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Astronomy & Astrophysics manuscript no. article   cESO 2014January 24, 2014

A new fitting-function to describe the time evolution of a galaxy’s

gravitational potential

Hans J.T. Buist and Amina Helmi

Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlandse-mail:  buist@astro.rug.nl

Received 15 November 2013  / Accepted 19 January 2014

ABSTRACT

We present a new simple functional form that may be used to model the evolution of a spherical mass distribution in a cosmologicalcontext. Two parameters control the growth of the system and this is modeled using a redshift dependent exponential for the scalemass and scale radius. In this new model, systems form inside out and the mass of a given shell can be set to never decrease, as

generally expected. This feature makes it more suitable for studying the smooth growth of galactic potentials or cosmological halosthan other parametrizations often used in the literature. This is further confirmed through a comparison to the growth of dark matterhalos in the Aquarius simulations.

Key words.   dark matter – galaxies: evolution

1. Introduction

The cosmological model predicts significant evolution in themass content of galaxies and of their dark matter halos throughcosmic time. This evolution may be directly measurable usingstellar streams, as these are typically sensitive probes of the grav-itational potential in which they are embedded  (Eyre & Binney

2009; Gómez & Helmi 2010). Clearly, to properly model thisdynamical evolution it is critical to use a physically motivatedrepresentation of the time-dependency of the host’s gravitationalpotential.

In the literature the time evolution of a galactic potential isoften parametrized through the evolution of its characteristic pa-rameters, such as total mass and scale radius. In the case of dark matter halos, the most often used parameters are its virial mass M vir  and concentration c. The virial mass is defined as the massenclosed within the virial radius  r vir, and at this radius the den-sity of the halo ρ(r vir)  = ∆× ρcrit, where ρcrit is the critical densityof the Universe, and the exact value of  ∆ depends on cosmology(Navarro et al. 1996, 1997). The evolution of  M vir   and   c   has

been thoroughly studied in cosmological numerical simulations(Bullock et al. 2001; Wechsler et al. 2002; Zhao et al. 2003b,a,2009;   Tasitsiomi et al. 2004; Boylan-Kolchin et al. 2010). Ithas been found that the evolution of the virial mass is well fit-ted with an exponential in redshift (but see also Tasitsiomi et al.2004; Boylan-Kolchin et al. 2010, for more complex functions),while the concentration appears to depend linearly on the expan-sion factor (Bullock et al. 2001; Wechsler et al. 2002), (see alsoZhao et al. 2003b,a, 2009). These relations work well in a statis-tical sense for an ensemble of halos, but do not always guaranteethat the evolution of an individual system is well representedand physical as we discuss below. Furthermore, it has recentlybeen pointed out that part of the evolution in mass, especially atlate times is driven by the definition of virial mass (through its

connection to the background cosmology) rather than to a trueincrease in the mass bound to the system (Diemer et al. 2013).

In this paper, we revisit the most widely used model for thetime evolution of dark matter halos (Sec. 2). Motivated by thefinding that for certain choices of the characteristic parameters,mass growth does not proceed inside out in this model, in Sec.  3we develop a new prescription for the time evolution of a gen-eral spherical mass distribution that does abide that property. InSec.   4  we compare this model to the growth of dark halos in

cosmological simulations and we summarize in Sec.  5.

2. Wechsler’s model of the evolution of NFW halos

The most commonly used approach to model the evolution of dark matter halos is that of  Wechsler et al. (2002). Cosmologicalsimulations show that halos follow characteristic density profilesknown as NFW after the seminal work of  Navarro et al. (1996).Generally, a two-parameter mass-profile may be expressed as

 M (r , t ) =  M s   f (r /r s) ,   (1)

with  r s(t ) the scale radius,  M s(t ) the mass contained within the

scale radius, and   f ( x) the functional form of the mass profile,with   f (1) =  1. The relative mass growth rate   ˙ M / M  is

