%auto-ignore \bibliographystyle{naturemag} \onecolumn \section*{\ \newline \Large Simulating the joint evolution of quasars, galaxies\vspace*{0.1cm}\newline and their large-scale distribution} %\baselineskip16pt \noindent{\sffamily Volker~Springel$^{1}$, % Simon~D.~M.~White$^{1}$, % Adrian~Jenkins$^{2}$, % Carlos~S.~Frenk$^{2}$, \newline% Naoki~Yoshida$^{3}$, % Liang~Gao$^{1}$, % Julio~Navarro$^{4}$, % Robert~Thacker$^{5}$, % Darren~Croton$^{1}$, \newline% John~Helly$^{2}$, % John~A.~Peacock$^{6}$, % Shaun~Cole$^{2}$, % Peter~Thomas$^{7}$, % Hugh~Couchman$^{5}$, \newline% August~Evrard$^{8}$, % J\"org~Colberg$^{9}$ \& % Frazer~Pearce$^{10}$} \ \\ \noindent% {\footnotesize\it% $^{1}${Max-Planck-Institute for Astrophysics, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany}\\ $^{2}${Inst. for Computational Cosmology, Dep. of Physics, Univ. of Durham, South Road, Durham DH1 3LE, UK}\\ $^{3}${Department of Physics, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan}\\ $^{4}${Dep. of Physics \& Astron., University of Victoria, Victoria, BC, V8P 5C2, Canada}\\ $^{5}${Dep. of Physics \& Astron., McMaster Univ., 1280 Main St. West, Hamilton, Ontario, L8S 4M1, Canada}\\ $^{6}${Institute of Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK}\\ $^{7}${Dep. of Physics \& Astron., University of Sussex, Falmer, Brighton BN1 9QH, UK}\\ $^{8}${Dep. of Physics \& Astron., Univ. of Michigan, Ann Arbor, MI 48109-1120, USA}\\ $^{9}${Dep. of Physics \& Astron., Univ. of Pittsburgh, 3941 O'Hara Street, Pittsburgh PA 15260, USA}\\ $^{10}${Physics and Astronomy Department, Univ. of Nottingham, Nottingham NG7 2RD, UK}\\ } \baselineskip26pt % On-and-half-space the manuscript. \setlength{\parskip}{12pt} \setlength{\parindent}{22pt}% \noindent{\bf The cold dark matter model has become the leading theoretical paradigm for the formation of structure in the Universe. Together with the theory of cosmic inflation, this model makes a clear prediction for the initial conditions for structure formation and predicts that structures grow hierarchically through gravitational instability. Testing this model requires that the precise measurements delivered by galaxy surveys can be compared to robust and equally precise theoretical calculations. Here we present a novel framework for the quantitative physical interpretation of such surveys. This combines the largest simulation of the growth of dark matter structure ever carried out with new techniques for following the formation and evolution of the visible components. We show that baryon-induced features in the initial conditions of the Universe are reflected in distorted form in the low-redshift galaxy distribution, an effect that can be used to constrain the nature of dark energy with next generation surveys.} Recent large surveys such as the 2 degree Field Galaxy Redshift Survey (2dFGRS) and the Sloan Digital Sky Survey (SDSS) have characterised much more accurately than ever before not only the spatial clustering, but also the physical properties of low-redshift galaxies. Major ongoing campaigns exploit the new generation of 8m-class telescopes and the Hubble Space Telescope to acquire data of comparable quality at high redshift. Other surveys target the weak image shear caused by gravitational lensing to extract precise measurements of the distribution of dark matter around galaxies and galaxy clusters. The principal goals of all these surveys are to shed light on how galaxies form, to test the current paradigm for the growth of cosmic structure, and to search for signatures which may clarify the nature of dark matter and dark energy. These goals can be achieved only if the accurate measurements delivered by the surveys can be compared to robust and equally precise theoretical predictions. Two problems have so far precluded such predictions: (i) accurate estimates of clustering require simulations of extreme dynamic range, encompassing volumes large enough to contain representative populations of rare objects (like rich galaxy clusters or quasars), yet resolving the formation of individual low luminosity galaxies; (ii) critical aspects of galaxy formation physics are uncertain and beyond the reach of direct simulation (for example, the structure of the interstellar medium, its consequences for star formation and for the generation of galactic winds, the ejection and mixing of heavy elements, AGN feeding and feedback effects \ldots) -- these must be treated by phenomenological