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trying to understand Primordial Nucleosynthesis?


Widdekind
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Soon after the Big Bang, light elements were synthesized, from primordial protons & neutrons, by "neutron capture" reactions. According to BBN theory:

 

As the universe expanded, it cooled. Free neutrons and protons are less stable than helium nuclei, and the protons and neutrons have a strong tendency to form helium-4 [via] the intermediate step of forming deuterium. At the time at which nucleosynthesis occured, the temperature was high enough for the mean energy per particle to be greater than the binding energy of deuterium; therefore any deuterium that was formed was immediately destroyed (a situation known as the deuterium bottleneck). Hence, the formation of helium-4 is delayed, until the universe becomes cool enough to form deuterium (at about T = 0.1 MeV [= 1 GK]), when there is a sudden burst of element formation. Shortly thereafter...the universe becomes too cool for any nuclear fusion to occur. At this point, the elemental abundances were fixed, and only change as some of the radioactive products of BBN (such as tritium) decayed...

 

Deuterium is in some ways the opposite of helium-4 in that while helium-4 is very stable and very difficult to destroy, deuterium is only marginally stable and easy to destroy. Because helium-4 is very stable, there is a strong tendency on the part of two deuterium nuclei to combine to form helium-4. The only reason BBN did not convert all of the deuterium in the universe to helium-4 is that the expansion of the universe cooled the universe, and cut this conversion short, before it could be completed. One consequence of this is that unlike helium-4, the amount of deuterium is very sensitive to initial conditions. The denser the universe is, the more deuterium gets converted to helium-4 before time runs out, and the less deuterium remains.

According to direct observations of space, there are (relatively) high levels of surviving D; and (relatively) low levels of Li:

 

BBNeta.gif

Thus, the following nuclear reactions occurred:

 

s197_1_e012i.small.jpg

but not:

 

s197_1_e013i.small.jpg

 

 

QUESTIONS:

 

Qualitatively, is the Lawson criterion applicable, i.e. the amount of fusion that occurred, when the primordial plasma was "singed" in BBN, was proportional, to the product, of baryon density during BBN, multiplied by the time of BBN, i.e. [math]n \tau[/math] ? (I understand, that the temperature regime, is plausibly constrained, 300 MK < T < 1000 MK.)

 

And so, if the baryon density was higher, then the time for fusion must have been lower ??

 

Could "inflation", or something similar, have "stretched out space", more swiftly than currently conceived, so that a hypothetically higher baryon density, e.g. [math]\rho \rightarrow \rho_c[/math], would have had less time to "cook" ?

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According to the Scientific American article Solving the Solar Neutrino Problem (to point to a specific, concrete, reference), neutrinos can affect crucial nuclear reactions, e.g. breaking apart D nuclei:

 

In deuteron breakup, the neutrino splits a deuterium nucleus, into its component proton, and neutron

Now, before BBN began, "hot" photons broke apart D nuclei, until T < 0.1 MeV, thereby delaying BBN. I understand, that in a plot of theoretical BBN elemental abundances, versus the baryon-to-photon ratio [math]\eta[/math], that

 

[math]\eta \leftrightarrow \rho_B \sim \rho_B \left( T_{BBN} \; \tau_{BBN} \right) \propto \eta = const.[/math]

i.e. the "Lawson triple product", where the temperature, and duration, of BBN are assumed, by assuming that BBN occurred, in a radiation-dominated epoch, i.e. a(t) = a0 t1/2; and by assuming that the temperature was dominated by red-shifting photons T(t) ~ a(t)-1. Thus, the "Lawson's triple product"

 

[math]\rho_B \; T_{BBN} \; \tau_{BBN} \sim \frac{1}{a^3} \frac{1}{a} a^2 = a^{-2}[/math]

Naively, and qualitatively, I understand, that if there were more baryons, i.e. [math]\rho_B \rightarrow \rho_c[/math]; then, to be consistent with observations (of abundances) & calculations (of [math]\eta[/math]), those baryons must have undergone BBN, later, for less time, at colder temperatures, after our universe had expanded more.