˙ M 

 M (r , t ) =

˙ M s

 M s−

r s

r sκ (r /r s) ,   (2)

where κ ( x) is the logarithmic slope of the mass profile

κ ( x) =  d log  f 

d log x.   (3)

Typically κ ( x) is a positive monotonically decreasing function of radius. For the NFW profile,   f ( x) is given by

 f ( x)  =  A( x)

 A(1);   A( x) =  log(1 + x) −

 x

1 + x.   (4)

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The relation between  M s   and virial mass  M vir(t ) for an NFWprofile is

 M s  = M vir

 f (c)  ,   (5)

where the concentration  c  ≡   r vir/r s. As discussed in the Intro-duction, the virial radius is defined as the radius in which thespherically averaged density reaches a certain threshold value

 ρc. For a given choice of  ρc, the virial radius and the virial massare related through

 M vir  =  4

3πρcr 

3vir  .   (6)

Therefore, only two parameters from the set { M s, r s, M vir, c} areneeded to fully specify the profile. According to  Wechsler et al.(2002) the virial mass evolves as

 M vir( z)  =  M O exp [−2ac( z − zO)] ,   (7)

with  M O   =   M vir( zO) and  zO   the epoch where the halo is "ob-served". The formation epoch ac  is arbitrarily defined to be theexpansion factor at which  d log M /d log a   =   2.   Wechsler et al.

(2002) found the concentration to evolve on average as

c(a) =  4.1 a

ac.   (8)

Halos with more quiescent histories are best described by theserelations, while violent mergers can lead to significant departuresfrom these smooth functions.

The Wechsler model can give realistic trajectories in M vir(t )and   c(t ) for many individual halos. However, Fig.   2  shows aproblem case. In this figure we have plotted the behaviour pre-dicted for a halo of 1014 M and  ac  = 0.15. The left panel showsthe mass growth rate   ˙ M / M  as a function of time for diff erentshells in the halo. Note that the mass growth rate of a given shell

decreases with time, as expected, but that for inner shells it be-comes even negative, indicating that the mass in the shell hasdecreased. At later times, the mass growth for the inner shellsseems to increase and even take over the behaviour at larger radii,as also evidenced in the central panel of this figure.

Such behaviour is unexpected in the ΛCDM cosmology, ashalos tend to form inside out   (Helmi et al. 2003;  Wang et al.2011), implying that the inner shells collapse first and shouldnot grow further at late times by smooth accretion (naturally oneexpects a redistribution of mass during major mergers). Let usnow look in more detail at what causes this odd unphysical be-haviour.

From Eq. (2), the condition that leads to a violation of ˙ M / M  >  0 at a given radius is

˙ M 

 M =

˙ M s

 M s− κ ( x)

r s

r s< 0  ,   (9)

which we can express in terms of  M vir(t ) and c(t ) using that

˙ M s

 M s=

˙ M vir

 M vir

− κ (c)c

c(10)

and

r s

r s=

  r vir

r vir

−c

c=

˙ M vir

3 M vir

−˙ ρc

3 ρc

−c

c.   (11)

This leads to

˙ M  M 

=˙ M vir

 M vir

1 − κ ( x)

3

+

 cc

(κ ( x) − κ (c)) + κ ( x)

3˙ ρc

 ρc

.   (12)

Let us analize the various terms in this equation. Since for anNFW it can be shown that κ ( x)  ≤  2, and for the Wechsler model,the first term

˙ M vir

 M vir

1 −

κ ( x)

3

 =  −2ac˙ z

1 −

κ ( x)

3

 >  0  .   (13)

On the other hand, since κ ( x) is a monotonically decreasing func-

tion, the second term

c

c

κ ( x) − κ (c))  =  −

˙ z

1 + z

κ ( x) − κ (c)

 >  0 (r  <  r rvir) ,   (14)

Finally the last term in Eq. (12) is always negative as ˙ ρc/ρc  <  0.This implies that there may be times and radii for which thecombination of the various terms is negative, leading to a de-crease in the mass contained within a given shell. The aboveanalysis clearly shows that the reason that the Wechsler modelhas this behaviour is because of pseudoevolution: it is the factthat the virial mass is defined with respect to the cosmologicalbackground and that this background evolves in time.

