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.

Wednesday, November 28, 2012

Hubble expansion and superluminality

why do we observe light from galaxies if their recession velocity exceeds the speed of light? at what redshift in a $\Lambda$CDM cosmology would a galaxy move as fast as light? why isn't that a problem with special relativity?

bonus question: can you show the property $\Gamma(x+1) = x\Gamma(x)$ of the $\Gamma$-function which is defined by $\Gamma(x)\equiv \int_0^\infty\mathrm{d}t\:t^{x-1}\mathrm{e}^{-t}$?

Wednesday, November 21, 2012

redundancy in the Friedmann equations

under which circumstances is the $\dot{a}$-Friedmann equation the integral of the $\ddot{a}$-equation? what acts as the integration constant? why would that particular variable not be able to introduce de/ac-celeration?

bonus question: why is (in contrast to real analysis) the derivative of a complex differentiable function always again complex differentiable?

Wednesday, November 14, 2012

mechanical similarity and planetary orbits

what would be Kepler's law of planetary motion (the cube of the orbit's major axis is proportional to the square of the orbital period) if the Newtonian gravitational potential was $\propto 1/r^2$ instead of $\propto 1/r$? can you generalise Kepler's law to arbitrary integer power laws for the potential, $\propto 1/r^n$?

would the two other laws by Kepler be affected (elliptical orbits and angular momentum conservation)?

bonus question: what's $\int_0^\infty\mathrm{d}x\:x^3\exp(-\alpha x^2)$?

Wednesday, November 7, 2012

cold core (aka cooling flow)

this week's question was suggested by R. Schmidt (a fellow cosmologist):
$X$-ray observations suggest that the centres of clusters of galaxies are sometimes colder than the rest of the cluster. what's the physical reason for these cold cores? Would cold cores show up in the thermal Sunyaev-Zel'dovich effect?

bonus question: can you show that $\ln\det(A) = \mathrm{tr}\ln(A)$, or equivalently $\exp\mathrm{tr}(A) = \mathrm{det}\exp(A)$ for any non-singular square matrix A?

bonus question 2: what do you think, do you have more or less friends than your friends have on average?

Wednesday, October 31, 2012

gravitationally bound systems

imagine you've assembled a self-gravitating bound equilibrated system from $n$ particles of mass $m$, such that the total mass is $M=nm$ and such that the system has a gravitational binding energy $E$. now you reassemble the system from $p$ times more particles of masses $m/p$ so that the total mass would be identical, $M = np \times m/p=nm$. would the gravitational binding still be $E$?

bonus question: why are the traces $\mathrm{tr}(A^n)$ of a square matrix $A$ taken to the power $n$ invariants? how are they related to the eigenvalues of $A$? would that be a viable way for computing the eigenvalues?

Wednesday, October 24, 2012

CMB and SNe

are cosmic microwave background measurements enough to claim spatial flatness and supernova measurements sufficient for claiming accelerated expansion of a FLRW-metric?

bonus question: if a set contains $n$ elements, how many subsets can you form? can you prove this relation with combinatorial arguments and with induction?

Wednesday, October 17, 2012

gravitational interaction of photons

why do gravitational interactions of (CMB) photons with the cosmic large-scale structure such as gravitational lensing or the integrated Sachs-Wolfe effect conserve the Planckian shape of the photon energy distribution?

bonus question: what's $\sum_{i=0}^n{n\choose i}$ with the binomial coefficient ${n\choose i}=n!/i!/(n-i)!$?

Wednesday, October 10, 2012

dark energy equation of state

why can the equation of state $w$ of dark energy in a scalar field model never be larger than $+1$ and never be more negative than $-1$? are there physical arguments why this would be weird? what would happen to distance measures and to structure growth if $w$ were outside this range?

bonus question: does the Higgs-mechanism in quantum field theory give rise to mass or to inertia?

second bonus question: why is dark matter called dark matter if it does not interact with light? if that was the reason, it should rather be called transparent matter, don't you agree?

Wednesday, October 3, 2012

gravity and dimensionality

would the description of gravity as a metric theory work in any number of dimensions?

bonus question: can you show without the rule of de l'Hospital or Taylor expansion that $\sin(x)/x=1$ for $x\rightarrow0$?

Wednesday, September 26, 2012

uncertainty and inflation

all fields associated with elementary particles fluctuate on the time scale $\Delta t$ by energies $\Delta E$ which can be estimated to be $>\hbar / \Delta t$ according to the uncertainty principle. Why isn't then the inflation field generating inflationary epochs every time (i.e. nowadays too!) if it acquires enough energy by statistical fluctuation?

Wednesday, September 19, 2012

planetary atmospheres

why don't planets lose their atmospheres to outer space? one would assume that the Boltzmann distribution would always allow a certain fraction of molecules to have kinetic energies large enough to overcome the gravitational potential.

bonus question: wouldn't the same thing apply to other gravitationally bound equilibrated systems such as open clusters, globular clusters and galaxies?

Wednesday, September 12, 2012

Copernican principle

what ways are there for testing the isotropy and homogeneity of the Universe on large scales? to what extend are those symmetry assumption tested and how would a violation impact on FLRW-models?

