exponentially accelerated cosmic expansion drives all objects away from us, and in the future we will think that there would just be the Virgo cluster as an island of galaxies in an otherwise empty Universe. when will that be? (it is funny to imagine that humans then would have historical information about the large-scale structure, but would they believe it?)

with that question we would like to wish our readers a merry Xmas and a happy New Year 2014 and we hope to see you again in January. if you want to contribute to CQW, please drop us a mail at cosmologyquestionoftheweek["AT"]gmail["DOT"]com.

## Wednesday, December 18, 2013

## Wednesday, December 11, 2013

### life in the early universe

in his paper "the habitable epoch in the early universe" A. Loeb makes the observation that the CMB in the early universe would have kept everything at temperatures allowing water to be liquid, and that during that time the Universe was a giant habitable zone. can you estimate how long (in years) this period lasted?

## Wednesday, December 4, 2013

### quantum microwave background

CQW celebrates its second anniversary!

would a classical description of the microwave background today be applicable? please justify your answer using the thermal wavelength.

bonus question: motion extremises the action in classical Lagrangian mechanics. is that extremum a maximum or a minimum?

would a classical description of the microwave background today be applicable? please justify your answer using the thermal wavelength.

bonus question: motion extremises the action in classical Lagrangian mechanics. is that extremum a maximum or a minimum?

## Wednesday, November 27, 2013

### classical photons

the flux density of a blackbody would have form

\begin{equation}

S(\omega) \propto \omega^3\exp\left(-\frac{\hbar\omega}{k_BT}\right)

\end{equation}

if the photons were classical distinguishable particles. can you derive Wien's displacement law and the Stefan-Boltzmann law and point out the differences to the Planck-spectrum?

\begin{equation}

S(\omega) \propto \omega^3\exp\left(-\frac{\hbar\omega}{k_BT}\right)

\end{equation}

if the photons were classical distinguishable particles. can you derive Wien's displacement law and the Stefan-Boltzmann law and point out the differences to the Planck-spectrum?

## Wednesday, November 20, 2013

### radiation density

can you express the density parameter $\Omega_\gamma$ of a Planck-distributed relativistic species as a function of temperature only?

## Wednesday, November 13, 2013

### slow acoustic waves

the scale of baryon acoustic oscillation is set by the distance a sound wave can travel from the end of inflation until recombination. it is very important in this context that the sound wave is driven by photon pressure which makes the wave so fast. could you estimate the size of the BAO-scale if the acoustic wave traveled as a normal sound wave, i.e. driven by adiabatic compression of an ideal gas?

## Wednesday, November 6, 2013

### natural scale of the Universe

can you express the Universe's current

in Planck-units?

- age $1/H_0$
- size $c/H_0$
- density $\rho_\mathrm{crit}=3H_0^2/(8\pi G)$
- temperature

in Planck-units?

## Wednesday, October 30, 2013

### galaxy rotation curves at large radii

can you give an expression $\upsilon_\mathrm{rot}(r)$ of how a galaxy rotation curve should look like far away from the centre of the galaxy?

## Wednesday, October 23, 2013

### cosmic neutrino background

can you estimate the amplitude of anisotropies in the cosmic neutrino background given the amplitude of CMB anisotropies?

## Wednesday, October 16, 2013

### shape of the Hubble-law

why does the Hubble law have to have the form $\vec{\upsilon} = H\vec{r}$? if it were of a different form, what would happen to the symmetry assumptions in the cosmological principle?

## Wednesday, October 9, 2013

### equation of state of baryons

can you estimate the equation of state parameter $w$ for the baryonic component in the Universe?

bonus question: how would $w$ change in time?

bonus question: how would $w$ change in time?

## Wednesday, October 2, 2013

### thermal wavelength

can you estimate the thermal wavelength of hydrogen atoms today? assume that there's no additional heat input so that you can predict the atom's temperature from the CMB temperature. are the atoms classical or does one have to use a quantum mechanical description?

## Wednesday, August 7, 2013

### Planck's summer: variations in $\Lambda$

sixth post by Youness in the "Planck's summer"-series on CQW:

The cosmological constant $\Lambda$, according to our current physical understanding, really constant in time or has its value changed? Think about the question from a quantum field theory perspective.

With that, we wish our readers a nice summer holiday and hope that you join our blog again in October! And many thanks to you, Youness, for providing this awesome set of questions.

The cosmological constant $\Lambda$, according to our current physical understanding, really constant in time or has its value changed? Think about the question from a quantum field theory perspective.

With that, we wish our readers a nice summer holiday and hope that you join our blog again in October! And many thanks to you, Youness, for providing this awesome set of questions.

