Please consider a spheroidal 'Dark Matter' (DM) halo, of total mass M; characteristic scale radius R; and, a power-law DM 'particle' Initial Mass Function (IMF), i.e. [math]N(m) \propto m^{-\alpha}[/math] (cp. galaxy luminosity function, star IMF). We shall require that the DM IMF be 'steep', i.e. [math]\alpha > 2[/math] (see below).

 

 

Normalizations:

 

W.h.t.:

 

[math]M = \int_{m_0}^{\infty} m N(m) dm[/math]

 

[math] = C \times \int_{m_0}^{\infty} m^{1-\alpha} dm[/math]

 

[math] = C \times \frac{1}{\alpha - 2} \frac{1}{m_0^{\alpha-2}}[/math]

 

[math]\therefore C = (\alpha-2) M m_0^{\alpha-2}[/math]

Thus:

 

[math]N = \int_{m_0}^{\infty} N(m) dm[/math]

 

[math] = C \times \int_{m_0}^{\infty} m^{-\alpha} dm[/math]

 

[math] = C \times \frac{1}{\alpha - 1} \frac{1}{m_0^{\alpha-1}}[/math]

 

[math]\therefore N = \frac{\alpha-2}{\alpha-1} \frac{M}{m_0}[/math]

Assuming the DM halo is 'virialized', i.e. <K> = 1/2 <U>, then w.h.t.:

 

[math]K \equiv m \, v(m)^2 = constant = \frac{G M_{<} \, m}{r(m)}[/math]

Now, compared to a given DM 'particle' of mass m, orbiting through the halo at radius r(m), all more massive DM 'particles' will reside at r < r(m), i.e. orbit interior to the given DM 'particle' being considered. And, similarly, all less massive DM 'particles' will reside at r > r(m). Therefore,

 

[math] M_{<} = \int_{m}^{\infty} m' N(m') dm'[/math]

 

[math] = C \times \frac{1}{\alpha - 2} \frac{1}{m^{\alpha-2}}[/math]

 

[math] = M \left( \frac{m_0}{m} \right)^{\alpha-2}[/math]

So, w.h.t.:

 

[math]r(m) = \frac{G M m_0}{K} \left( \frac{m_0}{m} \right)^{\alpha-1}[/math]

And, w.h.t.:

 

[math]R \equiv \frac{\int r(m) N(m) dm}{\int N(m) dm}[/math]

 

[math]= \frac{\alpha-1}{\alpha-2} \frac{m_0}{M} \frac{G M m_0}{K} (\alpha-2) M m_0^{\alpha-2} \times \int \left( \frac{m_0}{m} \right)^{\alpha-1} m^{-\alpha} dm[/math]

 

[math]= (\alpha-1) \frac{G M m_0}{K} \times \int \left( \frac{m_0}{m} \right)^{\alpha-1} \left( \frac{m_0}{m} \right)^{\alpha} \frac{dm}{m_0}[/math]

 

[math]= (\alpha-1) \frac{G M m_0}{K} \times \frac{1}{2 \alpha -2}[/math]

 

[math] = \frac{G M m_0}{2 K}[/math]

So, w.h.t.:

 

[math]r(m) = 2 R \left( \frac{m_0}{m} \right)^{\alpha-1}[/math]

 

[math] \equiv 2 R \mu^{1 - \alpha}[/math]

where we have defined the normalized DM 'particle' mass [math]\mu \equiv m/m_0[/math].

 

 

DM radial density profile

 

W.h.t.:

 

[math]\rho_{DM}® \equiv \frac{m N(m) dm}{4 \pi r(m)^2 dr}[/math]

 

[math] = \frac{(\alpha - 2) M m_0^{\alpha - 2} m^{1 - \alpha} dm}{16 \pi R^2 \left( \frac{m_0}{m} \right)^{2 (\alpha-1)} dr}[/math]

 

[math] = \frac{(\alpha - 2) M \mu^{1 - \alpha} \mu^{2 \alpha - 2} d\mu}{16 \pi R^2 dr}[/math]

 

[math] = \frac{(\alpha - 2) M \mu^{ \alpha - 1} d\mu}{16 \pi R^2 dr}[/math]

Now, w.h.t.:

 

[math]\frac{dr}{d\mu} = 2 R (1-\alpha) \mu^{-\alpha}[/math]

And therefore w.h.t.:

 

[math]\rho® = \frac{ (\alpha - 2) M}{16 \pi R^2} \frac{ \mu^{ \alpha - 1} } {dr/d\mu}[/math]

 

[math] = \frac{ (\alpha - 2) M }{ 16 \pi R^2 } \frac{ \mu^{ \alpha - 1}} {2 R (\alpha - 1) \mu^{-\alpha}}[/math]

 

[math] = \frac{ \alpha - 2 }{ \alpha - 1 } \frac{ M }{ 32 \pi R^3 } \mu^{ 2 \alpha - 1}[/math]

 

[math] = \frac{ \alpha - 2 }{ \alpha - 1 } \frac{ M }{ 32 \pi R^3 } \left( \frac{r}{2 R} \right)^{ -\left( \frac{2 \alpha - 1}{\alpha - 1} \right) }[/math]

This calculated radial density profile approaches the canonical [math]\rho_{DM} \propto r^{-2}[/math] for 'ultra-steep' DM IMF, i.e. [math]\alpha \rightarrow \infty[/math]. Note that the canonical profile is that of the 'singular iso-thermal sphere' (SIS) solution.

 

 

Question ?

 

Might this imply, that the DM IMF is 'steep', i.e. low-mass biased, i.e. [math]m_{DM} \ll m_{*}[/math], e.g. [math]m_{DM} \approx 0.08 \, M_{\odot}[/math] ?

According to observations, e.g. Vera Rubin, DM halos have [math]\rho_{DM} \propto r^{-2}[/math] radial density profiles. According to simulations, e.g. NFW, DM halos have (at large radii) [math]\rho_{DM} \propto r^{-3}[/math]. The above 'toy model' can succinctly account, for this discrepancy, between observation & simulation-theory. For, from the above analysis, the average mass of a 'DM particle' depends upon the steepness, of the DM IMF:

 

[math]<m> \equiv \frac{N}{M} = \frac{\alpha -1}{\alpha-2} m_0[/math]

And, computer simulations, at cosmological scales, utilize DM particles that are tens or hundreds of thousands, even millions, of solar masses. That 'infinite mass' limit corresponds to a 'flat' IMF, i.e. [math]\alpha \rightarrow 2[/math], according to which the predicted 'toy model' radial density profile [math]\rightarrow \propto r^{-3}[/math]. Conversely, the observed [math]\propto r^{-2}[/math] profile, corresponds to a 'steep' IMF, i.e. [math]\alpha \rightarrow 4[/math], having low mass DM particles, [math]<m> \approx 0.1 M_{\odot}[/math].

Per PP, "smaller-than-simulated" mass DM 'particles' could account, for recent observations, of DM in dwarf galaxies, having 'flat cores', not 'steep cusps' (SD 2011).

Good end of day;

No need to understand 'Dark Matter' halos , at least for our galaxy and the Messier 33 galaxy,

because if we correct use the gravitationnal law, we do not need Dark Matter for explain

what we observe(galactic rotation curve). Outside of our galaxy and outside of some else other,

i do not know.

 

All calculate is only for galactic bulb and disk, and for variable density for the disk, then take

all this contributions .

 

I reply and démontrate it to bwalter1 for his topic: Dark matter and time?

Here his topic and my reply:

 

http://www.sciencefo...dark-matter-and-time/