Science > Physics > Quantum Gravity Via Expansion-Contraction 20.4: Curvaton Non-Gaussianity in Primordial Perturbation
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Science > Physics |
| User: |
"OsherD" |
| Date: |
04 Sep 2006 09:55:33 AM |
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Quantum Gravity Via Expansion-Contraction 20.4: Curvaton Non-Gaussianity in Primordial Perturbation |
From Osher Doctorow
Misao Sasaki of Kyoto U. (Yukawa Institute for Theoretical Physics) and
Jussi Valiviita and David Wands of U. Portsmouth U.K. (Institute of
Cosmology and Gravitation) in "Non-Gaussianity of the primordial
perturbation in the curvaton model," astro-ph/0607627 v2 31 Aug 2006,
in their 16-page paper, calculate the Non-Gaussian (I capitalize N here
for emphasis) probability density function (pdf) of the primordial
curvature perturbation in the curvaton model.
Probable Influence/Causation (PI) is somewhat parallel to Shannon
entropy/information in regarding the Uniform and Exponential
probability distributions as more important than the Gaussian
("Normal") distribution. Actually, PI theory (or PI for short)
generalizes Shannon's results for continuous pdfs in the following way.
The Uniform pdf is a finite interval pdf (the pdf is 0 outside a
finite interval), and PI finds that all finite interval pdfs are
"maximum PDF-entropy" for all parameters (in this case, endpoints of
the interval) or characterizing constants unknown. The Exponential pdf
is asymmetric and restricted (nonzero) to the nonnegative real line,
and PI finds that all nonnegative real line (and thus asymmetric) pdfs
are "maximum PDF-entropy" for one unknown parameter. The
Normal/Gaussian pdf is symmetric on the whole real line (its graph
looks the same except for reflection on both sides of its maximum or
center) and PI finds that all symmetric real line pdfs are "maximum
PDF-entropy" for the case of no unknown parameters out of 2 (that is to
say, both characterizing constants of the pdf, such as the population
mean and variance, are known). But there is more
"Knowledge/Information" in unknown parameters, so the Normal/Gaussian
pdf has the lowest ranking of the "maximum PDF-entropy" pdfs, although
it is still better than remaining types of pdfs.
The paper by Sasaki et al (31 Aug 2006) uses the delta(N) formalism
which was also used in the paper by Sasaki and others cited in the
previous Section of this thread, and I'll try to discuss the new paper
more next time.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 20.4: Curvaton Non-Gaussianity in Primordial Perturbation |
04 Sep 2006 10:14:35 AM |
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From Osher Doctorow
Sasaki et al (31 Aug 2006) point out that most inflationary models
yield a nearly Gaussian distribution of primordial curvature
perturbations, with deviations from Gaussianity (from a Gaussian
distirbution) given in terms of nonlinearity parameter f_NL. The
nonlinearity is difficult to observe in single-field models of
inflation, but not in multi-field models of inflation. In the latter
scenario (multi-field), there is not only an inflation phi but also a
second weakly coupled light scalar field, the curvaton chi that was
"completely subdominant" (small) during inflation. In fact, they give
a simple example of the potential for the curvaton:
1) V = (1/2)M^2 phi^2 + (1/2)m^2 chi^2
with the energy density of phi driving inflation. At Hubble exit
during inflation, phi and chi acquire some "frozen in" classical
perturbations, but the observed cosmic microwave (CMB) and large-scale
structure (LSS) perturbations come from the curvaton instead of the
inflation. Here M is the inflaton mass, m the curvaton mass.
After the end of inflation, the inflaton decays into radiation
(relativistic particles) and the curvaton decays into radiation before
primordial nucleosynthesis.
The Hubble rate H decreases in time after inflation, eventually
becoming less than or equal to m in absolute value, at which time the
curvaton strats to oscillate about its minimum potential and then
behaves like pressureless dust with density inversely proportional to
volume and its energy density growing with respect to radiation
(proportional to or of the order of a^(-3) in the density and a^(-4) in
radiation). After this, the curvaton decays into ultra-relativistic
particles which leads to the radiation-dominated (adiabatic primordial
perturbations) era, which however creates a strongly Non-Gaussian
primordial curvature perturbation from an initially Gaussian one,
especially if the energy density of the curvaton is sub-dominant when
the curvaton decays.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Quantum Gravity Via Expansion-Contraction 20.4: Curvaton Non-Gaussianity in Primordial Perturbation |
04 Sep 2006 10:29:37 AM |
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From Osher Doctorow
Sasaki et al (31 Aug 2006) solve for the full Non-Gaussian primordial
curvature perturbation pdf in their Appendix. The pdf is the sum of
two terms, but in practice one term can be neglected, and the final
result is a product of of two quantities:
1) kuv where u and v are weighted means of exp(3w) and exp(-w) with w
the argument of the pdf, k being the type of constant found in the
Gaussian pdf
2) a Gaussian-like factor modified to include inside the exponential a
k2uv type expression like (1) but with different weights, where k2 is a
different constant
Osher Doctorow
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