Peak-background split

The peak-background split decomposes density fluctuations into long- and short-wavelength parts. A long overdensity lowers the effective local collapse threshold and thereby changes the abundance of halos.

For a universal abundance depending on halo peak height, the peak-background split gives a potential-response coefficient $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c{b}_L$, with $b_L$ the linear Lagrangian halo bias. If $\delta =\alpha (k)\phi$, the corresponding deterministic galaxy bias correction is $\Delta b=b_{\phi }/\alpha (k)$, proportional to $k^{-2}$ on sufficiently large scales. For a physical Eulerian tracer, $b_E=1+b_L$, so $b_{\phi }=2f_{\mathrm{NL}}{\delta }_c(b_E-1)$. The deterministic tracer power spectrum is $(b_E+\Delta b{)}^2{P}_m$; it remains nonnegative even if a negative correction changes the sign of the tracer-matter correlation.

Linear Lagrangian halo bias is the fractional response of halo abundance to a long-wavelength overdensity in the initial coordinates. For Press-Schechter halos, $b_L=({\nu }^2-1)/{\delta }_c$.

Mapping matter and halos from their initial positions adds the matter displacement contribution, giving $b=1+b_L$. Mass conservation requires the mass-weighted consistency relation $\int (dn/dM)(M/\bar{\rho })b(M)dM=1$.