estimating IGM density from cooling curve (?)

[Widdekind]

For T < 10⁴ K, the cooling curve of the ISM scales as ~T⁷ (Gaetz & Saltpeter 1983):

 

P

[ergs cm3 s-1]

= -10

^-23

x n

_h

x n

_e

x (T/10

⁴

K)

⁷

And, since z ~ 2, the IGM has cooled from 10,000 to 4,000 K:

 

From

z ~ 6 to 2

, Cluster gas continued to influence the IGM, maintaining its temperature through outflows. After

z ~ 2

, however, Clusters completely decoupled from the Hubble Flow, continuing to contract, compress, & heat Cluster gas. Meanwhile, the now isolated IGM, in the receding voids, began to cool radiatively. (Source:

Majestic Universe

[scientific American special report], pg. 8.)

So, we seek to calculate the cooling, of the IGM, as a function of time, in a Critical, matter-dominated cosmos. Consider, then, a comoving volume, expanding with the stretching of spacetime, having a volume V(t) = R(t)³.

 

 

 

Order-of-Magnitude Calculation

 

In the comoving volume, where the matter only loses energy, to radiation, through aforestated cooling, w.h.t.:

 

[math]E \approx \frac{\rho}{m_H} k T \times V(t) \approx \frac{M_{tot}}{m_H} k T[/math]

[ergs]

 

[math]\frac{dE}{dt} \approx -10^{-23} \left( \frac{\rho}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t) \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t)^{-1}[/math]

[ergs s

^-1

]

 

[math]\therefore \frac{dE}{dt} \approx \frac{M_{tot}}{m_H} k \frac{dT}{dt} \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right)^2 \left( \frac{T}{T_0} \right)^7 \times V(t)^{-1}[/math]

 

[math]k \frac{dT}{dt} \approx -10^{-23} \left( \frac{M_{tot}}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \frac{1}{R_0^3} \left( \frac{R_0}{R(t)} \right)^3 \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \left( \frac{R_0}{R(t)} \right)^3[/math]

Then, since we assume matter-dominated cosmology, R(t) ~ t^2/3:

 

[math]\therefore k \frac{dT}{dt} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( \frac{T}{T_0} \right)^7 \times \left( \frac{t_0}{t} \right)^2[/math]

Defining normalized variables:

 

[math]\therefore \frac{k T_0}{t_0} \frac{d\tau}{dx} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \tau^7 \times x^{-2}[/math]

This is easily integrated:

 

[math]\frac{k T_0}{t_0} \int_1^{0.4} \frac{d\tau}{\tau^7} \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \int_{z=2}^1 \frac{dx}{x^2}[/math]

 

[math]\frac{k T_0}{6 t_0} \left( 0.4^{-6} - 1 \right) \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 1 - x^{-1}|_{z=2} \right)[/math]

But, R(t) ~ (1+z)^-1, which, when combined with the above, gives x ~ (1+z)^-3/2:

 

[math]\frac{k T_0}{6 t_0} \left( 0.4^{-6} - 1 \right) \approx -10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 1 - 3^{3/2} \right) \approx 10^{-23} \left( \frac{\rho_0}{m_H} \right) \left( 3^{3/2} - 1 \right)[/math]

And, letting [math]\rho_0 \equiv \alpha \rho_{crit}[/math], w.h.t.:

 

[math]\alpha \approx 10^{23} \frac{k T_0}{6 t_0} \frac{m_H}{\rho_{crit}} \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1}[/math]

Now, for a Critical cosmology, t₀ = (2/3) H₀^-1. So, using the usual formula, for the Critical Density, w.h.t.:

 

[math]\therefore \alpha \approx 10^{23} \frac{m_H k T_0}{4} \frac{8 \pi G}{3 H_0} \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1} \approx 10^{23} \left( m_H k T_0 \right) \left( \frac{2 \pi G}{3 H_0}\right) \left( \frac{ 0.4^{-6} - 1 }{3^{3/2} - 1}\right) \approx 0.80 \, h_{72}^{-1}[/math]

According to this calculation, and assuming a matter-dominated Critical cosmology, the IGM has been cooling, since z ~ 2, from roughly 2 to 9 Gyr, as if its density were comparable to Critical. Thus, such a cosmology could account, for the cooling, observed in the IGM (which will soon undergo a "second Recombination" event, perhaps w/in roughly +1 Gyr +10 Tyr).

 

 

 

Numerical Factors

 

For a multi-component gas, the energy density is E = (3/2) n_tot k T. And, for 4,000 K < T < 10,000 K, and at ultra-low IGM gas pressures, Hydrogen is almost entirely ionized, whereas Helium is almost entirely neutral. So, ignoring the low IGM metallicity (Z ~ 0.007), w.h.t.:

 

[math]n_{tot} = n_H + n_{He} + n_Z + n_e \approx n_H + n_{He} + n_H = 2 n_H + n_{He} \approx \rho_{tot} \times \left( 2 \frac{X}{m_H} + \frac{Y}{4 m_H} \right)[/math]

Using canonical values, for the mass-fractions, X = 3/4 & Y = 1/4, w.h.t. [math]n_{tot} \approx \frac{\rho_{tot}}{m_H} \times 25/16[/math]. So, on the LHS, we have omitted numerical factors, of order unity, but specifically equal to 3/2 x 25/16 = 75/32. (Physically, hot, Hydrogen-ionized IGM, has ~2.5x more energy content, than was recognized above, due to having three degrees-of-freedom, and all of the extra Hydrogen-ionized electrons.)

 

Likewise, on the RHS, since all the radiative cooling comes from the (near-)fully ionized Hydrogen, recombining with its own electrons, we require the product of:

 

[math]n_H n_e \approx n_H^2 \approx \left( \frac{X \rho_{tot}}{m_H} \right)^2[/math]

So, on the RHS, we have omitted numerical factors, of order unity, but specifically equal to X² = 9/16. (Physically, hot, Hydrogen-ionized IGM, has only ~0.5x the cooling capacity, compared to what was recognized above, mainly due to all of the non-radiating mass 'locked away' in Helium.)

 

And, so, our original, order-of-magnitude estimate, for the fraction of Critical density in gas [math]\alpha[/math], we neglected to multiply by the ratio, of "LHS / RHS" = 75/32 x 16/9 = 75/18 ~ 4. (Physically, hot, Hydrogen-ionized IGM, has more energy content, and less cooling capacity, than was recognized above -- and, so, must be much more dense, to decrease its temperature, at the same rate.) Thus, more accurate physics seemingly suggests more strongly, that a matter-dominated Critical cosmology can account, or is even required to account, for observed IGM cooling, since z ~ 2 (~7 Gyr ?).

 

 

 

second Recombination

 

Sticking with the same physics, the IGM will cool to 3000 K ([math]\tau_f = 0.3[/math]), at some future time, when the relative age of the universe is [math]x = 1 + \delta x[/math]:

 

[math] 0.3^{-6} - 1 = 3^{3/2} - (1 + \delta x)[/math]

 

[math]\delta x = 0.3^{-6} - 3^{3/2} \approx 1400[/math]

Thus, given the expansion of space-time, and the IGM's already-quasi-cold condition, the IGM is could require trillions of years, to cool back down to 3000 K.

Edited May 31, 2011 by Widdekind