## Wednesday, July 31, 2013

### Planck's summer: varying electron masses

fifth post by Youness in the "Planck's summer"-series:

How can we test whether the electron mass changes with time (as compared to the other particle masses)? Give a list of astrophysical or cosmological observations that are sensitive to the electron mass (from the early to the late Universe). Do you have an idea how to measure the present time variation of the electron mass in the lab?

## Wednesday, July 24, 2013

### Planck's summer: observation of the CNB

fourth post in the "Planck's summer"-series by Youness

When did the cosmic neutrino background form, why is its temperature lower than that of the CMB and how could an experiment to detect it look like?

## Wednesday, July 17, 2013

### Planck's summer: gauge independence of cosmological surveys

third post in the "Planck's summer"-series by Youness:

Power spectra in linear perturbation theory depend on the gauge chosen. The differences are small on scales well within the horizon, but they grow as we go to very large scales. However, observers apparently measure galaxy positions unambiguously and calculate power spectra from this. How would you interpret their data on the largest scales?

## Wednesday, July 10, 2013

### Planck's summer: particle size matters!

second post in the "Planck's summer"-series by Youness:

Let us assume that dark matter is made of approximately collisionless particles evolving under the laws of gravity (just one type, for simplicity). Is there a minimum mass $m_\text{min}$ these particles need to have in order to fit observational constraints? Conversely, is there an upper limit $m_\text{max}$?

## Wednesday, July 3, 2013

### Planck's summer: CMB observations without CMB photons

CQW presents "Planck's summer", a series of posts composed by recent graduate Youness!

Imagine a planet with an atmosphere that is, for some reason, completely opaque for low-energy photons such that a direct ground-based detection of the CMB is impossible. Furthermore, the inhabitants of the planet do not yet possess the technology to build spacecraft that leaves the atmosphere. However, they are very good experimentalists on the ground. How could they test whether there is a CMB confirming a basic prediction of big bang cosmology?

## Wednesday, June 26, 2013

### CMB as an ideal black body

why is the cosmic microwave background such a perfect black body with a Planck-spectrum and why don't we see the recombination lines of the atomic levels?

maths bonus question: what's the reason why Christoffel symbols $\Gamma^{a}_{bc}$ are required to be symmetric in the lower indices?

maths bonus question: what's the reason why Christoffel symbols $\Gamma^{a}_{bc}$ are required to be symmetric in the lower indices?

## Wednesday, June 19, 2013

### CMB recombination

the CMB photosphere has a thickness of $\Delta z\simeq 100$ in redshift. how much time (in years) does it take for the atoms to recombine?

maths bonus: can you show that the Gauss-distribution is shape-invariant under convolution? what about the Cauchy-distribution?

maths bonus: can you show that the Gauss-distribution is shape-invariant under convolution? what about the Cauchy-distribution?

## Wednesday, June 12, 2013

### advection with the Hubble flow

what criterion would you impose that an object's redshift (for instance, a supernova) is really measuring the Universe's expansion and is following the Hubble flow?

bonus question: can you show that the determinant of a skew-symmetric matrix $A$, $A^t=-A$, is equal to zero if the dimension is odd?

bonus question: can you show that the determinant of a skew-symmetric matrix $A$, $A^t=-A$, is equal to zero if the dimension is odd?

## Wednesday, June 5, 2013

### supernovae at high redshift

if you observe a supernova at redshift $z=2$, does the lightcurve get time-dilated due to the Hubble expansion? if yes, by how much? at what redshift would you need to place a supernova if you want to stretch the lightcurve to last a year?

math bonus: just using a compass and a ruler, it is easily possible to construct $\sin(\alpha)$ and $\cos(\alpha)$ of an angle $\alpha$ (how?). can you construct $\sinh(\alpha)$ and $\cosh(\alpha)$ as well for arbitrary angles $\alpha$?

math bonus: just using a compass and a ruler, it is easily possible to construct $\sin(\alpha)$ and $\cos(\alpha)$ of an angle $\alpha$ (how?). can you construct $\sinh(\alpha)$ and $\cosh(\alpha)$ as well for arbitrary angles $\alpha$?

## Wednesday, May 29, 2013

### dark energy takes over...

what possibilities are there for defining the moment where the Universe switches from matter domination to dark energy domination? the two most obvious choices would be the instant $a$ where the density parameters are equal, $\Omega_m(a) = \Omega_\Lambda(a)$, or perhaps when the deceleration parameter changes sign, $q(a)=0$... how would you define it and what numerical values for $a$ would you get with the two above definitions?

math bonus question: can you derive the Taylor expansion of the logarithm using the geometric series of $1/(1+x)$?

