All Figures

$\begin{figure} \par\includegraphics[width=7.2cm,height=6.4cm,clip]{6832f1.ps}\end{figure}$ Figure 1: The self-energy terms $\Sigma _{0;ij}^+(k)$ of Eq. (60) as a function of wavenumber k. We display the four components $\Sigma _{0;11}^+$ (solid line), $\Sigma _{0;21}^+$ (dot-dashed line), $\Sigma _{0;12}^+$ (dotted line) and $\Sigma _{0;22}^+$ (dashed line). They are all negative except for $\Sigma _{0;11}^+$ which is positive at k < 0.2 h Mpc^-1 and negative at higher k. All terms are of the same magnitude and grow roughly as k² in agreement with Eq. (62).
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In the text

$\begin{figure} \par\includegraphics[width=7.2cm,height=6.6cm,clip]{6832f2.ps}\end{figure}$ Figure 2: The self-energy $\Sigma _0(k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=0.1,1 and $10 \times h$ Mpc^-1. The line styles are as in Fig. 1. The diagonal components vanish for $\eta _1<\eta _2$ whereas the off-diagonal components vanish for $\eta _1\leq \eta _2$ . All terms are negative except for $\Sigma _{0;11}$ which becomes positive at $\eta _1>-0.8$ for k=0.1 h Mpc^-1.
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In the text

$\begin{figure} \par\includegraphics[width=7.2cm,height=6.6cm,clip]{6832f3.ps}\end{figure}$ Figure 3: The self-energy $\Sigma _0(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=0.1,1 and $10 \times h$ Mpc^-1. The line styles are as in Fig. 1. All terms are negative except for $\Sigma _{0;11}$ which becomes positive at $\eta _2<-0.6$ for k=0.1 h Mpc^-1.
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In the text

$\begin{figure} \par\includegraphics[width=7.2cm,height=6.4cm,clip]{6832f4.ps}\end{figure}$ Figure 4: The self-energy $\Pi _0(k)$ as a function of wavenumber k. At high k all components are very close and positive. At $k\leq 0.06~h$ Mpc^-1 the off-diagonal components become negative.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.4cm,clip]{6832f5.ps}\end{figure}$ Figure 5: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot the linear response $R_{\rm L}$ which shows a simple exponential growth.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.4cm,clip]{6832f6.ps}\end{figure}$ Figure 6: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot the linear response $R_{\rm L}$ which shows a simple exponential growth.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f7.ps}\end{figure}$ Figure 7: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ , i.e. z₁=0,z₂=3. The horizontal lines are the linear response $R_{\rm L}$ of Eq. (24) which is independent of k.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.3cm,clip]{6832f8.ps}\end{figure}$ Figure 8: The density two-point correlation $G_{11}(k;\eta _1,\eta _2)$ as a function of time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We plot the logarithmic power $\Delta ^2=4\pi k^3G_{11}$ . We display the linear power $\Delta ^2_{\rm L}$ of Eq. (21) (dashed line), the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ of Eq. (81) (dotted line), the full non-linear power $\Delta ^2$ from Eq. (51) (solid line) and the contribution $\Delta ^2_i$ from Eq. (52) (dot-dashed line). Note that G₁₁ whence $\Delta ^2$ can be negative, as shown by the oscillations displayed in the figure.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f9.ps}\end{figure}$ Figure 9: The logarithmic power $\Delta ^2(k)$ of Eq. (19) at redshift z=0, that is for equal times $\eta _1=\eta _2=0$ (z₁=z₂=0). We display the linear power $\Delta ^2_{\rm L}$ (dashed line), the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), the full non-linear power $\Delta ^2$ from Eq. (51) (solid line) and the contribution $\Delta ^2_i$ from Eq. (52) (dot-dashed line). We also show the fit $\Delta ^2_{\rm nb}$ (upper solid line) from numerical simulations (Smith et al. 2003). All quantities (including $\Delta ^2_i$ which is the square of an oscillating function) are positive.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f10.ps}\end{figure}$ Figure 10: The logarithmic power $\Delta ^2(k)$ as in Fig. 9 but at redshift z=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f11.ps}\end{figure}$ Figure 11: The logarithmic power $\Delta ^2(k)$ as in Fig. 9 but at unequal times $(\eta _1=0,\eta _2=-1.4)$ (i.e. z₁=0,z₂=3).
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.2cm,clip]{6832f12.ps}\end{figure}$ Figure 12: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel), as in Fig. 5 but for the 2PI effective action method. For comparison we also plot the first half-oscillation of the response obtained in Sect. 5 from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 5.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.2cm,clip]{6832f13.ps}\end{figure}$ Figure 13: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot in the left panel the first half-oscillation of the response obtained in Sect. 5 from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 6.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f14.ps}\end{figure}$ Figure 14: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ . We also plot the first half-oscillation of the response obtained in Sect. 5 from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 7.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{6832f15.ps}\end{figure}$ Figure 15: The self-energy $\Sigma (k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). The curves labeled "sd'' are the prediction of the steepest-descent method as in Fig. 2.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,clip]{6832f16.ps}\end{figure}$ Figure 16: The self-energy $\Sigma (k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). The curves labeled "sd'' are the prediction of the steepest-descent method as in Fig. 3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f17.ps}\end{figure}$ Figure 17: The self-energy $\Sigma (k)$ as a function of wavenumber k at unequal redshifts z₁=0,z₂=3. The curves labeled "sd'' are the prediction of the steepest-descent method.
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In the text

