\section{\label{intro} Introduction}

The proximity effect between a ferromagnet and a superconductor has attracted
much attention recently due to its potential applications in such areas of
technology as magnetoelectronics \cite{Bauer} or quantum computing
\cite{Blatter}. The proximity effect in ferromagnet-superconductor (F-S)
structures is also important from the point of view of the scientific interest
as it allows the study of the interplay between ferromagnetism and
superconductivity \cite{Berk}.

Due to time reversal symmetry breaking the proximity effect in F-S structures
leads to new phenomena, not observed in usual normal metal - superconductor
(N-S) proximity systems. Those are: oscillations of the superconducting
transition temperature \cite{Wong}, density of states \cite{Kontos} and
superconducting pairing amplitude in F-S multilayers, Josephson $\pi$-junction
behavior in S-F-S heterostructures \cite{Ryazanov}, a giant mutual proximity
effect in S-F systems \cite{Petrashov}, spin valve \cite{Tagirov} or
spontaneous currents in F-S bilayers \cite{MK_1,MK_2,BLG}. For review of the
literature on those and related effects see \cite{Izyumov}.

We have recently examined F-S proximity system and found that a spontaneously
generated current flows on both sides of the interface \cite{MK_1,MK_2,BLG}.
Such a current, which flows in the ground state of the system is a hallmark of
the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) like state, originally predicted
for a bulk superconductor in a magnetic exchange field acting on spins only
\cite{FFLO}. It is the purpose of the present paper to see what the effect on
the FFLO state will have a normal metal sandwiched between ferromagnet and
superconductor (F-N-S structure). Will the spontaneous current still be
generated  ? If so, how will it be modified ? In the rest of the paper we show
that the presence of the normal metal has nontrivial consequences on the
generation of the spontaneous currents and on the FFLO state in general.

\section{\label{model} The Model}

Our model system is schematically shown in Fig. \ref{Fig1}.
\begin{figure}[h]
\begin{center}
 \resizebox{0.7\linewidth}{!}{
  \includegraphics{setup_1.eps}}
\end{center}
 \caption{\label{Fig1} Schematic view of our model system. The rows of empty
          circles ($n = -d_F,...,-d_N$) describe ferromagnetic layers, filled
	  ($n = -d_N,...,0$) - normal region, while shaded ($n = 1,...$) -
	  superconducting one. Directions of the magnetic field ${\bf B}$,
	  vector potential ${\bf A}$ and current ${\bf J}$ are also indicated.}
\end{figure}
It consists of a ferromagnet (layers $n = -d_F,...,-d_N$), a normal
(paramagnetic) region (layers $n = -d_N,...,0$) and a semi-infinite
superconductor (layers $n = 1,...$).

The model Hamiltonian is given by single orbital Hubbard model
\begin{eqnarray}
H = \sum_{ij\sigma} \left[ t_{ij} +
(\varepsilon_{i\sigma} - \mu) \delta_{ij} \right] c^+_{i\sigma} c_{j\sigma} +
\frac{1}{2} \sum_{i\sigma} U_i \hat n_{i\sigma} \hat n_{i -\sigma} ,
\label{Hamilt}
\end{eqnarray}
where, in the presence of a vector potential, the nearest neighbor hopping
integral is
$t_{ij} = -t e^{-ie \int^{{\bf r}_j}_{{\bf r}_i} {\bf A}({\bf r})
\cdot d{\bf r}}$. The site energy levels $\varepsilon_{i\sigma}$ are equal to
$\frac{1}{2} E_{ex} \sigma$ in the ferromagnet and $0$ in the normal and
superconducting region, and $\mu$ is the chemical potential. $U_i$ is the
on-site electron-electron interaction, which is assumed to be negative on
superconducting side and zero elsewhere, $c^+_{i\sigma}$ ($c_{i\sigma}$) are
the usual electron creation (annihilation) operators and
$\hat n_{i\sigma} = c^+_{i\sigma} c_{i\sigma}$ is the electron number operator.

We study the above model in the spin-polarized Hartree-Fock-Gorkov (SPHFG)
approximation, assuming the Landau gauge for the magnetic field
${\bf B} = (0, 0, B_z(x))$, thus ${\bf A} = (0, A_y(x), 0)$ (see Fig.
\ref{Fig1}). In the following we assume periodicity of our model in the
direction parallel to the interface and therefore we work in ${\bf k}$ space
in the $y$ direction but in real space in the $x$ direction.

As usual, self-consistency is assured by the relations determining SC order
parameter $\Delta_n$, current $J_{y \uparrow (\downarrow)}(n)$ and the vector
potential $A_y(n)$ on each layer $n$
\begin{eqnarray}
\Delta_n = U_n \chi_n =
- \frac{U_n}{\pi} \sum_{k_y} \int d\omega {\rm Im} G^{12}_{nn}(\omega, k_y)
f(\omega) ,
\label{Delta}
\end{eqnarray}
\begin{eqnarray}
J_{y\uparrow (\downarrow)}(n) =
-\frac{2 e t}{\pi} \sum_{k_y} sin(k_y - e A_y(n)) \int d\omega {\rm Im}
G^{11 (33)}_{nn}(\omega, k_y) f(\omega) ,
\label{current}
\end{eqnarray}
\begin{eqnarray}
A_y(n+1) - 2 A_y(n) + A_y(n-1) = -4 \pi J_y(n) ,
\label{vec_pot}
\end{eqnarray}
where $G^{\alpha\beta}_{nm}(\omega, k_y)$ is the $4 \times 4$ Nambu Green
function and $f(\omega)$ the Fermi distribution. Equation (\ref{vec_pot}) is a
lattice version of the Ampere's law
$\frac{d^2 A_y(x)}{d x^2} = - 4 \pi J_y(x)$. The above equations have to be
solved self-consistently, using the method described in \cite{MK_1}.

\section{\label{results} Results}

Figure \ref{Fig2} shows the spatial dependence of the normalized
superconducting pairing amplitude $\tilde \chi_n = \chi_n/\chi_{bulk}$ (left
panel) and spontaneous current $J_n$ (right panel) for a number of normal metal
$d_N$ layers ($d_F = 20 - d_N$).
\begin{figure}[h]
\begin{center}
 \resizebox{\linewidth}{!}{
  \includegraphics{f_pair.eps}
  \includegraphics{f_curr.eps}
  }
\end{center}
 \caption{\label{Fig2} Spatial dependence of the normalized superconducting
          pairing amplitude $\tilde \chi_n = \chi_n/\chi_{bulk}$ (left panel)
	  and spontaneous current $J_n$ (right panel) for a number of $d_F$ and
	  $d_N$ ($d_F + d_N = 20$) layers. The model parameters are:
	  $E_{ex} = 0.517$, $\Delta_S = 0.528$ and $T = 10^{-2}$ in units of
	  $t$.}
\end{figure}
As in our previous work on  F-S structures \cite{Buzdin_1,MK_1,MK_2,BLG}, the
superconducting pairing amplitude shows clear oscillations in the ferromagnet.
The period of the oscillations is related to the ferromagnetic coherence length
$\xi_F = 2t/E_{ex}$, i.e. $\chi_n \propto \frac{sin(n/\xi_F)}{(n/\xi_F)}$. No
such effect is observed in the N region. One could expect that the inclusion of
the normal metal suppresses the oscillations of $\chi_n$ in the F region.
Interestingly, the amplitude of the $\chi_n$ oscillations on the ferromagnetic
side remains almost unchanged. Only the phase of the oscillations can change.
Moreover, as for F-S bilayer \cite{MK_1}, the vanishing of the pairing
amplitude at $n = -d_F$ is related to the crossing of the Andreev bound states
(ABS) through the Fermi energy of the system. Furthermore, in such a situation,
spontaneous current is generated \cite{MK_1}. The current flowing produces a
magnetic field, which splits this zero energy ABS, thus lowering the total
energy of the system. The energy of this splitting is given in our model by
$\delta \approx 2 e t \bar{A}_y$, where $\bar{A}_y$ is the layer averaged
vector potential.

As we already mentioned, the zero energy ABS is responsible for the generation
of the spontaneous current. The typical distribution of such current, flowing
parallel to the interface, is shown in the right panel of Fig. \ref{Fig2}. The
current flows in the negative $y$ direction on the superconducting side and in
the positive direction in the whole $F-N$ region, giving a total current equal
to zero, as should be in the true ground state. Interestingly, the current in
the ferromagnet shows also oscillatory behavior, however, with a different
period than the pairing amplitude does. The oscillations of the current are
related to the oscillations of the density of states at the Fermi energy
\cite{MK_1,MK_2}. On the other hand, in the N region, the distribution of the
current is rather smooth, similarly to the pairing amplitude (compare the solid
and thin dotted lines in Fig. \ref{Fig2}).

The state with spontaneous current is a true ground state, as it has lower
energy than the state with no current. This can be seen in Fig. \ref{Fig3},
which shows the change in total energy of the system $\Delta E_{tot}$ between
the solution with spontaneous current and the one where the current is
constrained to be zero is shown.
\begin{figure}[h]
\begin{center}
 \resizebox{0.5\linewidth}{!}{
  \includegraphics{f_gsen.eps}
  }
\end{center}
 \caption{\label{Fig3} The difference in total energy between solutions with
          and without current.}
\end{figure}
Another important finding is that the system can be periodically switched
between the states with and without spontaneous current by increasing of the
thickness of the normal metal region. For $d_F + d_N = 20$ layers and
$E_{ex} = 0.517$ the period is equal to six (see Fig. \ref{Fig3}).

All above the results show that inclusion of the normal metal slab between the
ferromagnet and superconductor has non-trivial effects on the physics of the
proximity induced FFLO state. It cannot be argued that the normal metal simply
acts as interface transparency, as in our case the N slab is in the clean
(ballistic) regime, and thus does not suppress the proximity effect. However,
it leads to the suppression of the oscillatory behavior of the pairing
amplitude, spontaneous current and the density of states at the Fermi energy.
Moreover, it strongly modifies the positions of the ABS, which leads to the
periodical switching between the states with and without current. Interface
transparency also leads to such periodical switching of the current
\cite{MK_2}, however, at the same time it also kills the proximity effect,
suppressing the current and changing the period of the pairing amplitude
oscillations. No such effects are caused by the inclusion of the normal metal
slab. Perhaps the effect of the N slab would be more similar to the effect of
reduced interface transparency if the normal metal was in the dirty regime.

\section{\label{conclusions} Conclusions}
In conclusion we have studied the ground state properties of F-N-S proximity
system. We have observed oscillatory behavior of the superconducting pairing
amplitude in ferromagnet, but not in the normal region. Similarly to F-S
structures, we have found spontaneously generated currents flowing in the whole
F and N regions and within a distance of a few $\xi_S$ on superconducting side.
Interestingly, the system can be switched periodically (changing $d_N$) between
the states with and without the spontaneous current. All this suggests possible
realization of the proximity induced FFLO state in F-N-S heterostructure.

\section*{Acknowledgements}
This work has been partially supported by the grant no. PBZ-MIN-008/P03/2003,
and by the ESF network programme AQD JJ.


\section{Introduction}
There is mounting evidence today that gamma-ray bursts (GRBs) are associated
with collimated relativistic outflows or jets \citep[][]{rho99,kul99,gre03}.
It is still a matter of ongoing research, however, what types of internal
structures are actualized within these jets. Within the fireball framework,
for example, two different kinds of jet models seem to be compatible with the
observations, i.e., (i) the uniform "top-hat" jet model \citep[][]{rho99,
fra01,lam04}, in which all the hydrodynamical quantities (e.g., Lorentz factor,
energy density) are essentially the same within some well-defined opening angle
$\phi_j$ around the jet axis but sharply drop outside of $\phi_j$, and (ii)
the universal structured ("power-law" or "Gaussian") jet model \citep[][]{ros02,
zha02,kum03,zha04}, in which the hydrodynamical quantities are rather smooth
functions of the angle $\phi$ from the jet axis.\\
Here we analyze the implications of such jet structures for the acceleration
of energetic particles. We focus on shear acceleration as a promising mechanism
for converting the kinetic energy of the flow into nonthermal particles and
radiation. Such a mechanism has previously been successfully applied to the
relativistic jets in Active Galactic Nuclei \citep[][]{ost98,ost00,rie04,rie05a,
rie05b}.
Shear acceleration is based on the simple fact that particles may gain energy
by scattering off (small-scale) magnetic field irregularities with different
local velocities due to being systematically embedded in a collisionless shear
flow \citep[cf.][for a recent review]{rie05b}. In the case of nonrelativistic
shear acceleration it is straightforward to show that local power-law particle
momentum distributions $f(p) \propto p^{-\,(3 + \alpha)}$ can be generated,
assuming a momentum-dependent mean scattering time of the form $\tau \propto
p^{\alpha}$ with $\alpha>0$ \citep[cf.][]{berk81,rie05a}. Hence, for $\tau$
scaling with the gyro-radius, i.e., $\tau \propto p$, this results in a
power-law particle number density $n(p)\propto p^2\,f(p) \propto p^{-2}$, and
thus a synchrotron emissivity $j_{\nu}\propto \nu^{-1/2}$.

