\section{Introduction}

A central goal of modern cosmology is to uncover the physics that generated the primordial density perturbations and thereby seeded the large-scale structures (LSS) we see around us. 
The coherent nature of the cosmic microwave background (CMB) anisotropies suggests that the fluctuations were created at very early times, possibly during a period of inflation~\cite{inflation}.

One of the few observational probes that allows us access to the physics of that epoch is primordial non-Gaussianity~\cite{Komatsu:2009kd}.
At present, the best constraints on non-Gaussianity are coming from the CMB~(e.g.~\cite{Komatsu:2010hc}), but LSS is emerging as a promising complementary observable (e.g.~\cite{Desjacques:2010nn, Liguori:2010hx}).
Historically, the usefulness of LSS as a tool for early universe cosmology has been viewed with some suspicion, since non-linear evolution can itself produce significant non-Gaussianity even if the initial conditions were perfectly Gaussian.  
Disentangling any primordial non-Gaussianity from these late time effects always seemed like a messy business.
This attitude has changed somewhat when it was discovered that non-Gaussian initial conditions lead to a {\it scale-dependent} clustering of galaxies on large scales~\cite{Dalal:2007cu, Matarrese:2008nc}. 
In particular, it was shown that non-linear mode coupling induces a modulation of the local short-scale power $\sigma_8({{\boldsymbol{x}}})$ by the long-wavelength gravitational potential $\Phi({{\boldsymbol{x}}})$. 
This results in a biasing of halos (or galaxies) that is proportional to $\Phi$ rather than the dark matter density~$\delta$ (or $\nabla^2 \Phi$).
Crucially, the appearance of $\Phi$ rather than $\delta$ in the halo bias implies a specific form of scale-dependence that cannot be created dynamically (i.e.~by late time processes). This is the main reason that halo bias is such a robust probe of the initial conditions.

\begin{figure}[h!]
   \centering
       \includegraphics[width=9.cm]{3+4point2.pdf}
   \caption{The squeezed limit of the three-point function, $k_1 \to 0$, gives the dominant contribution to the scale-dependent halo bias. A stochastic form of scale-dependent halo bias arises if the four-point function is large in the collapsed limit, $k_{12} \equiv |{\boldsymbol{k}}_1+{\boldsymbol{k}}_2| \to 0$.}
  \label{fig:34}
\end{figure}



In this paper, we study {\it stochastic} halo bias on large scales.
The term `stochastic' here refers to the fact that
the halo over-density is not 100\% correlated to the matter over-density on large scales,
i.e.~the halo-halo power spectrum $P_{{\rm h} {\rm h}}(k)$ is boosted relative to the matter-halo power spectrum $P_{{\rm m}{\rm h}}(k)$. Formally,
this means that
\begin{equation}
P_{{\rm h}\h}(k) > b^2(k) P_{{\rm m}\m}(k) + \frac{1}{n_{{\rm h}}}\ ,
\end{equation} 
where $b(k) \equiv P_{{\rm m}{\rm h}}(k)/P_{{\rm m}\m}(k)$ is the halo bias, and $n_{{\rm h}}$ is the halo number density.
Large-scale stochastic bias arises in non-Gaussian models
when the small-scale power $\sigma_8({{\boldsymbol{x}}})$ varies from point to point, but in a way that isn't completely correlated with the local value of $\Phi({{\boldsymbol{x}}})$ and its derivatives.
This is most easily demonstrated in models with multiple fields, where the small-scale power may depend on fields that do not contribute to the (linearized) gravitational potential. 
Our goal in this paper is to provide an understanding of the origin of stochastic bias in a model-independent way.  In the absence of significant isocurvature perturbations, all the relevant information must be encoded in the correlation functions of gravitational potential $\Phi$.  It will be useful to define
\begin{align}
\hat{f}_{\mathsmaller{\rm NL}}&\equiv \frac{1}{4} \lim_{k_1 \to 0} \frac{\xi^{(3)}_{\Phi}({\boldsymbol{k}}_1,{\boldsymbol{k}}_2,{\boldsymbol{k}}_3)}{P_1 P_2} \ , \\
\hat{\tau}_{\mathsmaller{\rm NL}}  &\equiv \frac{9}{100} \lim_{k_{12}\to 0} \frac{\xi^{(4)}_\Phi({\boldsymbol{k}}_1,{\boldsymbol{k}}_2,{\boldsymbol{k}}_3,{\boldsymbol{k}}_4)}{P_1 P_3 P_{12}} \ ,
\end{align}
where $\langle \Phi_{{{\boldsymbol{k}}}_1} \cdots \Phi_{{{\boldsymbol{k}}}_N} \rangle_{\rm c} \equiv (2\pi)^3 \xi^{(N)}_{\Phi}({{\boldsymbol{k}}}_1, \cdots \hskip -1pt , {{\boldsymbol{k}}}_n)\, \delta_{\rm D}({{\boldsymbol{k}}}_1 + \cdots + {{\boldsymbol{k}}}_n)$ and $P_i \equiv \xi_\Phi^{(2)}(k_i)$.
This parametrizes the amplitude of the three-point function in the {\it squeezed limit}, $k_1 \ll {\rm min}\{k_2,k_3\}$, and the amplitude of the four-point function in the  {\it collapsed limit}, $k_{12} \equiv |{{\boldsymbol{k}}}_1 + {{\boldsymbol{k}}}_2| \ll {\rm min}\{k_i\}$.
As we will show, stochastic bias arises if the `collapsed four-point function' is {\it not} equal to the square of the `squeezed three-point function', i.e.~if $\hat{\tau}_{\mathsmaller{\rm NL}}  \ne (\frac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$. 
There exists a well-known theoretical constraint on the relative size of $\hat{\tau}_{\mathsmaller{\rm NL}}$ and $(\frac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$.
If only a single field (which may or may not be the inflaton) generates the primordial curvature perturbation and its non-Gaussianity, then $\hat{\tau}_{\mathsmaller{\rm NL}} = (\frac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$ \cite{Byrnes:2006vq} and the biasing is non-stochastic. On the other hand, if multiple coupled fields generate the non-Gaussianity, then $\hat{\tau}_{\mathsmaller{\rm NL}}$ can be larger\footnote{No matter how the fluctuations were created, the parameters have to satisfy the Suyama-Yamaguchi inequality $\hat{\tau}_{\mathsmaller{\rm NL}}  \geq (\tfrac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$ \cite{SY} (see also \cite{Sugiyama:2011jt, Lewis:2011au, Smith:2011if, Quasi4,Kehagias:2012pd}). This is easy to understand: we can think of $\hat{f}_{\mathsmaller{\rm NL}}$ as a measure of the large-scale correlation between the potential $\Phi$ and
the locally measured small-scale power, $\hat{f}_{\mathsmaller{\rm NL}} \sim \langle \Phi_\ell \Phi_s^2\rangle/\langle \Phi_\ell^2 \rangle \langle \Phi_s^2\rangle$. On the other hand,   $\hat{\tau}_{\mathsmaller{\rm NL}}$  is a measure of the
large-scale variance in the small-scale power, $\hat{\tau}_{\mathsmaller{\rm NL}} \sim \langle \Phi_s^2 \Phi_s^2 \rangle_{\rm c}/\langle \Phi_\ell^2 \rangle \langle \Phi_s^2\rangle^2$. The inequality  $\hat{\tau}_{\mathsmaller{\rm NL}}  \geq (\tfrac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$ then
arises simply as the condition that the correlation coefficient between the small-scale power and $\Phi$ must be between $-1$ and $1$.} than $(\frac{6}{5}\hat{f}_{\mathsmaller{\rm NL}})^2$ \cite{Suyama:2010uj, ByrnesSource} and the biasing will be stochastic.
We will discuss classes of inflationary theories that predict precisely this kind of observational signature~\cite{Tseliakhovich:2010kf,Chen:2009zp, Baumann:2011nk}.
This provides the opportunity of using scale-dependent stochastic bias\footnote{We should note that in this paper we are interested in large-scale stochastic bias. On small scales, non-linear evolution and astrophysical processes can create local stochasticity, which is not relevant in our study.}  as a probe of any early universe physics associated with a boosted collapsed four-point function---just like the non-stochastic scale-dependent bias is a powerful probe of the squeezed three-point function.

More generally, we find that the large-scale non-stochastic bias
can be written as a sum over $N$-point functions $\xi_\Phi^{(N)}({\boldsymbol{k}}_1, \cdots \hskip -1pt,{\boldsymbol{k}}_N)$ evaluated in the squeezed limit $k_1 \ll \min\{k_2,\cdots \hskip -1pt,k_N\}$.\footnote{More precisely, $k_1$ is fixed to the large scale $k$ where we are computing the bias, and $k_2,\cdots \hskip -1pt ,k_N$ are integrated
over a broad range of scales near the halo collapse scale $k_{\rm h} \sim \rho_{\rm m}^{1/3} M^{-1/3}$.}
The stochastic bias, on the other hand,
involves a double sum over $(M+N)$-point functions
$\xi_\Phi^{(M+N)}({\boldsymbol{k}}_1,\cdots \hskip -1pt,{\boldsymbol{k}}_{M+N})$ evaluated in the collapsed limit $|{\boldsymbol{k}}_1 + \cdots + {\boldsymbol{k}}_M| \ll \min\{k_i\}$.
Stochastic bias arises if any collapsed $(M+N)$-point function is boosted relative to the product of the corresponding
squeezed $(M+1)$-point and $(N+1)$-point functions.
In all physically interesting cases that we are aware of, this effect 
is due to the collapsed four-point function
being boosted relative to the square of the three-point function (i.e.~the case $M=N=2$). Therefore,
we will generally interpret stochastic
bias as a probe of the collapsed four-point function.
The main result of this paper is a general pair of formulas, eqs.~(\ref{equ:PmhFinal}) and~(\ref{equ:PhhFinal}), 
for the non-stochastic and stochastic parts of the bias, for completely general non-Gaussian initial conditions parametrized by the $N$-point cumulants $\xi_\Phi^{(N)}({\boldsymbol{k}}_1, \cdots\hskip -1pt, {\boldsymbol{k}}_N)$.

\vskip 6pt
The outline of the paper is as follows:
We will begin, in Section~\ref{sec:stochastic}, with a qualitative explanation of scale-dependent stochastic bias.
In Section~\ref{sec:barrier}, we will show how our intuitive understanding is borne out in the barrier crossing model of structure formation.
In Section~\ref{sec:examples}, we will illustrate these results with explicit examples.
In each case, we also derive our predictions in the peak-background split formalism. 
In a companion paper~\cite{equivalence}, we prove the mathematical equivalence of barrier crossing and peak-background split.
We present our conclusions in Section~\ref{sec:conclusions}.
Finally, Appendix~\ref{sec:convergence} discusses the convergence of the Edgeworth expansion for local non-Gaussianity.


