Galaxy Formation: Peaks Densities and Proto

Transcription

African Journal Of Mathematical Physics Volume 6 (2008) 145-156
Galaxy Formation: Peaks Densities and Proto-Haloes
El Yamani Diaf
Department of Physics, Multidisciplinary Faculty of Nador,
Univesrity Mohammed Premier, B.P N 300, Selouane, 27000 Nador, Morocco.
E-mail: [email protected]
Abstract
We investigate the connection between the peak of the initial mass (z = 50)
density smoothed with Gaussian window of radius RG = 1h−1 Mpc and the haloes
at present time traced back in time to their initial positions (proto-halo), using
a cosmological N-body simulation (GIF) with thousand of haloes. We find that
the alignation of the axes of Inertia tensor for the peaks and the proto-haloes are
correlated. This quantitative allows to understand the quantification of proto-haloes
and peaks density and agrees with the peak model describing galaxy formation.
key words: Cosmology: dark matter-Cosmology: Large Scale of the Universe Cosmology: theory-galaxies: formation-galaxies:haloes-galaxies: statistic
I. INTRODUCTION
Cold Dark Matter (CDM) has became the standard modern theory of cosmological structure
formation. Its prediction appears to be in good agreement with data on large scale, and it naturally
accounts for many properties of galaxies.
For the gravitational instability (GI) description of galaxy formation, galaxy angular momentum
arises from the tidal torquing during the early progalactic stages (Hoyle, 1949), and then analyzed
qualitatively by Peebles (1969). Gravitational torquing results whenever the inertia tensor of an
object is misaligned with the local gravitational shear tensor. In a paper by Porciani et al. (2001),
the authors used the tidal torque theory (TTT) approximation in order to understand the origin
of the halo angular momentum, by investigating the relation between the different components of
TTT . They found that the inertia and shear tensors are correlated in the alignations major axes.
The goal of this paper is focused on understanding how to identify proto-haloes in a given realization
of initial conditions; how these proto-haloes evolve via collapse and mergers to the present virialized
haloes, and how to quantify the degree of correlation between the inertia tensor of both the density
peaks and the proto-haloes. The application of TTT theory shows that the inertia tensor and
the shear tensor are uncorrelated (e.g, Hoffman 1986 a,b). Other applications have made the
assumption that the present-day halo angular momentum correlates with the shear tensor of the
initial conditions (LP00; Pen et al. 2000; Crittenden et al. 2001). This involves assuming a strong
correlation survives the non-linear evolution at late times. We show the validity of this assumption
in this case with the correlation function between the peaks and the proto-haloes. For this result,
we found that the inertia tensor of the density field around peaks and of the proto-haloes are
importantly correlated, this may show that the tidal field in much more important than the shape
0
c a GNPHE publication 2008, [email protected]
°
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El Yamani Diaf
of the density field which determine the direction of collapse.
The possibility that biasing could occur “naturally”;i.e through normal hierarchical clustering,
has been studied by Frenk et al. (1988, FWDE) for N-body simulation of small regions of the
Universe (∼ 14Mpc) at high resolution; they showed that massive haloes form preferentially in
regions of high density. This happens because of the higher background density accelerates the
formation of structure, including the formation of heavy haloes, when compared to regions of
average background density.
The aim this paper is to study the correspondence between the peaks in the linear density field
and the groups (haloes) that form out of that field. First, we study how well these peak particles
mimic the distribution of groups. Then, we determine if there is a correlation between the peaks
and groups mass.
The outline of this paper is as follows: In section 2, we describe the N-body simulation and
background of correlation function with smoothing density. We address to the code used for
this N-body simulation, the linear theory of inertia tensor and their properties. In section 3, we
investigate to the virial studies for the radius and mass and their correlation. In section 4, we
discuss our results and give the conclusion.