\section{Shear acceleration in structured GRB-type jets}
Let us consider an idealized radial, relativistic flow profile, which in four
vector notation is given by \citep[cf.][for the more general case]{rie05c}
\begin{equation}
   u^{\alpha}= \gamma_b\,\left(1,v_r(\phi)/c,0,0\right)\,,
\end{equation} where $\alpha=0,1,2,3$, and $\phi$ denotes the polar angle in spherical
coordinates, and $\gamma_b \equiv \gamma_b(\phi)=[1-v_r(\phi)^2/c^2]^{-1/2}$
is the bulk Lorentz factor of the flow. In the comoving frame, the related
gradual shear acceleration coefficient can be cast into the form \citep[e.g.,]
[eq.~3.27]{web89}
\begin{equation}
   < \dot{p}\,'\,> = \frac{1}{p\,'^{\,2}}\frac{\partial }{\partial p\,'}
                \left(p\,'^{\,4}\,\tau'\,\Gamma\right)
\end{equation} where $p'$ denotes the comoving particle momentum, $\tau' \simeq \lambda'
/c$ is the mean scattering time and $\Gamma$ is the relativistic shear coefficient.
We are interested in the strong scattering limit (corresponding to $\omega_g\,'
\,\tau' \ll 1$, with $\omega_g'$ being the relativistic gyro-frequency measured in
the comoving jet frame, i.e., assuming a sufficiently weak longitudinal mean
magnetic field and the presence of strong turbulence, so that collisions are
efficient enough to restore isotropy), where the shear coefficient is given by
$\Gamma =(c^2/30)\,\sigma_{\alpha \beta}\,\sigma^{\alpha \beta}$\citep[see][eq.~3.34]
{web89} and $\sigma_{\alpha \beta}$, with $\alpha,\beta=0,1,2,3$, is the usual
covariant fluid shear tensor \citep[cf.][for more details]{web89,rie04,rie05c}.
Using spherical coordinates and the velocity profile above, it can be shown
that the relativistic shear coefficient becomes \citep[cf.][]{rie05c}
\begin{equation}\label{shear-coeff}
  \Gamma = \frac{4}{45}\,\gamma_b^2 \left[\frac{v_r^2}{r^2}+\frac{3}{4\,r^2}\,
           \gamma_b^2\,\left(\frac{\partial v_r}{\partial \phi}\right)^2\right]\,,
\end{equation} which in the nonrelativistic limit ($\gamma_b \rightarrow 1$) reduces
to the (nonrelativistic) viscous transfer coefficient derived by \citet{ear88}
(their eq.~7) when the latter is expressed in spherical coordinates and the
corresponding velocity profile $\vec{v} = v_r(\phi)\,\vec{e}_r$ is applied.
The (comoving) timescale $t_{\rm acc} \simeq < p' >/< \dot{p}' >$ for the
shear flow acceleration of particles then becomes
\begin{equation}\label{tshear}
    t_{\rm acc}(r,\phi) \simeq \frac{45}{4 (4 + \alpha)}\,
               \frac{c}{\lambda'}\,
                \frac{r^2}{\gamma_b^2\,\left[v_r^2 + 0.75\,\gamma_b^2\,
                (\partial v_r/\partial \phi)^2\right]}\,,
\end{equation} where $r$ is the radial coordinate measured in the cosmological rest
frame (i.e., the supernova, collapsar or merger rest frame), and where a power-law
dependence $\tau'=\lambda'/c \propto p'^{\alpha}$ has been assumed. As the jet
flow is diverging, $t_{\rm acc}$ obviously increases with $r$ to the square.\\
In order to investigate the acceleration potential of structured GRB jets, two
applications appear particularly interesting \citep[e.g.,][]{zha02,kum03,zha04}:
a power-law model, in which $\gamma_b$ is a power-law function of $\phi$
outside a core of opening angle $\phi_c$, i.e., $\gamma_b(\phi) = 1 + (\gamma_{b0}
-1)(1+[\phi/\phi_c]^2)^{-b/2}$, with $1.5 < b \stackrel{<}{\scriptstyle \sim} 2$ \citep[cf.][]{zha02},
and a Gaussian model with $\gamma_b(\phi) = 1 + (\gamma_{b0} -1) \exp[-\phi^2/
2\phi_c^2]$, where $\gamma_{b0}$ denotes the Lorentz factor at the jet axis, and
typically $\phi_c = 0.1$ rad \citep[][]{zha04}. The shear acceleration timescale
(with the time and space coordinates measured in the cosmological rest frame and
particle Lorentz factor measured in the comoving jet frame) then becomes
\begin{equation}\label{tshear1}
  t_{\rm acc}(r,\phi) = \frac{45}{4 (4 + \alpha)}\,
      \frac{r^2}{c\,\lambda'\,\gamma_b(\phi)^2}\,\times
      \left\{ \begin{array}{rl}
      \left(\frac{v_r^2}{c^2}+\frac{3}{4}\,\frac{\left(\gamma_b(\phi)-1\right)}
      {\gamma_b(\phi)^2 \left(\gamma_b(\phi)+1\right)}\frac{b^2}{(1+\phi^2/\phi_c^2)^2}
      \,\frac{\phi^2}{\phi_c^4}\right)^{-1} & {\rm power-law} \\
      \left(\frac{v_r^2}{c^2}+\frac{3}{4}\,\frac{\left(\gamma_b(\phi)-1\right)}
      {\gamma_b^2(\phi)\left(\gamma_b(\phi)+1\right)}\frac{\phi^2}{\phi_c^4}\right)^{-1}
      & {\rm Gaussian\,model}
      \end{array}\,. \right.
\end{equation} Note that in contrast to nonrelativistic parallel shock acceleration,
$t_{\rm acc} \propto 1/\lambda$ as the probability of a particle sampling a higher
shear, and thus a more energetic scattering event increases with $\lambda$.
The general evolution of $t_{\rm acc}/t_0$ as a function of $\phi$, with $t_0 = 45\,r^2/[4
(4+\alpha)\,\lambda'\, c]$, is illustrated in Fig.~\ref{fig1}, indicating that a power-law
model becomes somewhat more favourable than a Gaussian model. Obviously, the shear
acceleration timescale in structured jets generally increases with $\phi$ due to the
decrease in $\gamma_b$, suggesting different maximum attainable energies in the local
comoving frame. Hence, at constant $r$ the higher energy particles and thus the higher
energy emission are expected to be naturally concentrated closer to the jet axis, i.e.,
toward smaller $\phi$ (provided the magnetic field is not very inhomogeneous across
the jet), enforcing the effect that observers looking at the same GRB from different
directions will see different gamma-ray light curves.

In general several constraints need to be satisfied in order to allow for
efficient particle acceleration in GRB jets, e.g., the particle acceleration process
may be limited by (i) radiative synchrotron losses, (ii) the transversal confinement
size, (iii) particle escape via cross-field diffusion or (iv) the duration of the jet
expansion. The first constraint implies that the acceleration timescale has to be smaller
than the comoving synchrotron cooling timescale, which, for particles with Lorentz
factors $\gamma'$ and isotropic pitch angle distribution, is given by
\begin{equation}\label{tcool}
  t_{\rm cool} =\frac{9\,m^3\,c^5}{4\,e^4}\frac{1}{\gamma'\,B'^2}
                 \nonumber
               = 4.8\cdot 10^{12} \,\left(\frac{1}{\gamma'}\right)\,
                  \left(\frac{m}{m_p}\right)^3
                   \left(\frac{10^3\;{\rm G}}{B'}\right)^{2}\;[{\rm s}]\,,
\end{equation} with $m_p$ being the proton mass. According to eq.~(\ref{tshear1}) the minimum
shear acceleration timescale is
\begin{equation}\label{tmin}
 t_{\rm acc} \sim \frac{5}{2}\,\frac{r^2}{c\,\lambda'\,\gamma_{b}^2}\,,
\end{equation} so that for a gyro-dependent particle mean free path $\lambda' = \xi\, r_g'$,
with $\xi < 1$, (as motivated by the observationally implied particle spectra with
momentum index $\sim -2$) the ratio of shear to cooling timescale becomes independent
of the particle Lorentz factor. Note that eq.~(\ref{tmin}) is also the timescale for a
uniform flow with $v_r$ independent of $\phi$ (cf. eq.~[\ref{tshear}])! For a given
magnetic field strength the acceleration process may thus work efficiently as long as
the velocity shear remains sufficiently high, i.e., as long as the radial coordinate
satisfies
\begin{equation}
 r < 4 \cdot 10^{15}\, \xi^{1/2}\left(\frac{m}{m_p}\right)^2
         \left(\frac{ 10^3\;{\rm G}}{B'}\right)^{3/2}
          \left(\frac{\gamma_{b0}}{300}\right)\;\,{\rm [cm]}\,,
\end{equation} suggesting that, in contrast to the acceleration of protons, efficient electron
acceleration is suppressed in the presence of high magnetic fields. It seems very likely,
however, that due to the expansion of the wind, the comoving magnetic field $B'$ will
depend inversely on $r$, i.e., $B' \propto 1/r^{\beta}$ with $\beta > 0$. In order to
study possible implications in more details, we may thus consider the following simple
parametrization for the commoving jet magnetic field strength, $B'= 1000\,b_0\,(10^{13}
\,{\rm cm}/r)^{\beta}$ G. For $\beta=1$ and $b_0=30$ this expression corresponds
to the lower limit required by \citet{wax95} in order to allow for efficient (second
order) Fermi-type acceleration of protons to ultra-high energies during expansion of
the wind (cf. his eq. [4a]). While it can be shown then that proton acceleration is
nearly unconstrained by radiative synchrotron losses, electron acceleration can only
work efficiently if the field decays rapidly enough and becomes comparatively weak,
e.g., for $\beta=2$ efficient electron acceleration may occur for $r \geq 10^{15}
\xi^{-1/4}\,b_0^{3/4}(300/\gamma_{b0})^{1/2}$ cm.\\
The second constraint requires the particle mean free path $\lambda'$ to be
smaller than the transversal width $R \sim r\,\phi_j$ of the jet (with $\phi_j >
\phi_c$ being the jet opening angle in the cosmological rest frame). For $\lambda' =
\xi\, r_g'$, $\xi < 1$ and $r_g'=\gamma' m\, c^2/(e\,B')$, this implies an upper
limit for the maximum possible (comoving) particle Lorentz factor given by
\begin{equation}
\gamma_{\rm max}' \sim 10^{9} \left(\frac{1}{\xi}\right) \left(\frac{m_p}
            {m}\right) \left(\frac{B'}{10^3\;{\rm G}}\right) \left(\frac{r}
            {10^{13}{\rm cm}}\right)\left(\frac{\phi_j}{0.3\,{\rm rad}}\right)\,.
\end{equation} Note again, that for $B'=1000\,b_0\,(10^{13}\,{\rm cm}/r)^{\beta}$ G, this
translates into maximum particle energies of $E \sim 2.8\cdot 10^{20}\,\xi^{-1}\,
b_0\,(10^{13}\,{\rm cm}/r)^{\beta-1}\,(\phi_j/0.3\,{\rm rad})\,(\gamma_{b}/300)$
eV in the observer frame. For a simple linear scaling, i.e., $\beta=1$, the maximum
energy would become independent of $r$, while for $\beta>1$ the higher energy
particles would originate on smaller scales.\\
The third constraint requires that the timescale for (cross-field) diffusion in the
comoving frame $t_{\rm cf} \sim R^2/\kappa'$, with $\kappa' \sim c\,\lambda'/3$,
is larger than $t_{\rm acc}$, i.e., $t_{\rm acc} \leq 3 \,(r^2\phi_j^2)
/(c\,\lambda')$. Comparison with eq.~(\ref{tshear1}) reveals that $t_{\rm acc}$
scales with $r$ and $\lambda'$ in the same way as $t_{\rm cf}$, indicating that
the third constraint may be easily satisfied as long as $\gamma_b(\phi)$ is larger
than a few, cf. Fig.~(\ref{fig1}).\\
Finally, the fourth constraint requires $t_{\rm acc}$ to be smaller than
the time $t_w' =\Delta'/c$ needed to transverse the overall radial (comoving)
width $\Delta'$ of the flow, which translates into a lower limit for the required
particle Lorentz factors. This width is essentially determined by the activity
of the inner engine driving the GRB and, when specified in the
observer frame, should be at least of order of the observed GRB duration $t_{\Delta}$
\citep[cf.][]{pir05}, i.e., $\Delta = t_{\Delta}\,c=\Delta'/\gamma_b$. Allowing
for expansion effects on larger scales with $\Delta' \stackrel{>}{\scriptstyle \sim} r/\gamma_b$ \citep[cf.]
[]{mes93}, we have $t_w' = {\rm max}\{\gamma_b\,t_{\Delta}, r/[\gamma_b\,c]\}$.
Using Eq.~(\ref{tmin}) one thus finds
\begin{equation}\label{constraint4}
\gamma_{\rm min}' \simeq 10^4 \,\xi^{-1} \left(\frac{m_p}
                  {m}\right)\left(\frac{B'}{10^3\;{\rm G}}\right)
                  \left(\frac{300}{\gamma_b}\right)^3\left(\frac{r}{10^{13}\,
                  {\rm cm}}\right)^2 \times {\rm min}\left\{\left( \frac{10\,{\rm sec}}
                  {t_{\Delta}}\right),\chi(r)\right\} \,,
\end{equation} where $\chi(r)= 2.7\cdot 10^3\,(\gamma_b/300)^4\,(10^{13}\,{\rm cm}/r)$. For the
chosen magnetic field parametrization $B'=1000\,b_0\,(10^{13}\,{\rm cm}/r)^{\beta}$ G,
this results in $\gamma_{\rm min}' \simeq 10^4\, \xi^{-1}\,(m_p/m)\,\,b_0
\,(r/10^{13}\,{\rm cm})^{2-\beta}$ $(300/\gamma_{b})^3 \times{\rm min}\{(10\,{\rm sec}/
t_{\Delta}), \chi(r)\}$, indicating that for $\beta=2$ the leading term becomes
independent of $r$, which may be favourable for efficient electron acceleration.
A more restrictive condition, roughly corresponding to $\gamma_{\rm min}'$
as implied by $\chi(r)$ in eq.~(\ref{constraint4}), is usually associated with
the requirement to overcome adiabatic losses.
Accordingly, shear acceleration can only act efficiently on particles that were
sufficiently accelerated through other processes. In the GRB context it is possible
that such particles can be provided, for example, by the mechanism responsible for
the prompt burst of emission (e.g., shock acceleration).