\section{Stochastic Bias} \label{sec:stochastic}

Galaxies reside in dark matter halos.
For Gaussian initial conditions and at long wavelengths, the fluctuations in the density of halos $\delta_{\rm h}$ can be expressed as an expansion in the linear matter density field $\delta$. At linear order, the two are simply related by a numerical factor---the bias  $b_g$---i.e.~$\delta_{\rm h} = b_g \delta$.  
This simple bias relation gets modified for non-Gaussian initial conditions, due to a coupling between short and long-wavelength modes. The short modes determine the collapse of dark matter halos, while long modes modulate the density on large scales, effectively raising or lowering the threshold for the formation of collapsed objects. A non-zero three-point function  affects the variance of the short modes, leading to a dependence of the number density of halos on the amplitude of the long modes.
For local non-Gaussianity\footnote{In real space, local non-Gaussianity is parametrized as $\Phi({{\boldsymbol{x}}}) = \phi({{\boldsymbol{x}}}) + f_{\mathsmaller{\rm NL}} (\phi^2({{\boldsymbol{x}}}) - \langle \phi^2 \rangle)$, where $\phi$ is Gaussian.} this leads to a dependence of the halo density on the long-wavelength gravitational potential $\Phi$ rather than the matter density $\delta \propto\nabla^2 \Phi$. This leads to a characteristic {\it scale-dependence} in the bias relation, $\Delta b \propto k^{-2}$~\cite{Dalal:2007cu}.
It is this scale-dependence that allows us to trust the large-scale bias as a probe of initial conditions. Crucially, the dependence of the halo density on $\Phi$ is {not} something that could be mimicked by local dynamics. Dynamical processes don't care about the local value of the potential, but are only sensitive to tidal forces which are proportional to $\nabla^2 \Phi$ and $\dot \Phi$ (essentially this is a consequence of the equivalence principle). Any dependence of the small-scale power on $\Phi$ itself can therefore only come from the initial conditions.
This is what makes scale-dependent bias such a promising probe of early universe physics, despite all the astrophysical uncertainties associated with galaxy formation.
 
{\it Stochastic} bias arises whenever the density of halos is not 100\% correlated with the potential $\Phi$ or its derivatives.
In order to develop some intuition, we now give a schematic derivation of the effect. In the next section, we will upgrade this to a more formal analysis in the barrier crossing approach.
 If we assume that the primordial perturbations are adiabatic, then the formation of halos can only depend on local physics of the fluctuations.  Nevertheless, long-wavelength variations of the number of halos may depend, not only on the local value of the linear density field, but on all of its local correlation functions.  Assuming only locality, we may therefore write the local halo number density as
\begin{equation}
n_{{\rm h}}({{\boldsymbol{x}}}) = \bar n_{\rm h}(\delta({{\boldsymbol{x}}})  ; [ \delta^n]({{\boldsymbol{x}}})) \ ,
\end{equation}
where $[ ... ]$ denotes an average over a small region of characteristic size $\ell$ that is centered around~${\boldsymbol{x}}$.  Long-wavelength fluctuations in the number of halos can then be understood as a Taylor expansion,
\begin{equation}
\delta_{{\rm h}}({{\boldsymbol{x}}}) \equiv \frac{\delta n_{\rm h}}{\bar n_{\rm h}} = b_g \delta({{\boldsymbol{x}}}) +  \beta [ \delta^2 ]({{\boldsymbol{x}}}) + \cdots \ , \label{equ:Taylor}
\end{equation}
where $b_g$ is the Gaussian bias and 
\begin{equation}
\beta \equiv   \frac{\partial\ln n_{\rm h}}{\partial   {[ \delta^2 ]}} \ .
\end{equation}  
It is easy to see (e.g.~by splitting all fields into long and short modes), that for local non-Gaussianity the short-scale power is modulated by the gravitational potential, $ [ \delta^2 ]\approx [ \delta^2 ]_g \,  (1 + 4 f_{\mathsmaller{\rm NL}} \Phi({{\boldsymbol{x}}}))$. 
This is the origin of scale-dependent bias in local non-Gaussianity.

Using the expansion (\ref{equ:Taylor}), we can also evaluate correlation functions between two spatially separated points ${{\boldsymbol{x}}}$ and ${{\boldsymbol{x}}}'$.  We use a prime to indicate that fields are evaluated at ${{\boldsymbol{x}}}'$, while fields without a prime are evaluated at ${{\boldsymbol{x}}}$.
The matter-halo correlation, in a large region of size $L \gg \ell$, then is
\begin{equation}\label{eqn:simplehm}
\frac{\langle \delta_{\rm h} \delta' \rangle}{\langle \delta \delta' \rangle} = b_g + \beta \frac{\langle [ \delta^2 ] \hskip 1pt \delta' \rangle}{\langle \delta \delta' \rangle} + \cdots \ ,
\end{equation}
while the halo-halo correlation is
\begin{equation}\label{eqn:simplehh}
\frac{\langle \delta_{\rm h} \delta_{\rm h}{}'  \rangle}{\langle \delta \delta' \rangle} = b_g^2 + 2 b_g \beta \frac{\langle [ \delta^2 ] \hskip 1pt \delta' \rangle}{\langle \delta \delta' \rangle} + \beta^2 \frac{\langle [ \delta^2 ] [ \delta^2 ]' \rangle}{\langle \delta \delta' \rangle} + \cdots \ .
\end{equation}
This leads to the possibility that the bias inferred from $\langle \delta_{\rm h} \delta \rangle$ is not equal to the bias inferred from $\langle \delta_{\rm h} \delta_{\rm h}{}' \rangle $.
We characterize this so-called stochasticity of the halo bias by the following parameter\footnote{In practice, we also have to subtract shot noise contributions from $\langle \delta_{\rm h} \delta_{\rm h}{}' \rangle$ and $\langle \delta_{\rm h} \delta' \rangle$---see \S\ref{ssec:barrier_crossing}.}
\begin{equation}
r \equiv \frac{\langle \delta_{\rm h} \delta_{\rm h}{}' \rangle}{\langle \delta \delta' \rangle} - \left( \frac{\langle \delta_{\rm h} \delta' \rangle}{\langle \delta \delta' \rangle}\right)^2  \ . \label{equ:rdef}
\end{equation}
Using eqs.~(\ref{eqn:simplehm}) and (\ref{eqn:simplehh}), we find
\begin{equation}
r = \beta^2 \left[ \frac{\langle [ \delta^2 ] [ \delta^2 ]' \rangle}{\langle \delta \delta' \rangle}  - \left( \frac{\langle [ \delta^2 ] \hskip 1pt \delta' \rangle}{\langle \delta \delta' \rangle}\right)^2 \, \right] + \cdots\ . \label{equ:rr}
\end{equation}
This simple argument gives reliable intuition for the origin of stochasticity.  Specifically, we see that if a local variation in the amplitude of $[ \delta^2 ]({{\boldsymbol{x}}})$ is uncorrelated with $\delta({{\boldsymbol{x}}}')$, then there is no extra contribution to the bias in eq.~(\ref{eqn:simplehm}).  Nevertheless, the halo-halo correlation function in eq.~(\ref{eqn:simplehh}) can still be modified by long-wavelength variations in $[ \delta^2 ]({{\boldsymbol{x}}})$.  
Moreover, the result (\ref{equ:rr}) makes it clear that stochasticity arises from a non-trivial four-point function of the primordial potential.  In fact, the real space correlation function $\langle [ \delta^2] ({{\boldsymbol{x}}}) [ \delta^2  ] ({{\boldsymbol{x}}}') \rangle$ relates to the collapsed limit of the four-point function in Fourier space, i.e.~$\lim_{|{{\boldsymbol{k}}}_{1}+{{\boldsymbol{k}}}_2| \to 0} \langle \Phi_{{{\boldsymbol{k}}}_1}  \Phi_{{{\boldsymbol{k}}}_2}   \Phi_{{{\boldsymbol{k}}}_3}  \Phi_{{{\boldsymbol{k}}}_4}\rangle$.


\section{ Predictions from Barrier Crossing}
\label{sec:barrier}

In this section, we give a formal derivation of stochastic bias using the classic barrier crossing method of Press and Schechter \cite{Press:1973iz}. 
Our goal is to obtain an expression for the stochasticity coefficient~(\ref{equ:rdef}) in terms of the cumulants of the smoothed density field. These in turn can be related to $N$-point functions of the primordial potential and hence contain information about the initial conditions.

\subsection{Definitions and Notation}

We begin with some basic definitions and a description of our notation.
Let $\hat \delta({\boldsymbol{x}}, z)$ denote the linear density field (to be distinguished by the hat from the non-linear density field $\delta$). 
The linearized Poisson equation relates $\hat \delta$ to the primordial potential $\Phi$,
\begin{equation}
\hat \delta_{{\boldsymbol{k}}}(z) = \alpha(k,z) \Phi_{{\boldsymbol{k}}}\ ,
\end{equation}
where
\begin{equation} \alpha(k,z) \equiv \frac{2}{3} \frac{k^2}{\Omega_m H_0^2} \, T(k) D(z) \ .\end{equation}
Here, $T(k)$ is the matter transfer function normalized such that $T(k) \rightarrow 1$ as $k \rightarrow 0$ and $D(z)$ is the linear growth factor (as function of redshift $z$), normalized so that $D(z) = (1+z)^{-1}$ in matter domination.  For notational simplicity, we will from now on suppress the redshift argument from all quantities.
We use $\delta_\mathsmaller{M}({\boldsymbol{x}})$ for the linear field smoothed with a top-hat window function with radius\footnote{The smoothing scale $R_\mathsmaller{M}$ corresponds to the comoving size of halos of mass $M$ in Lagrangian space.} $R_\mathsmaller{M} \equiv (3M/4 \pi \bar{\rho}_{\rm m})^{1/3}$, so that  
\begin{equation}
\delta_\mathsmaller{M}({{\boldsymbol{x}}}) = \int_{{\boldsymbol{k}}} e^{- i{{\boldsymbol{k}}} \cdot {{\boldsymbol{x}}}} \, W_\mathsmaller{M}(k) \hat \delta_{{\boldsymbol{k}}}  =  \int_{{\boldsymbol{k}}} e^{- i{{\boldsymbol{k}}} \cdot {{\boldsymbol{x}}}} \, \alpha_\mathsmaller{M}(k) \Phi_{{\boldsymbol{k}}} \ ,
\end{equation}
where $\int_{\boldsymbol{k}}\, (\cdot) \equiv \int \frac{{\rm d}^3 {\boldsymbol{k}}}{(2\pi)^3}\, (\cdot)$,
\begin{equation}\label{eqn:tophat}
W_\mathsmaller{M}(k) \equiv 3\, \frac{\sin(kR_\mathsmaller{M}) - kR_\mathsmaller{M} \cos(kR_\mathsmaller{M})}{(kR_\mathsmaller{M})^3}\ ,
\end{equation}
and $\alpha_\mathsmaller{M}(k) \equiv W_\mathsmaller{M}(k) \alpha(k)$. 
Let $\sigma_\mathsmaller{M} = \langle \delta^2_\mathsmaller{M} \rangle^{1/2}$  be the rms amplitude of the smoothed density field, and $\kappa_n(M)$ be its $n$-th non-Gaussian cumulant, 
\begin{equation}
\kappa_n(M) = \frac{\langle \delta_{\mathsmaller{M}}^n \rangle_{\rm c}}{\sigma_{\mathsmaller{M}}^n} \ ,  \label{eq:kappa_def}
\end{equation}
where the subscript `${\rm c}$' indicates the use of a connected correlation function.
Since $\delta_\mathsmaller{M}$ and $\sigma_\mathsmaller{M}$ are defined via linear theory, $\kappa_n(M)$ is independent of redshift. 
Similar definitions apply to the unsmoothed field $\hat \delta$, in which case we denote the variance and cumulants by $\hat \sigma$ and $\kappa_{\hat n}$.

