## Virial radius is defined. as the radius of a sphere that has a mean enclosed density of

180 times the mean density
3
23%
200 times the mean density
4
31%
180 times the critical density
2
15%
200 times the critical density
4
31%

Anze Slosar
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What should be the correct convention?
It makes about 50% difference for the two most extreme cases...

Gil Holder
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What happened to the old virial radius, defined through the spherical collapse model? That was the basis of the argument for using r_{180} for omega_m=1, but you end up with something close to 100 times the critical density for omega_m=0.3. It doesn't seem to have had much success in defining mass functions, but it always seemed like the one with the best physical foundation.

Tom Crawford
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As long as people agree on a consistent definition, the exact choice seems arbitrary --- until you want to compare with observations, at which point you want a quantity whose definition doesn't depend on other parameters (or depends on as few as possible). And if we're not going to tie the quantity directly to the spherical collapse overdensity (because that does vary with cosmology), then we might as well use a nice round number (rather than 178 or 180). Thus my vote for $200*\rho_c$.

Ilian Iliev
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Actually, the correct answer is probably "none of the above". The physically-meaningful mean density for halos is (roughly) proportional to the mean density at its time of formation. We have written some time ago a paper where we generalized the simple tophat model to something more physical, including centrally-concentrated density profile and dependence on Lambda (see http://adsabs.harvard.edu/cgi-bin/nph-b ... 4c09000496). This gave mean halo overdensity of ~130, weakly varying with Lambda. Numerical simulations also give more extended halos and lower mean density (see e.g. recent papers by Anatoly Klypin).

Sarah Bridle
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It is a bit confusing that these definitions are referred to as the "virial radius", when indeed Ilian and Tom point out the two opposing uses: "something related to virialisation" vs "a convenient definition of the mass of a halo".
I agree with Tom that it is handy to use a round number, and good if it doesn't change too much with cosmology.

Ilian Iliev
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The appropriate definition apparently depends on the problem. If one needs merely to locate a halo and give it some rough mass, then almost any resonable definition would do. For things that depend on the inner halo parts (e.g. X-ray emission) even higher overdensities would be approporiate. But if one wants to know what mass is virialized and the total mass of a halo, then the boundary moves quite a bit out and the overdensity is lower (maybe ~100 or even less, this is still not settled).

Sarah Bridle
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Sheth & Tormen give their mass function (astro-ph/9901122), but I can't see from their paper how they define the mass of a halo. Am I missing it?

Jenkins et al. mention "The mean overdensity was set to ... 324 in the ... LambdaCDM" which sounds promising, but still doesn't seem to contain all the information.
In particular, is this number relative to the critical density or to the mean density? (Should it be obvious? 324 seems v different to the numbers Ilian is giving..)

And do Sheth & Tormen define halo mass the same as Jenkins et al anyway? Thanks very much for any clues as I would like to be able to use the Sheth & Tormen mass function correctly!

Pirin Erdogdu
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Sheth and Tormen (1999) paper defines the critical value of the initial overdensity as $\delta_c=1.68647$ (page 3, equations 9 and 10). Thus, their $\delta_{vir}$ must be $\approx$180-200 (mean or critical density, it does not matter since this value is for an Einstein-de Sitter universe). Or?

Ilian Iliev
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This is a rather complex question, which is often glossed over and not really discussed much. The definition of what is a halo in N-body simulations depends on the halo finder used. Two popular ones are the friends-of-friends (FOF) and spherical overdensity, but there are also others. Each of these has free parameters, the linking length in the first case and the overdensity in the second. The overdensity is of course what we are discussing here, while the linking length (usually chosen to be 0.2) is only loosely connected to a particular overdensity. The value of the free parameter is chosen rather empirically so as to give reasonable results for the halos. The halo mass depends on the value chosen, so there is obviously a certain freedom here. It should also be noted that while these methods agree fairly well neither method finds the halos perfectly and each has its difficulties in some situations.

If I am not mistaken, the Sheth and Tormen model does not point to a particular overdensity, but is just calibrated against numerical simulations, so has much the same freedoms the N-body simulations have.