models whose form and parameters are adjusted by trial and error as part of the overall data-modelling process. We have developed a framework which combines very large computer simulations of structure formation with post-hoc modelling of galaxy formation physics to offer a practical solution to these two entwined problems. During the past two decades, the cold dark matter (CDM) model, augmented with a dark energy field (which may take the form of a cosmological constant `$\Lambda$'), has developed into the standard theoretical paradigm for galaxy formation. It assumes that structure grew from weak density fluctuations present in the otherwise homogeneous and rapidly expanding early universe. These fluctuations are amplified by gravity, eventually turning into the rich structure that we see around us today. Confidence in the validity of this model has been boosted by recent observations. Measurements of the cosmic microwave background (CMB) by the WMAP satellite\cite{Bennett2003} were combined with the 2dFGRS to confirm the central tenets of the model and to allow an accurate determination of the geometry and matter content of the Universe about $380\,000$ years after the Big Bang\cite{Spergel2003}. The data suggest that the early density fluctuations were a Gaussian random field, as predicted by inflationary theory, and that the current energy density is dominated by some form of dark energy. This analysis is supported by the apparent acceleration of the current cosmic expansion inferred from studies of distant supernovae\cite{Riess1998,Perlmutter1999}, as well as by the low matter density derived from the baryon fraction of clusters\cite{White1993}. While the initial, linear growth of density perturbations can be calculated analytically, the collapse of fluctuations and the subsequent hierarchical build-up of structure is a highly nonlinear process which is only accessible through direct numerical simulation\cite{Davis1985}. The dominant mass component, the cold dark matter, is assumed to be made of elementary particles that currently interact only gravitationally, so the collisionless dark matter fluid can be represented by a set of discrete point particles. This representation as an N-body system is a coarse approximation whose fidelity improves as the number of particles in the simulation increases. The high-resolution simulation described here -- dubbed the {\it Millennium Simulation} because of its size -- was carried out by the Virgo Consortium, a collaboration of British, German, Canadian, and US astrophysicists. It follows $N= 2160^3\simeq 1.0078\times 10^{10}$ particles from redshift $z=127$ to the present in a cubic region $500\,h^{-1}{\rm Mpc}$ on a side, where $1+z$ is the expansion factor of the Universe relative to the present and $h$ is Hubble's constant in units of $100\,{\rm km\,s^{-1}Mpc^{-1}}$. With ten times as many particles as the previous largest computations of this kind\cite{Colberg2000,Evrard2002,Wambsganss2004} (see Supplementary Information), it offers substantially improved spatial and time resolution within a large cosmological volume. Combining this simulation with new techniques for following the formation and evolution of galaxies, we predict the positions, velocities and intrinsic properties of all galaxies brighter than the Small Magellanic Cloud throughout volumes comparable to the largest current surveys. Crucially, this also allows us to establish evolutionary links between objects observed at different epochs. For example, we demonstrate that galaxies with supermassive central black holes can plausibly form early enough in the standard cold dark matter cosmology to host the first known quasars, and that these end up at the centres of rich galaxy clusters today. \begin{figure*} \noindent\hspace*{-0.5cm}% \resizebox{17.0cm}{!}{\includegraphics{fig1.eps}} % \caption{The dark matter density field on various scales. Each individual image shows the projected dark matter density field in a slab of thickness $15\,h^{-1}{\rm Mpc}$ (sliced from the periodic simulation volume at an angle chosen to avoid replicating structures in the lower two images), colour-coded by density and local dark matter velocity dispersion. The zoom sequence displays consecutive enlargements by factors of four, centred on one of the many galaxy cluster halos present in the simulation.