3. Alternative model

We explore now a diff erent way of modeling the evolution of a mass profile, by directly focusing on the growth of the scaleradius  r s(t ) and the scale mass  M s(t ). This model must satisfythat the mass growth is non-negative at all times and at everyradius

dM 

dt ≥ 0,  ∀ r , t  ,   (15)

Also important is that the mass grows inside out, which meansthat the relative mass growth rate should increase with radius

∂

∂r 

  ˙ M 

 M 

 ≥  0,  ∀ r , t  .   (16)

Essential in the determination of whether these conditionsare satisfied is the logarithmic slope  κ ( x) in Eq.  (2). Becauseκ ( x) is a monotonically decreasing positive function, we can seta lower limit on the mass growth:

˙ M 

 M ≥

˙ M s

 M s− κ max

r s

r s,   (17)

where κ max  is the maximum value of  κ ( x). For NFW and Hern-quist profiles   κ max   =   2, for Jaff e profiles   κ max   =   1, while for

the Plummer and Isochrone potentials  κ max   =   3. The condition˙ M / M  =  0 would lead to a solution of the form

 M s(t ) =  M s,O

r s(t )

r s,O

κ max

,   (18)

with r s,O and  M s,O the scale mass and scale radius at some epoch zO. Motivated by this we propose that in general, a solution of Eq.( 2) should be of the form

 M s(t ) =  M s,O

r s(t )

r s,O

γ 

,   (19)

where γ > 0 is a constant. This functional form is also supported

by the work of  Zhao et al. (2003b,a), who using N-body cosmo-logical simulations have shown that a strong correlation between

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5 2 1 0.5 0.2

108

109

1010

1011

1012

 

ag  = 0.8;γ  = 2.0

r = 1 kpcr = 2 kpcr = 5 kpc

r = 10 kpcr = 20 kpcr = 50 kpc

0 4 8 12

108

109

1010

1011

1012

t  (Gyr)

    M            (   r ,    t            )            [    M     

            ]

ag  = 0.2;γ  = 2.0

5 2 1 0.5 0.2

z

 

ag  = 0.8;γ  = 4.0

Mvir

0 4 8 12

ag  = 0.2;γ  = 4.0

Fig. 3.   Mass growth history for various radial shells, normalized

to the final value as in Fig.   3,  for diff erent values of  ag   and  γ . Theformation epoch ag  changes the growth of the individual shells and theoverall growth pattern of the halo. This can also be seen in the half masstime of the shells indicated with a plus on every curve. The parameterγ   mostly influences how individual shells grow. This is seen as wechange γ  from 2 to 4 (left vs right panels)- only a small e ff ect is seen inthe overall growth of the halo, but the inner shells grow on much shortertimescales than the outer shells.

this slowing down is faster for the inner most shells. The halothus forms inside out as expected (central panel), and there isno mass exchange / decrease between neighbouring shells. In thisfigure we have also plotted the behaviour of  M vir(t ), determined

using Eqs. (1) and (6). Satisfyingly we note that the evolution of the virial mass for our model is very similar to that of  Wechsleret al. (2002).

In Fig.   3  we show the eff ect of changing the slope   γ   andthe formation epoch  ag. For example, for smaller  ag   (bottompanels) the mass growth occurs earlier, and the growth rate todayis lower. The parameter   γ   does not alter so much the overallevolution of the halo, but changes the (relative) growth patternfor the individual radial shells. For example, for   γ   =   2 (leftpanels) the inner shells grow much faster than the outer shells,while for   γ   =   4 (right panels) the diff erence in mass growthbetween the diff erent shells is much smaller.