II. SMOOTHING DENSITY AND CORRELATION FUNCTION
A. Gaussian Smoothing
We start by smoothing the density in initial condition (z = 50) with Gaussian window RG =
1h−1 Mpc. In particular, we select the same number of proto-haloes and peaks using the following
criteria: The number of particles in proto-haloes Np ≥ 173 and peak density fluctuations threshold
δ ≥ δmin = 0.05. We identify peaks in the IC density field smoothed, by grid points with values of
δ which are maximal within a 3 × 3 × 3 cells surrounding them. The number of proto-halos selected
here is 405 with (masses greather then 173h−1 M¯ ) within a slice with thickness ∆z = 3h−1 Mpc.
B. N-body Simulation and Inertia Tensor
The N-body simulation analyzed here is a part of GIF project (e.g., Kauffmann et al. 1999) using the the parallel, adaptive particle-particle/particle-mesh (AP 3 M) code developed by the Virgo
consortium (Pearce et al. 1995; Pearce & Couchman 1997). The simulation analyzed here is a
ΛCDM scenario, with Ωm = 0.3, Ωλ = 0.7 and Hubble constant H0 = 100h km s−1 Mpc−1 with
h = 0.7. The power spectrum of initial density fluctuations is of CDM with shape parameter
Γ = 0.21.
The simulation was performed in a periodic cubic box of side 141.3 h−1 Mpc, with the mass represented by 2563 particles of 1.4 x 1010 h−1 M¯ each.
i)- Inertia Tensor
For each proto-haloes we can define the Lagrangian inertia tensor by direct summation over its N
particles of mass m by
Iij = m
N
X
0
Xi
(n)
0
Xj
(n)
(II.1)
n=1
0
where X (n) is the position of the n-th particle with respect to the halo centre of mass and m is
the mass of each particle in halos. We study the shape of the proto-haloes in the Lagrangian space
using the inertia tensor I.
The eigenvalues i1 ≥ i2 ≥ i3 are in the increasing order and also such perfect geometrical is given
by this following cases:
• perfect sphere has: e = p = 0.
• flat circular disc has: e = 1/4 and p = −1/4.
• thin filament has: e = 1/2 and p = 1/2.
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El Yamani Diaf
ii)- Properties of proto-haloes: Prolateness and ellipticity.
We describe the shape of proto-haloes by the following three parameters: the trace τ = i1 + i2 + i3 ,
the ellipticity, e = (i1 − i3 )/2τ , and the prolateness p = (i1 − 2i2 + i3 )/2τ , Porciani et al. (2001). In
Fig. 1, we can see the joint distribution of ellipticity and prolateness of proto-haloes. In the Fig. 2
and Fig. 3, we show the distribution of the shape parameters e and p of the proto-haloes near and
far from peaks. Then we find that on average the shape parameters are slightly smaller near peaks.
This means that the probability distribution of prolateness and the ellipticity are important with
an average and standard values. In order to make sure for this small difference is systematic, we
calculate the cumulative distribution function, and apply the the Kolmogorov-Smirnov test ( see,
Fig. 4, and Fig. 5, that is showed there is a difference between this two quantity which means
that there is a such correlation between the proto-haloes and peaks.)
The variance of this property is given by
σp = (< p2 > − < p >2 )1/2 , σe = (< e2 > − < e >2 )1/2
(II.2)
FIG. 1. The prolateness and ellipticity of the inertia tensor of proto-haloes (red) d ≤ 1h−1 Mpc from the
peaks and (blue) with d ≥ 2h−1 Mpc.
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El Yamani Diaf
FIG. 2. The shape of proto-haloes near peaks. The probability distribution of prolateness (red) and
ellipticity (blue) of the inertia tensor of proto-haloes within a distance of r = Rs = 1h−1 Mpc from a peak
with number of particles Np ≥ 100.