} \label{FigDMDist} \end{figure*} \subsubsection*{Dark matter halos and galaxies} The mass distribution in a $\Lambda$CDM universe has a complex topology, often described as a ``cosmic web'' \cite{Bond1996}. This is visible in full splendour in Fig.~\ref{FigDMDist} (see also the corresponding Supplementary Video). The zoomed out panel at the bottom of the figure reveals a tight network of cold dark matter clusters and filaments of characteristic size $\sim 100\,h^{-1} {\rm Mpc}$. On larger scales, there is little discernible structure and the distribution appears homogeneous and isotropic. Subsequent images zoom in by factors of four onto the region surrounding one of the many rich galaxy clusters. The final image reveals several hundred dark matter substructures, resolved as independent, gravitationally bound objects orbiting within the cluster halo. These substructures are the remnants of dark matter halos that fell into the cluster at earlier times. \begin{figure} \hspace*{-1.0cm}% \resizebox{16.0cm}{!}{\includegraphics{fig2.eps}} \caption{ \baselineskip20pt Differential halo number density as a function of mass and epoch. The function $n(M,z)$ gives the comoving number density of halos less massive than $M$. We plot it as the halo multiplicity function $M^2\rho^{-1}\,{\rm d}n/{\rm d}M$, where $\rho$ is the mean density of the universe. Groups of particles were found using a friends-of-friends algorithm\cite{Davis1985} with linking length equal to 0.2 of the mean particle separation. The fraction of mass bound to halos of more than 20 particles (vertical dotted line) grows from $6.42\times 10^{-4}$ at $z=10.07$ to 0.496 at $z=0$. Solid lines are predictions from an analytic fitting function proposed in previous work\cite{Jenkins2001}, while the dashed lines give the Press-Schechter model\cite{Press1974} at $z=10.07$ and $z=0$. \label{FigMassFunc} } \end{figure} The space density of dark matter halos at various epochs in the simulation is shown in Fig.~\ref{FigMassFunc}. At the present day, there are about 18 million halos above a detection threshold of 20 particles; 49.6\% of all particles are included in these halos. These statistics provide the most precise determination to date of the mass function of cold dark matter halos\cite{Jenkins2001,Reed2003}. In the range that is well sampled in our simulation ($z \le 12$, $M\ge 1.7\times 10^{10}\,h^{-1}{\rm M}_\odot$), our results are remarkably well described by the analytic formula proposed by Jenkins et al.\cite{Jenkins2001} from fits to previous simulations. Theoretical models based on an ellipsoidal excursion set formulation\cite{Sheth2002} give a less accurate, but still reasonable match. However, the commonly used Press-Schechter formula\cite{Press1974} underpredicts the high-mass end of the mass function by up to an order of magnitude. Previous studies of the abundance of rare objects, such as luminous quasars or clusters, based on this formula may contain large errors\cite{Efstathiou1988}. We return below to the important question of the abundance of quasars at early times. To track the formation of galaxies and quasars in the simulation, we implement a semi-analytic model to follow gas, star and supermassive black hole processes within the merger history trees of dark matter halos and their substructures (see Supplementary Information). The trees contain a total of about 800 million nodes, each corresponding to a dark matter subhalo and its associated galaxies. This methodology allows us to test, during postprocessing, many different phenomenological treatments of gas cooling, star formation, AGN growth, feedback, chemical enrichment, etc. Here, we use an update of models described in\cite{Springel2001b,Kauffmann2000} which are similar in spirit to previous semi-analytic models\cite{WhiteFrenk1991,Kauffmann1993,Cole1994,Baugh1996,Sommerville1999,Kauffmann1999}; the modelling assumptions and parameters are adjusted by trial and error in order to fit the observed properties of low redshift galaxies, primarily their joint luminosity-colour distribution and their distributions of morphology, gas content and central black hole mass. Our use of a high-resolution simulation, particularly our ability to track the evolution of dark matter substructures, removes much of the uncertainty of the more traditional semi-analytic approaches based on Monte-Carlo realizations of merger trees. Our technique provides accurate positions and peculiar velocities for all the model galaxies. It also enables us to follow the evolutionary history of individual objects and thus to investigate the relationship between populations seen at different epochs. It is the ability to establish such evolutionary connections that makes this kind of modelling so powerful for interpreting observational data. \subsubsection*{The fate of the first quasars} Quasars are among the most luminous objects in the Universe and can be detected at huge cosmological distances. Their luminosity is thought to be powered by accretion onto a central, supermassive black hole. Bright quasars have now been discovered as far back as redshift $z=6.43$ (ref.