4. Comparison with Simulations

To further justify our model, we study now how it fairs in de-scribing the behaviour of the Milky Way-like dark matter ha-los from the Aquarius project simulations (Springel et al. 2008).This is a suite of six cosmological dark matter N-body simula-tions ran at a variety of resolutions. Here we use an intermediateresolution level, in which the Aquarius Milky Way-like (or main)halos have ∼ 5×106 particles, and we focus on the behaviour Aq-A-4 to Aq-E-4 (the sixth halo, Aq-F experiences a major mergerat low redshift and hence is not considered in our analysis). Foreach main halo, we have computed the spherically averaged den-sity profile and fitted the NFW functional form with parameters

 M s and r s for output redshifts z  <  6 (this is to avoid the epoch of major merger activity).

In this way we determine the behaviour of  M s   and  r s   withredshift, to which we fit our model and determine the slope  ag

and γ/ag  in Eqs. (22) and (23).The mass growth history of the halos together with the re-

sults of our fitting procedure are shown in Fig.   4.   The valuesof   ag   for halos Aq-A and Aq-C are very low (0.05 and 0.04,respectively), implying an early formation, while Aq-B, Aq-Dand Aq-E having formed later, have larger  ag   (0.14, 0.10, and

0.09, respectively). This is consistent with the results of  Boylan-Kolchin et al. (2010), who studied the mass accretion history of Milky Way-mass halos in the Millenium-II cosmological simu-lations. These authors found that halo D and E follow the medianhistory, while halo A and C form earlier. Halo B was found tocatch up with the median growth around  z  ≈  2. In view of this,we may argue that a value  ag   ∼   0.1 roughly corresponds to themedian mass accretion history of Milky Way-mass halos.

Note that the values of the slope  γ  determined in our fits areclose to 2, the  κ max  for the NFW potential, but generally tend tobe slightly smaller, which would imply a mass decrease for someshells. Figure  4  shows that halo Aq-A depicts this behaviourearly in its history. Nonetheless, we find that for all halos, setting

γ   =  2 also produces a reasonable fit to the evolution history in M s   and  r s, as can be seen in from the gray dotted line in thisfigure.

In order to relate our results more closely to the  Wechsleret al. (2002) model, we determined the  ac  values for halos Aq-A to Aq-E, and found these to be in the range of 0.1 to 0.2,therefore appearing at the lower end of the distribution given byWechsler et al. in their fig. 8. This could be due to the morequiescent merger history of the Aq-halos.

5. Summary and Conclusions

We have studied the evolution of a spherical mass distribution ina cosmological context. Since we expect galaxies (and their ha-

los) to grow inside out, inner shells should form earlier than outerones, and in the smooth accretion regime, their mass should notdecrease with time. Motivated by violations of these conditionsfound in some models often used in the literature for certain re-gions of parameter space, we have presented an alternative wayof modelling the mass evolution of a spherical potential. In ourmodel, we let the mass at the scale radius and the scale radiusitself grow exponentially with redshift, but relate them in such away that the condition of no mass decrease at any radius can al-ways be satisfied. The setup is quite general, and can be appliedto any spherical density profile. It depends on two parameterswhich can be chosen to obtain a variety of growth histories andprovide more control on the growth rate of shells at di ff erent

radii.In comparison to previous work, our model does not diff erstrongly from that presented by Zhao et al. (2003b,a, 2009) whoalso use a power law relation between scale mass and scale ra-dius to describe the growth of dark matter halos in cosmologicalsimulations, except that we have found a general condition un-der which a system following a given density profile will growinside out.  Tasitsiomi et al. (2004) have found that some accre-tion histories were better fitted using a power law of the scalefactor a, instead of an exponential in redshift  z, and proposed amass accretion history which combines both the power-law rela-tion and the exponential behaviour. Boylan-Kolchin et al. (2010)proposed a slight modification to this model, which gives evenbetter fits to the MS-II simulations. The addition of an extra pa-

rameter allows the halos to grow earlier and reach a much lowergrowth rate at later times.

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