Wednesday, December 19, 2012

virial theorem in clusters

what would be the virial theorem for a different gravitational law, i.e. $\Phi\propto 1/r^n$ instead of the Newtonian potential? Could you explain Zwicky's conjecture for the presence of dark matter in a galaxy cluster just by postulating a different value for $n$ in the potential while assuming that only light-producing matter is gravitating, and what would that slope need to be?

bonus question: why does the covariance matrix $C$ of a multivariate Gaussian probability density
\begin{equation}
p(\vec{x})\mathrm{d}\vec{x} = \frac{1}{\sqrt{(2\pi)^n\mathrm{det}(C)}}\exp\left(-\frac{1}{2}\vec{x}^tC^{-1}\vec{x}\right)\mathrm{d}\vec{x}
\end{equation}
have to be positive definite (2 reasons)?

fun bonus question: aren't the factorials somehow the opposites of prime numbers?

with that, CQW wishes all readers a happy holiday season and a successful year full of new discoveries. we'll be on vacation until 09.Jan.2013.

Wednesday, December 12, 2012

peculiar motion and the CMB

the CMB-dipole is usually interpreted as motion dipole, i.e. generated by the earth's movement relative to the CMB and would in this interpretation not contain any cosmological information. are there other imprints in the CMB due to peculiar motion of the observer?

bonus question: how large would seasonal variations of the CMB motion dipole be?

bonus question 2: can you estimate the order of magnitude of $(10^{10}!)$?

Wednesday, December 5, 2012

inflationary dynamics

CQW celebrates its first anniversary (with 10.000+ views, 130 followers on twitter and 100+ likes on facebook - thank you very much, cosmologists!) with a guest post by Shaun Hotchkiss (from the blog trenches of discovery):

the amplitude of the power spectrum of primordial fluctuations arising from inflation is proportional to $1/\dot{\phi}^2$. Therefore, as $\dot{\phi}$ decreases, the power spectrum increases. However, in the limit of a massless scalar field (i.e. a perfectly flat potential), $V(\phi)$ is constant and thus so is the energy density. What reconciles this apparent problem?

bonus question: can you show that $\Delta \phi = 0$ is solved by the Newton-potential and $(\Delta+m^2)\phi = 0$ is solved by the Yukawa-potential (both in 3d spherical coordinates)? What's the length scale of the Yukawa-potential in this example? why is there no length scale in the Newton-potential? why are both solutions isotropic?

please take part in the poll asking for the cosmological reason of the apocalypse in two weeks.