~\cite{Fan2003}), and are believed to harbour central black holes of mass a billion times that of the sun. At redshift $z\sim 6$, their comoving space density is estimated to be $\sim (2.2 \pm 0.73)\times 10^{-9}\,h^3{\rm Mpc}^{-3}$ (ref.~\cite{Fan2004}). Whether such extreme rare objects can form at all in a $\Lambda$CDM cosmology is an open question. A volume the size of the Millennium Simulation should contain, on average, just under one quasar at the above space density. Just what sort of object should be associated with these ``first quasars'' is, however, a matter of debate. In the local universe, it appears that every bright galaxy hosts a supermassive black hole and there is a remarkably good correlation between the mass of the central black hole and the stellar mass or velocity dispersion of the bulge of the host galaxy\cite{Tremaine2002}. It would therefore seem natural to assume that at any epoch, the brightest quasars are always hosted by the largest galaxies. In our simulation, `large galaxies' can be identified in various ways, for example, according to their dark matter halo mass, stellar mass, or instantaneous star formation rate. We have identified the 10 `largest' objects defined in these three ways at redshift $z=6.2$. It turns out that these criteria all select essentially the same objects: the 8 largest galaxies by halo mass are identical to the 8 largest galaxies by stellar mass, only the ranking differs. Somewhat larger differences are present when galaxies are selected by star formation rate, but the 4 first-ranked galaxies are still amongst the 8 identified according to the other 2 criteria. \begin{figure} \vspace*{-1.0cm}\hspace*{-0.3cm}% \resizebox{8.2cm}{!}{\includegraphics{fig3a.eps}} % \resizebox{8.2cm}{!}{\includegraphics{fig3b.eps}}\vspace*{0.05cm}\\% \hspace*{-0.3cm}% \resizebox{8.2cm}{!}{\includegraphics{fig3c.eps}} % \resizebox{8.2cm}{!}{\includegraphics{fig3d.eps}}\\% \caption{Environment of a `first quasar candidate' at high and low redshifts. The two panels on the left show the projected dark matter distribution in a cube of comoving sidelength $10\,h^{-1}{\rm Mpc}$, colour-coded according to density and local dark matter velocity dispersion. The panels on the right show the galaxies of the semi-analytic model overlayed on a gray-scale image of the dark matter density. The volume of the sphere representing each galaxy is proportional to its stellar mass, and the chosen colours encode the restframe stellar $B-V$ colour index. While at $z=6.2$ (top) all galaxies appear blue due to ongoing star formation, many of the galaxies that have fallen into the rich cluster at $z=0$ (bottom) have turned red. \label{FigFirstQuasar}} \end{figure} In Figure~\ref{FigFirstQuasar}, we illustrate the environment of a ``first quasar'' candidate in our simulation at $z=6.2$. The object lies on one of the most prominent dark matter filaments and is surrounded by a large number of other, much fainter galaxies. It has a stellar mass of $6.8\times 10^{10}\,h^{-1}{\rm M}_\odot$, the largest in the entire simulation at $z=6.2$, a dark matter virial mass of $3.9\times 10^{12}\,h^{-1}{\rm M}_\odot$, and a star formation rate of $235\, {\rm M_\odot yr^{-1}}$. In the local universe central black hole masses are typically $\sim 1/1000$ of the bulge stellar mass\cite{Merrit2001}, but in the model we test here these massive early galaxies have black hole masses in the range $10^8 - 10^9{\rm M}_\odot$, significantly larger than low redshift galaxies of similar stellar mass. To attain the observed luminosities, they must convert infalling mass to radiated energy with a somewhat higher efficiency than the $\sim 0.1\,c^2$ expected for accretion onto a {\em non-spinning} black hole. Within our simulation we can readily address fundamental questions such as: ``Where are the descendants of the early quasars today?'', or ``What were their progenitors?''