Ultimately, we will be interested in two-point clustering statistics.
Let ${{\boldsymbol{x}}}$ and ${{\boldsymbol{x}}}'$ be two points separated by a distance $r \equiv |{{\boldsymbol{x}}}-{{\boldsymbol{x}}}'|$. Moreover, let a prime indicate that the field is evaluated at~${{\boldsymbol{x}}}'$, e.g.~$\delta_{\mathsmaller{M}}' \equiv \delta_{\mathsmaller{M}}({{\boldsymbol{x}}}')$. Fields without a prime are evaluated at ${{\boldsymbol{x}}}$.
The joint cumulants are then defined by  
\begin{align}
\kappa_{\hat m,n}(r,M) &\ \equiv\ \frac{\langle \hat \delta^m (\delta_{\mathsmaller{M}}')^n\rangle_{\rm c}}{\hat \sigma^m\sigma_{\mathsmaller{M}}^n}\ , \label{equ:kk1} \\
\kappa_{m,n}(r,M,\bar M) &\ \equiv\ \frac{\langle (\delta_{\mathsmaller{M}})^m (\delta_{\bar \mathsmaller{M}}')^n\rangle_{\rm c}}{\sigma_{\mathsmaller{M}}^{m} \sigma_{\bar \mathsmaller{M}}^n}  \ . \label{equ:kk2}
\end{align}
These cumulants can be related to $N$-point functions of the gravitational potential,
\begin{equation}
\langle \Phi_{{{\boldsymbol{k}}}_1}  \Phi_{{{\boldsymbol{k}}}_2} \cdots  \Phi_{{{\boldsymbol{k}}}_N}\rangle_{\rm c} = (2\pi)^3 \delta_{\rm D}({{\boldsymbol{k}}}_{1 2\dots N}) \, \xi_\Phi^{(N)}({{\boldsymbol{k}}}_1, {{\boldsymbol{k}}}_2, \dots, {{\boldsymbol{k}}}_N)\ ,
\end{equation}
where ${{\boldsymbol{k}}}_{1 2\dots N} \equiv {{\boldsymbol{k}}}_{1} + {{\boldsymbol{k}}}_{2} + \cdots + {{\boldsymbol{k}}}_{N}$.



\subsection{Edgeworth Expansions}
\label{sec:edge}

The probability density functions (PDFs) of weakly non-Gaussian random variables have well-defined Edgeworth expansions (for a review see e.g.~\cite{Bernardeau:2001qr}). 
Consider first the variables $\delta_{\mathsmaller{M}}$ and $\delta'_{\mathsmaller{M}}$.
It will be convenient to define the rescaled fields 
\begin{equation}
\nu \equiv  \frac{\delta_{\mathsmaller{M}}}{\sigma_{\mathsmaller{M}}} \qquad {\rm and} \qquad \nu \hskip 1pt{}' \equiv \frac{\delta_{\mathsmaller{M}}'}{\sigma_\mathsmaller{M}}\ , 
\end{equation}
with 
$\langle \nu \rangle = \langle \nu \hskip 1pt{}' \rangle = 0$ and $\langle \nu^2 \rangle = \langle (\nu \hskip 1pt{}')^2 \rangle = 1$. 
The cumulants in eqs.~(\ref{eq:kappa_def}) and (\ref{equ:kk2}) then become $\kappa_n = \langle \nu^n\rangle_{\rm c}$ and $\kappa_{m,n}=\langle \nu^m (\nu \hskip 1pt{}')^n\rangle_{\rm c}$. 
The Edgeworth expansion for the marginal PDF is
\begin{align}
p(\nu) &= \exp \left( \sum_{n \ge 3} \frac{(-1)^n}{n!}  \kappa_n  \frac{\partial^n}{\partial \nu^n} \right) p_g(\nu) \ , \qquad {\rm where} \quad p_g(\nu) \equiv \frac{1}{\sqrt{2\pi}} e^{- \frac{1}{2} \nu^2}\ .
\end{align}
The first few terms can be written as
\begin{align}
p(\nu)  & \ = \ \left( 1+ \frac{\kappa_3}{3!} H_3(\nu) + \frac{\kappa_4}{4!} H_4(\nu) + \cdots \right) p_g(\nu) \ , \label{equ:p1}
\end{align}
where the functions $H_n(\nu)$ are Hermite polynomials
\begin{equation}
H_n(\nu) \equiv (-1)^n e^{\frac{1}{2} \nu^2} \frac{d^n}{d\nu^n} e^{-\frac{1}{2} \nu^2 } \ .
\end{equation}
Similarly, the Edgeworth expansion for the joint PDF i
\begin{align}
p(\nu,\nu \hskip 1pt{}') = \exp \left( \kappa_{1,1} \frac{\partial^2 }{\partial \nu \partial \nu \hskip 1pt{}'} + \sum_{m+n \ge 3} \frac{(-1)^{m+n}}{m! n!} \kappa_{m,n} \frac{\partial^{m+n}}{\partial \nu^m \partial (\nu \hskip 1pt{}')^n} \right) p_g(\nu) p_g(\nu \hskip 1pt{}')\ . \label{equ:edge1}
\end{align}
In Appendix~\ref{sec:convergence}, we discuss the convergence properties of this expansion.
In the next section, we will use it to compute halo-halo correlations. 

The matter-halo case is completely analogous: to construct the joint PDF of the variables $\hat\delta$ and $\delta'_{\mathsmaller{M}}$, we define
rescaled variables $\hat\nu = \hat\delta/\hat\sigma$ and $\nu' = \delta'_{\mathsmaller{M}}/\sigma_{\mathsmaller{M}}$.  The joint PDF $p(\hat\nu,\nu')$ is then given
by the Edgeworth series~(\ref{equ:edge1}) with the cumulant $\kappa_{m,n}$ replaced by $\kappa_{\hat m,n}$.


\subsection{Barrier Crossing}
\label{ssec:barrier_crossing}

In the simplest version of the barrier crossing formalism~\cite{Press:1973iz}, halos of mass $\geq M$ are identified with regions where the \textit{linearly} evolved smoothed density field exceeds a constant threshold value~$\delta_c$ for collapse. The halo number density $n_{\rm h}({\boldsymbol{x}})$ is then given by
\begin{equation}
n_{\rm h}({{\boldsymbol{x}}}) = 2\hskip 1pt \frac{\bar{\rho}_{\rm m}}{M} \, \Theta(\delta_\mathsmaller{M}({{\boldsymbol{x}}}) - \delta_c)\ ,   \label{eq:nh_barrier}
\end{equation}
with $\Theta$ the Heaviside step function.
It has been shown numerically that $\delta_c \approx 1.42$ produces good results~\cite{Grossi:2009an}, but for our analytical calculations we don't need to specify a particular value for $\delta_c$.
The fraction of space occupied by regions above the collapse threshold is 
\begin{equation}
f(M)  
=
 \int_{\nu_c}^{\infty} [{\rm d} \nu]\ p(\nu)\ ,
\end{equation}
where
$\nu_c(M) \equiv \delta_c/\sigma_\mathsmaller{M}$.
Using the Edgeworth expansion (\ref{equ:p1}), we find\footnote{In our notation the halo mass function is $d n_{\rm h}/d M = -\hskip 1pt \bar{\rho}_{\rm m}/M (d f / d M)$.}
\begin{equation}
f(M) = \frac{1}{2} \, {\rm erfc} \left( \frac{\nu_c}{\sqrt{2}} \right) + p_g(\nu_c) \left[ \frac{\kappa_3(M)}{3!} H_2(\nu_c) + \frac{\kappa_4(M)}{4!} H_3(\nu_c) + \cdots \right]\ . \label{equ:x0}
\end{equation}
When interpreting calculations in the barrier crossing model, it must be kept in mind that eq.~(\ref{eq:nh_barrier}) for $n_{\rm h}({\boldsymbol{x}})$ is the number density of halos in {\em Lagrangian} space.  Our convention throughout this paper is that the power spectra $P_{{\rm m}{\rm h}}(k)$ and $P_{{\rm h}\h}(k)$ are always computed in Lagrangian space.  In particular, $b_g$ denotes the Lagrangian bias.  The relevant quantity to compare to observations or simulations is the {\it Eulerian} bias 
which, to lowest order, is given by $b_g^{\mathsmaller{\rm E}} = 1 + b_g$.

The barrier crossing model also neglects shot noise contributions which arise from the finite halo number
density $n_{\rm h}$.  Throughout this paper, $P_{{\rm h}\h}$ always denotes the halo-halo power spectrum after subtracting
the shot noise contribution $1/n_{\rm h}$.
(There are also shot noise, or one-halo, contributions to the matter-halo power spectrum $P_{{\rm m}{\rm h}}$, which are usually negligible, but are a leading source of stochastic bias in the Gaussian case \cite{Hamaus:2010im,Smith:2010gx}.)

\subsubsection{Matter-Halo Correlations}

The correlation between the halo field at
${\boldsymbol{x}}'$ and the dark matter field at ${\boldsymbol{x}}$ is given by
\begin{align}
\hat \xi(r,M) &\ =\ \frac{M}{2 \bar{\rho}_{\rm m}} \int_{- \infty}^\infty [{\rm d} \hat \delta] \int_{-\infty}^\infty [{\rm d} \delta_{\mathsmaller{M}}']  \, \hat \delta({{\boldsymbol{x}}}) \, n_{\rm h}({\boldsymbol{x}}')\, p(\hat \delta, \delta_{\mathsmaller{M}}') \ . \label{equ:hatxi1}
\end{align}
In the rescaled variables $\hat \nu \equiv \hat \delta({\boldsymbol{x}})/\hat \sigma$ and $\nu \hskip 1pt{}' \equiv \delta_\mathsmaller{M}({\boldsymbol{x}}')/\sigma_\mathsmaller{M}$, this becomes
\begin{align}
\hat \xi(r,M) &\ =\ \hat \sigma \int_{- \infty}^\infty [{\rm d} \hat \nu] \int_{\nu_c}^\infty [{\rm d} \nu \hskip 1pt{}']  \, \hat \nu \, p(\hat \nu,\nu \hskip 1pt{}') \ . \label{equ:hatxi2}
\end{align}
It will be convenient to work in momentum space via $\hat \xi(k, M) = \int {\rm d}^3 {{\boldsymbol{r}}} \, e^{i {{\boldsymbol{k}}}\cdot {{\boldsymbol{r}}} } \hat \xi(r, M)$.  To describe the correlations of halos in the mass bin $[M,M+{\rm d} M]$, we take derivatives with respect to $M$.
The matter-halo power spectrum is then given by
\begin{equation}
P_{\rm mh}(k,M) = \frac{d \hat \xi(k,M)}{d M} \, \left(\frac{d f(M)}{d M}\right)^{-1}\ . \label{equ:Pmh}
\end{equation}
To compute the correlation function~(\ref{equ:hatxi2}), we
substitute the Edgeworth expansion (\ref{equ:edge1}) for $p(\hat \nu, \nu \hskip 1pt{}')$.
Only terms with exactly one $\hat \nu$-derivative survive the integration, and we therefore find
\begin{equation}
\hat \xi(k,M) =  \hat \sigma\, p_g(\nu_c) \left[ \kappa_{\hat 1,1} + \frac{H_1(\nu_c)}{2!} \kappa_{\hat 1,2} +  \frac{H_2(\nu_c)}{3!} \kappa_{\hat 1,3} + \frac{H_3(\nu_c)}{3!} \kappa_{\hat 1,1} \star \kappa_3 + \cdots \right] \ , \label{equ:hatxi3}
\end{equation}
where $\star$ denotes a convolution.  
We see that the matter-halo correlations, or equivalently the non-stochastic part of the halo bias, only depend on the following cumulants 
\begin{equation}
\kappa_{\hat 1,n}(k,M) = \hat \sigma^{-1} \sigma_{\mathsmaller{M}}^{-n}  \left(\, \prod_{i=1}^n \int_{{{\boldsymbol{q}}}_i} \alpha_\mathsmaller{M}(q_i) \right) \alpha(k) \langle \Phi_{{\boldsymbol{k}}} \Phi_{{{\boldsymbol{q}}}_1} \cdots \Phi_{{{\boldsymbol{q}}}_n} \rangle_{\rm c} \ .
\end{equation}
Moreover, we note that the large-scale limit, $\lim_{k\to 0} \kappa_{\hat 1,n}(k,M) $, is determined by the {squeezed limit} of the primordial $(n+1)$-point function \cite{Desjacques:2011mq}, 
\begin{equation}
\lim_{k\to 0}  \xi_\Phi^{(n+1)}({{\boldsymbol{k}}}, {{\boldsymbol{q}}}_1, \cdots \hskip -1pt ,{{\boldsymbol{q}}}_n ) \ .
\end{equation}
The explicit form of the cumulants $\kappa_{\hat 1,n \ge 2}$ depends on the type of non-Gaussianity. We compute some examples in Section~\ref{sec:examples}.