FIG. 3. The shape of proto-haloes far the peaks. The probability distribution of prolateness (red) and
ellipticity (blue) of the inertia tensor of proto-haloes with a distance of r ≥ Rs = 2h−1 Mpc from all the
peaks with same number of particles.
with < p > and < e > are the standard deviation for the prolateness and ellipticity respectively.
The location of the peak is then fine-tuned by a small shift ∆~r towards the peak using a 2nd-order
Taylor expansion,
1
δmodel (~r) = δ0 + Aα rα + Bαβ rα rβ
2
(II.3)
that we locally fit to the values of δ(~r) at the 33 grid points. The fit is done actually by minimizing
~ model = 0 at
residuals with respect to the 10 parameters δ0 , A and Bαβ = Bβα . By requiring ∇δ
the peak, we get the coordinates for the location of the peak within the grid cell,
1 ~ −1 ~
A
∆~r = − B
2
We define a local inertia tensor of a peak by
Z
p
Iαβ
≡ ρ̄
(II.4)
d3 r[1 + δ(~r)]rα rβ
(II.5)
r<R
where R is a small radius around the peak. Using the model fitted locally at the peak positions,
we get
p
Iαβ
∝ Bαβ + const.
Thus the eigenvectors of the inertia tensor coincide with the eigenvectors of Bαβ ≡
148
(II.6)
∂2δ
∂rα ∂rβ .
El Yamani Diaf
FIG. 4. A comparison between the cumulative distribution function of the ellipticity of proto-haloes
near peaks, within a distance r ≤ 1h−1 Mpc from a peak (e1 , black) and proto-haloes far from peaks, with
nearest peak at a distance r ≥ 2h−1 Mpc (e2 ,red). A Kolmogorov-Smirnov test shows that the probability
that these two samples given their seizes were withdraw from the same distribution is 0.25.
FIG. 5. A comparison between the cumulative distribution function of the prolateness of proto-haloes
near peaks, within a distance r ≤ 1h−1 Mpc from a peak (e1 , black) and proto-haloes far from peaks, with
nearest peak at a distance r ≥ 2h−1 Mpc (e2 ,red). A Kolmogorov-Smirnov test shows that the probability
that these two samples given their seizes were withdraw from the same distribution is 0.14.
Before the calculation of I. We select the same number of proto-haloes and peaks using the
criteria cited in §2.1.
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El Yamani Diaf
III. VIRIAL RADIUS STUDIES: ALIGNMENT OF PEAKS AND PROTO-HALOES
The density at the redshift z = 0 (today) is δv,0 being related to the density in the initial
condition (z = 50) assuming a spherical collapse.
δv,z=0 = δc ≡ δcollapse = 1.686
(III.1)
δv,0
1.686
=
.
(1 + z)
51
(III.2)
which implies that
δv,IC(z=50) =
and
ρ̄(R) =
M (< R)
4
3
3 πR
(III.3)
60
40
20
0
0
20
40
60
FIG. 6. This plot show the virial radius of peaks in the case where we are assuming that the shape near
peaks is spherical symetric.
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El Yamani Diaf
FIG. 7. The average and the standard deviation for the prolateness and ellipticity of mass within a
distance of virial radius Rv ≤ 1h−1 Mpc from a peak.
FIG. 8. The average and the standard deviation for the prolateness and ellipticity of mass within a
distance of the virial radius Rv ≥ 2h−1 Mpc.
