All Figures

$\begin{figure} \par\includegraphics[angle=-90,width=7.8cm,clip]{4945fig1.ps} \end{figure}$ Figure 1: The variation of the function f(M) with log₁₀(halo mass) as a function of the power-spectrum shape parameter $\Gamma$ , with h=0.7and $\sigma _8 = 0.74$ . Curves from top to bottom have $\Gamma = 0.07$ ,0.14, 0.21, 0.28, 0.35, 0.7; solid curves show the calculated function, dashed curves show the fitting function, Eq. (12).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fig2.ps} \end{figure}$ Figure 2: Comparison of the mean mass accretion rate calculated analytically (solid curves) and measured from Monte-Carlo realisations (solid symbols), shown as a function of redshift for a variety of masses.
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fig3.ps} \end{figure}$ Figure 3: Comparison of f(M) (solid curve) with that deduced from the Monte-Carlo realisations by factoring out the linear dependence on M and ${\rm d}\delta_{\rm c}\protect/{\rm d}t$ .
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=7.8cm,clip]{4945fig4.ps} \end{figure}$ Figure 4: Comparison of the mean accretion rate from this paper (dotted lines) with the two alternative numerical fits proposed by Van den Bosch (2002) (solid lines) and by Wechsler et al. (2002) with parameter values estimated by Van den Bosch (2002) (dot-dashed lines). See the text for more details of the calculation of these curves, which are shown for values of $\log_{10}(M(z)/{M_\odot})=8,10,12,14$ . The Van den Bosch (2002) curves becomes unphysical at low redshift and depart significantly from the analytic results. Both numerical fits become unphysical at high redshift and are truncated outside the region of validity, as discussed in the text.
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In the text

$\begin{figure} \par\includegraphics[angle=270,width=7.8cm,clip]{4945fig5.ps} \end{figure}$ Figure 5: The bolometric luminosity density deduced from the best-fit model of Ueda et al. (2003), integrating over the range $10^{40}<L_{\rm X}<10^{48}$ erg s^-1 and applying the bolometric correction of Marconi et al. (2004) and correction for Compton-thick AGN of Ueda et al. (2003) (see text) (dashed curve). Also shown are uncertainties estimated from refitting to the binned data of Ueda et al. (2003) (points with error bars: see text; horizontal bars indicate the range of redshifts included in each point). For comparison, we show the equivalent calculation from the pure-luminosity-evolution fit of Richards et al. (2005) to the optical 2SLAQ QSO survey in the redshift range 0.4<z<2.1 (lower dashed curve). The luminosity density expected in PCE is shown for two cases: (i) no evolution in the comoving black hole mass density (dotted upper curve); (ii) evolution in the comoving black hole mass density that tracks the evolution of massive dark halos with $M_{\rm H}>10^{11.5}~{M_\odot}$ (solid curve). Both curves assume average radiative efficiency $\langle\epsilon\rangle=0.04$ (see text). Note that the PCE calculation predicts a value for the luminosity density at z=0, dependent only on $\langle\epsilon\rangle$ and cosmological parameters, with the higher redshift evolution further dependent only on choice of dark halo mass: the normalisation is not allowed to float arbitrarily.
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In the text

$\begin{figure} \par\includegraphics[angle=270,width=7.8cm,clip]{4945fig6.ps} \end{figure}$ Figure 6: Ratio of mean accretion rate to the limiting Eddington rate, $\lambda$ , as a function of (1+z) over the range 0<z<20, plotted logarithmically to demonstrate the near power-law dependence. Results are shown for $M_{\rm H}= 10^{12.5} ~h^{-1}~{M_\odot}$ and mean radiative efficiency $\langle\epsilon\rangle=0.04$ for: (a) $\Lambda$ CDM cosmology, $\Omega _0=0.3$ , $\Omega _{\Lambda }=0.7$ , H₀ = 73 km s^-1 Mpc^-1, $\sigma _8 = 0.74$ (solid curve); (b) Einstein-de Sitter cosmology with parameters as in (a) except $\Omega _0=1$ , $\Omega _{\Lambda }=0$ (dashed curve).
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In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fA1.ps} \end{figure}$ Figure A.1: Plot showing the mismatch between contours of constant $\delta _{\rm c}$ (black lines) and the lines along which $\Omega _M$ and $\Omega _\Lambda$ vary with the evolution of the universe (grey lines). The contours of constant $\delta _{\rm c}$ are plotted as a ratio of $\delta _{\rm EdS}$ , the critical overdensity for collapse at redshift zero in an Einstein-de Sitter cosmology. Because these lines cross, the evolution of $\delta _{\rm c}(a)$ is not solely dependent on the linear growth factor.

In the text

$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fA2.ps} \end{figure}$ Figure A.2: The percentage error in approximating the time derivative of $\delta _{\rm c}(a)$ as $\delta _{\rm c}(a)=\delta _{\rm EdS}D/D(a)$ , where D(a) is the linear growth factor. Open symbols incorporate the approximation for D(a) given by Eq. (A.6), while solid symbols calculate D(a) by numerically integrating Eq. (A.4). Results are shown for three cosmological models: for an Einstein-de Sitter cosmology, both approximations reduce to the exact behaviour.

In the text

$\begin{figure} \par\includegraphics[angle=-90,width=7.8cm,clip]{4945fig1.ps} \end{figure}$	Figure 1: The variation of the function f(M) with log₁₀(halo mass) as a function of the power-spectrum shape parameter $\Gamma$ , with h=0.7and $\sigma _8 = 0.74$ . Curves from top to bottom have $\Gamma = 0.07$ ,0.14, 0.21, 0.28, 0.35, 0.7; solid curves show the calculated function, dashed curves show the fitting function, Eq. (12).
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fig2.ps} \end{figure}$	Figure 2: Comparison of the mean mass accretion rate calculated analytically (solid curves) and measured from Monte-Carlo realisations (solid symbols), shown as a function of redshift for a variety of masses.
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$\begin{figure} \par\includegraphics[width=7.8cm,clip]{4945fig3.ps} \end{figure}$	Figure 3: Comparison of f(M) (solid curve) with that deduced from the Monte-Carlo realisations by factoring out the linear dependence on M and ${\rm d}\delta_{\rm c}\protect/{\rm d}t$ .
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