In summary, if the jet magnetic field is sufficiently weak (e.g., $b_0 \sim 1$)
and/or decays rapidly enough (say, e.g., with $\beta = 2$), both the first and
the fourth constraint may be satisfied even for electrons, so that the maximum
energy is essentially determined by the third constraint. Particles may then, for
example, be accelerated efficiently at distances larger than $r \sim 10^{12}\,
(m_p/m)$ cm (cf. constraint [i]), with possible maximum energies, measured in the
observer frame, of less than $\sim 10^{18}$ eV for electrons and $\sim 10^{21}$ eV
for protons, and successively (i.e., linearly for $\beta=2$) decreasing with time.
For a high (e.g., $b_0 \sim 10$) and slowly decaying (say, e.g., $\beta \leq 1$)
jet magnetic field, on the other hand, efficient shear acceleration of electrons is
virtually excluded, while protons again may well be accelerated up to energies
$>10^{20}$ eV.

\section{Comparison with shock acceleration}
It is widely believed that shock-accelerated electrons are responsible for the
observed prompt GRB and afterglow emission via synchrotron radiation processes
\citep[e.g.,][]{pir05}. In particular, diffusive electron acceleration at midly
relativistic internal shocks (with Lorentz factor $\Gamma_s \sim$ a few), caused
by velocity variations in the relativistic outflow is usually thought to be
behind the powerful burst of $\gamma$-rays \citep[][]{ree94}. In the case of
nonrelativistic shocks Fermi acceleration leads to power-law particle spectra
$N(\gamma) \propto \gamma^{-s}$, which are only dependent on the shock compression
ratio $\rho=u_1/u_2$ (where $1 < \rho \leq 4$), i.e., $s=(\rho+2)/(\rho-1)$, so
that for strong shocks ($\rho=4$) the famous $s=2$ result is obtained \citep[e.g.,]
[]{dru83,kir99}. In general, the acceleration timescale for diffusive shock
acceleration depends on both the upstream and downstream residence times. For
an unmodified nonrelativistic shock, one thus obtains \citep[e.g.,][]{dru83,jok87}
\begin{equation}
  t_{\rm acc}=\frac{3}{u_1-u_2}\left(\frac{\kappa_1}{u_1}+\frac{\kappa_2}{u_2}\right)
             =\frac{3\,\rho}{(\rho-1)}\frac{\left(\kappa_1+\rho\,\kappa_2\right)}
                {u_s^2}\,,
\end{equation} with $u_1$ and $u_2$ the upstream and downstream flow velocities measured in
the shock frame, respectively, $\kappa_1$ and $\kappa_2$ the upstream and downstream
(spatial) diffusion coefficients, and $u_s$ the shock speed as measured in the upstream
frame. If the acceleration process operates at nearly the Bohm limit [i.e., $\kappa
\simeq \lambda\,c/3 = \eta\, r_g\,c/3$, with $\eta \simeq O(1)$], one finds
\begin{equation}\label{tacc2}
 t_{\rm acc} \stackrel{>}{\scriptstyle \sim} 6\, \eta\,r_g\,\frac{c}{u_s^2}\,,
\end{equation} assuming $\kappa_1 \simeq \kappa_2$. When $u_s\sim c$ this is comparable to
the Larmor time which is a result also predicted by the simulations of \citet{lem03}.
It has been often suggested that the nonrelativistic limit can also be approximately
applied to midly relativistic shocks in GRBs, proposing, for example, that efficient
proton acceleration to ultra-high energies might be possible \citep[e.g.,][]{wax04}.
Equating the acceleration timescale (eq.~[\ref{tacc2}]), replacing $r_g$ by $r_g'$ and
$u_s$ by $u_s'$ as the (comoving) internal shock speed, with the cooling timescale
eq.~(\ref{tcool}), using $B'=1000\,b_0\,(10^{13}\,{\rm cm}/r)^{\beta}$ G and $u_s'
=\beta_s\,c$ with $\beta_s < 1$, gives
\begin{equation}\label{c_shock1}
 \gamma_{\rm max}' \simeq 2.5\cdot 10^9\,\eta^{-1/2}\,b_0^{-1/2}
                    \left(\frac{\beta_s}{0.9}\right)\,\left(\frac{m}{m_p}\right)\,
                    \left(\frac{r}{10^{13}\,{\rm cm}}\right)^{\beta/2}\,.
\end{equation} Note that magnetic field amplification in the vicinity of a shock may yield a
field value $b_0$ well above the usual (background) flow magnetic field, thus leading
to a somewhat more restrictive condition. In addition, the acceleration timescale
also has to be smaller than the comoving time $\sim r/{c\,\gamma_b}$ needed to
transverse the width of a shell, which gives
\begin{equation}\label{c_shock2}
\gamma_{\rm max}' \stackrel{<}{\scriptstyle \sim} 1.5 \cdot 10^6\,\eta^{-1}\,b_0\, \left(\frac{\beta_s}{0.9}
                  \right)^2\,\left(\frac{m_p}{m}\right)\,\left(\frac{300}{\gamma_b}
                  \right)\,\left(\frac{10^{13}\,{\rm cm}}{r}\right)^{\beta-1}\,.
\end{equation}
A more detailed comparison of shear with shock acceleration may perhaps be reached
by analyzing a critical particle Lorentz factor $\gamma_{\rm eq}'\propto b_0\,
r^{1-\beta}\,(m_p/m)$, defined by equating eq.~(\ref{tmin}) with eq.~(\ref{tacc2}),
above which shear acceleration may become more efficient than shock acceleration.
Several applications with respect to the acceleration of protons are illustrated in
Fig.~(\ref{fig2}), suggesting the possibility that under a reasonable range of
conditions shear acceleration in GRB jets may become more relevant for the production
of UHE cosmic rays than shock-type acceleration processes. Concerning electron
acceleration in weak magnetic fields (e.g., $b_0 \leq 1$, $\beta=2$, $r \stackrel{>}{\scriptstyle \sim}
10^{15}$ cm), on the other hand, shock-type processes would allow for a much
quicker particle energization (but would also be more severely limited by
eq.~[\ref{c_shock1}] and eq.~[\ref{c_shock2}]) and shear effects would only
become dominant above $\gamma' \sim 10^8\,(10^{15}\,{\rm cm}/r)$.

\section{Conclusions}
Using an idealized model we have analyzed the possible role of shear acceleration
for the energization of particles in relativistic GRB-type jets. Our results suggest
that efficient electron acceleration on scales $r\stackrel{>}{\scriptstyle \sim} 10^{15}$ cm, with maximum
energy decreasing with distance, may be possible in the presence of weak magnetic
fields, assuming that high energy seed particles are provided by the mechanism
responsible for the prompt burst of emission. This may result in a weak and long
duration component in the GRB emission. Protons, on the other hand, may reach UHE
energies $>10^{20}$ eV under a broad range of conditions.

\acknowledgments
We are grateful to R.~Blandford for discussions and the referee for very useful
comments. FMR acknowledges support through a Marie-Curie Individual and a
Cosmogrid Fellowship.


\section{Introduction}

How long do perturbations ``remember" their initial state? What sets the validity of linear perturbation theory? The answer to these questions is important because, for example, measures of the power spectrum in the linear regime are being used~\cite{WMAP-SDSS} and planned to be used at higher redshifts~\cite{zeq3} to determine cosmological parameters. In the linear regime the fluctuations being observed contain the full information in the initial conditions~\footnote{Fundamentally, this is also true at the nonlinear level. The nonlinear evolution of collisionless dark matter is described by the Vlasov equation, which fully preserves information since the initial phase-space density {\em is conserved} along particle trajectories, the characteristic curves of the Vlasov-Poisson system. In practice we don't have full access to phase-space densities $f({\hbox{\bf x}},{\hbox{\bf p}})$ and we try to learn about initial conditions from statistics of the evolved system. In this case whatever information is available is encoded in all the correlation functions of (moments with respect to ${\hbox{\bf p}}$ of) $f$. For a discussion of information in gravitational clustering from a different point of view see~\cite{infoP}.}. The breakdown of linear theory means that, unless we can somehow model deviations from it, only a very restricted range of scales is reliable for cosmological parameter estimation.

In this paper we study the propagator of density and velocity fields, the main ingredient that enters into a well-controlled extension of perturbation theory into the nonlinear regime (see our companion paper~\cite{paper1}, hereafter paper I). We shall see that the propagator measures the memory of initial conditions and its decay at high wavenumbers $k$ is nearly Gaussian with a characteristic scale that provides a clean definition of the scale at which linear theory breaks down. The end result of our study is an analytic expression for the fully nonlinear propagator of density and velocity fields that can be used in the formalism developed in paper I to calculate the nonlinear power spectrum~\cite{paper3},  and other statistics.

The most common criterion used in the literature to decide the validity of linear perturbation theory is to calculate the linear power spectrum and look for the scale at which the {\em rms} fluctuations are unity. This is intuitively reasonable, but in reality is very unreliable. The reason is that nonlinear corrections depend sensitively on the shape of the initial power spectrum (see e.g.~Fig.~12 in ~\cite{review}), not just on its amplitude. A better approach is to see how well does the final configuration of density and velocity fields looks like evolved from initial conditions by just pure linear theory. One statistical way of doing this is by calculating the cross-correlation coefficient $r_a$ between initial and final conditions as a function of scale,

\begin{equation}
r_a(k) \equiv \frac{X_a(k)}{\sqrt{P_a(k)P_0(k)}},
\label{crosscorr}
\end{equation}

where $\mathop{\langle} \Psi_a({\hbox{\bf k}}) \delta_0({\hbox{\bf k}}') \mathop{\rangle}= X_a(k)\, \delta_{\rm D}({\hbox{\bf k}}+{\hbox{\bf k}}')$ is the cross-spectrum between initial (described by the density $\delta_0$, velocities are assumed in the linear growing mode) and final conditions (denoted by $ \Psi_a$, with  $a=1$ for density and $a=2$ for velocity divergence field, see Eq.(\ref{2vector}) for more details). In Eq.~(\ref{crosscorr}) $P_0$ denotes the initial power spectrum, and $P_a$ the final one. Figure~\ref{figure1} shows $r_a(k)$ at redshift $z=0,3$ for density and velocity fields. Note that, as expected, the cross-correlation decays slower for $z=3$ than $z=0$, indicating that linear theory is valid over a larger range of scales at $z=3$ than $z=0$. However, the characteristic scale of decay does not differ by as much as one would guess from the condition that the fluctuations be of order unity. Indeed, if one estimates the nonlinear scale $k_{\rm nl}$ from the linear spectrum, $4\pi k_{\rm nl}^3 P_L(k_{\rm nl})=1$, $k_{\rm nl}(z=0)=0.21 \, h \, {\rm Mpc}^{-1}$ and $k_{\rm nl}(z=3)=4.02 \, h \, {\rm Mpc}^{-1}$, a ratio of about 20, whereas the characteristic scale of decay of cross-correlations between initial and final fields (though it agrees with $k_{\rm nl}$ at $z=0$) only shifts by a factor of about $2-3$ at $z=3$. Therefore, the ``non-linear scale" can be misestimated by a large factor if $4\pi k_{\rm nl}^3 P_L(k_{\rm nl})=1$ is used at high redshifts.

\begin{figure*}
\begin{center}
\begin{tabular}{cc}
{\includegraphics[width=0.5\textwidth]{crosscorr3.eps}}&
{\includegraphics[width=0.5\textwidth]{crosscorr0.eps}}
\end{tabular}
\caption{Cross-correlation coefficients between initial and final conditions $r_a$ as a function of wavenumber $k$ for density ($a=1$, solid lines) and velocity divergence ($a=2$, dashed lines) fields at redshift $z=3$ (left panel) and $z=0$ (right panel). Initial conditions are at redshift $z=35$.}
\label{figure1}
\end{center}
\end{figure*}

In practice, the cross-correlation coefficient between initial and final conditions is not entirely satisfactory as a measure of deviations from linear evolution, since the denominator in Eq.~(\ref{crosscorr}) is also sensitive to nonlinearities through $P_a(k)$, and this can lead to cancellations or enhacements. In particular, from Fig.~\ref{figure1} it appears that the density becomes nonlinear at slightly larger scales (lower $k$) than the velocities; however, this turns out to be incorrect, being more a reflection of the fact that nonlinear corrections to $P_a(k)$ are mostly positive for the density ($a=1$) and negative for the velocities ($a=2$). As we shall discuss in detail below, this problem can be avoided by the use of the propagator of density and velocity fields rather than the cross-correlation coefficient, the propagator $G_{ab}$ is defined by~\cite{Sco00,paper1}

\begin{equation}
G_{ab}(k,\eta)\ \delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}') \equiv \left\langle \frac{\delta \Psi_a({\hbox{\bf k}},\eta)}{\delta \phi_b({\hbox{\bf k}}')}\right\rangle,
\label{NLProp}
\end{equation}
where $\Psi_a$ represents the final conditions and $\phi_a$ the initial conditions, and again $a=1$ corresponds to density and $a=2$ to velocity divergence fields. As we show below, for Gaussian initial conditions the propagator is proportional to the cross-correlation coefficient with a proportionality constant that gets rid of the dependence on $P_a(k)$, avoiding the problem with $r_a$. It is obvious from the definition in Eq.~(\ref{NLProp}) that if linear theory is valid the derivative yields a scale-independent result (the growth factor, for growing-mode initial conditions), and any dependence on scale is due to nonlinearities. As $k$ increases nonlinear effects become important and the final state does not ``remember" the initial one, driving the derivative to zero on average.  In this sense, the propagator can be though as a measure of the memory that perturbations have to their initial conditions. The characteristic scale of decay of the propagator is the cleanest measure of the scale where nonlinearities become important.