Keeping only linear terms\footnote{In Appendix~\ref{sec:convergence}, we explain that the lowest order cumulants usually dominate and that products of cumulants are suppressed.} in eq.~(\ref{equ:hatxi3}), we get \begin{equation} 
\hat \xi(k,M) = p_g(\nu_c) ( \kappa_{\hat 1, 1} \hat \sigma \sigma_\mathsmaller{M}) \left[\frac{1}{\sigma_\mathsmaller{M}} + \sum_{n=2}^\infty \frac{H_{n-1}(\nu_c)}{n!} \, f_{\hat 1,n} + \cdots \right] \ ,  \label{eq:xi_mh_linear}
\end{equation}
where we defined 
\begin{equation}
f_{\hat 1, n}(k,M) \equiv \frac{\kappa_{\hat 1, n}(k,M)}{\kappa_{\hat 1, 1}(k,M) \, \sigma_\mathsmaller{M}}\ . \label{equ:f1n}
\end{equation}
It was convenient to factor out the Gaussian term $\kappa_{\hat 1,1}  \hat \sigma \sigma_\mathsmaller{M}$, since at long wavelengths it becomes the matter power spectrum
\begin{equation}
\kappa_{\hat 1, 1} \, \hat \sigma \sigma_\mathsmaller{M} =  \int_{{\boldsymbol{q}}} \alpha(k) \alpha_\mathsmaller{M}(q) \langle \Phi_{{\boldsymbol{k}}} \Phi_{{\boldsymbol{q}}} \rangle =  W_\mathsmaller{M}(k)  P_{{\rm m} {\rm m}}(k) \ \xrightarrow{k\to 0} \ P_{{\rm m} {\rm m}}(k)\ .
\end{equation}
Evaluating eq.~(\ref{equ:Pmh}), we find
\begin{align}
P_{\rm mh}(k)  &\ \xrightarrow{k\to 0} \  P_{{\rm m} {\rm m}}(k) \left[ \, b_g + \sum_{n=2}^\infty \left( \beta_n +  \tilde \beta_n \frac{d }{d \ln \sigma_\mathsmaller{M}} \right) f_{\hat 1,n} + \cdots  \right]\ , \label{equ:PmhFinal}
\end{align}
where
\begin{align}
b_g &\equiv  \frac{1}{\sigma_\mathsmaller{M}}\frac{\nu_c^2 -1}{\nu_c}\quad , \quad \beta_{n} \equiv \frac{H_n(\nu_c)}{n!} \quad {\rm and} \quad \tilde \beta_{n} \equiv \frac{H_{n-1}(\nu_c)}{n!\,\nu_c}\ .
\end{align}
The ellipses in eq.~(\ref{equ:PmhFinal}) stand for terms that are non-linear in the cumulants.
For local non-Gaussianity, the derivative terms $df_{\hat 1,n}/d\ln \sigma_\mathsmaller{M} $ will be negligible, but in principle, we can keep them (and sometimes we have to). 

Our expression~(\ref{equ:PmhFinal}) agrees with the general formula for the non-stochastic bias given in \cite{Desjacques:2011mq}; however,
ref.~\cite{Desjacques:2011mq} implicitly found that non-Gaussianity cannot generate large-scale stochastic bias. In the next section, we will find the
opposite conclusion. The disagreement is easy to understand: Ref.~\cite{Desjacques:2011mq} claims after their eq.~(40) that contributions to $P_{{\rm h}\h}$
from cumulants $\kappa_{m,n}$ with $m,n\ge 2$ must approach a constant as $k\rightarrow 0$. This is not true for general non-Gaussian initial conditions and exceptions to that statement are precisely what causes the effects discuss in this paper.


\subsubsection{Halo-Halo Correlations}
\label{sec:hh}

Next, we consider the correlation between the halo fields at ${\boldsymbol{x}}$ and ${\boldsymbol{x}}'$,
\begin{align}
\xi(r,M,\bar M) &\ =\  \frac{M \bar M}{4 \bar{\rho}_{\rm m}^2} \int_{-\infty}^\infty [{\rm d} \delta_{{\mathsmaller{M}}}] \int_{-\infty}^\infty [{\rm d} \delta_{{\bar\mathsmaller{M}}}']  \, n_{\rm h}({\boldsymbol{x}}) n_{\rm h}({\boldsymbol{x}}')\, p(\delta_{{\mathsmaller{M}}}, \delta_{{\bar \mathsmaller{M}}}') \ .
\end{align}
In the rescaled variables $\nu \equiv \delta_\mathsmaller{M}({\boldsymbol{x}})/\sigma_\mathsmaller{M}$ and $\nu \hskip 1pt{}' \equiv \delta_{\bar \mathsmaller{M}}({\boldsymbol{x}}')/\sigma_{\bar \mathsmaller{M}}$,  this becomes
\begin{equation}
\xi(r, M, \bar M) \ = \ \int_{\nu_c}^\infty [{\rm d} \nu] \int_{\bar \nu_c}^\infty [{\rm d} \nu \hskip 1pt{}'] \ p(\nu,\nu \hskip 1pt{}')\ .  \label{equ:xi1}
\end{equation}
Notice that, in principle, we have allowed for two distinct mass thresholds, $M$ and $\bar M$.
The power spectrum of halos in the mass bins $[M, M+{\rm d} M]$ and $[\bar M, \bar M + {\rm d} \bar M]$ then is
\begin{equation}
P_{{\rm h} \bar {\rm h}}(k) = \frac{d^2 \xi(k,M,\bar M)}{d M d \bar M} \, \left(\frac{d f(M)}{d M} \frac{d f(\bar M)}{d \bar M}\right)^{-1} \ .
\end{equation}
For simplicity, we will restrict the following presentation to correlations of equal mass halos, $M= \bar M$. The power spectrum for a narrow mass bin around $M$ is then given by
\begin{equation}
P_{{\rm h} {\rm h}}(k) = \frac{d^2 \xi(k,M, \bar{M})}{d M d \bar{M}} \bigg |_{\bar{M} = M} \, \left(\frac{d f(M)}{d M}\right)^{-2}  \ .\label{equ:Phh}
\end{equation}
To compute the correlation function~(\ref{equ:xi1}), we
substitute the Edgeworth expansion (\ref{equ:edge1})
 for $p(\nu,  \nu \hskip 1pt {}')$,
\begin{align}
\xi(k,M, \bar M) &= p_g(\nu_c) p_g(\bar \nu_c) \left[ \kappa_{1, 1} +  \frac{1}{2} \big( \kappa_{2, 1} H_1(\nu_c) + \kappa_{ 1, 2} H_1(\bar \nu_c) \big) +  \frac{1}{6} \big( \kappa_{3, 1} H_2(\nu_c) + \kappa_{ 1, 3} H_2(\bar \nu_c) \big) \nonumber \right. \\
 &\left. \hspace{4cm}+  \frac{1}{4} \kappa_{2,2} H_1(\nu_c) H_1(\bar \nu_c) +  \frac{1}{2} \kappa_{1,1} \star \kappa_{1,1} + \ \cdots \right] \ . \label{equ:xi2}
\end{align}
The form of higher-order cumulants, such as $\kappa_{1,2}$, $\kappa_{1,3}$ and $\kappa_{2,2}$, again depends on the type of non-Gaussianity. 
We compute some examples in Section~\ref{sec:examples}.

Keeping only the terms linear in $\kappa_{m,n}$ (this approximation will be justified in Appendix~\ref{sec:convergence}) in eq.~(\ref{equ:xi2}), we find 
\begin{align}
\xi(k,M,\bar M) &\ =\ p_g(\nu_c) p_g(\bar \nu_c) (\kappa_{1,1} \sigma_\mathsmaller{M} \sigma_{\bar \mathsmaller{M}})\left[ \frac{1}{\sigma_\mathsmaller{M} \sigma_{\bar \mathsmaller{M}}} +  \sum_{n=2}^\infty \left(\frac{1}{\sigma_\mathsmaller{M}}  \frac{H_{n-1}(\bar \nu_c) }{n!}  \, f_{ 1, n}  + \frac{1}{\sigma_{\bar \mathsmaller{M}}}   \frac{H_{n-1}(\nu_c) }{n!}  \, f_{ n, 1}\right)  \nonumber \right. \\
& \left. \hspace{5.1cm} +\sum_{m=2}^\infty \sum_{n=2}^\infty  \frac{H_{m-1}(\nu_c)}{m!}  \frac{H_{n-1}(\bar \nu_c)}{n!} f_{m,n}\ +\ \cdots \ \right] \ , \label{equ:xi3}
\end{align}
where
\begin{align}
f_{1,n}(k,M,\bar M) &\equiv \frac{\kappa_{1,n}(k,M,\bar M)}{\kappa_{1,1}(k,M,\bar M) \sigma_{\bar\mathsmaller{M}}} \ \ \ \, \, \qquad \mbox{for $n \ge 1$}\ , \\
f_{m,n}(k,M,\bar M) & \equiv \frac{\kappa_{m,n}(k,M,\bar M)}{\kappa_{1,1}(k,M,\bar M) \sigma_{\mathsmaller{M}} \sigma_{\bar \mathsmaller{M}}} \qquad \mbox{for $m,n \ge 2$}\ .
\end{align}
We again factored out the Gaussian contribution, $\kappa_{1,1} \, \sigma_\mathsmaller{M} \sigma_{\bar \mathsmaller{M}}  \ \xrightarrow{k\to 0} \ P_{{\rm m} {\rm m}}(k)$.
Note that $f_{1,n} = f_{\hat 1,n}$ in the large scale limit $k \ll R_{\mathsmaller{M}}^{-1}$, where $f_{\hat 1,n}$ was defined in eq.~(\ref{equ:f1n}).
Substituting (\ref{equ:xi3}) into (\ref{equ:Phh}), we get
\begin{align}
P_{{\rm h} {\rm h}}(k) &\ \xrightarrow{k\to 0} \  P_{{\rm m} {\rm m}}(k) \left[\,  b_g^2 + 2 b_g \sum_{n=2}^\infty \left( \beta_n + \tilde \beta_n \frac{\partial}{\partial \ln \sigma_\mathsmaller{M}} \right)  f_{1,n}\right. \nonumber \\
& \hspace{2cm} \left. + \sum_{m=2}^\infty \sum_{n=2}^\infty 
   \left( \beta_m + \tilde\beta_n\, \frac{\partial}{\partial\ln\sigma_{\mathsmaller{M}}} \right)
   \left( \beta_m + \tilde\beta_n\, \frac{\partial}{\partial\ln\sigma_{\bar\mathsmaller{M}}} \right) f_{m,n} +\ \cdots\ \right]\ . \label{equ:PhhFinal}
\end{align}
Note that, while in the end we always take $M=\bar M$ in this paper, $M$ and $\bar M$ are independent variables
when calculating partial derivatives of $f_{m,n}(M,\bar M)$.


