Extended Press-Schechter Formalism Research Articles

The abundance of dark matter haloes down to Earth mass

ABSTRACT We use the voids-within-voids-within-voids (VVV) simulations, a suite of successive nested N-body simulations with extremely high resolution (denoted, from low to high resolution, by L0 to L7), to test the Press–Schechter (PS), Sheth–Tormen (ST), and extended Press–Schechter (EPS) formulae for the halo abundance over the entire mass range, from minihaloes of 10−6 M⊙, to cluster haloes of 1015 M⊙, at different redshifts, from z = 30 to the present. We find that at z = 0 and z = 2, ST best reproduces the results of L0, which has the mean cosmic density (overdensity δ = 0), at 1011−15 M⊙. The higher resolution levels (L1–L7) are biased underdense regions (δ < −0.6). The EPS formalism takes this into account since it gives the mass function of a region conditioned, in this case, on having a given underdensity. EPS provides good matches to these higher levels, with deviations ≲20 per cent, at 10−6−12.5 M⊙. At z ∼ 7−15, the ST predictions for L0 and the EPS for L1–L7 show somewhat larger deviations from the simulation results. However, at even higher redshifts, z ∼ 30, EPS fits the simulations well again. We confirm our results by picking more subvolumes from the L0 simulation, finding that our conclusions depend only weakly on the size and overdensity of the region. The good agreement of EPS with the higher level simulations implies that PS (or ST) gives an accurate description of the total halo mass function in representative regions of the universe.

Ghostly Galaxies: Accretion-dominated Stellar Systems in Low-mass Dark Matter Halos

Wide-area deep imaging surveys have discovered large numbers of extremely low surface brightness (LSB) dwarf galaxies, which challenge galaxy formation theory and, potentially, offer new constraints on the nature of dark matter. Here we discuss one as-yet-unexplored formation mechanism that may account for a fraction of LSB dwarfs. We call this the “ghost galaxy” scenario. In this scenario, inefficient radiative cooling prevents star formation in the “main branch” of the merger tree of a low-mass dark matter halo, such that almost all its stellar mass is acquired through mergers with less massive (but nevertheless star-forming) progenitors. Present-day systems formed in this way would be “ghostly” isolated stellar halos with no central galaxy. We use merger trees based on the extended Press–Schechter formalism and the Copernicus Complexio cosmological N-body simulation to demonstrate that mass assembly histories of this kind can occur for low-mass halos in ΛCDM, but they are rare. They are most probable in isolated halos of present-day mass ∼4 × 109 M ⊙, occurring for ∼5% of all halos of that mass under standard assumptions about the timing and effect of cosmic reionization. The stellar masses of star-forming progenitors in these systems are highly uncertain; abundance-matching arguments imply a bimodal present-day mass function having a brighter population (median M ⋆ ∼ 3 × 106 M ⊙) consistent with the tail of the observed luminosity function of ultradiffuse galaxies. This suggests that observable analogs of these systems may await discovery. We find that a stronger ionizing background (globally or locally) produces brighter and more extended ghost galaxies.

Constraints on dark matter-neutrino scattering from the Milky-Way satellites and subhalo modeling for dark acoustic oscillations

The elastic scattering between dark matter (DM) and radiation can potentially explain small-scale observations that the cold dark matter faces as a challenge, as damping density fluctuations via dark acoustic oscillations in the early universe erases small-scale structure. We study a semi-analytical subhalo model for interacting dark matter with radiation, based on the extended Press-Schechter formalism and subhalos' tidal evolution prescription. We also test the elastic scattering between DM and neutrinos using observations of Milky-Way satellites from the Dark Energy Survey and PanSTARRS1. We conservatively impose strong constraints on the DM-neutrino scattering cross section of σ DM–ν,n ∝ En ν (n = 0,2,4) at 95% confidence level (CL), σ DM–ν,0 < 10-32 cm2 (m DM/ GeV), σ DM–ν,2 < 10-43 cm2 (m DM/ GeV)(Eν /Eν 0)2 and σ DM–ν,4 < 10-54 cm2 (m DM /GeV)(Eν /Eν 0)4, where Eν 0 is the neutrino energy and Eν 0 is the average momentum of relic cosmic neutrinos today, Eν 0 ≃ 6.1 K. By imposing a satellite forming condition, we obtain the strongest upper bounds on the DM-neutrino cross section at 95% CL, σ DM–ν,0 < 4 × 10-34 cm2 (m DM/ GeV), σ DM–ν,2 < 10-46 cm2 (m DM/ GeV)(Eν /Eν 0)2 and σ DM–ν,4 < 7 × 10-59 cm2 (m DM/GeV)(Eν /Eν 0)4.