This paper is organized as follows. In the next section we review the basics of perturbation theory, see~\cite{review,paper1} for more details. Section~\ref{CalcProp} describes how we arrive at the analytic expression for the nonlinear propagator in RPT. Section~\ref{Measure} presents how to measure the propagator in numerical simulations and compares the results with RPT. Section~\ref{conclude} presents our conclusions.

\section{Equations of Motion and Perturbation Theory}

In this section we briefly review perturbation theory (PT) in a form useful to facilitate the process of resummation (see \cite{paper1} for more details). The equations of motion for the evolution of dark matter can be written in a compact way by introducing the two-component ``vector''

\begin{equation}
\Psi_a({\hbox{\bf k}},\eta) \equiv \Big( \delta({\hbox{\bf k}},\eta),\ -\theta({\hbox{\bf k}},\eta)/{\cal H} \Big),
\label{2vector}
\end{equation}
where the index $a=1,2$ selects the density or velocity components, with $\delta({\hbox{\bf k}})$ being the Fourier transform of the density contrast  $\delta ({\hbox{\bf x}},\tau)=\rho({\hbox{\bf x}})/\bar \rho - 1$ and similarly for the peculiar velocity divergence $\theta\equiv\nabla\cdot{\bf v}$. ${\cal H}\equiv {d\ln a /{d\tau}}$ is the conformal expansion rate
with  $a(\tau)$ the cosmological scale factor and $\tau$ the conformal time. The time variable $\eta$ is defined from the scale factor by

\begin{equation}
\eta\equiv\ln a(\tau),
\label{timevariable}
\end{equation}
corresponding to the number of e-folds of expansion. Here we consider a cosmology where $\Omega_m=1$ and $\Omega_{\Lambda}=0$, see section~\ref{cosmodep} for the more general case. The equations of motion in Fourier space can then be written as (we henceforth use the convention that repeated Fourier arguments are integrated over)
\begin{equation}
\partial_{\eta} \Psi_a({\hbox{\bf k}},\eta) + \Omega_{ab} \Psi_b({\hbox{\bf k}},\eta) =
 \gamma_{abc}^{(\rm s)}({\hbox{\bf k}},{\hbox{\bf k}}_1,{\hbox{\bf k}}_2) \ \Psi_b({\hbox{\bf k}}_1,\eta) \ \Psi_c({\hbox{\bf k}}_2,\eta),
\label{eom}
\end{equation}
where
\begin{equation}
\Omega_{ab} \equiv \Bigg[
\begin{array}{cc}
0 & -1 \\ -3/2 & 1/2
\end{array}        \Bigg],
\end{equation}
and the symmetrized {\it vertex} matrix $\gamma_{abc}^{(\rm s)}$ describes the non linear interactions between different Fourier modes and is given by

\begin{eqnarray}
\gamma_{121}^{({\rm s})}({\hbox{\bf k}},{\hbox{\bf k}}_1,{\hbox{\bf k}}_2)&=&\delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}_1-{\hbox{\bf k}}_2) \  {({\hbox{\bf k}}_1+{\hbox{\bf k}}_2) \cdot {\hbox{\bf k}}_1 \over{2 k_1^2}},
\label{ga121} \nonumber \\
\gamma_{222}^{({\rm s})}({\hbox{\bf k}},{\hbox{\bf k}}_1,{\hbox{\bf k}}_2)&=&\delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}_1-{\hbox{\bf k}}_2) \ {|{\hbox{\bf k}}_1+{\hbox{\bf k}}_2|^2 ({\hbox{\bf k}}_1 \cdot {\hbox{\bf k}}_2 )\over{2 k_1^2 k_2^2}},
\label{ga222}
\label{vertexdefinition}
\end{eqnarray}
$\gamma_{abc}^{(\rm s)}({\hbox{\bf k}},{\hbox{\bf k}}_i,{\hbox{\bf k}}_j)=\gamma_{acb}^{(\rm s)}({\hbox{\bf k}},{\hbox{\bf k}}_j,{\hbox{\bf k}}_i)$, and $\gamma$ is zero otherwise, $\delta_{\rm D}$ denotes the Dirac delta distribution. The formal integral solution to Eq. (\ref{eom}) is given by (see \cite{Sco98,Sco00,paper1} for a detailed derivation)

\begin{equation}
\Psi_a({\hbox{\bf k}},\eta) = g_{ab}(\eta) \ \phi_b({\hbox{\bf k}}) + \int_0^\eta  d\eta' \ g_{ab}(\eta-\eta') \ \gamma_{bcd}^{(\rm s)}({\hbox{\bf k}},{\hbox{\bf k}}_1,{\hbox{\bf k}}_2)\ \Psi_c({\hbox{\bf k}}_1,\eta') \Psi_d({\hbox{\bf k}}_2,\eta'),
\label{eomi}
\end{equation}
where  $\phi_a({\hbox{\bf k}})\equiv\Psi_a({\hbox{\bf k}},\eta=0)$ denotes the initial conditions, set when the linear growth factor $a(\tau)=1$ and $\eta=0$. The {\em linear propagator} $g_{ab}(\eta)$ is given by

\begin{equation}
g_{ab}(\eta) = \frac{{\rm e}^{\eta}}{5}
\Bigg[ \begin{array}{rr} 3 & 2 \\ 3 & 2 \end{array} \Bigg] -
\frac{{\rm e}^{-3\eta/2}}{5}
\Bigg[ \begin{array}{rr} -2 & 2 \\ 3 & -3 \end{array} \Bigg],
\label{prop}
\end{equation}
for $\eta\geq 0$, whereas $g_{ab}(\eta) =0$ for $\eta<0$ due to causality, and $g_{ab}(\eta) \rightarrow \delta_{ab}$ as $\eta\rightarrow 0^{+}$. The {\em linear propagator} is the Green's function of the linearized version of Eq.~(\ref{eom}) and describes the standard linear evolution of the density and velocity fields from their initial state.

An explicit expression for $\Psi_a({\hbox{\bf k}},\eta)$ can be written in the form of a series expansion

\begin{eqnarray}
\Psi_a({\hbox{\bf k}},\eta)=\sum_{n=1}^{\infty} \Psi^{(n)}_a({\hbox{\bf k}},\eta),
\label{seriesexp}
\end{eqnarray}
with
\begin{equation}
\Psi_a^{(n)}({\hbox{\bf k}},\eta)= \int \delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}_{1\ldots n})\ {\cal F}_{a a_1 a_2 \ldots a_n}^{(n)}({\hbox{\bf k}}_1,\ldots,{\hbox{\bf k}}_n;\eta)\ \phi_{a_1}({\hbox{\bf k}}_1) \ldots \phi_{a_n}({\hbox{\bf k}}_n),
\label{seriesol}
\end{equation}
where ${\hbox{\bf k}}_{1\ldots n} \equiv {\hbox{\bf k}}_1+ \ldots + {\hbox{\bf k}}_n$. Replacing Eq.~(\ref{seriesexp}) and~(\ref{seriesol}) into  Eq.~(\ref{eomi}) determines the recursion relations satisfied by the kernels ${\cal F}_{a a_1 a_2 \ldots a_n}^{(n)}$.

In the standard form of Perturbation Theory (see~\cite{review} for a review), only the fastest growing mode is considered at each order. In contrast with this, Eq.~(\ref{seriesol}) gives the full time dependence, including all transients from initial conditions \cite{Sco98,paper1}. This is important since it plays a key role in allowing the process of resummation of the non linear propagator. See paper~I for a detailed discussion of this point.

It is worth noticing that so far we have not yet assumed any particular {\it kind} of initial conditions. From a physical point of view, the most interesting initial configurations are those where $\delta({\hbox{\bf k}},\eta=0)$ and $\theta({\hbox{\bf k}},\eta=0)$ are proportional Gaussian random fields, and thus we can write $\phi_a({\hbox{\bf k}})=u_a \delta_0({\hbox{\bf k}})$, with $\delta_0({\hbox{\bf k}})$ Gaussian distributed. Of particular importance when setting initial conditions in simulations are the {\em growing mode initial conditions}, for which $u=(1,1)$. Only the case $\phi_a({\hbox{\bf k}})=u_a \delta_0({\hbox{\bf k}})$ will be considered in this paper. Replacing this relation in Eq.~(\ref{seriesol}) we obtain a simplified version of it

\begin{equation}
\Psi_a^{(n)}({\hbox{\bf k}},\eta)= \int \delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}_{1\ldots n})\, {\bar {\cal F}}_a^{(n)}({\hbox{\bf k}}_1,\ldots,{\hbox{\bf k}}_n;\eta) \ \delta_0({\hbox{\bf k}}_1) \ldots \delta_0({\hbox{\bf k}}_n),
\label{seriesolsimp}
\end{equation}
where $\bar {\cal F}_a^{(n)} \equiv {\cal F}_{a a_1 \ldots a_n}^{(n)} \,u_{a_1}\ldots u_{a_{2n}}$. The recursion relations for the kernels $\bar {\cal F}_a^{(n)}$ are given in \cite{paper1}.

\section{Calculating the Propagator}
\label{CalcProp}

\subsection{The Propagator as a Memory of Initial Conditions}
\label{memIC}

Non linear interactions lead to deviations from the linear evolution described by the first term in the r.h.s of Eq.~(\ref{eomi}). It is possible to interpret this effect as a modification of the linear propagator defined in Eq.~(\ref{prop}), this leads to a {\it propagator renormalization}. The study of this renormalized quantity, the non linear propagator, is the main goal of this paper.

From its definition,  Eq.~(\ref{NLProp}), we see that the non linear propagator quantifies the dependence of the density and velocity fields on their initial values, in an {\it ensemble average} sense, and as we now show it has a very simple physical interpretation as a measure of memory of initial conditions. With the help of Eq.~(\ref{seriesexp}) we write Eq.~(\ref{NLProp}) as a series expansion

\begin{equation}
G_{ab}(k,\eta) = g_{ab}(\eta) + \sum_{n=2}^{\infty} \left\langle \frac{\delta \Psi^{(n)}_a({\hbox{\bf k}},\eta)}{\delta \phi_b({\hbox{\bf k}})}\right\rangle,
\label{PropagatorExpansion}
\end{equation}
where we have explicitly separated the linear part from the non-linear contributions. Now we want to make use of Eq.~(\ref{seriesol}) for $\Psi_a^{(n)}$, assuming Gaussian initial conditions. This implies that the statistical properties of the fields $\phi_a({\hbox{\bf k}})$ are completely characterized by the two-point correlator

\begin{equation}
\mathop{\langle} \phi_a({\hbox{\bf k}}) \ \phi_b({\hbox{\bf k}}') \mathop{\rangle} = \delta_{\rm D}({\hbox{\bf k}}+{\hbox{\bf k}}') \
u_a u_b\, P_0(k),
\label{2pt}
\end{equation}
where $P_0(k)$ denotes the initial power spectrum of density fluctuations. Therefore, for Gaussian initial conditions, only the odd terms in the series for $G_{ab}$ will give nonzero contributions after taking the functional derivative on $\Psi^{(n)}$ and performing the ensemble average. Replacing Eq.~(\ref{seriesol}) into Eq.~(\ref{NLProp}), we arrive at

\begin{equation}
G_{ab}(k,\eta) = g_{ab}(\eta) + \sum_{n=1}^{\infty} (2 n +1)!! \int \bar {\cal F}_{ab}^{(2n+1)}({\hbox{\bf k}},{\hbox{\bf k}}_1,-{\hbox{\bf k}}_1,\ldots,{\hbox{\bf k}}_n,-{\hbox{\bf k}}_n;\eta) P_0(k_1) \ldots  P_0(k_n),
\label{Gkernels}
\end{equation}
with
\begin{equation}
\bar {\cal F}_{ab}^{(2n+1)}({\hbox{\bf k}},{\hbox{\bf k}}_1,-{\hbox{\bf k}}_1,\ldots,{\hbox{\bf k}}_n,-{\hbox{\bf k}}_n;\eta) \equiv {\cal F}_{a b a_1 \ldots a_{2n}}^{(2n+1)}({\hbox{\bf k}},{\hbox{\bf k}}_1,-{\hbox{\bf k}}_1,\ldots,{\hbox{\bf k}}_n,-{\hbox{\bf k}}_n;\eta)\ u_{a_1}\ldots u_{a_{2n}} .
\label{Fbar}
\end{equation}

As pointed out in the introduction, the non linear propagator is closely related to the cross-correlation between final and initial conditions (assumed to be Gaussian). To see this, we multiply Eq.~(\ref{seriesol}) by $\phi_c({\hbox{\bf k}}')$, and ensemble averaging assuming Gaussian initial conditions it follows that

\begin{equation}
G_{ab}(k,\eta)\,\langle\phi_b({\hbox{\bf k}})\phi_c({\hbox{\bf k}}')\rangle=\langle\Psi_a({\hbox{\bf k}},\eta)\,\phi_c({\hbox{\bf k}}')\rangle.
\label{Gcorr}
\end{equation}
Therefore, while in the non linear case it is not true anymore that $\Psi_a({\hbox{\bf k}},\eta)=G_{ab}(k,\eta)\phi_b({\hbox{\bf k}})$, since $G_{ab}$ is not the Green's function of Eq.~(\ref{eom}), the nonlinear propagator plays the role of a Green's function in two-point sense, $\langle \Psi_a \phi_b \rangle = G_{ac}\, \langle \phi_c\phi_b\rangle$.
This is why $G_{ab}$ plays an essential role in calculating different correlation functions at the non linear level~\cite{paper1}. Also note that Eq.~(\ref{Gcorr}) makes explicit that $G_{ab}$ cannot depend upon the direction of ${\hbox{\bf k}}$, since both the initial and final configurations are statistically homogeneous and isotropic.