\subsection{Stochastic Halo Bias}

We now combine the above results to evaluate the stochasticity coefficient
\begin{equation}
r \equiv \frac{P_{\rm hh} }{P_{\rm mm}} - \left( \frac{P_{\rm mh}}{P_{\rm mm}}\right)^2 \ ,
\end{equation}
where, as usual, it is understood that shot noise is subtracted from $P_{{\rm h} {\rm h}}$  and $P_{{\rm m} {\rm h}}$.
Substituting eqs.~(\ref{equ:PmhFinal}) and (\ref{equ:PhhFinal}), we find
\begin{equation}
r \xrightarrow{k\to 0} \sum_{m=2}^\infty \sum_{n=2}^\infty
   \left( \beta_m + \tilde\beta_m \frac{\partial}{\partial\ln\sigma_{\mathsmaller{M}}} \right)
   \left( \beta_n + \tilde\beta_n \frac{\partial}{\partial\ln\sigma_{\bar\mathsmaller{M}}} \right) f_{m,n}
  - \left[ \, \sum_{n=2}^\infty \left( \beta_n + \tilde\beta_n \frac{\partial}{\partial\ln\sigma_\mathsmaller{M}} \right) f_{n,1} \right]^2 \ . \label{equ:master}
\end{equation}
We note that cumulants $\kappa_{m,n}(k)$ with $m,n\ge 2$ contribute to the halo-halo power spectrum but not the matter-halo power spectrum~(\ref{equ:PmhFinal}),
so stochastic halo bias is sourced by these cumulants.  These cumulants can be written in terms of the $(m+n)$-point functions of the gravitational potential,
\begin{align}
\kappa_{m,n}(k,M,\bar M) & \ \xrightarrow{k\to 0}\ \frac{1}{\sigma_{\mathsmaller{M}}^m \sigma_{\bar\mathsmaller{M}}^n} \left( \prod_{i=1}^{m-1} \int_{{\boldsymbol{q}}_i} \alpha_\mathsmaller{M}(q_i) \right) \left( \prod_{j=1}^{n -1} \int_{{\boldsymbol{q}}_j'} \alpha_{\bar \mathsmaller{M}}(q_j') \right) \alpha_\mathsmaller{M}(q) \,\alpha_{\bar \mathsmaller{M}}(q')
\nonumber \\
& \hspace{1cm}\ \ \times 
  \xi_\Phi^{(m+n)}\left( {\boldsymbol{q}}_1,\cdots \hskip -1pt ,{\boldsymbol{q}}_{m-1},- {\boldsymbol{q}}+{\boldsymbol{k}},{\boldsymbol{q}}'_1,\cdots \hskip -1pt,{\boldsymbol{q}}'_{n-1},-{\boldsymbol{q}}' -{\boldsymbol{k}} \right) \ ,
\end{align}
where ${\boldsymbol{q}} \equiv \sum_{i=1}^{m-1}{\boldsymbol{q}}_i$ and ${\boldsymbol{q}}' \equiv \sum_{j=1}^{n-1}{\boldsymbol{q}}_j'$.
We see that, in general, large-scale stochastic bias arises whenever an $(m+n)$-point function $\xi_\Phi^{(m+n)}({\boldsymbol{k}}_1,\cdots \hskip -1pt,{\boldsymbol{k}}_{m+n})$ 
is boosted in the collapsed limit $|\sum_{i=1}^m {\boldsymbol{k}}_i| \rightarrow 0$, relative to the product of the corresponding squeezed $(m+1)$-point and $(n+1)$-point
functions. 
In the next section, we will compute eq.~(\ref{equ:master}) for a few interesting examples.
In most cases, we will get stochastic bias from the case $m=n=2$,
i.e.~a collapsed four-point function $\lim_{|{\boldsymbol{k}}_1+{\boldsymbol{k}}_2|\rightarrow 0} \xi^{(4)}_\Phi({\boldsymbol{k}}_1,{\boldsymbol{k}}_2,{\boldsymbol{k}}_3,{\boldsymbol{k}}_4)$ 
which is larger than the square of the squeezed three-point function $\lim_{k_1\rightarrow 0} \xi^{(3)}_\Phi({\boldsymbol{k}}_1,{\boldsymbol{k}}_2,{\boldsymbol{k}}_3)$.






\newpage
\section{Examples}
\label{sec:examples}

In this section, we discuss several physical mechanisms that lead to 
 stochastic halo bias.
For each example, we will derive the result in two different ways:
\begin{enumerate}
\item[1)] using a peak-background split (PBS) method;
\item[2)] using the barrier crossing analysis of the previous section.
\end{enumerate}
We demonstrate explicitly that both approaches lead to the same answers.


\subsection{$\tau_{\rm NL}$ Cosmology} 
\label{sec:tnl}

A simple phenomenological way to get a boosted collapsed limit for the four-point function is the following generalization of the local ansatz to multiple fields
\begin{equation}
\Phi = A_i \phi_i + B_{ij} \left( \phi_i \phi_j - \langle \phi_i \phi_j \rangle \right)\ ,
\end{equation}
with the Einstein summation convention understood.
This structure arises, for example, in the curvaton model of \cite{Tseliakhovich:2010kf} (see also \cite{Smith:2010gx}), 
\begin{equation}\label{eqn:taunlmodel}
\Phi = \phi + \psi + f_{\mathsmaller{\rm NL}}(1+\Pi)^2 \, (\psi^2 - \langle \psi^2 \rangle)\ , \qquad {\rm where} \quad \frac{P_\phi}{P_\psi} \equiv \Pi\ .
\end{equation}
Here, $\phi$ and $\psi$ are uncorrelated Gaussian random fields with power spectra that are proportional to each other. 
The three- and four-point functions take the local form
\begin{eqnarray} \xi^{(3)}_{\Phi}({\boldsymbol{k}}_1, {\boldsymbol{k}}_2, {\boldsymbol{k}}_3) &=& f_{\mathsmaller{\rm NL}} \big[P_1 P_2 + \mathrm{5 \, perms.} \big] + \mathcal{O}(f_{\mathsmaller{\rm NL}}^3)\ , \label{equ:tnl3}\\
\xi^{(4)}_{\Phi}({\boldsymbol{k}}_1, {\boldsymbol{k}}_2, {\boldsymbol{k}}_3, {\boldsymbol{k}}_4)& = & 2\left (\tfrac{5}{6} \right )^2 \tau_{\mathsmaller{ \rm NL}} \big[P_1P_2 P_{13} + \mathrm{23 \, perms.} \big] + \mathcal{O}(\tau_{\mathsmaller{ \rm NL}}^2) \ , \label{equ:tnl4}
\end{eqnarray}
where we have defined $P_i \equiv P_\Phi(k_i)$ and $P_{ij} \equiv P_\Phi(|{{\boldsymbol{k}}}_i + {{\boldsymbol{k}}}_j|)$. However, unlike the single-field local ansatz, now $\tau_{\mathsmaller{ \rm NL}}$ need not be equal to $\left ( \frac{6}{5} f_{\mathsmaller{\rm NL}} \right )^2 $. Instead, the ansatz (\ref{eqn:taunlmodel}) implies $\tau_{\mathsmaller{ \rm NL}} \equiv \left ( \frac{6}{5} f_{\mathsmaller{\rm NL}} \right )^2 (1+\Pi)$, in agreement with the Suyama-Yamaguchi inequality, $\tau_{\mathsmaller{ \rm NL}} \ge \left ( \frac{6}{5} f_{\mathsmaller{\rm NL}} \right )^2$~\cite{SY} (see also~\cite{Sugiyama:2011jt, Lewis:2011au, Smith:2011if, Quasi4, Kehagias:2012pd}). 
The following limits will be useful in computing the cumulants required in the barrier crossing calculation:
\begin{align}
\lim_{k_1 \to 0 } \xi^{(3)}_{\Phi}({{\boldsymbol{k}}}_1, {{\boldsymbol{k}}}_2, {{\boldsymbol{k}}}_3) &= 4 f_{\mathsmaller{\rm NL}} P_1 P_2 \label{equ:A1}\ , \\
\lim_{k_1 \to 0 }   \xi^{(4)}_{\Phi}({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3,{{\boldsymbol{k}}}_4) &=   8 \left( \tfrac{5}{6} \right) ^2 \tau_{\mathsmaller{ \rm NL}} \, P_1 \big[ P_2 P_3 +  P_2 P_4 +  P_3 P_4 \big] \ , \label{equ:A2} \\
\lim_{k_{12} \to 0 }   \xi^{(4)}_{\Phi}({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3,{{\boldsymbol{k}}}_4) &= 16 \left( \tfrac{5}{6} \right) ^2 \tau_{\mathsmaller{ \rm NL}} \, P_{12} P_1 P_3\ . \label{equ:A3} \end{align}
However, before we discuss the explicit barrier crossing result, we present an alternative derivation using the peak-background split approach.

\subsubsection{Peak-Background Split}
\label{sssec:tnl_pbs}

PBS is a heuristic procedure for predicting the large-scale clustering statistics of dark matter halos.
All fields are split into long and short modes---i.e.~the Gaussian fields in eq.~(\ref{eqn:taunlmodel}) are written as $\phi = \phi_s + \phi_\ell$ and $\psi = \psi_s + \psi_\ell$. 
The short scales ($\lesssim R_\mathsmaller{M} \lesssim 10$~Mpc$/h$) determine halo formation, while the long scales ($\gtrsim 100$ Mpc$/h$) are the ones on which we want to measure the clustering of halos. Long modes are therefore always much larger than the Lagrangian size of the halos that we consider, i.e.~$R_\mathsmaller{M}  k_{\ell} \ll 1 $.
The precise split into long and short modes isn't important for physical observables, as long as it satisfies the above constraints.