Prospects for future binary black hole gravitational wave studies in light of PTA measurements

NANOGrav and other Pulsar Timing Arrays (PTAs) have discovered a common-spectrum process in the nHz range that may be due to gravitational waves (GWs): if so, they are likely to have been generated by black hole (BH) binaries with total masses &gt; 109 M⊙. Using the Extended Press-Schechter formalism to model the galactic halo mass function and a simple relation between the halo and BH masses suggests that these binaries have redshifts z = 𝒪(1) and mass ratios ≳10, and that the GW signal at frequencies above 𝒪(10) nHz may be dominated by relatively few binaries that could be distinguished experimentally and would yield observable circular polarization. Extrapolating the model to higher frequencies indicates that future GW detectors such as LISA and AEDGE could extend the PTA observations to lower BH masses ≳103 M⊙.

Evolution of the mass relation between supermassive black holes and dark matter halos across the cosmic time

ABSTRACT A positive observational proof suggests that most galaxies contain a central supermassive black hole (SMBH) with mass in the range of 106 M⊙–1010 M⊙. It is suggested that the mass of SMBHs is proportionally related to that of the dark matter (DM) halo, even at z = 6. This implies that these SMBHs could coevolve with the host DM halos. Here, we investigate the mass evolution of SMBHs in a hierarchical structure formation by building halo merger trees using the extended Press–Schechter formalism. An SMBH with a mass that follows various power-law relations with the DM halo mass is assigned as an initial condition. Assuming that the mass growth of all black holes is due to mergers, we obtain the relation between SMBH and DM halo masses at the present epoch. By requiring that the mass of the SMBHs at z = 0 should not be greater than the currently observed SMBH-DM halo mass relation, a lower bound on the mass of the DM halo that can contain a SMBH can be imposed at z = 6 as Mlim &amp;gt; 3.6 × 1010 M⊙ × (1.4 − n)2.3, where n is the power-law index of the SMBH-DM halo mass relation at z = 6. Because we only consider mergers for the mass evolution of SMBHs, this model is simplistic and should underestimate the mass of SMBHs relative to the mass of the DM halo at the present epoch. Our work aims to constrain the SMBH-DM mass relation at z = 6 and not build a new model to explain the observations

Warm dark matter constraints using Milky Way satellite observations and subhalo evolution modeling

Warm dark matter (WDM) can potentially explain small-scale observations that currently challenge the cold dark matter (CDM) model, as warm particles suppress structure formation due to free-streaming effects. Observing small-scale matter distribution provides a valuable way to distinguish between CDM and WDM. In this work, we use observations from the Dark Energy Survey and PanSTARRS1, which observe 270 Milky-Way satellites after completeness corrections. We test WDM models by comparing the number of satellites in the Milky Way with predictions derived from the Semi-Analytical SubHalo Inference ModelIng (SASHIMI) code, which we develop based on the extended Press-Schechter formalism and subhalos' tidal evolution prescription. We robustly rule out WDM with masses lighter than 4.4 keV at 95% confidence level for the Milky-Way halo mass of $10^{12} M_\odot$. The limits are a weak function of the (yet uncertain) Milky-Way halo mass, and vary as $m_{\rm WDM}>3.6$-$5.1$ keV for $(0.6$-$2.0) \times 10^{12} M_\odot$. For the sterile neutrinos that form a subclass of WDM, we obtain the constraints of $m_{\nu_s}>11.6$ keV for the Milky-Way halo mass of $10^{12} M_{\odot}$. These results based on SASHIMI do not rely on any assumptions of galaxy formation physics or are not limited by numerical resolution. The models, therefore, offer a robust and fast way to constrain the WDM models. By applying a satellite forming condition, however, we can rule out the WDM mass lighter than 9.0 keV for the Milky-Way halo mass of $10^{12} M_\odot$.