For the particular case when $\phi_b({\hbox{\bf k}})=u_b \delta_0({\hbox{\bf k}})$ and growing mode initial conditions, $u=(1,1)$, it follows that

\begin{equation}
\delta_{\rm D}({\hbox{\bf k}}+{\hbox{\bf k}}')\ [ G_{a1}(k,\eta)+G_{a2}(k,\eta) ]
=\frac{\langle\Psi_a({\hbox{\bf k}},\eta)\,\delta_0({\hbox{\bf k}}') \rangle}{P_0(k)}.
\label{propcrosscorrelation}
\end{equation}
or, in terms of the cross-correlation coefficient, Eq.~(\ref{crosscorr})

\begin{equation}
G_{a1}(k,\eta)+G_{a2}(k,\eta) = {\rm r}_a(k,\tau) \sqrt{\frac{P_a(k,\tau)}{P_0(k)}}.
\label{relation}
\end{equation}

Therefore, the nonlinear propagator quantifies the memory of a density and velocity divergence field to its initial conditions as a function of scale and time,  a direct measure of the correlation between the final and initial configurations.

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.8\textwidth]{figura5.eps}
\caption{Diagrams for the non linear propagator $G(k,\eta)$ up to two loops.}
\label{figure3}
\end{center}
\end{figure}

\subsection{Diagrammatic Representation}
\label{diagrammaticrepresentation}
In order to calculate analytically the terms in the series expansion of Eq.~(\ref{NLProp}), it is useful to resort to a diagrammatic representation (see paper I for more details). For Gaussian initial conditions, only the odd terms in the expansion for $G_{ab}$ are non zero, thus, we define

\begin{equation}
\delta G^{(n)}_{ab}(k,\eta)\ \delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}') \equiv \left\langle \frac{\delta \Psi^{(2n+1)}_a({\hbox{\bf k}},\eta)}{\delta \phi_b({\hbox{\bf k}}')}\right\rangle,
\end{equation}
so that Eq.~(\ref{NLProp}) becomes,
\begin{equation}
G_{ab}(k,\eta)=g_{ab}(\eta)+\sum_{n=1} \delta G^{(n)}_{ab}(k,\eta).
\label{volvioelaireacondicionado!!}
\end{equation}

In principle, an analytic expression for $\delta G_{ab}^{(n)}$ can be obtained, given the kernel $\bar{\mathcal F}^{(n)}_{ab}$, by integrating out Eq.~({\ref{Gkernels}). However, this path becomes very cumbersome with increasing $n$. In order to overcome this difficulty, and simplify the process of resummation, we developed in paper~I~\cite{paper1} a systematic approach in which the $n$th term in the series expansion of Eq.~(\ref{volvioelaireacondicionado!!}) is represented by a Feynman diagram with $n$ loops. Then, a set of rules enabled us to establish a one to one correspondence between diagrams and the corresponding integral contributing to $\delta G_{ab}^{(n)}(k,\eta)$. All the diagrams for the nonlinear propagator up to two loops are shown in Fig.~\ref{figure3}. Notice that the tree level diagram corresponds to the linear propagator $g_{ab}$ defined in Eq.(\ref{prop}).

A detailed description of the procedure to draw the diagrams, for both $\Psi^{(n)}_a$ and $\delta G^{(n)}_{ab}$, and the rules to write down the corresponding integrals is given in paper~I, here we only give a brief summary. The symbol $\otimes$ represents the initial power spectrum $u_a P_0(q) u_b$ [see Eq.~(\ref{2pt})], from where two lines emerge carrying opposite wavenumbers, say ${\hbox{\bf q}}$ and $-{\hbox{\bf q}}$. Each line finishes in a vertex, and represents a linear propagation $g_{ab}(s_i)$ from the initial conditions ($\eta=0$) to an arbitrary {\it time of interaction} $s_i$ at the vertex. At each vertex two modes interact, say ${\hbox{\bf k}}_1$ and ${\hbox{\bf k}}_2$, to give a nonlinear contribution to a third one, ${\hbox{\bf k}}$, with ${\hbox{\bf k}}={\hbox{\bf k}}_1+{\hbox{\bf k}}_2$ (conservation of momentum). Each vertex in a diagram represents a vertex matrix $\gamma^{(s)}_{abc}({\hbox{\bf k}},{\hbox{\bf k}}_1,{\hbox{\bf k}}_2)$, defined in Eq.~(\ref{vertexdefinition}). The lines between vertices symbolize linear propagations between the corresponding times, e.g. $g(s_i-s_j)$. In addition, each diagram has an {\it outgoing} and an {\it incoming} line, identifiable with arrows. They represent, respectively, a linear propagator $g(\eta-s_1)$ and $g(s_{2n})$, where $s_1$ and $s_{2n}$ are the times corresponding to the first and last vertices of a general diagram with $n$ loops. Finally, all the intermediate wave vectors running through the loops are integrated over as well as all the time variables $s_j$, each between $[0,\eta]$.

\subsection{The One-Loop Propagator}
\label{onelooppropagator}

The first non-linear correction in Eq.~(\ref{volvioelaireacondicionado!!}) follows simply from applying these rules and is represented by diagram $2$ in Fig.~\ref{figure3}. It reads,

\begin{eqnarray}
\delta G^{(1)}_{ab}(k,\eta)=4 \int_0^{\eta} ds_1 \int_0^{s_1} ds_2 \int d^3q \ \ g_{ac}(\eta-s_1)& \gamma^{(s)}_{cde}({\hbox{\bf k}},{\hbox{\bf q}},{\hbox{\bf k}}-{\hbox{\bf q}})& g_{df}(s_1)u_f\,g_{eg}(s_1-s_2) \nonumber \\ &
\gamma^{(s)}_{ghi}({\hbox{\bf k}}-{\hbox{\bf q}},-{\hbox{\bf q}},{\hbox{\bf k}})& g_{hj}(s_2)u_j\,g_{ib}(s_2)\,P_0(q) \ \ \ \
\label{OL1}
\end{eqnarray}
where the $\delta_{\rm D}$ in the definition of $\gamma^{(s)}$ in Eq.~(\ref{vertexdefinition}) has been integrated out, making explicit the conservation of momentum at each vertex. With the help of Eq.~(\ref{vertexdefinition}) and assuming growing mode initial conditions, is possible to write Eq.~(\ref{OL1}) as

\begin{equation}
\delta G^{(1)}= \int_0^{\eta} ds_1 \int_0^{s_1} ds_2 \int_{-1}^1 dx \int_0^{\infty} 2\pi q^2 dq \ \ \frac{k x}{q}\Big(1-\frac{k x}{q}\Big)\,g(\eta-s_1)\,\Gamma\,g(s_1-s_2)\Gamma^{-1}\,g(s_2) e^{s_1+s_2} P_0(q)
\label{OL2}
\end{equation}
with
\begin{equation}
\Gamma \equiv \Bigg[
\begin{array}{cc}
x\, k/q+x\, q/k-2x^2 & 1-x\, q/k \\ 0 &  1-x\, k/q
\end{array}        \Bigg],
\end{equation}
and $x={{\hbox{\bf k}}}\cdot{{\hbox{\bf q}}}/k\, q$. Equation (\ref{OL2}) can be integrated out to give,

\begin{equation}
\delta G^{(1)}_{ij}(k,a)=\sum_m \, f^{(m)}_{ij}(k) \, a^m
\label{fns}
\end{equation}
where here $m$ takes the values  $m=\{3,2,1,1/2,-1/2,-3/2\}$ and $a(\tau)=\exp(\eta)$, Eq.~(\ref{timevariable}). It is possible to show that only $4$ functions $f^{(m)}_{ij}$ are independent, out of the $24$ in the set. We take these four to be,
\begin{equation}
f\equiv\frac{5}{3}f^{(3)}_{11} \ \ ,\ \ g\equiv\frac{5}{2}f^{(-3/2)}_{11} \ \ ,\ \ h\equiv\frac{5}{2}f^{(1/2)}_{11} \ \ ,\ \ i\equiv\frac{5}{3}f^{(1)}_{11}.
\end{equation}
The normalization is such that in the large-$k$ limit $f,g,h,i \, \rightarrow -\frac{1}{2} k^2\sigma_v^2$, with $\sigma_v^2 \equiv \frac{1}{3} \int P_0(q) d^3q/q^2$, while they all vanish as $k \rightarrow 0$. Their explicit expressions are

\begin{eqnarray}
f(k)&=&\int\frac{1}{504 k^3 q^5}
\left[6k^7q-79k^5q^3+50q^5k^3-21kq^7+\frac{3}{4}(k^2-q^2)^3(2k^2+7q^2)\ln \frac{|k-q|^2}{|k+q|^2}\right]P_0(q) \, d^3q, \nonumber \\
g(k)&=&\int\frac{1}{168k^3q^5}
\left[6k^7q-41k^5q^3+2k^3q^5-3kq^7+\frac{3}{4}(k^2-q^2)^3(2k^2+q^2)\ln\frac{|k-q|^2}{|k+q|^2}\right]P_0(q)\, d^3q,     \nonumber \\
h(k)&=&\int\frac{1}{24k^3q^5}
\left[6k^7q+k^5q^3+9kq^7+\frac{3}{4}(k^2-q^2)^2(2k^4+5k^2q^2+3q^4)\ln\frac{|k-q|^2}{|k+q|^2}\right]P_0(q) \,d^3q,   \nonumber    \\
i(k)&=&\int\frac{-1}{72k^3\,q^5}
\left[6k^7q+29k^5q^3-18k^3q^5+27kq^7+\frac{3}{4}(k^2-q^2)^2(2k^4+9k^2q^2+9q^4)\ln\frac{|k-q|^2}{|k+q|^2}\right] P_0(q) \, d^3q,  \nonumber \\
\label{functionsofk}
\end{eqnarray}
Equation (\ref{fns}) can now be written in terms of these independent functions in a remarkably symmetric way

\begin{eqnarray}
\delta G_{11}^{(1)}(k,a)&=& \frac{3}{5} a \,\alpha(a)\ f(k) - \frac{3}{5} \beta(a)\ i(k) - \frac{2}{5} \gamma(a)\ h(k) + \frac{2}{5} a^{-3/2} \delta(a)\ g(k), \nonumber \\
\delta G_{12}^{(1)}(k,a)&=& \frac{2}{5} a \,\alpha(a)\ f(k) - \frac{2}{5} \beta(a)\ h(k) + \frac{2}{5} \gamma(a)\ h(k) - \frac{2}{5} a^{-3/2} \delta(a)\ f(k), \nonumber \\
\delta G_{21}^{(1)}(k,a)&=& \frac{3}{5} a \,\alpha(a)\ g(k) - \frac{3}{5} \beta(a)\ i(k) + \frac{3}{5} \gamma(a)\ i(k) - \frac{3}{5} a^{-3/2} \delta(a)\ g(k), \nonumber \\
\delta G_{22}^{(1)}(k,a)&=& \frac{2}{5} a \,\alpha(a)\ g(k) - \frac{2}{5} \beta(a)\ h(k) - \frac{3}{5} \gamma(a)\ i(k) + \frac{3}{5} a^{-3/2} \delta(a)\ f(k),
\label{proponeloop1}
\end{eqnarray}
with
\begin{eqnarray}
\alpha(a)=a^2-\frac{7}{5}a+\frac{2}{5}a^{-3/2},\,\beta(a)=\frac{3}{5}a^2-a+\frac{2}{5}a^{-1/2},\,\gamma(a)=\frac{2}{5}a^{2}-a^{1/2}+\frac{3}{5}a^{-1/2},\,\delta(a)=\frac{2}{5}a^{7/2}-\frac{7}{5}a+1.
\label{functionstime}
\nonumber
\end{eqnarray}

The low-$k$ limit of the one-loop propagator, $G^{(1)}=g+\delta G^{(1)}$, can be computed straightforwardly from Eqs.~(\ref{functionsofk}) and (\ref{proponeloop1}), reading~\cite{Sco00}

\begin{eqnarray}
G_{ab}^{(1)}(k,a) = g_{ab}(a) - (k \sigma_v)^2 a^3 \Bigg[
\begin{array}{cc} 61/350 & 61/525 \\ 27/50 & 9/25 \end{array}
\Bigg],
\label{lowklimit}
\end{eqnarray}
where we kept only the fastest growing mode. Using Eqs.~(\ref{prop}) and (\ref{lowklimit}) we obtain for growing-mode initial conditions

\begin{eqnarray}
G_\delta & \equiv & G_{11}+G_{12} \approx a\, \Big[1-\frac{61}{210}(k \sigma_v a)^2 \Big],
\label{Gde} \\
G_\theta & \equiv & G_{21}+G_{22} \approx a\, \Big[1-\frac{9}{10}(k \sigma_v a)^2 \Big],
\label{Gthe}
\end{eqnarray}
where we have defined the {\em density propagator} $G_\delta$ and the {\em velocity-divergence propagator} $G_\theta$ for growing mode initial conditions. We see that the one-loop corrections define a characteristic scale for which the propagator deviates significantly from its linear ($k$-independent) value. We shall come back to this important point below after we implement the process of resummation.