$\begin{figure} \par\includegraphics[width=8cm,height=7.4cm,clip]{6832f18.ps}\end{figure}$ Figure 18: The density two-point correlation $G_{11}(k;\eta _1,\eta _2)$ as a function of time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We plot the logarithmic power $\Delta ^2=4\pi k^3G_{11}$ . We display the full non-linear power $\Delta ^2$ from Eq. (51) (solid line) and the contribution $\Delta ^2_i$ from Eq. (52) (dot-dashed line), as well as the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ of Eq. (81) (dotted line) and the prediction of the steepest-descent method ("sd'', dotted line) also shown in Fig. 8.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f19.ps}\end{figure}$ Figure 19: The logarithmic power $\Delta ^2(k)$ at redshift z=0, that is at equal times z₁=z₂=0. We display the full non-linear power $\Delta ^2$ from Eq. (51) (solid line) and the contribution $\Delta ^2_i$ from Eq. (52) (dot-dashed line), as well as the linear power $\Delta ^2_{\rm L}$ (dashed line), the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ of Eq. (81) (dotted line), the prediction of the steepest-descent method ("sd'', dotted line) also shown in Fig. 9 and the fit $\Delta ^2_{\rm nb}$ from numerical simulations (Smith et al. 2003). All quantities (including $\Delta ^2_i$ ) are positive.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f20.ps}\end{figure}$ Figure 20: The logarithmic power $\Delta ^2(k)$ at redshift z=3, that is at equal times z₁=z₂=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f21.ps}\end{figure}$ Figure 21: The logarithmic power $\Delta ^2(k)$ at unequal times $(\eta _1=0,\eta _2=-1.4)$ (i.e. z₁=0,z₂=3).
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.5cm,clip]{6832f22.ps}\end{figure}$ Figure 22: The self-energy $\Pi (k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 and $10 \times h$ Mpc^-1.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f23.ps}\end{figure}$ Figure 23: The self-energy $\Pi (k)$ as a function of wavenumber k at equal redshifts z₁=z₂=0. We also plot the prediction of the steepest-descent method (curves labeled "sd'').
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f24.ps}\end{figure}$ Figure 24: The self-energy $\Pi (k)$ as a function of wavenumber k at unequal redshifts z₁=0,z₂=3. We also plot the prediction of the steepest-descent method (curves labeled "sd'').
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f25.ps}\end{figure}$ Figure 25: The logarithmic power $\Delta ^2(k)$ at redshift z=0. We display the ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ , where $\Delta ^2_{\rm L}$ is the linear power, for the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (upper solid line) from numerical simulations (Smith et al. 2003), the result of the steepest-descent method ("sd'', dotted line) shown in Fig. 9 and the approximate power $\Delta ^2_{\rm app}$ (solid line) obtained from the approximation (109).
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f26.ps}\end{figure}$ Figure 26: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 25 but at redshift z=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f27.ps}\end{figure}$ Figure 27: The logarithmic power $\Delta ^2(k)$ at redshift z=0. We display the ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ , where $\Delta ^2_{\rm L}$ is the linear power, for the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (upper solid line) from numerical simulations (Smith et al. 2003), the result of the 2PI effective action method ("2PI'', dotted line) shown in Fig. 19 and the approximate power $\Delta ^2_{\rm app}$ (solid line) obtained from the approximation (118).