Muscle-ups: improved approximations of the matter field with the extended Press–Schechter formalism and Lagrangian perturbation theory

ABSTRACT Lagrangian algorithms to simulate the evolution of cold dark matter (CDM) are invaluable tools to generate large suites of mock halo catalogues. In this paper, we first show that the main limitation of current semi-analytical schemes to simulate the displacement of CDM is their inability to model the evolution of overdensities in the initial density field, a limit that can be circumvented by detecting halo particles in the initial conditions. We thus propose ‘MUltiscale Spherical Collapse Lagrangian Evolution Using Press–Schechter’ (muscle-ups), a new scheme that reproduces the results from Lagrangian perturbation theory on large scales, while improving the modelling of overdensities on small scales. In muscle-ups, we adapt the extended Press and Schechter formalism to Lagrangian algorithms of the displacement field. For regions exceeding a collapse threshold in the density smoothed at a radius R, we consider all particles within a radius R collapsed. Exploiting a multiscale smoothing of the initial density, we build a halo catalogue on the fly by optimizing the selection of halo candidates. This allows us to generate a density field with a halo mass function that matches one measured in N-body simulations. We further explicitly gather particles in each halo together in a profile, providing a numerical, Lagrangian-based implementation of the halo model. Compared to previous semi-analytical Lagrangian methods, we find that muscle-ups improves the recovery of the statistics of the density field at the level of the probability density function, the power spectrum, and the cross-correlation with the N-body result.

Toward a General Parameterization of the Warm Dark Matter Halo Mass Function

Abstract Studies of flux anomalies statistics and perturbations in stellar streams have the potential to constrain models of warm dark matter (WDM), including sterile neutrinos. Producing these constraints requires a parameterization of the WDM mass function relative to that of the cold dark matter (CDM) equivalent. We use five WDM models with half-mode masses, M hm = [1.3,35] × 108 , spread across simulations of the Local Group, lensing ellipticals and the z = 2 universe, to generate such a parameterization: we fit parameters to a functional form for the WDM-to-CDM halo mass function ratio, , of . For M X ≡ virial mass of central halos we obtain α = 2.3, β = 0.8, and γ = −1.0, and this fit is steeper than the extended Press–Schechter formalism predicts. For M X ≡ mass of subhalos we instead obtain α = 4.2, β = 2.5, and γ = −0.2; in both mass definitions the scatter is ∼20%. The second fit typically underestimates the relative abundance of z = 2 WDM subhalos at the tens of percent level. We caution that robust constraints will require bespoke simulations and a careful definition of halo mass, particularly for subhalos of mass &lt;108 .

Halo Substructure Boosts to the Signatures of Dark Matter Annihilation

The presence of dark matter substructure will boost the signatures of dark matter annihilation. We review recent progress on estimates of this subhalo boost factor—a ratio of the luminosity from annihilation in the subhalos to that originating the smooth component—based on both numerical N-body simulations and semi-analytic modelings. Since subhalos of all the scales, ranging from the Earth mass (as expected, e.g., the supersymmetric neutralino, a prime candidate for cold dark matter) to galaxies or larger, give substantial contribution to the annihilation rate, it is essential to understand subhalo properties over a large dynamic range of more than twenty orders of magnitude in masses. Even though numerical simulations give the most accurate assessment in resolved regimes, extrapolating the subhalo properties down in sub-grid scales comes with great uncertainties—a straightforward extrapolation yields a very large amount of the subhalo boost factor of ≳100 for galaxy-size halos. Physically motivated theoretical models based on analytic prescriptions such as the extended Press-Schechter formalism and tidal stripping modeling, which are well tested against the simulation results, predict a more modest boost of order unity for the galaxy-size halos. Giving an accurate assessment of the boost factor is essential for indirect dark matter searches and thus, having models calibrated at large ranges of host masses and redshifts, is strongly urged upon.