\subsection{Resummation in the large-$k$ limit}
\label{resummationlargek}

\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.9\textwidth]{figura2.eps}
\caption{Tree diagrams for $\Psi_a({\hbox{\bf k}},\eta)$, up to 4 vertices, that lead to the non linear propagator's resummable subset.}
\label{tree}
\end{center}
\end{figure}

As we discussed in paper~I, the non linear propagator plays an essential role in renormalized perturbation theory, since it allows a well-defined expansion for the power spectrum and higher-order correlation functions into the nonlinear regime. Therefore, one of the main objectives of this paper is to find an adequate analytical description for $G_{ab}$. In order to achieve this, we would  need to sum the infinite number of terms defining $G_{ab}$ in Eq.~(\ref{volvioelaireacondicionado!!}), equivalent to adding up an infinite number of  diagrams, shown in Fig.~\ref{figure3} up to two loops.

Although it is very likely that this resummation cannot be done exactly,  we shall see that reasonable physical arguments can be put forward to simplify the task enough so that it becomes doable. The first step is to realize that since the propagator measures how memory to initial conditions is lost, the next best thing to having an exact calculation of it, is to include in the calculation those processes that lead to the {\em slowest} decay, or memory loss, since they are the ones that establish the dominant behavior.

Different ``processes" first appear at two-loops, where there is more than one diagram contributing.
The two-loop diagrams in Fig.~\ref{figure3} can be organized according to how many interactions (vertices) take place along the principal path of the diagram (that connects the beginning to the end without going through initial conditions). For example diagrams 3,4,5 have all four interactions along the principal path, diagrams 8,9,10 have three and diagrams 6,7,11 have only two interactions along the principal path. Diagrams that have all interactions along the principal path are special in the sense that the principal path interacts only with waves coming directly from the initial conditions in their growing mode. In addition, in the high-$k$ limit each of these interactions results in a wave still in the growing mode, since as we show below in this case $\gamma_{abc}u_b u_c \propto u_a$ and this keeps repeating itself interaction after interaction along the principal path. For the other diagrams, on the contrary, the interactions outside the principal path do not preserve the linear growing mode in the high-k limit (since they involve momenta that cannot be compared to $k$). Therefore,  {\em when resummed} these should be suppressed compared to the resummation of the ones where all interactions happen along the principal path, since they should cross-correlate less with initial conditions. One exception to this is when the time between final and initial conditions is small enough, in which case there is no time for decorrelation and the distinction we made is not significant.

The dominant subset as defined above can be resummed exactly in the high-$k$ limit. In general, the subset has $(2n-1)!!$ diagrams with $n$ loops, and it gives rise to diagram 1 (tree level), diagram 2 (one loop) and diagrams 3, 4 and 5 (two loops) in Fig.~\ref{figure3}. The infinite number of diagrams in this subset originate in tree diagrams for $\Psi_a({\hbox{\bf k}},\eta)$, as described in paper~I, with a very simple topology. They all start in two Fourier modes that evolve linearly from the initial conditions in growing mode, with $u_a=(1,1)$, interact (represented by a vertex), and give rise to a third mode. This mode, in turn, evolves and interacts with a fourth mode that had grown linearly until that time, i.e. with no previous interaction. This process continues until the final time $\eta$, with $2n$ interactions needed to describe the $n$-loop diagrams. In Fig.~\ref{tree} we show the trees with 0, 2 and 4 vertices that will originate the {\em tree}, {\em one-loop} and {\em two-loop} diagrams respectively. {\em In contrast with the rest of the diagrams, the ones belonging to this subset have the simplest ramification possible and thus most direct connection to the initial conditions}. As discussed above, we expect this subset to be the dominant, although we cannot prove it rigorously since we could not sum up the remaining subset and show it is indeed subdominant.

The expression for $\Psi_a^{(n)}$, relevant to compute $\delta G_{ab}^{(n)}$ in Eq.~(\ref{volvioelaireacondicionado!!}), can be written recursively as,

\begin{equation}
\Psi^{(n)}_a({\hbox{\bf k}}',s)=2\int_0^s ds_n\,g_{ab}(s-s_n) \gamma^{(s)}_{bcd}({\hbox{\bf k}}',{\hbox{\bf q}}_n,{\hbox{\bf k}}+{\hbox{\bf q}}_{n-1\ldots1})\Psi_c^{(1)}({\hbox{\bf q}}_n,s_n)\Psi_d^{(n-1)}({\hbox{\bf k}}+{\hbox{\bf q}}_{n-1\ldots1},s_n)
\label{rec1}
\end{equation}
with ${\hbox{\bf k}}'={\hbox{\bf k}}+{\hbox{\bf q}}_{n\ldots1}$ and

\begin{equation}
\Psi^{(1)}_a({\hbox{\bf q}},s)=g_{ab}(s)\,\phi_b({\hbox{\bf q}})=e^{s}\,u_a\,\delta_0({\hbox{\bf q}})
\label{linearmode}
\end{equation}
The factor of $2$ in Eq.~(\ref{rec1}) is because all the branchings in Fig.~\ref{tree} are asymmetric if $n>2$. This should be replaced by 1 for $n=2$. Next, we need to take the functional derivative of Eq.~(\ref{rec1}) with respect to $\phi_b({\hbox{\bf k}})$. An important condition, in order to recover the desired diagrams, is that the functional derivative must be taken only on any of the two linear modes $\Psi^{(1)}$ that precede the very first vertex in Fig.~\ref{tree}. In other words, the derivative must be iterated through Eq.~(\ref{rec1}) until $n=2$. After using  Eq.~(\ref{rec1}) for $\Psi_a^{(2n+1)}$, taking the derivative as described and using Eq.~(\ref{linearmode}), we arrive at

\begin{eqnarray}
\frac{\delta \Psi^{(2n+1)}_a({\hbox{\bf k}}',\eta)}{\delta \phi_b({\hbox{\bf k}})}=2^{2n}\int_0^{\eta} ds_{2n} \int_0^{s_{2n}} ds_{2n-1} \ldots \int_0^{s_2} ds_1 \left[g(\eta-s_{2n})\,\bar\gamma^{(s)}({\hbox{\bf k}}+{\hbox{\bf q}}_{2n\ldots1},{\hbox{\bf q}}_{2n},{\hbox{\bf k}}+{\hbox{\bf q}}_{2n-1\ldots1})\right. \nonumber \\
\left. g(s_{2n}-s_{2n-1})\ldots g(s_2-s_1)\,\bar\gamma^{(s)}({\hbox{\bf k}}+{\hbox{\bf q}}_1,{\hbox{\bf q}}_1,{\hbox{\bf k}})\,g(s_1)\right]_{ab} \delta_0({\hbox{\bf q}}_{2n})\ldots\delta_0({\hbox{\bf q}}_1) \, e^{s_{2n}+\ldots+s_1},
\label{rec2}
\end{eqnarray}
where $\bar\gamma$ is a $2\times2$ matrix defined as $\bar\gamma^{(s)}_{ac}\equiv\gamma^{(s)}_{abc}\, u_b$, and ${\hbox{\bf k}}'={\hbox{\bf k}}+{\hbox{\bf q}}_{2n\ldots1}$. In the large-k limit, where the external momentum ${\hbox{\bf k}}$ is much larger than any internal ${\hbox{\bf q}}_i$, we can expand $\bar\gamma$ to leading order, with the help of Eq.~(\ref{vertexdefinition}), as

\begin{equation}
\bar\gamma^{(s)}_{ab}({\hbox{\bf k}}+{\hbox{\bf q}}_{n\ldots1},{\hbox{\bf q}}_n,{\hbox{\bf k}}+{\hbox{\bf q}}_{n-1\ldots1})\approx\frac{1}{2} \frac{k}{q_n} x_n \, \delta_{ab}
\label{VE}
\end{equation}
where $x_n= {\hbox{\bf k}} \cdot {\hbox{\bf q}}_n / k\,q_n$, and $\delta_{ab}$ is the $2\times2$ identity matrix. Equation~(\ref{VE}) shows that in the high-$k$ limit $\gamma_{abc}u_b u_c \propto u_a$ at each vertex for this subset of diagrams, as discussed above. A great simplification of Eq.~(\ref{rec2}) follows from the fact that $\bar\gamma$ is proportional to the identity matrix in the large-k limit. Replacing Eq.~(\ref{VE}) into Eq.~(\ref{rec2}), and using that $g(u-s)g(s-z)=g(u-z)$, we obtain

\begin{eqnarray}
\left\langle\frac{\delta \Psi^{(2n+1)}_a({\hbox{\bf k}}',\eta)}{\delta \phi_b({\hbox{\bf k}})} \right\rangle =g_{ab}(\eta) \frac{{(a(\eta)-1)}^{2n}}{(2n)!}\int d^3q_{2n}\ldots d^3q_1\,\frac{k^{2n}}{q_{2n}\ldots q_1}\,x_{2n}\ldots x_1\,\langle\delta_0({\hbox{\bf q}}_{2n})\ldots\delta_0({\hbox{\bf q}}_1)\rangle
\label{rec3}
\end{eqnarray}
where we have integrated the $2n$ time variables according to,

\begin{eqnarray}
\int_0^{\eta} ds_{2n} \int_0^{s_{2n}} ds_{2n-1} \ldots \int_0^{s_2} ds_1 \, e^{s_{2n}+\ldots+s_1} = \frac{{(a(\eta)-1)}^{2n}}{(2n)!}
\nonumber
\end{eqnarray}
with $\eta=\ln(a)$. The ensemble average of the $2n$ Gaussian fields leads to $(2n-1)!!$ equal contributions to Eq.~(\ref{rec3}), corresponding to all different pairings of the fields $\delta_0$. In addition, for any given pairing there are $n$ equal contributions, each given by

\begin{equation}
\int\int \frac{k^2}{q_i q_j}\, x_i x_j \, \langle \delta_0({\hbox{\bf q}}_i)\delta_0({\hbox{\bf q}}_j)\rangle \, d^3q_i d^3q_j = \frac{-k^2}{3} \int \frac{P_0(q)}{q^2} d^3 q \equiv -k^2 \sigma_v^2
\label{ec4}
\end{equation}
Replacing Eq.~(\ref{ec4}) to the $n$th power into Eq.~(\ref{rec3}), and the result into Eq.(\ref{volvioelaireacondicionado!!}) we get

\begin{equation}
G_{ab}(k,a)=g_{ab}(a)+ g_{ab}(a) \sum_{n=1} \frac{(2n-1)!!}{(2n)!} \left[ -k^2 \sigma_v^2 (a-1)^2 \right]^n
\end{equation}
This series can be summed up to give
\begin{equation}
G_{ab}(k,a)=g_{ab}(a) \exp(-k^2 \sigma_v^2 (a-1)^2/2),
\label{largeKresult}
\end{equation}
which shows that we expect the propagator to decay as a Gaussian in the large-$k$ limit.

\subsection{The Propagator in Renormalized Perturbation Theory}
\label{propRPT}

We would like now to extend the result in Eq.~(\ref{largeKresult}) to obtain an analytic result for the nonlinear propagator than can be used at any $k$. The Gaussian dependence on $k$ is also recovered, as shown  in paper~I, for the {\em exact} density propagator $G_{\delta}=G_{11}+G_{12}$  in the Zel'dovich approximation, which has a Gaussian dependence on $k$, {\em for all} $k$, $G^{{\rm ZA}} _{\delta}(k,a)= a\, {\rm e}^{-\frac{1}{2} k^2 \sigma_v^2 a^2}$ for growing-mode initial conditions.

Based on these considerations we assume that the nonlinear propagator has a nearly Gaussian behavior, with a characteristic scale of decay that can be then determined from knowing the behavior of the propagator for low-$k$, the one-loop result in Eq.~(\ref{proponeloop1}).