. By tracking the merging history trees of the host halos, we find that all our quasar candidates end up today as central galaxies in rich clusters. For example, the object depicted in Fig.~\ref{FigFirstQuasar} lies, today, at the centre of the ninth most massive cluster in the volume, of mass $M=1.46\times10^{15}\,h^{-1}{\rm M}_\odot$. The candidate with the largest virial mass at $z=6.2$ (which has stellar mass $4.7\times 10^{10}\,h^{-1}{\rm M}_\odot$, virial mass $4.85\times 10^{12}\,h^{-1}{\rm M}_\odot$, and star formation rate $218\, {\rm M_\odot yr^{-1}}$) ends up in the second most massive cluster, of mass $3.39\times10^{15}\,h^{-1}{\rm M}_\odot$. Following the merging tree backwards in time, we can trace our quasar candidate back to redshift $z=16.7$, when its host halo had a mass of only $1.8\times 10^{10}\,h^{-1}{\rm M}_\odot$. At this epoch, it is one of just 18 objects that we identify as collapsed systems with $\ge 20$ particles. These results confirm the view that rich galaxy clusters are rather special places. Not only are they the largest virialised structures today, they also lie in the regions where the first structures developed at high redshift. Thus, the best place to search for the oldest stars in the Universe or for the descendants of the first supermassive black holes is at the centres of present-day rich galaxy clusters. \subsubsection*{The clustering evolution of dark matter and galaxies} The combination of a large-volume, high-resolution N-body simulation with realistic modelling of galaxies enables us to make precise theoretical predictions for the clustering of galaxies as a function of redshift and intrinsic galaxy properties. These can be compared directly with existing and planned surveys. The 2-point correlation function of our model galaxies at redshift $z=0$ is plotted in Fig.~\ref{FigClustering} and is compared with a recent measurement from the 2dFGRS\cite{Hawkins2003}. The prediction is remarkably close to a power-law, confirming with much higher precision the results of earlier semi-analytic\cite{Kauffmann1999,Benson2000} and hydrodynamic\cite{Weinberg2004} simulations. This precision will allow interpretation of the small, but measurable deviations from a pure power-law found in the most recent data\cite{Padilla2003,Zehavi2004}. The simple power-law form contrasts with the more complex behaviour exhibited by the dark matter correlation function but is really no more than a coincidence. Correlation functions for galaxy samples with different selection criteria or at different redshifts do not, in general, follow power-laws. \begin{figure} \vspace*{-0.0cm}\ \\% \resizebox{14.0cm}{!}{\includegraphics{fig4.eps}}\\ \caption{Galaxy 2-point correlation function at the present epoch. Red symbols (with vanishingly small Poisson error-bars) show measurements for model galaxies brighter than $M_K = -23$. Data for the large spectroscopic redshift survey 2dFGRS\cite{Hawkins2003} are shown as blue diamonds. The SDSS\cite{Zehavi2002} and APM\cite{Padilla2003} surveys give similar results. Both, for the observational data and for the simulated galaxies, the correlation function is very close to a power-law for $r\le 20\, h^{-1}{\rm Mpc}$. By contrast the correlation function for the dark matter (dashed line) deviates strongly from a power-law. \label{FigClustering}} \end{figure} \begin{figure} \hspace*{-1.0cm}\ \resizebox{17cm}{!}{\includegraphics{fig5.eps}} \caption{Galaxy clustering as a function of luminosity and colour. In the panel on the left, we show the 2-point correlation function of our galaxy catalogue at $z=0$ split by luminosity in the bJ-band (symbols). Brighter galaxies are more strongly clustered, in quantitative agreement with observations\cite{Norberg2001} (dashed lines). Splitting galaxies according to colour (right panel), we find that red galaxies are more strongly clustered with a steeper correlation slope than blue galaxies. Observations\cite{Madgwick2003} (dashed lines) show a similar trend, although the difference in clustering amplitude is smaller than in this particular semi-analytic model. \label{FigClusteringSubsamples}} \end{figure} Although our semi-analytic model was not tuned to match observations of galaxy clustering, in not only produces the excellent overall agreement shown in Fig.