Physical Properties of a Coma-analog Protocluster at z = 6.5

Abstract We present evidence for the discovery of a protocluster of starburst galaxies (Lyα emitters, or LAEs) near the end of the epoch of reionization. The recent trend in the search for high-redshift protoclusters focuses on utilizing bias tracers, such as luminous starburst galaxies, as signposts of overdensities. Thus, we conducted a photometric selection of LAE candidates around a pair of spatially close, luminous LAEs at z = 6.5 in the Subaru/XMM-Newton Deep Survey field, using OSIRIS in its imaging mode at the 10.4 m Gran Telescopio Canarias in La Palma, Spain. The spectroscopic follow-up was done with OSIRIS in its multiobject spectroscopy capability. We have spectroscopically confirmed 10 LAEs via their recognizable Lyα emission feature. The redshifts of these LAEs shed light on their 3D distributions within the observing window defined by the photometric selection. We have derived the galaxy number density contrast of δ gal = 3.18 − 1.99 + 3.47 , which led to the expected mass of the overdensity of 8.40 − 1.39 + 2.98 × 10 14 M ⊙ . We also found evidence for the presence of a virialized core with M 200 = 4.06 − 1.90 + 2.77 × 10 13 M ⊙ within this overdensity. Based on the extended Press–Schechter formalism, this overdensity would continue to grow in the linear regime and collapse to form a galaxy cluster at z coll = 0.84 − 0.43 + 0.57 . By the time this protocluster reaches z = 0, it will be a massive cluster of galaxies with mass 1.54 − 0.69 + 1.12 × 10 15 M ⊙ , comparable to the Coma cluster. Thus, our careful analysis has pointed to evidence that this protocluster would evolve into a Coma-analog cluster in the present-day universe.

Bimodal Formation Time Distribution for Infall Dark Matter Halos

Abstract We use a 200 a-side N-body simulation to study the mass accretion history (MAH) of dark matter halos to be accreted by larger halos, which we call infall halos. We define a quantity to characterize the MAH of infall halos, where and are the accretion and formation redshifts, respectively. We find that, at given , their MAH is bimodal. Infall halos are dominated by a young population at high redshift and by an old population at low redshift. For the young population, the distribution is narrow and peaks at about 1.2, independent of , while for the old population, the peak position and width of the distribution both increase with decreasing and are both larger than those of the young population. This bimodal distribution is found to be closely connected to the two phases in the MAHs of halos. While members of the young population are still in the fast accretion phase at z peak, those of the old population have already entered the slow accretion phase at . This bimodal distribution is not found for the whole halo population, nor is it seen in halo merger trees generated with the extended Press–Schechter formalism. The infall halo population at are, on average, younger than the whole halo population of similar masses identified at the same redshift. We discuss the implications of our findings in connection to the bimodal color distribution of observed galaxies and to the link between central and satellite galaxies.

The stellar mass, star formation rate and dark matter halo properties of LAEs at z ∼ 2

Abstract We present average stellar population properties and dark matter halo masses of z ∼ 2 Lyα emitters (LAEs) from spectral energy distribution fitting and clustering analysis, respectively, using ≃ 1250 objects ($\mathit {NB387}\le 25.5$) in four separate fields of ≃ 1 deg2 in total. With an average stellar mass of 10.2 ± 1.8 × 108 M⊙ and star formation rate of 3.4 ± 0.4 M⊙ yr−1, the LAEs lie on an extrapolation of the star-formation main sequence (MS) to low stellar mass. Their effective dark matter halo mass is estimated to be $4.0_{-2.9}^{+5.1} \times 10^{10}{\,\,}M_{\odot }$ with an effective bias of $1.22^{+0.16}_{-0.18}$, which is lower than that of z ∼ 2 LAEs (1.8 ± 0.3) obtained by a previous study based on a three times smaller survey area, with a probability of 96%. However, the difference in the bias values can be explained if cosmic variance is taken into account. If such a low halo mass implies a low H i gas mass, this result appears to be consistent with the observations of a high Lyα escape fraction. With the low halo masses and ongoing star formation, our LAEs have a relatively high stellar-to-halo mass ratio (SHMR) and a high efficiency of converting baryons into stars. The extended Press–Schechter formalism predicts that at z = 0 our LAEs are typically embedded in halos with masses similar to that of the Large Magellanic Cloud (LMC); they will also have similar SHMRs to the LMC, if their star formation rates are largely suppressed after z ∼ 2 as some previous studies have reported for the LMC itself.

Testing the conditional mass function of dark matter haloes against numerical N-body simulations

We compare the predicted conditional mass function (CMF) of dark matter halos from two theoretical prescriptions against numerical N-body simulations, both in overdense and underdense regions and at different Eulerian scales ranging from $5$ to $30\,h^{-1}\,$Mpc. In particular, we consider in detail a locally-implemented rescaling of the unconditional mass function (UMF) already discussed in the literature, and also a generalization of the standard rescaling method described in the extended Press-Schechter formalism. First, we test the consistency of these two rescalings by verifying the normalization of the CMF at different scales, and showing that none of the proposed cases provides a normalized CMF. In order to satisfy the normalization condition, we include a modification in the rescaling procedure. After this modification, the resulting CMF generally provides a better description of numerical results. We finally present an analytical fit to the ratio between the CMF and the UMF (also known as the matter-to-halo bias function) in underdense regions, which could be of special interest to speed-up the computation of the halo abundance when studying void statistics. In this case, the CMF prescription based on the locally-implemented rescaling provides a slightly better description of the numerical results when compared to the standard rescaling.