where M is the mass inside the sphere (volume) of radius R, is the mass selected inside the
sphere of radius less than R, (ie; the virial mass)
ρ̄ =
Mtot
N.mp
=
L3
v
ρ̄(Rv ) = (1 + δv,IC )ρ̄ = (1 +
(III.4)
M (< Rv )
1.686
)ρ̄ = 4 3
51
3 πRv
(III.5)
and
Z
Z
d3 rρ(~r) =
M (< Rv ) =
|~
r<R|
d3 r[1 + δ(~r)]ρ̄
|~
r <R|
151
(III.6)
El Yamani Diaf
where mp is particles masses of haloes. Here we can use the data of GIF simulation, so the
integration is just the sum of particles numbers, so
X
M (< Rv ) = ρ̄a3
(1 + δi )
(III.7)
ri <Rv
L
and the a = Ngrid is the grid cell size. The radius of each peak Rv is defined, as the radius within
the average density contrast is
ρ̄(Rv )
= (1 + δv,IC )
ρ̄
(III.8)
The density field around each peak is fitted to
δ(r) = Aij rij + B i ri + δ0
(III.9)
FIG. 9. The Kolmogorov-Smirnov test show how much is the relation between the prolateness and the
different nearest distance from the peaks such that d ≤ Rs = 1h−1 Mpc and d ≥ 2h−1 Mpc for the virial
radius.
FIG. 10. The Kolmogorov-Smirnov test show how much is the relation between the ellipticity and its
different nearest distance from the peaks such that d ≤ Rs = 1h−1 Mpc and d ≥ 2h−1 Mpc for the virial
radius.
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El Yamani Diaf
FIG. 11. This figure show the distribution of the absolute values of the cosine angle between the directions of eigenvectors of the inertia tensor for pairs of the peak and its closest proto-haloes (d ≤ 0.1h−1 Mpc).
The correlation between these two quantity is important.
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El Yamani Diaf
60
40
20
0
0
20
40
60
FIG. 12. This figure shows the projection of the proto-haloes particle (black circle) and peaks (red circle)
with the major axes of both them (arrow black for the proto-haloes and red for the peaks). This is done
in the slice with the thickness ∆z = 2h−1 Mpc.
where ri are with respect to the peak’s center (the point with the maximal density), δ0 is the
maximal density at the peak’s center, Bi = 0 by definition and the Aij is the tensor which is
proportional to the local Inertia tensor.
In this case, we are going to select just a specific volume for the peaks such that we assuming that
the density profiles around peaks are spherically symmetric. In Fig. 6, we show the virial radius
of each peaks. The different virial radii vary with masses; ie, the larger volume corresponds to
the higher value of density peaks (overdensity or filament) and the smaller one correspond to the
(underdensity)(voids). The figures,7, 8, 9, 10 show respectively the shapes of proto-haloes nearest
to the peaks for the different distance (d ≤ 1h−1 Mpc, d ≥ 2h−1 Mpc from the peaks) and the
cumulative distribution figured by the Kolmogov-Smirnov test. In this case, we are not finding the
big difference for the virial radius in the general cases discussed above as figured in figures 2, 3, 4
and 5.
Let i1 , i2 and i3 be the eigenvectors of peak’s Aij in the order of increasing eigenvalues and t1 ,t2
and t3 the eigenvectors of proto-haloes in the same order.
The Fig. 11, show the distribution of absolute values of cosine of the angle between the directions
of the principal axes of the inertia tensor for the peaks (iα ) and the inertia tensor for the protohaloes (tβ ), with α, β = 1, 2, 3. The corresponding eigenvalues are ordered as i1 ≥ i2 ≥ i3 and
t1 ≥ t2 ≥ t3 . We found that all the correlation of the inertia tensor for the peaks and proto-haloes
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El Yamani Diaf
are important.
FIG. 13. This figure show the distribution of the absolute values of the cosine angle between the directions of eigenvectors of the inertia tensor for pairs of the peak (d ≤ 1h−1 Mpc) in the virial radius and its
closest proto-haloes.
i)- Correlation between the Peaks in the Virial Radius and the Proto-Haloes
As we see in the Fig, 12, there is an alignment of peaks (red points and dot) with the proto-haloes
(black arrow and dot). The doted points and arrow represent the positions and the directions of the
major axes about the centres of mass for the peaks and proto-haloes. The arrows are proportional
to the projection of a unit vectors along the major axis on the plane and the slice is of thickness
∆z = 2h− 1Mpc. We found that the principal axis of the inertia tensor of proto-haloes and peaks
at the centre of the panel lines almost aligned.
ii)- Correlation between Proto-Haloes and Density Peak to the Virial Radius
Other studies, to see the positions and the major axes of proto-haloes, is to Calculate the major
axes of proto-haloes and the density to the virial radius. In Fig. 13, we show the positions and the
major axes of Inertia tensor for the proto-haloes and the density to virial radius. In this case, we
show that there is a certain alignment between the proto-halos and the density peak virial radius.