Wednesday, September 5, 2012

atmospheric pressure on Venus

the atmospheric pressure on the surface of Venus is almost 100 times larger than on Earth, despite the fact that the gravitational acceleration is 10% less. how is that possible?

bonus question: why is the pressure on Mars so low? (actual numbers are $p=6\times10^{-3}$ bar at a third of Earth's gravity)

Wednesday, July 11, 2012

Sunyaev-Zel'dovich effect

the thermal Sunyaev-Zel'dovich effect is usuall derived for Compton-interactions of the CMB photons with a thermal gas of electrons - is the assumption of thermality necessary?

bonus: what's happening if the electrons become relativistic?

extra bonus: can you express $i^{+i}\pm i^{-i}$ in terms of trigonometric functions?

Wednesday, July 4, 2012

non-trivial topology of the Universe

this week we have a guest post by Shaun Hotchkiss
(a fellow cosmologist and blogger at trenches of discovery)

if the universe was a torus with a radius less than the current particle horizon, what sort of characteristic signatures would we expect to see in the large scale structure and/or the CMB?

and for a change, a bonus question which tackles the sociology of science (also proposed by Shaun): what do you think is the reason why scientists are underrepresented in political institutions?

Wednesday, June 27, 2012

baryonic acoustic oscillations

in what way do BAOs work as a geometrical distance measurement? What happens to BAOs in nonlinear structure formation? What assumptions concerning biasing does one do?

bonus question: is there an intuitive argument why the BAOs get destroyed in nonlinear structure formation?

extra bonus: what's $\log_i(\sqrt[i]{i})$ with $i = \sqrt{-1}$

Wednesday, June 20, 2012

haloes of all sizes

what are the largest and smallest objects (a.k.a. dark matter haloes) that can form in a given cosmology? which processes determine their sizes?

bonus question: what's $\log_i(\mathrm{e})$ with $i=\sqrt{-1}$

Wednesday, June 13, 2012

static universe

would it be possible to construct a FLRW-cosmology with no time dependence? what would be needed for such a cosmology? what's a sensible starting point for constructing this cosmology? would the solution be stable?

bonus question: what's $|i!|$ with $i=\sqrt{-1}$?

Wednesday, June 6, 2012

gravitational lensing

what matter distribution generates a lensing effect where the lensing convergence is equal to the shear amplitude for any position?

bonus question: can the image of a lensed galaxy be rotated in gravitational lensing? if yes, under which circumstances?

Wednesday, May 30, 2012

black body in 4 dimensions

what would be a black body's flux-frequency relation in 4 spatial dimensions? in $n$ spatial dimensions? what would be equivalent relations for the Wien displacement law and the Stefan-Boltzmann law?

bonus question: what's $\mathrm{d}/\mathrm{d}x\: \mathrm{arcosh}(x)$?

Wednesday, May 23, 2012

selection bias - cosmo CSI

during the 1960's, criminal investigators were puzzled by an observation that in many countries with a seaside a considerable number of dismembered feet in sports sneakers were washed ashore. a logical explanation seemed to be a serial killer murdering joggers, but the truth is different and involves statistics, and the biasedness of observations: what would be your guess?

Wednesday, May 16, 2012

time-variation of the CMB

does the pattern of hot and cold patches of the CMB change in time? if yes, on what time scale?

bonus question: which parts of the CMB spectrum will gain in amplitude and which effects will loose in their importance?

Thursday, May 10, 2012

CDM particle properties

what statements about the properties of the dark matter particles can we make by observing cosmic structures?

Thursday, May 3, 2012

negative redshifts

would negative cosmological redshifts (aka blueshifts  ;) ) be useful to describe events in the future evolution of the Universe? if yes, what'd be the largest (most negative) redshift which can be defined in a sensible way?

Thursday, April 26, 2012

fine structure constant

what would be different in cosmology if the fine structure constant would not be 1/137 but 1/138?

bonus points: why do we know that it has has stayed constant in the course of cosmic history?

Thursday, April 19, 2012

Olbers' paradoxon

classic cosmology question: why is it dark at night? what are minimal assumptions for answering this, and how would you reply with the knowledge of a FLRW-cosmology? are there other (astrophysical) explanations you can think of? could you construct an alternative whacky cosmology (i.e. non-FLRW) that would resolve the paradoxon?

bonus question: what's $(\pi+20)^i$ with $i=\sqrt{-1}$? surprised?

Thursday, April 12, 2012

to infinity and beyond...

suppose you've got a spaceship with amazing engines that are able to provide an acceleration of $1g$ measured inside the rocket, such that you would not notice any difference to living on Earth.

what's the time measured on clocks inside the rocket that would pass while travelling
1. to the centre of the Milky Way?
2. to the most distant quasars we see today?
3. the Hubble distance?