Solving the small-scale structure puzzles with dissipative dark matter

Small-scale structure is studied in the context of dissipative dark matter, arising for instance in models with a hidden unbroken Abelian sector, so that dark matter couples to a massless dark photon. The dark sector interacts with ordinary matter via gravity and photon-dark photon kinetic mixing. Mirror dark matter is a theoretically constrained special case where all parameters are fixed except for the kinetic mixing strength, ϵ. In these models, the dark matter halo around spiral and irregular galaxies takes the form of a dissipative plasma which evolves in response to various heating and cooling processes. It has been argued previously that such dynamics can account for the inferred cored density profiles of galaxies and other related structural features. Here we focus on the apparent deficit of nearby small galaxies (``missing satellite problem"), which these dissipative models have the potential to address through small-scale power suppression by acoustic and diffusion damping. Using a variant of the extended Press-Schechter formalism, we evaluate the halo mass function for the special case of mirror dark matter. Considering a simplified model where Mbaryons ∝ Mhalo, we relate the halo mass function to more directly observable quantities, and find that for ϵ ≈ 2 × 10−10 such a simplified description is compatible with the measured galaxy luminosity and velocity functions. On scales Mhalo ≲ 108 M⊙, diffusion damping exponentially suppresses the halo mass function, suggesting a nonprimordial origin for dwarf spheroidal satellite galaxies, which we speculate were formed via a top-down fragmentation process as the result of nonlinear dissipative collapse of larger density perturbations. This could explain the planar orientation of satellite galaxies around Andromeda and the Milky Way.

The accretion history of dark matter haloes – I. The physical origin of the universal function

Understanding the universal accretion history of dark matter halos is the first step towards determining the origin of their structure. We use the extended Press-Schechter formalism to derive the halo mass accretion history from the growth rate of initial density perturbations. We show that the halo mass history is well described by an exponential function of redshift in the high-redshift regime. However, in the low-redshift regime the mass history follows a power law because the growth of density perturbations is halted in the dark energy dominated era due to the accelerated expansion of the Universe. We provide an analytic model that follows the expression $M(z) = M_{0}(1+z)^{af(M_{0})}e^{-f(M_{0})z}$, where $M_{0} = M(z=0)$, $a$ depends on cosmology and $f(M_{0})$ depends only on the linear matter power spectrum. The analytic model does not rely on calibration against numerical simulations and is suitable for any cosmology. We compare our model with the latest empirical models for the mass accretion history in the literature and find very good agreement. We provide numerical routines for the model online.

Galaxy formation as a cosmological tool – I. The galaxy merger history as a measure of cosmological parameters

As galaxy formation and evolution over long cosmic time-scales depends to a large degree on the structure of the universe, the assembly history of galaxies is potentially a powerful approach for learning about the universe itself. In this paper we examine the merger history of dark matter halos based on the Extended Press-Schechter formalism as a function of cosmological parameters, redshift and halo mass. We calculate how major halo mergers are influenced by changes in the cosmological values of $\Omega_{\rm m}$, $\Omega_{\Lambda}$, $\sigma_{8}$, the dark matter particle temperature (warm vs. cold dark matter), and the value of a constant and evolving equation of state parameter $w(z)$. We find that the merger fraction at a given halo mass varies by up to a factor of three for halos forming under the assumption of Cold Dark Matter, within different underling cosmological parameters. We find that the current measurements of the merger history, as measured through observed galaxy pairs as well as through structure, are in agreement with the concordance cosmology with the current best fit giving $1 - \Omega_{\rm m} = \Omega_{\rm \Lambda} = 0.84^{+0.16}_{-0.17}$. To obtain a more accurate constraint competitive with recently measured cosmological parameters from Planck and WMAP requires a measured merger accuracy of $\delta f_{\rm m} \sim 0.01$, implying surveys with an accurately measured merger history over 2 - 20 deg$^{2}$, which will be feasible with the next generation of imaging and spectroscopic surveys such as Euclid and LSST.