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f28.ps}\end{figure}$ Figure 28: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 27 but at redshift z=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f29.ps}\end{figure}$ Figure 29: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ , for the mixed approaches where G₀ is used in the computation of ${\Sigma }$ .
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f30.ps}\end{figure}$ Figure 30: The logarithmic power $\Delta ^2(k)$ at redshift z=0, that is at equal times z₁=z₂=0. We display the linear power $\Delta ^2_{\rm L}$ (dashed line), the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (lower solid line) from numerical simulations (Smith et al. 2003), the mixed approach $\Delta ^2_{\Pi _0}$ based on $\Pi _0$ (dashed line) and the mixed approach $\Delta ^2_{\Pi }$ where $\Pi$ is coupled to G (upper solid line).
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f31.ps}\end{figure}$ Figure 31: The logarithmic power $\Delta ^2(k)$ at redshift z=3, that is at equal times z₁=z₂=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f32.ps}\end{figure}$ Figure 32: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ at redshift z=0 of the logarithmic power $\Delta ^2$ to the linear power $\Delta ^2_{\rm L}$ . We display the ratio $\Delta ^2/\Delta ^2_{\rm L}$ for the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (lower solid line) from numerical simulations (Smith et al. 2003), the mixed approach $\Delta ^2_{\Pi _0}$ based on $\Pi _0$ (dashed line) and the mixed approach $\Delta ^2_{\Pi }$ where $\Pi$ is coupled to G (upper solid line).
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f33.ps}\end{figure}$ Figure 33: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 32 but at redshift z=3.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f34.ps}\end{figure}$ Figure 34: The non-linear response function $R(k;\eta _1,\eta _2)$ given by the direct steepest-descent method as a function of wavenumber k for a $\Lambda$ CDM universe, at redshifts z₁=0,z₂=3. The horizontal lines are the linear response $R_{\rm L}$ of Eq. (24) which is independent of k.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f35.ps}\end{figure}$ Figure 35: The non-linear response function $R(k;\eta _1,\eta _2)$ given by the 2PI effective action method as a function of wavenumber k for a $\Lambda$ CDM universe, at redshifts z₁=0,z₂=3. We also plot the first half-oscillation of the response obtained from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 34.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f36.ps}\end{figure}$ Figure 36: The logarithmic power $\Delta ^2(k)$ at redshift z=0 for a $\Lambda$ CDM universe. We display the linear power $\Delta ^2_{\rm L}$ (dashed line), the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (lower solid line) from numerical simulations (Smith et al. 2003), the steepest-descent prediction (dotted line "sd'') and the 2PI effective action result (solid line "2PI'').
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f37.ps}\end{figure}$ Figure 37: The logarithmic power $\Delta ^2(k)$ at redshift z=3 for a $\Lambda$ CDM universe.
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f38.ps}\end{figure}$ Figure 38: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ at redshift z=0 of the logarithmic power $\Delta ^2$ to the linear power $\Delta ^2_{\rm L}$ for a $\Lambda$ CDM universe. We display the ratio $\Delta ^2/\Delta ^2_{\rm L}$ for the usual one-loop perturbative result $\Delta ^2_{\rm 1 loop}$ (dotted line), a fit $\Delta ^2_{\rm nb}$ (lower solid line) from numerical simulations (Smith et al. 2003), the steepest-descent prediction (dashed line "sd'') and the 2PI effective action result (solid line "2PI'').
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In the text