Wednesday, April 11, 2012

CMB dipole - fact or fiction?

the cosmic microwave background has a dipolar structure which is interpreted in terms of the Milky Way moving relative to the CMB, inducing Doppler-shifts in the photon energies and boosing the flux. would it be possible to distinguish between a motion dipole and a cosmological dipole on the last-scattering surface? why is the Doppler-boosted photon spectrum still a Planck-spectrum with the correct Wien- and Stefan-Boltzmann-laws?

bonus question: what's $\int\mathrm{d}x\:x\ln(x)$?

Thursday, April 5, 2012

digits in physical constants

have a look at a table of physical constants, which are numbers spanning a wide range, and count the occurrence of the digit "1", of the digit "2" and so on. you will find, that the digit "1" occurs more often than "2", with decreasing frequency down to "9". is that odd?

Thursday, March 29, 2012

CMB temperature

if the ionisation temperature of hydrogen is $T=10^4$K, the CMB decoupled at a redshift of $z=10^3$, and if for photons the temperature drops proportional to $1/a$, shouldn't the CMB have a temperature of $10$K instead of $3$K?

bonus question: would it be possible to generalise the hyperbolic functions to a basis $a\neq e=\exp(1)$ in a way that $\mathrm{asinh}(x)$ behaves asymptotically as $\log_{a}(x)$ for $x\gg 1$?

Thursday, March 22, 2012

age of the Universe

imagine you're stuck in a discussion with a creationist: what evidence is there that the Universe is more than ten billion years old? of course there are multiple answers, try to think of as many as you can.

bonus question (a tough one, this time): what's the derivative $\frac{\mathrm{d}}{\mathrm{d}x}x^{x^x}$?

Thursday, March 15, 2012

time-dependence of cosmological constants

in cosmology, we define all quantities by their values today. would we need to revise their numerical values in the future? if yes, on what time scale?

bonus question: what's larger, $|\pi^i|$ or $|i^\pi|$, with $i=\sqrt{-1}$ (principal values only)?

Wednesday, March 14, 2012

Mercury's orbit

can one explain the perihelion precession of Mercury's orbit with Newtonian gravity? what would be needed for this explanation? please keep in mind that at the time of Einstein's derivation (happy birthday, Prof. Einstein!) there was no real observational need for general relativity.

for bonus points: what's the order of magnitude of the perinigricon (point of closest approach to the black hole) precession of the stars orbiting the black hole at the centre of the Milky Way?

Wednesday, March 7, 2012

precision cosmology

a question for the era of precision cosmology... future surveys such as EUCLID and PLANCK are designed to yield almost $10^{-4}$ precision on the cosmological parameters $\Omega_m$ and $H_0$ (in a reasonably simple model). at the same time table-top experiments constrain the gravitational constant $G$ to a similar precision (it's a bit scary to realise that $G$ is the least known constant of Nature with only 4 significant digits!). would it make sense to improve cosmological measurements without improving on the gravitational constant?

bonus question: what's the asymptotic behaviour of $\frac{d}{dx}\left(\frac{1}{x}\right)^\frac{1}{x}$ for large $x$?

Thursday, March 1, 2012

lensing on an expanding metric

in relation to the last question: why is it possible to treat gravitational lensing on an expanding background metric as if it was Minkowskian, weakly perturbed by the gravitational potential? if this is related to a property of photons, to which other particle trajectories would this apply?

bonus question: what's $\sqrt[i]{i}$ with $i=\sqrt{-1}$?

Saturday, February 25, 2012

Minkowski metric

why is the Minkowski metric not the solution to the Einstein equations for empty space? (or equivalently, why is the empty space of special relativity not the empty space of general relativity?)

Monday, February 20, 2012

voids

inside a void, does time pass faster or slower? by how much?

bonus points: what's $\cos(i)$ with $i = \sqrt{-1}$?

Friday, February 17, 2012

Schwarzschild radius

estimate the Schwarzschild-radius $r_S$ of the Universe!

Friday, February 10, 2012

distance measures

which of the cosmological distance measures are additive, $D(z_1,z_2) + D(z_2,z_3) = D(z_1,z_3)$ and which ones are monotonic, $dD/dz > 0$?

and for bonus points, like always: what's $i^i$ with $i = \sqrt{-1}$?

Friday, February 3, 2012

Gaussianity

why does structure formation in the linear regime conserve Gaussianity? does it conserve other statistical properties of the initial density field?

Friday, January 27, 2012

redshift space

what's the finger-of-god-effect and the Kaiser-effect in galaxy surveys?

Friday, January 20, 2012

SZ-effect

in the thermal Sunyaev-Zel'dovich effect there's an interaction between a galaxy cluster's electrons and the CMB photons. Does a galaxy cluster become colder in time due to the thermal SZ-effect?

for bonus points: does a cluster get slowed down by the kinetic SZ-effect? if yes, relative to which frame of reference?

Friday, January 13, 2012

dark energy

what do the cosmological constant and dark energy have in common? why are they different? why is it wrong to say that Lambda is just DE with $w=-1$? could there be a cosmology with Lambda and DE at the same time?

Friday, January 6, 2012

Hubble function

in cosmology we can measure the Hubble function $H(a)$ and the growth function $D_+(a)$. do they carry the same information? could one compute $D_+$ with the knowledge of only $H$ and vice versa? are there cosmologies where this is NOT possible?