The long-wavelength modes alter the statistical properties of the small-scale fluctuations. For instance,
to lowest order, the locally measured small-scale power is
$\sigma = \bar \sigma \left[1 + 2f_{\mathsmaller{\rm NL}} (1+\Pi) \psi_\ell \right]$, and the locally measured halo number density is
\begin{equation}
n_{\rm h}({{\boldsymbol{x}}}) = \bar n_{\rm h}\left( \delta_c-\delta_\ell \, ; \bar\sigma  \left[1 + 2 f_{\mathsmaller{\rm NL}} (1+\Pi) \psi_\ell \right]\right)\ .
\end{equation}
Taylor expanding this expression, we get
\begin{equation}
\delta_{\rm h} \equiv \frac{\delta n_{\rm h}}{\bar n_{\rm h}} = b_g \delta_\ell + \beta_f(1+\Pi) f_{\mathsmaller{\rm NL}} \psi_\ell\ \ ,
\end{equation}
where
\begin{equation}
b_g \equiv \frac{\partial \ln n_{\rm h}}{\partial \delta_\ell}\quad {\rm and} \quad \beta_f \equiv 2 \frac{\partial \ln n_{\rm h}}{\partial \ln \sigma} \ . \label{equ:coefs}
\end{equation}
Hence, we find
\begin{equation}
P_{\rm mh} = \left( b_g + \beta_f \frac{f_{\mathsmaller{\rm NL}}}{\alpha(k)}\right) P_{\rm mm}\ ,
\end{equation}
and
\begin{align}
P_{\rm hh} 
&= \left( b_g^2 + 2 b_g \beta_f \frac{f_{\mathsmaller{\rm NL}}}{\alpha(k)} + \beta_f^2 \frac{\left(\tfrac{5}{6}\right)^2 \tau_{\mathsmaller{ \rm NL}}}{\alpha^2(k)} \right) P_{\rm mm} \ .
\end{align}
This leads to large-scale halo stochasticity of the form
\begin{equation}
r = \left( \left(\tfrac{5}{6}\right)^2 \tau_{\mathsmaller{ \rm NL}} - f_{\mathsmaller{\rm NL}}^2 \right) \frac{\beta_f^2}{\alpha^2(k)}\ . \label{equ:rPBS}
\end{equation}
As $\Pi \rightarrow 0$, this reduces to the classic $f_{\mathsmaller{\rm NL}}$ model, with $\tau_{\mathsmaller{ \rm NL}} = \left(\frac65 f_{\mathsmaller{\rm NL}} \right)^2$ and hence no stochasticity.


\subsubsection{Barrier Crossing}
\label{sssec:tnl_barrier_crossing}

Next, we  show that eq.~(\ref{equ:rPBS}) can be reproduced precisely from the barrier crossing analysis of the previous section. In Appendix~\ref{sec:convergence}, we show that only the lowest-order cumulants will be significant.
Here, we calculate the relevant cumulants explicitly:
Using eq.~(\ref{equ:A1}),
we get
\begin{equation}
f_{1,2} = f_{\hat 1,2} \ \xrightarrow{k\to 0} \ 4\, \frac{f_{\mathsmaller{\rm NL}}}{\alpha(k)}\ . \label{equ:316}
\end{equation}
The order-of-magnitude estimates in Appendix~\ref{sec:convergence} suggest that this will be the dominant contribution. In particular, we expect, $f_{1,2} > f_{1,3}$. We can confirm this explicitly.
Using eq.~(\ref{equ:A2}),
we get 
\begin{equation}
f_{1,3} = f_{\hat 1,3} \ \xrightarrow{k\to 0} \ 4\, \frac{ \left( \tfrac56 \right)^2 \tau_{\mathsmaller{ \rm NL}} }{ \alpha(k) } \cdot \kappa_{3}^{(f_{\mathsmaller{\rm NL}} = 1)}\ ,  \label{equ:318}
\end{equation}
where
\begin{equation}
\kappa_{3}^{(f_{\mathsmaller{\rm NL}} = 1)}(M) \equiv \frac{6}{\sigma_{\mathsmaller{M}}^{3}} \int \limits_{{{\boldsymbol{q}}}_1} \int \limits_{{{\boldsymbol{q}}}_2}  \alpha_\mathsmaller{M}(q_1) \alpha_\mathsmaller{M}(q_2)  \alpha_\mathsmaller{M}(q_{12})  P_{\Phi}(q_1) P_{\Phi}(q_2) \ . \label{equ:k1}
\end{equation}
Since $\kappa_3^{(f_{\mathsmaller{\rm NL}}=1)}$ is of order $\Delta_\Phi$, we see that the condition $f_{1,3} \ll f_{1,2}$ is equivalent to
$f_{\mathsmaller{\rm NL}}(1+\Pi) \Delta_\Phi \ll 1$.  This latter condition is always satisfied if all fields are weakly coupled.\footnote{In
more detail, to show that $f_{\mathsmaller{\rm NL}} (1+\Pi) \Delta_\Phi \ll 1$, we
argue as follows.  Assuming that the field $\psi$ is not strongly coupled, the dimensionless non-Gaussianity parameter
$f_{\mathsmaller{\rm NL}}^{(\psi)} \Delta_\psi = f_{\mathsmaller{\rm NL}} (1+\Pi)^{3/2} \Delta_\Phi$ must be $\lsim 1$.  Therefore
\begin{equation}
f_{\mathsmaller{\rm NL}} (1+\Pi) \Delta_\Phi = \Big[ f_{\mathsmaller{\rm NL}} \Delta_\Phi \Big]^{1/3} \cdot \Big[ f_{\mathsmaller{\rm NL}} (1+\Pi)^{3/2} \Delta_\Phi \Big]^{2/3} \lsim [10^{-3}]^{1/3} \cdot [1] = 10^{-1}\ ,
\end{equation}
where the bound on the first factor is the current observational bound $f_{\mathsmaller{\rm NL}} \lsim 10^2$.}

Finally, using eq.~(\ref{equ:A3}),
we get 
\begin{equation}
f_{2,2} \xrightarrow{k\to 0} 16\, \frac{ \left( \tfrac56 \right)^2 \tau_{\mathsmaller{ \rm NL}}}{\alpha^2(k)}\ .
\end{equation}
Substituting the above into eq.~(\ref{equ:master}) gives
\begin{equation}
r = \left( \left(\tfrac{5}{6}\right)^2 \tau_{\mathsmaller{ \rm NL}} - f_{\mathsmaller{\rm NL}}^2 \right) \frac{\beta_f^2}{\alpha^2(k)}\ ,
\end{equation}
where we have used the relation $\beta_f = 4 \beta_2 = 2(\nu_c-1)$,
which can be derived by evaluating the derivative $\beta_f = 2 \hskip 1pt \partial\ln n_{\rm h}/\partial\ln\sigma$ in the barrier crossing model~\cite{Slosar:2008hx}.
Comparing with eq.~(\ref{equ:rPBS}), we find that barrier crossing and peak-background split give consistent answers.


\subsection{$g_{\rm NL}$ Cosmology} 
\label{sec:gnl}


As our next example, we consider a cubic form of local non-Gaussianity.\footnote{We should say from the outset 
that the large-scale stochasticity in the $g_{\mathsmaller{\rm NL}}$ model will be too small to be observationally relevant.
Although the non-stochastic and stochastic contributions to $P_{{\rm h}\h}(k)$ will turn out to be parametrically identical
($\sim g_{\mathsmaller{\rm NL}}^2 \Delta_\Phi^2 P_\Phi(k)$), the non-stochastic contribution is typically larger by a constant factor $\approx 10^4$.
Nevertheless, the $g_{\mathsmaller{\rm NL}}$ example provides an interesting check of our formalism.}
In this case, the non-Gaussian potential is parametrized by the expansion
\begin{equation}
\Phi = \phi + g_{\mathsmaller{\rm NL}} \left(\phi^3 - 3 \langle \phi^2 \rangle \phi \right)\ .
\end{equation}
The power spectrum of the non-Gaussian field is
\begin{equation}
P_\Phi(k) = P_\phi(k) + g_{\mathsmaller{\rm NL}}^2 P_{\phi^3}(k)\ ,
\end{equation}
where
\begin{equation}
P_{\phi^3}(k) \equiv 6 \int_{{{\boldsymbol{q}}_1}} \int_{{{\boldsymbol{q}}_2}} P_\phi(q_1) P_\phi(q_2) P_\phi(|{{\boldsymbol{k}}}-{{\boldsymbol{q}}_1}-{{\boldsymbol{q}}_2}|)\ .
\end{equation}
We note that for scale-invariant initial conditions, $(k^3/2\pi^2)P_\phi(k) = \Delta_\phi^2 $, the power spectrum $P_{\phi^3}$ is infrared divergent. If the IR divergence is regulated by putting the fields in a finite box with length $L$, then the power spectrum diverges as 
\begin{equation}
P_{\phi^3}(k) \, \sim\, 18 \hskip 1pt \Delta_{\phi}^4 \ln^2(k L) P_\phi (k) \ .
\end{equation}
On large scales, the matter power spectrum therefore is
\begin{equation}
P_{{\rm m} {\rm m}}(k) \simeq \alpha^2(k) P_\Phi(k) = P_{g}(k) \left( 1+ 18 \hskip 1pt g_{\mathsmaller{\rm NL}}^2\Delta_{\phi}^4 \ln^2(k L) \right) \ ,
\end{equation}
where we defined $P_{g}(k) \equiv \alpha^2(k) P_\phi(k)$.
Current observational constraints imply that $|g_{\mathsmaller{\rm NL}} \Delta_\phi^2| \ll 1$.
To obtain answers to zeroth or first order in $g_{\mathsmaller{\rm NL}} \Delta_\phi^2$, it suffices to set  $P_{{\rm m} {\rm m}} \simeq P_{g}$.
 
For the barrier crossing analysis,
we require the following higher-order correlation functions
\begin{eqnarray} 
\xi^{(4)[{\rm tree}]}_{\Phi}({\boldsymbol{k}}_1, {\boldsymbol{k}}_2, {\boldsymbol{k}}_3, {\boldsymbol{k}}_4) & = & g_{\mathsmaller{\rm NL}} \big[P_1 P_2 P_3 + \mathrm{23 \, perms.} \big] + \mathcal{O}(g_{\mathsmaller{\rm NL}}^2)\ , \label{equ:A2g} \\
\xi^{(4)[{\rm loop}]}_{\Phi}({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3,{{\boldsymbol{k}}}_4) &= & 9 \hskip 1ptg_{\mathsmaller{\rm NL}}^2  \big[P_{1} P_2 P_{\phi^2}({k}_{13}) + 11 \, {\rm perms.} \big]\ , \label{equ:A3g} \\
\xi^{(6)}_{\Phi}({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3,{{\boldsymbol{k}}}_4, {{\boldsymbol{k}}}_5, {{\boldsymbol{k}}}_6) &= & 36\hskip 1pt g_{\mathsmaller{\rm NL}}^2 \big[ P_1 P_2 P_3 P_4 P_{125} + 89\, {\rm perms.} \big]\ , \label{equ:A4g}
 \end{eqnarray}
where $k_{ij}=|{\boldsymbol{k}}_i+{\boldsymbol{k}}_j|$, $P_i = P_\phi(k_i)$, $P_{ijk} = P_\phi(|{\boldsymbol{k}}_i+{\boldsymbol{k}}_j+{\boldsymbol{k}}_k|)$,  and
\begin{equation}
P_{\phi^2}(k) \equiv 2 \int_{{{\boldsymbol{q}}}} P_\phi(q) P_\phi(|{{\boldsymbol{k}}}-{{\boldsymbol{q}}}|) \ \sim\ 4 \hskip 1pt \Delta_\phi^2 \ln(kL) P_\phi(k)\ .
\end{equation}
Note that odd-point correlation functions $\xi^{(2N+1)}_{\Phi}$ are zero due to the $\Phi \rightarrow -\Phi$ symmetry.
Next, we will derive the stochastic halo bias both in peak-background split and in barrier crossing.  