A suppressed contribution of low-mass galaxies to reionization due to supernova feedback

Motivated by recent observations of the star formation rate density function out to z~7, we describe a simple model for the star formation rate density function at high redshift based on the extended Press-Schechter formalism. This model postulates a starburst following each major merger, lasting for a time t_SF and converting at most f_star of galactic gas into stars. We include a simple physical prescription for supernovae feedback that suppresses star formation in low mass galaxies. Constraining t_SF and f_star to describe the observed star formation rate density at high redshifts, we find that individual starbursts were terminated after a time t_SF~10^7 years. This is comparable to the main-sequence lifetimes of supernova progenitors, indicating that high redshift starbursts are quenched once supernovae feedback had time to develop. High redshift galaxies convert ~10% of their mass into stars for galaxies with star formation rates above ~1 solar mass per year, but a smaller fraction for lower luminosity galaxies. Our best fit model successfully predicts the observed relation between star formation rate and stellar mass at z>~4, while our deduced relation between stellar mass and halo mass is also consistent with data on the dwarf satellites of the Milky Way. We find that supernovae feedback lowers the efficiency of star formation in the lowest mass galaxies and makes their contribution to reionization small. As a result, photo-ionization feedback on low mass galaxy formation does not significantly affect the reionization history. Using a semi-analytic model for the reionization history, we infer that approximately half of the ionizing photons needed to complete reionization have already been observed in star-forming galaxies.

Dark matter halo merger histories beyond cold dark matter – I. Methods and application to warm dark matter

We describe a methodology to accurately compute halo mass functions, progenitor mass functions, merger rates and merger trees in non-cold dark matter universes using a self-consistent treatment of the generalized extended Press-Schechter formalism. Our approach permits rapid exploration of the subhalo population of galactic halos in dark matter models with a variety of different particle properties or universes with rolling, truncated, or more complicated power spectra. We make detailed comparisons of analytically derived mass functions and merger histories with recent warm dark matter cosmological N-body simulations, and find excellent agreement. We show that, once the accretion of smoothly distributed matter is accounted for, coarse-grained statistics such as the mass accretion history of halos can be almost indistinguishable between cold and warm dark matter cases. However, the halo mass function and progenitor mass functions differ significantly, with the warm dark matter cases being strongly suppressed below the free-streaming scale of the dark matter. We demonstrate the importance of using the correct solution for the excursion set barrier first-crossing distribution in warm dark matter - if the solution for a flat barrier is used instead the truncation of the halo mass function is much slower, leading to an overestimate of the number of low mass halos.

星系中心大黑洞的宇宙学演化

In the cosmological model of the hierarchical structure formation of dark matter halos, the black holes residing at the center of galaxies continue merging and accreting, following the merging of the dark matter halos. Based on the extended Press-Schechter formalism, we have developed a Monte Carlo algorithm with the self-adaptive time step, which could be used to reconstruction the merging tree of dark matter halos. Having introduced the evolutionary history of massive black holes, especially the two kinds of accretion modes: the direct accretion modes which includes the standard thin accretion disk model and the magnetohydrodynamic accretion disk model and the chaotic accretion model, we investigate the cosmological evolution of the spin and masses of massive black holes. Comparing with the observed luminosity function of active galactic nuclei, it is found that the magnetohydrodynamic accretion disk model describe the growth of massive black holes better than the other accretion models. We have also study the evolution of the binaries of massive black holes, and find that the circum-disk of the black hole binaries significantly affect the distribution of their mass ratio.

Stochastic gravitational wave background originating from halo mergers

The stochastic gravitational wave background (GWB) from halo mergers is investigated by a quasi-analytic method. The method we employ consists of two steps. The first step is to construct a merger tree by using the Extended Press-Schechter formalism or the Sheth & Tormen formalism, with Monte-Carlo realizations. This merger tree provides evolution of halo masses. From $N$-body simulation of two-halo mergers, we can estimate the amount of gravitational wave emission induced by the individual merger process. Therefore the second step is to combine this gravitaional wave emission to the merger tree and obtain the amplitude of GWB. We find $\Omega_{GW}\sim 10^{-19}$ for $f\sim 10^{-17}-10^{-16}$ Hz, where $\Omega_{GW}$ is the energy density of the GWB. It turns out that most of the contribution on the GWB comes from halos with masses below $10^{15} M_\odot$ and mergers at low redshift, i.e., $0<z<0.8$.