~\ref{FigClustering}, but also reproduces the observed dependence of clustering on magnitude and colour in the 2dFGRS and SDSS\cite{Norberg2001,Zehavi2002,Madgwick2003}, as shown in Figure~\ref{FigClusteringSubsamples}. The agreement is particularly good for the dependence of clustering on luminosity. The colour dependence of the slope is matched precisely, but the amplitude difference is greater in our model than is observed\cite{Madgwick2003}. Note that our predictions for galaxy correlations split by colour deviate substantially from power-laws. Such predictions can be easily tested against survey data in order to clarify the physical processes responsible for the observed difference. In contrast to the near power-law behaviour of galaxy correlations on small scales, the large-scale clustering pattern may show interesting structure. Coherent oscillations in the primordial plasma give rise to the well-known acoustic peaks in the CMB\cite{deBernardis2000,Mauskopf2000,Spergel2003} and also leave an imprint in the linear power spectrum of the dark matter. Detection of these ``baryon wiggles'' would not only provide a beautiful consistency check for the cosmological paradigm, but could also have important practical applications. The characteristic scale of the wiggles provides a ``standard ruler'' which may be used to constrain the equation of state of the dark energy\cite{Blake2003}. A critical question when designing future surveys is whether these baryon wiggles are present and are detectable in the {\em galaxy} distribution, particularly at high redshift. On large scales and at early times, the mode amplitudes of the {\it dark matter} power spectrum grow linearly, roughly in proportion to the cosmological expansion factor. Nonlinear evolution accelerates the growth on small scales when the dimensionless power $\Delta^2(k) = k^3 P(k)/(2\pi^2)$ approaches unity; this regime can only be studied accurately using numerical simulations. In the Millennium Simulation, we are able to determine the nonlinear power spectrum over a larger range of scales than was possible in earlier work\cite{Jenkins1998}, almost five orders of magnitude in wavenumber $k$. At the present day, the acoustic oscillations in the matter power spectrum are expected to fall in the transition region between linear and nonlinear scales. In Fig.~\ref{FigWiggles}, we examine the matter power spectrum in our simulation in the region of the oscillations. Dividing by the smooth power spectrum of a $\Lambda$CDM model with no baryons\cite{Bardeen1986} highlights the baryonic features in the initial power spectrum of the simulation, although there is substantial scatter due to the small number of large-scale modes. Since linear growth preserves the relative mode amplitudes, we can approximately correct for this scatter by scaling the measured power in each bin by a multiplicative factor based on the initial difference between the actual bin power and the mean power expected in our $\Lambda$CDM model. This makes the effects of nonlinear evolution on the baryon oscillations more clearly visible. As Fig.~\ref{FigWiggles} shows, nonlinear evolution not only accelerates growth but also reduces the baryon oscillations: scales near peaks grow slightly more slowly than scales near troughs. This is a consequence of the mode-mode coupling characteristic of nonlinear growth. In spite of these effects, the first two ``acoustic peaks'' (at $k\sim 0.07$ and $k\sim 0.13\,h\,{\rm Mpc}^{-1}$, respectively) in the dark matter distribution do survive in distorted form until the present day (see the lower right panel of Fig.~\ref{FigWiggles}). \begin{figure} \begin{center} \vspace*{-1.6cm}\hspace*{-1.0cm}% \resizebox{15.8cm}{!}{\includegraphics{fig6.eps}}\vspace*{-1.0cm}% \end{center} \caption{ Power spectra of the dark matter and galaxy distributions in the baryon oscillation region. All measurements have been divided by a linearly evolved, CDM-only power spectrum\cite{Bardeen1986}. Red circles show the dark matter, and green squares the galaxies. Blue symbols give the actual realization of the initial fluctuations in our simulation, which scatters around the mean input power (black lines) due to the finite number of modes. Since linear growth preserves relative mode amplitudes, we correct the power in each bin to the expected input power and apply these scaling factors at all other times. At $z=3.06$, galaxies with stellar mass above $5.83\times 10^9\,h^{-1}{\rm M}_\odot$ and space-density of $8\times 10^{-3}\,h^{3}{\rm Mpc}^{-3}$ were selected. Their large-scale density field is biased by a factor $b=2.7$ with respect to the dark matter (the galaxy measurement has been divided by $b^2$). At $z=0$, galaxies brighter than $M_B = -17$ and a space density higher by a factor $\sim 7.2$ were selected. They exhibit a slight antibias, $b=0.92$. Corresponding numbers for $z=0.98$ are $M_B = -19$ and $b=1.15$. \label{FigWiggles}} \end{figure} Are the baryon wiggles also present in the galaxy distribution? Fig.