\subsubsection{Peak-Background Split}
\label{sssec:gnl_pbs}

The PBS analysis proceeds as before.
Splitting the Gaussian potential into long and short modes, $\phi = \phi_\ell + \phi_s$, we find that the locally measured small-scale power is
$\sigma = \bar \sigma \left[1+ 3 g_{\mathsmaller{\rm NL}} \left( \phi_\ell^2 - \langle \phi_\ell^2 \rangle \right) \right]$. 
Moreover, the locally measured value of $f_{\mathsmaller{\rm NL}}$ is
$f_{\mathsmaller{\rm NL}}^{\rm eff} = 3 g_{\mathsmaller{\rm NL}} \phi_\ell $ \cite{Smith:2011ub}. 
The
halo number density therefore is
\begin{equation}
n_{\rm h}({{\boldsymbol{x}}}) = \bar n_{\rm h}\left( \delta_c-\delta_\ell \, ; \bar\sigma \left[1+ 3 g_{\mathsmaller{\rm NL}} \left( \phi_\ell^2 - \langle \phi_\ell^2 \rangle \right) \right] ; f_{\mathsmaller{\rm NL}}^{\rm eff}\right)\ ,
\end{equation}
where $\delta_\ell \simeq \alpha(k_\ell) \phi_\ell$.  Taylor expanding this expression, we find
\begin{equation}
\delta_{\rm h}  = b_g \delta_\ell + \tfrac{3}{2}\beta_f g_{\mathsmaller{\rm NL}}  \left( \phi_\ell^2 - \langle \phi_\ell^2 \rangle \right) + \beta_g g_{\mathsmaller{\rm NL}} \phi_\ell\ ,
\end{equation}
where $b_g$ and $\beta_f$ are the same as in (\ref{equ:coefs}), and
\begin{equation}
 \beta_g \equiv 3 \frac{\partial \ln n_{\rm h}}{\partial f_{\mathsmaller{\rm NL}}}\ .
\end{equation}
It follows that
\begin{equation}
P_{\rm mh} = b_g P_{\rm mm} + \beta_g g_{\mathsmaller{\rm NL}} P_{{\rm m} \phi} = \left( b_g + \beta_g \frac{g_{\mathsmaller{\rm NL}}}{\alpha(k)}\right) P_{\rm mm}\ ,
\end{equation}
and 
\begin{equation}
P_{{\rm h} {\rm h}} = \left( b_g + \beta_g \frac{g_{\mathsmaller{\rm NL}}}{\alpha(k)} \right)^2 P_{\rm mm} + \frac94 \hskip 1pt \beta_f^2 g_{\mathsmaller{\rm NL}}^2 P_{\phi^2} \ .
\end{equation}
This implies a large-scale halo stochasticity of the form
\begin{equation}
r = \frac{9}{4}\hskip 1pt \beta_f^2 g_{\mathsmaller{\rm NL}}^2 \frac{P_{\phi^2}}{P_{\rm mm}} \ . \label{equ:PBSx}
\end{equation}


\subsubsection{Barrier Crossing}
\label{sssec:gnl_barrier_crossing}

We now show that the same result is obtained from barrier crossing.
In Appendix~\ref{sec:convergence}, we argue that only the first few cumulants need to be taken into account. It is straightforward to compute them explicitly.
From eqs.~(\ref{equ:A2g}) and (\ref{equ:A4g}),
we get
\begin{align}
f_{1, 3}(k) &\ \xrightarrow{k\to 0} \ \frac{3 g_{\mathsmaller{\rm NL}}}{\alpha(k)} \, \kappa_{3}^{(f_{\rm NL} = 1)}\qquad {\rm and} \qquad 
f_{3,3}(k) = [f_{1, 3}(k)]^2\ .
\end{align}
This only contributes to the non-stochastic bias.
However, since $f_{\hat 1,2} = 0$, stochastic bias arises from $f_{2,2}$.
First, we note that the tree-level four-point function (\ref{equ:A2g}) leads to a very small and scale-independent contribution to $f_{2,2}$\,:
\begin{align}
f_{2,2}^{[{\rm tree}]}(k,M,\bar M) &\ \xrightarrow{k\to 0} \ \frac{12 \hskip 1pt g_{\mathsmaller{\rm NL}}}{P_{{\rm m}\m}(k)} \left(
  \frac{1}{\sigma_{\mathsmaller{M}}^2} \int_{{\boldsymbol{q}}} \alpha_{\mathsmaller{M}}^2(q) P_\phi^2(q) +
  \frac{1}{\sigma_{\bar\mathsmaller{M}}^2} \int_{{\boldsymbol{q}}} \alpha_{\bar\mathsmaller{M}}^2(q) P_\phi^2(q) \right)\ .
\end{align}
Plugging into eq.~(\ref{equ:master}) and noting that $\beta_2 = \frac{1}{4}\beta_f$ and $\tilde\beta_2 = \frac{1}{2}$,
we get a small scale-dependent contribution to the large-scale stochastic bias
\begin{equation}
r_{[{\rm tree}]} = \frac{3}{2} \frac{g_{\mathsmaller{\rm NL}}}{P_{{\rm m}\m}(k)} \left( \beta_f^2 + 2 \beta_f \right) 
  \left( \frac{1}{\sigma_{\mathsmaller{M}}^2} \int_{{\boldsymbol{q}}} \alpha_{\mathsmaller{M}}^2(q) P_\phi^2(q) \right)  \ . \label{eq:gnl_rtree}
\end{equation}
In practice, this contribution to the large-scale stochasticity can't be used as a probe of initial conditions, since a
contribution to $r$ with $r \propto 1/k$
(or equivalently a contribution to $P_{{\rm h}\h}(k)$ which approaches
a constant as $k\rightarrow 0$) is degenerate with other sources of stochasticity such as second-order Gaussian bias.
Finally, the one-loop four-point function (\ref{equ:A3g}) leads to the following contribution to $f_{2,2}$\,:
\begin{align}
f_{2, 2}^{[{\rm loop}]}(k) &\ \xrightarrow{k\to 0} \ 36 \, g_{\mathsmaller{\rm NL}}^2 \cdot \frac{P_{\phi^2}(k)}{P_{{\rm m} {\rm m}}(k)}\ .
\end{align}
The corresponding stochasticity parameter is
\begin{equation}
r_{[{\rm loop}]} = \frac{9}{4}\hskip 1pt \beta_f^2 g_{\mathsmaller{\rm NL}}^2 \frac{P_{\phi^2}(k)}{P_{\rm mm}(k)} \propto \frac{1}{k^4}\ ,  \label{eq:gnl_rloop}
\end{equation}
in agreement with the PBS predictions (\ref{equ:PBSx}).

\subsection{Quasi-Single-Field Inflation}

Our last example is quasi-single field inflation (QSFI) \cite{Chen:2009zp}.  These models involve extra massive scalar degrees of freedom during inflation.  In the simplest examples, a single scalar field $\sigma$ of mass\footnote{We note that extra scalars with masses close to the Hubble scale $H$ are a natural prediction of supersymmetric theories of inflation (see~\cite{Baumann:2011nk} for further discussion).} $m^2 \leq \tfrac{9}{4} H^2$ mixes with the fluctuation of the inflaton\footnote{Recall that $\delta \phi$ in spatially flat gauge is proportional to the curvature perturbation, $\zeta \equiv \frac{H}{\dot \phi} \delta \phi$.} $\delta \phi$.
The mixing communicates non-Gaussianity from the hidden (isocurvature) sector to the observable (adiabatic) sector. As we now show, it also leads to a significant stochasticity in the halo bias.

\newpage
\subsubsection{Boosted Four-Point Function}

Again, we need the squeezed and collapsed limits of the primordial correlation functions\footnote{See \cite{Baumann:2011nk} for an intuitive explanation of the scalings in eqs.~(\ref{equ:A1a}) and (\ref{equ:A3a}).}~\cite{Chen:2009zp, Baumann:2011nk,Quasi4}: 
\begin{align}
\lim_{k_1 \to 0 } \xi^{(3)}_{\Phi}({{\boldsymbol{k}}}_1, {{\boldsymbol{k}}}_2, {{\boldsymbol{k}}}_3) &= 4 f_{\mathsmaller{\rm NL}} \left( \frac{k_1}{k_2}\right)^{\Delta} P_1 P_2 \label{equ:A1a}\ , \\
\lim_{k_{12} \to 0 }   \xi^{(4)}_{\Phi}({{\boldsymbol{k}}}_1,{{\boldsymbol{k}}}_2,{{\boldsymbol{k}}}_3,{{\boldsymbol{k}}}_4) &= \  16 \left( 
\tfrac{5}{6}\right)^2 \tau_{\mathsmaller{ \rm NL}}   \left( \frac{k_{12}^2}{k_{1} k_{3}}\right)^{\Delta} P_1 P_3 P_{12} \ , \ \  \label{equ:A3a} \end{align}
where we defined the parameter 
\begin{equation}
\Delta \equiv \frac{3}{2} - \sqrt{\frac{9}{4} - \frac{m^2}{H^2} }\ . 
\end{equation}
The non-trivial momentum scaling of eqs.~(\ref{equ:A1a}) and (\ref{equ:A3a}) is a remarkable signature of extra Hubble mass scalars during inflation~\cite{Chen:2009zp,Baumann:2011nk, Sefusatti:2012ye, Norena:2012yi}.
Moreover, if the mixing between $\sigma$ and $\phi$ (or $\zeta$) is parametrized by a small dimensionless number $\varepsilon < 1$, then 
\begin{equation}
\tau_{\mathsmaller{ \rm NL}} \sim  \varepsilon^{-2} \, (\tfrac{6}{5}f_{\mathsmaller{\rm NL}})^2 \ >\  (\tfrac{6}{5}f_{\mathsmaller{\rm NL}})^2 \ .
\end{equation}
The enhancement of $\tau_{\mathsmaller{ \rm NL}}$ arises because the trispectrum is generated by the exchange of the $\sigma$-field which is only weakly coupled to $\zeta$. 
The size of the four-point function $\langle \zeta^4 \rangle$ can be estimated
from the square of the three-point function $ \langle \zeta ^2 \sigma \rangle$ at horizon crossing,
\begin{equation}
\langle\zeta^4 \rangle \sim \langle \zeta ^2 \sigma \rangle^2 \sim \varepsilon^{-2}\hskip 2pt  \frac{\langle \zeta^3 \rangle^2}{\langle \zeta^2 \rangle}\ .
\end{equation} 
 The boost of $\tau_{\mathsmaller{ \rm NL}}$ is the result of the small correlation between the curvature fluctuation and the massive field, $\varepsilon \ll 1$. The precise dependence of $f_{\mathsmaller{\rm NL}}$ and $\tau_{\mathsmaller{ \rm NL}}$ on the fundamental parameters of the QSFI Lagrangian can be found in \cite{Quasi4}.