A note regarding the previous post - the critical density &#948;c is not an initial overdensity, but is the value of the linear-theory overdensity reached when the nonlinear top-hat pertubation collapses to infinite density. The value &#948;c=1.68 is correct for Einstein-de Sitter model, while for LambdaCDM it is slightly lower.

Pirin Erdogdu
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Sorry for being a bit unclear in my previous post.

If I am not mistaken, the Sheth and Tormen model does not point to a particular overdensity, but is just calibrated against numerical simulations, so has much the same freedoms the N-body simulations have.
Sheth and Tormen (ST99, 1999) find a fit to $\nu f(\nu)$ and $\nu\equiv[\delta_c(z)/\sigma(m)]^2$
where (I quote from their paper) $\delta_c(z)$ is the critical value of the initial overdensity which is required to collapse at $z$, computed using the spherical collapse model and $\sigma(m)$ is the value of the rms fluctuation in spheres which on average contain mass $m$ at inital time, extrapolated using linear theory to $z$.

In section 3 of ST99, it reads:
We measure mass functions in the simulations in the usual way, using spherical overdensity group finder (see Tormen 1998 for details).
And in Section 2.2 of Tormen (1998, MNRAS, 297, 648), it reads:
We defined [dark matter] lumps by an overdensity criterion and included all particles within a sphere of mean overdensity $\delta_v=178$, centred on the particle with lowest potential energy. The value $\delta_v$ corresponds to the viral overdensity in the model of a spherical top-hat collapse in an Einstein- de Sitter universe.
It seems to me that they have a particular virial radius for the simulations ($\delta_v=178$). Any thoughts on this would be much appreciated.

Ilian Iliev
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The value 178 (or more precisely 18$\pi^2$) is the standard value coming from the classic top-hat model. As far as I am aware, simulations do not point specifically to this value. In the spherical overdensity halo finder it is a free parameter, which could be varied (within some limits). If one picks 200 or 130, this would be just as reasonable. As I mentioned before, some newer simulations that were specifically looking for the halo boundaries (virialization shocks in gas/caustics in dark matter) were finding rather smaller overdensities, maybe ~100 or less.

Sarah Bridle
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Thanks very much indeed for your replies. To be honest, at the moment I'm not too bothered about the definition of the virial radius (as defined by turn-around), but just want to know what "mass" is in the mass function I get from the Sheth-Tormen equations (i.e. once they had selected a halo in their simulations, out to what radius did they sum up their mass?).

Thanks for the Tormen (astro-ph/9802290) ref, which suggests use of 178 in the EdS simulations in that paper. Does anyone know if 178 was also used in Sheth & Tormen?

If indeed Sheth & Tormen use 178 then they must have had to make a decision when applying this to the non-EdS simulations they use in their paper: is this 178 relative to the mean density at the halo redshift, or the critical density at the halo redshift?

Thanks very much for any more comments.

Anne Green
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I'm not an expert, but I think that mass is usually taken to be the mass within the virial radius, so it does boil down to what definition is used for the virial radius.

Bryan and Norman (astro-ph/9710107) have fitting functions (which are often cited) for the dependence of the virial over density, defined in terms of the collapse of a spherical perturbation, on cosmology (eq. 6).

A related problem that's been bothering me lately is the time dependence of a halo's properties (radius and mass) if you use the "standard" overdensity of 200; as the Universe evolves the background density
(total or matter) decreases so that the virial radius, and hence the nominal mass, of a given halo increases even if it remains physically unchanged. I'd be interested to know if anyone has any thoughts on this problem.

Niayesh Afshordi
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A related problem that's been bothering me lately is the time dependence of a halo's properties (radius and mass) if you use the "standard" overdensity of 200; as the Universe evolves the background density
(total or matter) decreases so that the virial radius, and hence the nominal mass, of a given halo increases even if it remains physically unchanged. I'd be interested to know if anyone has any thoughts on this problem.

This is correct. At the same time the NFW concentration parameter increases, so that the profile remains the same, although the mass is increasing. This is also why smaller haloes that form earlier have a larger concentration parameter.

Anne Green
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