In order to do this and obtain a well defined nonlinear propagator we first rewrite $\delta G^{(1)}$ in Eq.~(\ref{proponeloop1}) in a way similar to the linear propagator. That is, we group the contributions to each component of $\delta G^{(1)}$ in two terms, one ``growing'' (with an overall factor $a$), and one ``decaying'' (with an overall $a^{-3/2}$). The only guidelines for defining each term are that the most growing contribution (i.e. $\alpha$) must belong to the growing part. In addition, both terms must be always negative, in particular in the limits of large $a$ (for all scales) and large-$k$ (for all times). This guarantees the nonlinear propagator will decay in the long-time or large-$k$ limit.

From their definition in Eq.~(\ref{functionsofk}), it is easy to check that $f,g,i$ are always negative and diverge monotonously to $-\infty$ as $k$ increases, as $-k^2 \sigma^2_v(\tau)/2$. Moreover, $h(k)>f(k)>g(k), i(k)$ for all $k$. Provided with this and the definitions of $\alpha$, $\beta$, $\gamma$ and $\delta$ in Eq.~(\ref{functionstime}), it is possible to conclude that the only feasible way to satisfy our requirements is to group things in the following way,

\begin{eqnarray}
g_{11}(a)+\delta G_{11}^{(1)}(k,a)&=& \frac{3}{5} \, a + \frac{2}{5} \, a^{-3/2} + \frac{3}{5} \, a \, [\,\alpha(a) \, f(k) - \beta_g(a) \, i(k)\,] + \frac{2}{5} \, a^{-3/2} \, [\,\delta(a) \, g(k) - \gamma_d(a) \, h(k)\,] \nonumber \\
g_{12}(a)+\delta G_{12}^{(1)}(k,a)&=& \frac{2}{5} \, a - \frac{2}{5} \, a^{-3/2} + \frac{2}{5} \, a \, [\,\alpha(a) \, f(k) - \beta_g(a) \, h(k)\,] - \frac{2}{5} \, a^{-3/2} \, [\,\delta(a) \, f(k) - \gamma_d(a) \, h(k)\,] \nonumber \\
g_{21}(a)+\delta G_{21}^{(1)}(k,a)&=& \frac{3}{5} \, a - \frac{3}{5} \, a^{-3/2} + \frac{3}{5} \, a \, [\,\alpha(a) \, g(k) + \gamma_g(a) \, h(k)\,] - \frac{3}{5} \, a^{-3/2} \, [\,\delta(a) \, g(k) + \beta_d(a) \, i(k)\,]\nonumber \\
g_{22}(a)+\delta G_{22}^{(1)}(k,a)&=& \frac{2}{5} \, a + \frac{3}{5} \, a^{-3/2} + \frac{2}{5} \, a \, [\,\alpha(a) \, g(k) - \frac{3}{2} \gamma_g(a) \, i(k)\,] + \frac{3}{5} \, a^{-3/2} \, [\,\delta(a) \, f(k) - \frac{2}{3} \beta_d(a) \, h(k)\,] \nonumber,  \\
\label{proponeloop}
\end{eqnarray}
 where we have redefined $\beta$ and $\gamma$ in Eq.~(\ref{functionstime}) through  $a\,\beta_g(a)=a^{-3/2} \beta_d(a)=\beta(a)$, and $a\,\gamma_g(a)=a^{-3/2} \gamma_d(a)=\gamma(a)$. Next we regard  Eq.~(\ref{proponeloop}) as a nearly Gaussian expansion of the full result and write

\begin{eqnarray}
g_{11}+\delta G^{(1)}_{11}\,\rightarrow\,G_{11}(k,a)&=&\frac{3}{5} \, a \, {\rm e}^{(\alpha(a) f(k) - \beta_g(a) i(k))} + \frac{2}{5} \, a^{-3/2} \, {\rm e}^{(\delta(a) g(k) - \gamma_d(a) h(k))} \nonumber ,\\
g_{12}+\delta G^{(1)}_{12}\,\rightarrow\,G_{12}(k,a)&=&\frac{2}{5} \, a \, {\rm e}^{(\alpha(a) f(k) - \beta_g(a) h(k))} - \frac{2}{5} \, a^{-3/2} \, {\rm e}^{(\delta(a) f(k) - \gamma_d(a) h(k))}  \nonumber ,\\
g_{21}+\delta G^{(1)}_{21}\,\rightarrow\,G_{21}(k,a)&=&\frac{3}{5} \, a \, {\rm e}^{(\alpha(a) g(k) + \gamma_g(a) h(k))} - \frac{3}{5} \, a^{-3/2} \, {\rm e}^{(\delta(a) g(k) + \beta_d(a) i(k))} \nonumber ,\\
g_{22}+\delta G^{(1)}_{22}\,\rightarrow\,G_{22}(k,a)&=&\frac{2}{5} \, a \, {\rm e}^{(\alpha(a) g(k) - (3/2) \gamma_g(a) i(k))} + \frac{3}{5} \, a^{-3/2} \, {\rm e}^{(\delta(a) f(k)-(2/3) \beta_d(a) h(k))}.
\label{model3one}
\end{eqnarray}
This is our prediction for the nonlinear propagator in RPT that we will contrast against N-body simulations below. Note that there are no free parameters, given an initial spectrum and cosmological parameters we can fully predict $G_{ab}(k,\eta)$. Before we do so, however, we must take into account how our calculation is changed when we go beyond the $\Omega_m=1$, $\Omega_\Lambda=0$ model we have considered so far.

\subsection{Dependence on $\Omega_m$ and $\Omega_\Lambda$}
\label{cosmodep}

The dependence on cosmological parameters ignored so far can be taken into account, to a very good approximation. First, if we redefine

\begin{equation}
\Psi_a({\hbox{\bf k}},\eta) \equiv \Big( \delta({\hbox{\bf k}},\eta),\ -\theta({\hbox{\bf k}},\eta)/{\cal H} f \Big),
\label{2vector2}
\end{equation}
where $f \equiv  {\rm d}\ln D_+/{\rm d}\ln a$, with $D_+$ the linear growth factor in the appropriate cosmology, and
\begin{equation}
\eta\equiv\ln D_+(\tau),
\label{timevariable2}
\end{equation}
the equations of motion take the same form as in Eq.~(\ref{eom}) with the only change~\cite{SCFFHM98}

\begin{equation}
\Omega_{ab} \equiv \Bigg[
\begin{array}{cc}
0 & -1 \\ -3\Omega_m/2f^2 & 1/2
\end{array}        \Bigg],
\label{Omega2}
\end{equation}

These results are so far exact. The problem in Eq.~(\ref{Omega2}) is that both $\Omega_m$ and $f$ are functions of time, therefore the Laplace transform solution in Eq.~(\ref{eomi}) is not correct anymore. Although the growing mode will be correct, due to the change in Eqs.~(\ref{2vector2}-\ref{timevariable2}), the decaying mode is not, since in general it is given by the Hubble constant $H$ which does not scale as $D_+^{-3/2}$ in the general case. However it is well known in the standard formulation of perturbation theory, that the dependence of the perturbative solutions in the values of $\Omega_m$ and $\Omega_{\Lambda}$ is extremely weak, once the linear growth factors have been scaled out~\cite{Bouchet,Bernardeu,review}. In other words, almost all of the information about the cosmological parameters is encoded in the linear growth factor $D_{+}(\tau)$. The reason is that during most of the time evolution $\Omega_m/f^2 \approx 1$~\cite{SCFFHM98}. Therefore, when we compare the propagator in RPT against simulations for the $\Lambda$CDM model in the next section, we will simply replace $a(\tau)$ in Eq.~(\ref{model3one}) by the corresponding linear growth factor $D_+(\tau,\Omega_m,\Omega_{\Lambda})$.

\section{Measuring the Propagator in Numerical Simulations}
\label{Measure}

We now describe how to measure the propagator in numerical simulations. The simulations we use here correspond to three different box sizes, $L_{\rm box}=100$, $239.5$, $479 \, h^{-1} \, {\rm Mpc}$ with a number of particles $N_{\rm par}=200^3$, $256^3$, $512^3$, respectively. They all correspond to the $\Lambda$CDM model with $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ and $\sigma_8=0.9$ at redshift $z=0$. The small and medium simulations have an input power spectrum with shape parameter $\Gamma=0.21$, the large simulation has instead an input power spectrum that includes the full transfer function with $\Omega_b=0.04$ in baryons. We have 24 realizations of the small simulation (see~\cite{karim} for a description of these), and only one realization of the medium and large simulations. In the following we present results from the large simulation, except for section~\ref{voldep} where we explore the dependence on the simulation volume.

The propagator has never been measured in a numerical simulation before. Here we explore two methods, one based on an implementation of the numerical derivative involved in its definition, Eq.~(\ref{NLProp}), the other based on the relationship between the propagator and the cross-correlation coefficient, Eq.~(\ref{relation}). We show that both methods give the same result.

\subsection{Computing the Functional derivative numerically}
\label{fdn}

According to Eq.~(\ref{NLProp}), the propagator involves a functional derivative of the final conditions with respect to the initial conditions, which formally is given by

\begin{equation}
\frac{\delta \Psi_a({\hbox{\bf k}})}{\delta \phi_b({\hbox{\bf k}}')} = \lim_{\epsilon \rightarrow 0}\  \frac{\Psi_a[\phi_b({\hbox{\bf k}})+\epsilon \, \delta_{\rm D}({\hbox{\bf k}}-{\hbox{\bf k}}')] - \Psi_a[\phi_b({\hbox{\bf k}})]}{\epsilon}.
\end{equation}

In order to measure $G_{ab}$ for a given Fourier mode ${\hbox{\bf k}}_i$, based on this definition we would need to evolve two simulations whose initial configurations of density and velocity fields are equal, except at the value of the Fourier coefficient $\phi_b({\hbox{\bf k}}_i)$, where they differ by a small amount $\epsilon$ (but consistent with Gaussian initial conditions). Next, the difference of the final Fourier coefficients $\Psi_a({\hbox{\bf k}}_i)$ must be divided by the initial difference $\epsilon$. This needs to be done several times for each Fourier mode, to obtain an expectation value, and repeated for each scale of interest. Computationally, this clearly represents a nearly impossible task.

Instead, we proceed based on an assumption that resembles the notion of ``ergodicity'', namely, that all Fourier modes ${\hbox{\bf k}}_i$, within a bin of wave vectors of magnitude around  $|{\hbox{\bf k}}|$, can be thought as different realizations of the same Fourier mode. This is the same idea one uses in calculating the power spectrum from a single realization, here also making use of the fact that $G_{ab}$ depends only on the magnitude of ${\hbox{\bf k}}$ due to statistical homogeneity and isotropy (see section~\ref{memIC}), as does the power spectrum. Therefore, given one realization of the simulation we compute the difference between $\Psi_a({\hbox{\bf k}}_i)$ and the value of the field $\Psi_a$ at another Fourier mode ${\hbox{\bf k}}_j$, within the same bin determined by $|{\hbox{\bf k}}|$. Then, we divide the result by their difference at the initial conditions, $\phi_b({\hbox{\bf k}}_i)-\phi_b({\hbox{\bf k}}_j)$. Finally, we average over all pairs of modes within each bin.

All the simulations we use were set up using growing mode initial conditions, where $\delta=\theta=\delta_0$ initially. As a result of this, it is only possible to measure a ``directional functional derivative'' $\langle \delta \Psi_a / \delta \phi_b \rangle \cdot u_b=G_{a1}+G_{a2} $, which following Eqs.~(\ref{Gde}-\ref{Gthe}) above we shall call the density ($G_\delta=G_{11}+G_{12}$) and velocity divergence ($G_\theta=G_{21}+G_{22}$) propagators.

In summary, given the Fourier coefficients of the final density ($a=1$) or velocity divergence fields ($a=2$), and the initial conditions $\delta_0({\hbox{\bf k}})$, we estimate the non linear propagators as

\begin{eqnarray}
G_{a1}(k,\eta)+G_{a2}(k,\eta)=\frac{1}{N(N-1)} \sum_{{\bf k}_i} \sum_{{\bf k}_j \neq {\bf k}_i } \frac{ \Psi_a({{\bf k}_i,\eta})- \Psi_a({{\bf k}_j},\eta)}{\delta_0({{\bf k}_i})- \delta_0({{\bf k}_j})},
\label{funcderivative}
\end{eqnarray}
where the sum runs over pairs of Fourier modes in the  bin of magnitude $|{\hbox{\bf k}}|$. The result is real because the sum involves pairs of Fourier modes with opposite directions.