\subsubsection{Barrier Crossing}

In QSFI, the higher-order $N$-point functions are suppressed by factors of the power spectrum, just as in our previous examples.  The dominant contributions to the large-scale structure signal therefore arise from the squeezed limit of the three-point function and the collapsed limit of the four-point function.  The relevant cumulants are
\begin{align}
\kappa_{\hat{1},2} (k) &\, \xrightarrow{k\to 0}\, \frac{4 f_{\mathsmaller{\rm NL}}}{\hat \sigma} (k R_\mathsmaller{M})^{ \Delta} \, \frac{P_{\rm mm} (k)}{\alpha(k)} \,\frac{\Sigma^2_{\mathsmaller{M}}(\Delta) }{\sigma_{\mathsmaller{M}}^2 }\ ,
\end{align}
and 
\begin{align}
\kappa_{2,2} (k) &\, \xrightarrow{k\to 0}\,  16 \left( 
\tfrac{5}{6}\right)^2 \tau_{\mathsmaller{ \rm NL}} (k^2 R_\mathsmaller{M} R_{\bar \mathsmaller{M}})^{\Delta} \, \frac{P_{\rm mm}(k)}{\alpha^2(k)} \, \frac{\Sigma^2_{\mathsmaller{M}}(\Delta)}{\sigma_{\mathsmaller{M}}^2 }\frac{\Sigma^2_{\bar \mathsmaller{M}}(\Delta)}{\sigma_{\bar \mathsmaller{M}}^2 } \ .
\end{align}
Here, we have defined
\begin{equation}
\Sigma^2_\mathsmaller{M}(\Delta) \equiv \int \frac{{\rm d}^3 k_s}{(2\pi)^3} \, W^2_{\mathsmaller{M}}(k_s)(k_s R_\mathsmaller{M}) ^{- \Delta } P_{\rm mm}(k_s) \ , \label{equ:Sigma}
\end{equation}
where the integration variable, $k_s$, is one of the short momenta and $R_\mathsmaller{M}$ is the smoothing scale defined by eq.~(\ref{eqn:tophat}). By definition, $\Sigma_\mathsmaller{M}(0) = \sigma_\mathsmaller{M}$.  
In the limit $\Delta \to 0$, we recover the results of the $\tau_{\mathsmaller{ \rm NL}}$~model.  Therefore, we find
\begin{equation}
f_{\hat 1, 2} = 4 f_{\mathsmaller{\rm NL}}  \, \frac{(k R_\mathsmaller{M})^{ \Delta}}{\alpha(k)} \,\frac{\Sigma^2_{\mathsmaller{M}}(\Delta) }{\sigma_{\mathsmaller{M}}^2 } \qquad {\rm and} \qquad f_{2,2}= \frac{\tau_{\mathsmaller{ \rm NL}}}{(\tfrac{6}{5} f_{\mathsmaller{\rm NL}})^2} \, f_{\hat 1, 2}(M) f_{\hat 1, 2}(\bar M) \ . \label{equ:fs}
\end{equation}

\begin{figure}[h!]
   \centering
       \includegraphics[height=6.7cm]{QSFIv4.pdf}
   \caption{Numerical evaluation of eqs.~(\ref{equ:Sigma}) and (\ref{equ:fs}). For $\Delta \gtrsim 1.0$, the cumulant $f_{\hat 1,2}$ depends significantly on the halo mass scale~$M$. This is in contrast to local non-Gaussianity, which corresponds to the limit $\Delta \to 0$.}
  \label{fig:QSFI}
\end{figure}

\noindent
To obtain the large-scale stochasticity, we substitute the cumulants into eq.~(\ref{equ:master}),
\begin{align}
r \ =\ \left( \beta_2 + \tilde\beta_2 \frac{\partial}{\partial\ln\sigma_{\mathsmaller{M}}} \right)
     \left( \beta_2 + \tilde\beta_2 \frac{\partial}{\partial\ln\sigma_{\bar\mathsmaller{M}}} \right)
     f_{2,2} \, -\,  \left[ \left( \beta_2 + \tilde\beta_2 \frac{\partial}{\partial\ln\sigma_{\mathsmaller{M}}} \right)
     f_{\hat 1,2} \right]^2\ .
\end{align}
Because the cumulants depend explicitly on $R_\mathsmaller{M}$, we 
 have to be concerned that the derivatives with respect to $\sigma_\mathsmaller{M}$ may this time not be negligible. 
Indeed, numerical evaluation of the integral shows significant $\sigma_\mathsmaller{M}$ dependence of $f_{\hat 1, 2}$ (see fig.~\ref{fig:QSFI}).
Keeping the derivative terms, we get 
\begin{align}
r &\ =\ \left( \left ( \tfrac{5}{6} \right )^2 \tau_{\mathsmaller{ \rm NL}} -   f_{\mathsmaller{\rm NL}} ^2 \right) \frac{k^{2\Delta}}{\alpha^2(k)} 
      \left[ \left( \beta_f + 2\, \frac{d}{d\ln\sigma_{\mathsmaller{M}}} \right) \frac{R_{\mathsmaller{M}}^{\Delta}\hskip 1pt \Sigma_{\mathsmaller{M}}^2(\Delta)}{\sigma_{\mathsmaller{M}}^2} \right]^2 \nonumber \\
  &\ \propto \ \frac{\left ( \tfrac{5}{6} \right )^2 \tau_{\mathsmaller{ \rm NL}} -   f_{\mathsmaller{\rm NL}} ^2 }{k^{4 - 2 \Delta}} \ . \label{equ:rQSFI}
\end{align}
The characteristic momentum scaling of eq.~(\ref{equ:rQSFI}) and the natural boost of $\tau_{\mathsmaller{ \rm NL}}$ makes halo stochasticity an interesting probe of quasi-single-field inflation.

\newpage
\section{Conclusions}
\label{sec:conclusions}

What was the number of light degrees of freedom during inflation? 
And, what were their interactions?
The great virtue of primordial non-Gaussianity is that it is sensitive to these basic questions about the physics of inflation.
In particular, it is well-known that the {squeezed} limit of the primordial {three-point} function, 
\begin{equation}
\lim_{k_1 \to 0} \langle \Phi_{{\boldsymbol{k}}_1} \Phi_{{\boldsymbol{k}}_2} \Phi_{{\boldsymbol{k}}_3} \rangle\ ,
\end{equation}
can only be large if more than one light field was dynamically relevant during inflation~\cite{Maldacena:2002vr,Creminelli:2004yq}. Remarkably, this statement is independent of the details of the Lagrangian for the inflaton field and its initial conditions.
Measurements of the squeezed limit therefore have the potential to rule out all models of single-field inflation \cite{Maldacena:2002vr,Creminelli:2004yq}. 
Moreover, the precise scaling in the squeezed limit is sensitive to the details of the mass spectrum~\cite{Chen:2009zp, Quasi4}, allowing a test of extra Hubble mass fields, such as those generically expected in supersymmetric theories~\cite{Baumann:2011nk}.
Having a large three-point function in the squeezed limit modulates the two-point function of halos and therefore leads to scale-dependent bias~\cite{Dalal:2007cu}. In the future, this effect may well be our most sensitive probe of the squeezed limit. 

In this paper, we have discussed a {\it stochastic} form of scale-dependent halo bias.
This effect arises if the {collapsed} limit of the primordial {four-point} function,
\begin{equation}
\lim_{k_{12} \to 0} \langle \Phi_{{\boldsymbol{k}}_1} \Phi_{{\boldsymbol{k}}_2} \Phi_{{\boldsymbol{k}}_3}  \Phi_{{\boldsymbol{k}}_4} \rangle_{\rm c} \ ,
\end{equation}
is larger than the square of the squeezed limit of the three-point function.
More generally, stochastic bias arises whenever a suitable collapsed limit of an $(M+N)$-point function is larger than the
product of the associated squeezed $(M+1)$-point and $(N+1)$-point functions, where $M,N\ge 2$.  
The key tool for obtaining this result, and a main result of this paper, is a pair
of formulas, eqs.~(\ref{equ:PmhFinal}) and~(\ref{equ:PhhFinal}), for the matter-halo and halo-halo power spectra in a general
non-Gaussian model parametrized by the $N$-point functions of the primordial potential.

In non-Gaussian models which generate significant stochastic halo bias, the results of this paper
are important even at a qualitative level.
As a concrete example, it should be possible to measure $f_{\mathsmaller{\rm NL}}$ and $\tau_{\mathsmaller{ \rm NL}}$ independently using stochastic bias.
This can be done either by measuring multiple tracer populations and directly estimating large-scale stochasticity
(which has the advantage of eliminating sample variance), or from a single tracer population by measuring $P_{{\rm h}\h}(k)$
and using the functional form
\begin{equation}
P_{{\rm h}\h}(k) = b_g^2 \left( 1 + f_{\mathsmaller{\rm NL}} \frac{2 \delta_c}{\alpha(k)} + \tau_{\mathsmaller{ \rm NL}} \frac{(\frac{5}{6})^2 \delta_c^2}{\alpha^2(k)} \right)
\end{equation}
to fit for $b_g$, $f_{\mathsmaller{\rm NL}}$ and $\tau_{\mathsmaller{ \rm NL}}$ independently.
Recently, ref.~\cite{Biagetti:2012xy} showed that if only non-stochastic bias is considered, the leading contribution from $\tau_{\mathsmaller{ \rm NL}}$ is small
(in our language, this corresponds to the ${\mathcal O}(\tau_{\mathsmaller{ \rm NL}})$ contribution to $\kappa_{1,3}$) and it is difficult to separate $f_{\mathsmaller{\rm NL}}$ and $\tau_{\mathsmaller{ \rm NL}}$,
so stochastic bias has an important qualitative effect.
As another example, in quasi-single field inflation, the stochastic bias is larger than the non-stochastic bias
by a large factor (parametrically $\varepsilon^{-2}$), leading to a similarly large enhancement in signal-to-noise when stochastic bias
is considered.
We defer quantitative forecasts incorporating stochastic bias for future work.

In general, there is no stochastic bias if only a single field (which may or may not be the inflaton) 
generates the primordial curvature perturbation and its non-Gaussianity~\cite{Suyama:2010uj}.
Measuring stochastic halo bias would therefore teach us about the effective number of degrees of freedom that generated the primordial fluctuations and its higher-order correlations. In particular, stochasticity is sensitive to what we may call ``hidden sector non-Gaussianity", i.e. situations in which two fields generate the curvature perturbation, but only one (hidden) field is responsible for its non-Gaussianity.
In this paper, we have derived this effect for general non-Gaussian initial conditions. We have also applied our formalism to a number of explicit examples, such as curvaton models \cite{Tseliakhovich:2010kf} and quasi-single field inflation~\cite{Chen:2009zp}.
We have shown that halo bias, in principle, gives us information about the soft limits of both the primordial three-point function and the four-point function. It is therefore a valuable tool in the quest to uncover the physics that created the initial perturbations.


\acknowledgments
We thank Valentin Assassi, Eugene Lim, Marilena LoVerde, Marcel Schmittfull, David Spergel and Matias Zaldarriaga for helpful discussions.
D.B.~gratefully acknowledges support from a Starting Grant of the European Research Council (ERC STG grant 279617). S.F.~acknowledges support from a fellowship at the Department of Astrophysical Sciences of Princeton University.
The research of D.G.~is supported by the DOE under grant number DE-FG02-90ER40542 and the Martin A.~and Helen Chooljian Membership at the Institute for Advanced Study.
K.M.S.~was supported by a Lyman Spitzer fellowship in the Department of Astrophysical Sciences at Princeton University.
Research at Perimeter Institute is supported by the Government of Canada
through Industry Canada and by the Province of Ontario through the Ministry of Research \& Innovation.

\newpage