$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f39.ps}\end{figure}$ Figure 39: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 38 but at redshift z=3.
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In the text

$\begin{figure} \par\includegraphics[width=7.2cm,height=6.6cm,clip]{6832f3.ps}\end{figure}$	Figure 3: The self-energy $\Sigma _0(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=0.1,1 and $10 \times h$ Mpc^-1. The line styles are as in Fig. 1. All terms are negative except for $\Sigma _{0;11}$ which becomes positive at $\eta _2<-0.6$ for k=0.1 h Mpc^-1.
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$\begin{figure} \par\includegraphics[width=7.2cm,height=6.4cm,clip]{6832f4.ps}\end{figure}$	Figure 4: The self-energy $\Pi _0(k)$ as a function of wavenumber k. At high k all components are very close and positive. At $k\leq 0.06~h$ Mpc^-1 the off-diagonal components become negative.
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$\begin{figure} \par\includegraphics[width=8cm,height=7.4cm,clip]{6832f5.ps}\end{figure}$	Figure 5: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot the linear response $R_{\rm L}$ which shows a simple exponential growth.
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$\begin{figure} \par\includegraphics[width=8cm,height=7.4cm,clip]{6832f6.ps}\end{figure}$	Figure 6: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot the linear response $R_{\rm L}$ which shows a simple exponential growth.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f7.ps}\end{figure}$	Figure 7: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ , i.e. z₁=0,z₂=3. The horizontal lines are the linear response $R_{\rm L}$ of Eq. (24) which is independent of k.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f10.ps}\end{figure}$	Figure 10: The logarithmic power $\Delta ^2(k)$ as in Fig. 9 but at redshift z=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f11.ps}\end{figure}$	Figure 11: The logarithmic power $\Delta ^2(k)$ as in Fig. 9 but at unequal times $(\eta _1=0,\eta _2=-1.4)$ (i.e. z₁=0,z₂=3).
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$\begin{figure} \par\includegraphics[width=8cm,height=7.2cm,clip]{6832f13.ps}\end{figure}$	Figure 13: The non-linear response $R(k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). We also plot in the left panel the first half-oscillation of the response obtained in Sect. 5 from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 6.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f14.ps}\end{figure}$	Figure 14: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ . We also plot the first half-oscillation of the response obtained in Sect. 5 from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 7.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{6832f15.ps}\end{figure}$	Figure 15: The self-energy $\Sigma (k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). The curves labeled "sd'' are the prediction of the steepest-descent method as in Fig. 2.
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$\begin{figure} \par\includegraphics[width=7cm,clip]{6832f16.ps}\end{figure}$	Figure 16: The self-energy $\Sigma (k;\eta _1,\eta _2)$ as a function of backward time $\eta _2$ , for $\eta _1=0$ (i.e. z₁=0) and wavenumbers k=1 ( left panel) and $10 \times h$ Mpc^-1 ( right panel). The curves labeled "sd'' are the prediction of the steepest-descent method as in Fig. 3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f17.ps}\end{figure}$	Figure 17: The self-energy $\Sigma (k)$ as a function of wavenumber k at unequal redshifts z₁=0,z₂=3. The curves labeled "sd'' are the prediction of the steepest-descent method.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f20.ps}\end{figure}$	Figure 20: The logarithmic power $\Delta ^2(k)$ at redshift z=3, that is at equal times z₁=z₂=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f21.ps}\end{figure}$	Figure 21: The logarithmic power $\Delta ^2(k)$ at unequal times $(\eta _1=0,\eta _2=-1.4)$ (i.e. z₁=0,z₂=3).
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$\begin{figure} \par\includegraphics[width=7cm,height=6.5cm,clip]{6832f22.ps}\end{figure}$	Figure 22: The self-energy $\Pi (k;\eta _1,\eta _2)$ as a function of forward time $\eta _1$ , for $\eta _2=-1.4$ (i.e. z₂=3) and wavenumbers k=1 and $10 \times h$ Mpc^-1.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f23.ps}\end{figure}$	Figure 23: The self-energy $\Pi (k)$ as a function of wavenumber k at equal redshifts z₁=z₂=0. We also plot the prediction of the steepest-descent method (curves labeled "sd'').
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f24.ps}\end{figure}$	Figure 24: The self-energy $\Pi (k)$ as a function of wavenumber k at unequal redshifts z₁=0,z₂=3. We also plot the prediction of the steepest-descent method (curves labeled "sd'').
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f26.ps}\end{figure}$	Figure 26: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 25 but at redshift z=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f28.ps}\end{figure}$	Figure 28: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 27 but at redshift z=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f29.ps}\end{figure}$	Figure 29: The non-linear response function $R(k;\eta _1,\eta _2)$ as a function of wavenumber k, at times $\eta _1=0,\eta _2=-1.4$ , for the mixed approaches where G₀ is used in the computation of ${\Sigma }$ .
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f31.ps}\end{figure}$	Figure 31: The logarithmic power $\Delta ^2(k)$ at redshift z=3, that is at equal times z₁=z₂=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f33.ps}\end{figure}$	Figure 33: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 32 but at redshift z=3.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f34.ps}\end{figure}$	Figure 34: The non-linear response function $R(k;\eta _1,\eta _2)$ given by the direct steepest-descent method as a function of wavenumber k for a $\Lambda$ CDM universe, at redshifts z₁=0,z₂=3. The horizontal lines are the linear response $R_{\rm L}$ of Eq. (24) which is independent of k.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f35.ps}\end{figure}$	Figure 35: The non-linear response function $R(k;\eta _1,\eta _2)$ given by the 2PI effective action method as a function of wavenumber k for a $\Lambda$ CDM universe, at redshifts z₁=0,z₂=3. We also plot the first half-oscillation of the response obtained from the direct steepest-descent method (curves labeled "sd'') which was shown in Fig. 34.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f37.ps}\end{figure}$	Figure 37: The logarithmic power $\Delta ^2(k)$ at redshift z=3 for a $\Lambda$ CDM universe.
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$\begin{figure} \par\includegraphics[width=7cm,height=6.2cm,clip]{6832f39.ps}\end{figure}$	Figure 39: The ratio $\Delta ^2(k)/\Delta ^2_{\rm L}(k)$ as in Fig. 38 but at redshift z=3.
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