~\ref{FigWiggles} shows that the answer to this important question is `yes'. The $z=0$ panel shows the power spectrum for all model galaxies brighter than $M_B = -17$. On the largest scales, the galaxy power spectrum has the same shape as that of the dark matter, but with slightly lower amplitude corresponding to an ``antibias'' of 8\%. Samples of brighter galaxies show less antibias while for the brightest galaxies, the bias becomes slightly positive. The figure also shows measurements of the power spectrum of luminous galaxies at redshifts $z=0.98$ and $z=3.06$. Galaxies at $z=0.98$ were selected to have a magnitude $M_B<-19$ in the restframe, whereas galaxies at $z=3.06$ were selected to have stellar mass larger than $5.83\times 10^9\,h^{-1}{\rm M}_\odot$, corresponding to a space density of $8\times 10^{-3}\,h^{3}{\rm Mpc}^{-3}$, similar to that of the Lyman-break galaxies observed at $z\sim 3$\cite{Adelberger1998}. Signatures of the first two acoustic peaks are clearly visible at both redshifts, even though the density field of the $z=3$ galaxies is much more strongly biased with respect to the dark matter (by a factor $b=2.7$) than at low redshift. Selecting galaxies by their star formation rate rather than their stellar mass (above $10.6\,{\rm M_\odot yr^{-1}}$ for an equal space density at $z=3$) produces very similar results. Our analysis demonstrates conclusively that baryon wiggles should indeed be present in the galaxy distribution out to redshift $z=3$. This has been assumed but not justified in recent proposals to use evolution of the large-scale galaxy distribution to constrain the nature of the dark energy. To establish whether the baryon oscillations can be measured in practice with the requisite accuracy will require detailed modelling of the selection criteria of an actual survey and a thorough understanding of the systematic effects that will inevitably be present in real data. These issues can only be properly addressed by means of specially designed mock catalogues constructed from realistic simulations. We plan to construct suitable mock catalogues from the Millennium Simulation and make them publicly available. Our provisional conclusion, however, is that the next generation of galaxy surveys offers excellent prospects for constraining the equation of state of the dark energy. N-body simulations of CDM universes are now of such size and quality that realistic modelling of galaxy formation in volumes matched to modern surveys has become possible. Detailed studies of galaxy and AGN evolution exploiting the unique dataset of the Millennium Simulation therefore enable stringent new tests of the theory of hierarchical galaxy formation. Using the simulation we demonstrated that quasars can plausibly form sufficiently early in a $\Lambda$CDM universe to be compatible with observation, that their progenitors were already massive by $z \sim 16$, and that their $z=0$ descendents lie at the centres of cD galaxies in rich galaxy clusters. Interesting tests of our predictions will become possible if observations of the black hole demographics can be extended to high redshift, allowing, for example, a measurement of the evolution of the relationship between supermassive black hole masses and the velocity dispersion of their host stellar bulges. We have also demonstrated that a power-law galaxy autocorrelation function can arise naturally in a $\Lambda$CDM universe, but that this suggestively simple behaviour is merely a coincidence. Galaxy surveys will soon reach sufficient statistical power to measure precise deviations from power-laws for galaxy subsamples, and we expect that comparisons of the kind we have illustrated will lead to tight constraints on the physical processes included in the galaxy formation modelling. Finally, we have demonstrated for the first time that the baryon-induced oscillations recently detected in the CMB power spectrum should survive in distorted form not only in the nonlinear dark matter power spectrum at low redshift, but also in the power spectra of realistically selected galaxy samples at $0