 

Could "deuterium breakup", caused by neutrinos, have further delayed BBN, so that instead of BBN occurring from +100-1000s, at temperatures 0.1-0.03 MeV, BBN occurred only later, for less time, in colder conditions ?? Naively,

 

[math]\Lambda = \int d\Lambda = \int \rho T dt = 2 \rho_0 T_0 t_0 \int a^{-3} a^{-1} a da = \rho_0 T_0 t_0 \left(\frac{1}{a_i^2} - \frac{1}{a_f^2} \right) \propto \eta = const.[/math]

I understand, that the only "free parameters" (without invoking new physics), are [math]\rho_0[/math] & [math]a_i[/math], i.e. comparing various scenarios,

 

[math]\rho_0 \left(\frac{1}{a_i^2} - \frac{1}{a_f^2} \right) = \rho_0' \left(\frac{1}{a_i'^2} - \frac{1}{a_f^2} \right)[/math]

 

[math]\left( \frac{1}{x^2} - 1 \right) = \frac{\rho_0'}{\rho_0} \left( \frac{1}{x'^2} - 1 \right)[/math]

where [math]x \equiv a_i / a_f \approx 1/3[/math] from the temperature data. If so, then "demanding" that the baryon density be ~20x higher, i.e. close to critical density, would "require":

 

[math]10 \approx 20 \left( \frac{1}{x'^2} - 1 \right)[/math]

 

[math]\frac{3}{2} \approx \frac{1}{x'^2}[/math]

 

[math]x' \approx \frac{4}{5}[/math]

i.e. BBN could only have occurred, from T = 0.04-0.03 MeV, corresponding to t = 700-1000s, i.e. a delay of ~10 minutes. Could a dense "fog" of neutrinos, breaking up deuterium nuclei, have accomplished such a delay ??

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i.e. BBN could only have occurred, from T = 0.04-0.03 MeV, corresponding to t = 700-1000s, i.e. a delay of ~10 minutes. Could a dense "fog" of neutrinos, breaking up deuterium nuclei, have accomplished such a delay ??

 

I have constructed a model of inflationary big bang expansion of the universe from simple assumptions similar to Alan Guth's model. But, he does't provide a model that stretches all the way from the first plausible and treatable instant to the present and beyond as my model does. And, by means of a single adjustable parameter, I can mimic acceleration, steady expansion or deceleration.

 

In the case of acceleration, my model naturally passes through a short pause of perhaps about 100 - 200 minutes or maybe more, not exactly as your calculation suggests, but the duration of this period is subject to the choice of somewhat arbitrary initial conditions. This period also provides time for equilibration of temperature and density differences in the inflationary expansion. Because the universe was still very small, when all the regions or arbitrarily small parcels were still in causal contact, this pause is essential to the inflationary scenario. This pause is necessary and most fortunate because this simple model does not provide for a lag or "induction" period initially, as Guth assumed, even though the model is a fully exponential growth process.

 

When I saw this expansion rate curve come down from a maximum, turn right around and then go through a minimum and turning around again to resume its upward trajectory, I was amazed that such a simple equation could produce such odd behavior. I wondered if there was any consequence other than to offer a time period long enough to allow equilibration. Now, I see that it may have been crucial to BBN.

 

My model is posted on my website, www.lonetree-pictures.net , but this site is temporarily down because the index.htm main page has become corrupt and I cannot access it until I figure a way to replace the bad code on this page. In addition, my website host has issues recognizing my password for uploading. So, it is difficult to replace the corrupt page. But, I can post the details on my blog http://neocosmology.blogspot.com . So, this has just become a priority on my agenda. As I recall, I posted it there a long time ago, so if you check older posts, it may show up. In the meantime, I have posted images on FotoThing.com: http://www.fotothing.com/Gak/ , images 94 to 96.

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