speed counterterm in large-scale structure. Short-scale stochastic contributions start at $k^4$ by mass and momentum conservation.

A bias expansion expresses a tracer density, such as a galaxy or halo density, as a renormalized sum of local and nonlocal operators built from matter density, tidal fields, derivatives, and stochastic fields. Its coefficients encode unresolved tracer formation physics.

Galaxy bias describes how fluctuations in galaxy number density differ from fluctuations in the underlying matter density. On sufficiently large scales, the leading deterministic relation is ${\delta }_g=b_1{\delta }_m$.

Number conservation under the displacement from Lagrangian coordinates to Eulerian coordinates givesThe added one is the linear matter-density response of the coordinate Jacobian.

The simplest nonlinear local bias model isFor Gaussian matter fluctuations, its leading galaxy three-point function is proportional to $b_1^2{b}_2$.

The galaxy bispectrum is the connected momentum-space three-point function of the galaxy density contrast after removing its momentum-conserving delta function.