\subsection{Computing the Propagator from the cross-correlation}
\label{crosscorrelationmemoryfunction}

\begin{figure*}
\begin{center}
\begin{tabular}{cc}
{\includegraphics[width=0.5\textwidth]{DerVsXden.eps}}&
{\includegraphics[width=0.5\textwidth]{DerVsXvel.eps}}
\end{tabular}
\caption{The density (left panel) and velocity divergence (right panel) propagators at redshift $z=0$ from initial conditions at $z=5$ (corresponding to $D_+=4.68$). The symbols represent measurement in numerical simulations, from implementation of the functional derivative (crosses) and from the relation to the cross-correlation coefficient (circles). The solid lines show the predictions of one-loop PT, Eq.~(\ref{proponeloop1}).}
\label{DerVsX}
\end{center}
\end{figure*}

An alternative approach is to take advantage of the relationship in Eq.~(\ref{propcrosscorrelation}) between the propagator and the cross-correlation coefficient. This leads to a straightforward method to measure the density and velocity propagators directly from simulations as

\begin{eqnarray}
G_{a1}(k,\eta)+G_{a2}(k,\eta)=\frac{1}{P_0(k)}\frac{1}{N} \sum_{{\bf k}_i} \Psi_a({{\bf k}_i,\eta})\delta_0({-{\bf k}_i}),
\label{crosscorrmethod}
\end{eqnarray}
where the sum runs over Fourier modes in the $|{\hbox{\bf k}}|$ bin. Notice that the two methods proposed to measure $G_{a1}+G_{a2}$ in Eqs~(\ref{funcderivative}) and (\ref{crosscorrmethod}) are intrinsically very different. One is rooted in the definition of the non linear propagator as a functional derivative while the other exploits the relation between $G_{ab}$ and the cross-correlation coefficient. Nevertheless, Fig.~\ref{DerVsX} shows that the two methods give essentially the same answer. Since the cross-correlation method is much faster (being of order $N$ as opposed to $N^2$) we shall use that in the following. The only limitation of Eq.~(\ref{crosscorrmethod}) is that it assumes Gaussian initial conditions, however, as long as the initial redshift is large enough this should not be a problem, as Fig.~\ref{DerVsX} shows for $z_{\rm initial}=5$.

Figure~\ref{DerVsX} also includes a comparison with the one-loop propagators from Eq.~(\ref{proponeloop1}), showing that in the low-$k$ limit the one-loop result does very well in describing the decay of the nonlinear propagators. We also see that the one-loop propagator becomes negative (and very quickly  large as $k$ increases), which is unphysical given its physical interpretation as measuring the memory to initial conditions. This makes clear the main point of RPT discussed in paper~I: unless the propagator is resummed to recover the right behavior as $k \rightarrow \infty$, the loop expansion in standard PT gives rise to large negative contributions which hinder the convergence of PT as nonlinear scales are probed. However, from Fig.~\ref{DerVsX} we see that the one-loop corrections are not too far off from describing the full decay of the propagator, therefore an approximate resummation as described  in section~\ref{propRPT}, stands a good chance of providing a good description.

\subsection{The Propagator in RPT vs. N-Body Simulations}

\begin{figure*}
\begin{center}
\begin{tabular}{cc}
{\includegraphics[width=0.5\textwidth]{RPTvsNBden.eps}}&
{\includegraphics[width=0.5\textwidth]{RPTvsNBvel.eps}}
\end{tabular}
\caption{Predictions for the propagator of density (left panel) and velocity divergence (right panel) from Eq.~(\ref{model3one}) against measurements in N-body simulations. The three cases correspond (from left to right in each panel) to $z=0,2,5$, with the initial conditions at $z_{\rm initial}=35$. The predictions shown in solid lines {\em have no free parameters}.}
\label{RPTvsNB}
\end{center}
\end{figure*}

Figure \ref{RPTvsNB} shows the comparison between the nonlinear propagator from Eq.~(\ref{model3one}) and measurements in the large numerical simulation, for various redshifts $z=0,2,5$ (corresponding to linear growth factors $D_+=28,11.8,5.9$) from initial conditions at $z_{\rm initial}=35$. The left panel shows the density propagator, whereas the right panel corresponds to the velocity divergence propagator. From this we see that, despite the approximations involved in the resummation process, Eq.~(\ref{model3one}) describes the dependence of the non linear propagator with scale and time remarkably well {\em without introducing any free parameters}. The reason why we can predict the behavior of the nonlinear propagator down to scales in the nonlinear regime can be summarized as,

\begin{itemize}

\item[i)] The one-loop correction already describes the onset of the decay of the propagator very well (Fig.~\ref{DerVsX}),

\item[ii)] In the high-$k$ and long-time limit one expects the subset of diagrams with all interactions along the principal path to dominate over the rest. These can be resummed exactly, Eq.~(\ref{largeKresult}), leading to Gaussian behavior,

\item[iii)] In order to match these two results and thus complete the resummation for all $k$, we make the assumption that the one-loop result is a nearly Gaussian expansion of the renormalized propagator. Then, as discussed in section~\ref{propRPT} there is a unique way to recover the latter from it's one-loop expansion satisfying the right asymptotics, leading to Eq.~(\ref{model3one}).

\end{itemize}

\subsection{Dependence of Propagators on the Simulation Volume}
\label{voldep}

\begin{figure*}
\begin{center}
\begin{tabular}{cc}
{\includegraphics[width=0.5\textwidth]{PropNBLboxDen.eps}}&
{\includegraphics[width=0.5\textwidth]{PropNBLboxVel.eps}}
\end{tabular}
\caption{The dependence of the measured density (left) and velocity divergence (right) propagators on the volume of the simulation used. The symbols denote measurements of the propagator at $z=0$ from initial conditions at $z_{\rm initial}=5$ for different simulation volumes, $L_{\rm box}=100 \, h^{-1} \, {\rm Mpc}$ (pentagons), $L_{\rm box}=239.5 \, h^{-1} \, {\rm Mpc}$ (squares), $L_{\rm box}=479 \, h^{-1} \, {\rm Mpc}$ (triangles).}
\label{PropLbox}
\end{center}
\end{figure*}

There is one important aspect of the prediction from the RPT propagator that we have ignored so far, namely, its  dependence on the large-scale modes. This is important because numerical simulations artificially truncate large-scale modes due to their finite volume. The predictions from Eqs.~(\ref{proponeloop1}),~(\ref{largeKresult}), and~(\ref{model3one}) indicate that the dependence on the large-scale modes must be significant, since the characteristic scale of decay is determined by $\sigma_v^{-1}$, which is quite sensitive to the smallest wavenumber $k_{\rm box}$ available in a simulation volume. In fact, the analytic predictions presented already in all the figures use the exact value $k_{\rm box}=2\pi/L_{\rm box} =0.013 \, h \, {\rm Mpc}^{-1}$ corresponding to the large simulation box. From Eqs.~(\ref{Gde}-\ref{Gthe}) we see that one can define the characteristic scale of decay from the low-$k$ expansion of the density and velocity propagators as

\begin{eqnarray}
\label{kdecayden}
k_\delta(z)&=&\sqrt{105/61}\ \sigma_v^{-1}(z), \\
\label{kdecayvel}
k_\theta(z)&=&\sqrt{5/9}\ \sigma_v^{-1}(z), \\
\label{svz}
\sigma_v^2(z) &=& (4\pi/3) D_+^2(z) \int_{k_{\rm box}}^\infty P_0(k) dk,
\end{eqnarray}
which is roughly the scale at which $G_\delta/D_+$ and $G_\theta/D_+$ have respectively dropped to $e^{-1/2}\approx 0.6$.

Figure~\ref{PropLbox} shows the dependence of the propagators on the simulation volume, confirming that there is a significant dependence, in agreement with the estimates from Eqs.~(\ref{kdecayden}-\ref{kdecayvel}). A larger volume leads to a stronger decay due to the fact that more modes are present and the increased interactions lead to faster loss of memory of the initial conditions. A similar situation holds for the cross-correlation coefficient between initial and final conditions presented in Fig.~\ref{figure1}. In other words, the larger the volume allows interactions among more modes which leads to a larger scatter of the final densities and velocities about the linearly extrapolated initial conditions, which suppresses $r_a$.

Why is such a strong dependence on the simulation volume not seen in the power spectrum measurements? The reason is that in a finite volume the propagator decays slower, leading to a fictitiously wider validity of linear perturbation theory, but the truncation of low-$k$ modes also suppresses the mode-coupling terms that should enhance the power, therefore these two effects compensate each other. {\em However}, that means that the dependence of the power spectrum on cosmological parameters will be incorrect at the transition scale, since these two contributions have a different dependence on them (their ratio being proportional to the linear power spectrum) and therefore cannot completely cancel.

The importance of this finite-volume effect is stronger for steep spectra, in fact one should correct the $L_{\rm box}=479 \, h \, {\rm Mpc}^{-1}$ measurements in Fig.~\ref{PropLbox} by a shift of about $4\%$ to the right according to Eqs.~(\ref{kdecayden}-\ref{svz}) to take into account the difference between the initial spectrum of this simulation as opposed to the $L_{\rm box}=100,239.5 \, h^{-1} \, {\rm Mpc}$ ones, but this does not alter the conclusions significantly.

Moreover, using Eqs.~(\ref{kdecayden}-\ref{svz}) one concludes that the decay length of the propagators in the $L_{\rm box}=479 \, h \, {\rm Mpc}^{-1}$ case is still off by about $4\%$ compared to the infinite volume case that will show a stronger decay. An important lesson from this is that simulations designed to study the high-redshift Universe that generally use smaller boxes can misestimate nonlinear effects due to lack of large-scale modes very easily, as the requirement to get the decay scale of the propagator correctly  {\em is independent of redshift} and, for example, to get $k_\delta,k_\theta$ accurate to $10\%$ requires a simulation of about $L_{\rm box}=240 \, h^{-1} \, {\rm Mpc}$, which can be challenging for high-redshift numerical studies.

\begin{figure}[t!]
\includegraphics[width=0.5\textwidth]{nlscale.eps}
\caption{Comparison between the nonlinear scale defined from the linear power spectrum $k_{\rm nl}(z)$ (short-dashed lines) and those derived from the characteristic scale of decay of the nonlinear propagator $k_\delta(z)$ (solid lines), $k_\theta(z)$ (long-dashed lines) given by Eqs.~(\ref{kdecayden}-\ref{kdecayvel}). At high redshift $k_{\rm nl}(z)$ significantly underestimates the importance of nonlinearities.}
\label{NLscale}
\end{figure}

\section{Conclusions}
\label{conclude}

In paper~I~\cite{paper1} we developed renormalized perturbation theory (RPT), in which nonlinear evolution in the growth of large-scale structure is  represented by Feynman diagrams constructed in terms of three objects: the initial conditions (e.g. perturbation spectrum), the vertex (describing non-linearities) and the propagator (describing linear evolution). We showed that loop corrections to the linear power spectrum organize themselves into two classes of diagrams: one corresponding to mode-coupling effects, the other to a renormalization of the propagator. The nonlinear propagator that results  quantifies the deviation from linear evolution of individual Fourier modes, and allows a well-defined perturbation theory that can probe the nonlinear regime.

In this paper we studied the propagator in detail and showed that its decay into the nonlinear regime can be thought as measuring the ``memory'' of perturbations to their initial conditions, being directly proportional to the cross-correlation coefficient between final and initial configurations.  We developed an analytical description of the nonlinear propagator by summing its perturbation series using physically motivated approximations. The results are in remarkable agreement with measurements in numerical simulations all the way into the nonlinear regime.  We also described how to measure the propagator in numerical simulations using two very different algorithms that gave consistent answers. The analytic results show that the propagator is rather sensitive to the finite volume of simulations, and this is indeed in agreement with measurements in simulations of different volume. The requirement of getting the right dependence of the propagator on scale turns out to be a stringent one for N-body simulations, particularly at high redshift where typically smaller boxes are used.

Coming back to the questions raised in the introduction about the validity of linear perturbation theory, we can give a quantitative answer for a robust definition of the nonlinear scale by looking at the characteristic scale of decay of the propagator of density and velocity divergence fields $k_\delta,k_\theta$ given by Eqs.~(\ref{kdecayden}-\ref{kdecayvel}). Figure~\ref{NLscale} shows the evolution of $k_\delta,k_\theta$ as a function of redshift and compared to the nonlinear scale $k_{\rm nl}$ defined from the linear power spectrum, $4\pi k_{\rm nl}^3 P_L(k_{\rm nl})=1$. Although the different definitions agree for the density field at $z=0$, at high redshift the nonlinear scale $k_{\rm nl}$ typically used in the literature can be more than an order of magnitude off from $k_\delta,k_\theta$, which we argued here is the cleanest definition of at what scale nonlinearities become important.

Since the non linear propagator plays an essential role in the calculation of  correlations functions, the next obvious step is to use the results presented here to calculate the non linear power spectrum. In the framework of RPT, once the nonlinear propagator is known, the power spectrum is given as a mode-coupling series~\cite{paper1}. Although in principle many terms are needed in order to cover a large range of scales, the study of how baryon wiggles are affected by nonlinear evolution requires only a few terms~\cite{paper3}. Already, the dependence of the propagator on scale found here provides a simple estimate of the scale at which  baryon wiggles should be significantly suppressed (remaining only as $e^{-1/2} \approx 0.6$ of their linear amplitude over a smooth component), given by $k_\delta(z),k_\theta(z)$ in Fig.~\ref{NLscale}.

\acknowledgments

We thank G. Gabadadze, A. Gruzinov, D.W. Hogg, S. Pueblas, and E. Sefusatti,  for useful discussions. The small-box numerical simulations used here where run at the NYU Beowulf cluster supported by NSF grant PHY-0116590. We thank K. Benabed for his help on that. The medium and large simulations were carried out by the Virgo Supercomputing Consortium using computers based at the Computing Centre of the Max-Planck Society in Garching and at the Edinburgh parallel Computing Centre. The data are publicly available at {\tt http://www.mpa-garching.mpg.de/NumCos}.