This alignment means that there is a such quantity of peaks and proto-haloes on formation in such
correspondence.
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El Yamani Diaf
These results are roughly in (not good) correspondence between the peaks in the initial density
field and the collapsed groups. The result has been found in this case show that there is a important
correlation between the peaks density and the proto-halors in the IC condition. This means that
just a few fraction of the peaks in the linear density field and the groups (haloes) that form out of
that field are in correspondence.
IV. CONCLUSION AND DISCUSSION
In this paper, we have studied the correlation of the peaks and proto-haloes using thousand
of haloes from the data of N-body simulation for the ΛCDM cosmology. The other way to this
correlation is Inertia Tensor. It turns out that there is also a correlation between these two
quantities such that ∼ 50% of the proto-haloes containing a peaks within a distance corresponding
to ∼ 1h−1 Mpc The distributions of the ellipticity and prolateness show that a such correlation in
the elongation of the Inertia tensor axes is important, Fig 1, 2 and 3.
Regarding the virial radius, the distribution of absolute values of the cosine angle between the
directions of the Inertia tensor for pairs of the peaks and its closet proto-haloes are smaller. This
means that the correlation in this case is also important, Fig 11. From Fig 12 and 13. We conclude
that there is a certain correlation between the peaks and proto-haloes because the alignment of
their major axes of Inertia tensor is not perfect which is different to results of Porciani et al, (2001).
In our case, we have found no really strong correlation between the peaks and the proto-haloes in
the initial condition. Dynamical friction phenomena and existence of black holes do not participate
to our galaxy formation at redshift z=0.
Acknowledgments: The author would like to thanks Prof. A. Dekel and Prof. E. Rabinovici
for their encouragement and helps. We also thank A. Eldar and C. Porciani who collaborated
on part of this work and for fruitful discussions. Many thanks to the International Centre for
Theoretical Physics, ICTP, for support.
V. REFERENCES
C. Porciani, A. Dekel, Y. Hoffman, astro-ph/0105165.
Crittenden R. G., Natarajan P., Pen U., Theuns T., 2001, ApJ, in press, astro-ph/0009052.
Frenk C. S., White S., D., M., Efstathiou G., 1988, ApJ, 327, 507.
Heavens A., Peacock J. A., 1998, MNRAS, 232, 339.
Hoffman Y., 1986a, ApJ, 301, 65.
Hoffman Y., 1986b, ApJ, 308, 493.
Hoyle F., 1949, Burgers J. M., van de Hulst H. C., eds., in Problems of Cosmical Aerodynamics, Central
Air Documents Office, Dayton, p. 195.
Kauffmann G., Colberg J. M., Diaferio A., White S. D. M., 1999, MNRAS, 303, 188.
Lee J., Pen U., 2000, ApJ, 532, L5, (LP00).
Pearce F. R.., Couchman H. P. M., 1997, astro-ph/9703183.
Pearce F. R.., Couchman H. P. M., Jenkins A. R., Thomas P. A., 1995, Hydra-Resolving a Parallel Nightmare, in Dynamic Load Balancing on MPP systems.
Peebles P., J., E., 1969, ApJ, 155, 393.
Pen U., Lee J., & Seljak U., 2000, ApJ, 543, 107.
Steinmetz M., Bartelmann M., 1995, MNRAS, 272, 570.
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Galaxy Formation: Peaks Densities and Proto

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