## Wednesday, May 22, 2013

### size of the Universe

can you estimate a minimum size of the Universe $\chi_K$ given current bounds on the curvature paramter $\Omega_K$ and compare this length scale to the size $\chi_H=c/H_0$ of the observable Universe?

maths bonus question: what's the slope of $x^{\sin(x)^x}$ at $x=0$ and where's the first minimum in the positive $x$-range?

physics bonus question: why do neutrons in neutron stars not decay? with neutron decay half-time of about a quarter of an hour, neutron stars should decay rather quickly!

maths bonus question: what's the slope of $x^{\sin(x)^x}$ at $x=0$ and where's the first minimum in the positive $x$-range?

physics bonus question: why do neutrons in neutron stars not decay? with neutron decay half-time of about a quarter of an hour, neutron stars should decay rather quickly!

## Wednesday, May 15, 2013

### tests for FLRW-cosmologies

is there an observational test of verifying that we live in a FLRW-universe, i.e. of all (which?) symmetry assumptions behind choosing the Robertson-Walker-metric? is this a sufficient condition?

physics bonus question: can we test if the cosmological fluids are ideal?

maths bonus question: can you prove (by induction) the Bernoulli inequality, namely that

\begin{equation}

(1+x)^n\geq 1+nx

\end{equation}

for all real numbers $x\geq-1$?

physics bonus question: can we test if the cosmological fluids are ideal?

maths bonus question: can you prove (by induction) the Bernoulli inequality, namely that

\begin{equation}

(1+x)^n\geq 1+nx

\end{equation}

for all real numbers $x\geq-1$?

## Wednesday, May 8, 2013

### energy input into the CMB

the CMB is produced in thermal equilibrium.... would it be possible to notice if there was some extra energy put into it? what consequences could this have, depending on the way it was done?

physics bonus question: can you derive the energy-time-uncertainty from the momentum-position-uncertainty in quantum mechanics?

maths bonus question: what are the asymptotes of $\exp(\tanh(x))$ for $x\rightarrow\pm\infty$ and what is the slope at $x=0$?

physics bonus question: can you derive the energy-time-uncertainty from the momentum-position-uncertainty in quantum mechanics?

## Wednesday, May 1, 2013

### inflationary perturbations

guest post by Shaun Hotchkiss (from the blog trenches of discovery):

if, through the CMB and large scale structure, we can study the primordial density perturbation over a range of scales corresponding to five orders of magnitude (i.e. $k_\mathrm{max} = 10^5 k_\mathrm{min}$), what fraction $\Delta \phi/\phi$ of the inflationary potential does this probe (if inflation is true and the inflationary potential is $V(\phi)=m^2\phi^2$)?

bonus question: where's the first minimum of $x^{\sinh(x)}$, $x>0$?

bonus question: where's the first minimum of $x^{\sinh(x)}$, $x>0$?

## Wednesday, April 24, 2013

### invariants of the lensing Jacobian

the lensing Jacobian $\mathcal{A}=\partial\vec{\beta}/\partial\vec{\theta}$ describes how the lensing deflection angle $\beta$ changes with position $\theta$. how many invariants (under orthogonal transformations) of $\mathcal{A}$ can you construct? what are they and what is their physical interpretation?

bonus question: can you show that $\exp(x)^{\sin(x)}$ is an even function and that the derivative is an odd function?

bonus question: can you show that $\exp(x)^{\sin(x)}$ is an even function and that the derivative is an odd function?

## Wednesday, April 17, 2013

### $\Lambda$ as a source of energy

this week, CQW has a guest question by U. Bastian:

is it possible to tap dark energy as a source of (mechanical) energy? could you set up a machine that would accomplish this?

bonus question: assume you're trying to compute the expectation value $\langle x\rangle$ of a random distribution $p(x)\mathrm{d}x$, which has the cumulative distribution $P(x)$. you could write:

\begin{equation}

\langle x\rangle = \int\mathrm{d}x\:xp(x) = \int\mathrm{d}x\:x\frac{\mathrm{d}}{\mathrm{d}x}P(x) = xP(x) - \int\mathrm{d}x\: P(x)

\end{equation}

by integration by parts, where all integration boundaries are taken to be $-\infty\ldots+\infty$. both expressions in the final expression are not finite. where's the mistake?

is it possible to tap dark energy as a source of (mechanical) energy? could you set up a machine that would accomplish this?

bonus question: assume you're trying to compute the expectation value $\langle x\rangle$ of a random distribution $p(x)\mathrm{d}x$, which has the cumulative distribution $P(x)$. you could write:

\begin{equation}

\langle x\rangle = \int\mathrm{d}x\:xp(x) = \int\mathrm{d}x\:x\frac{\mathrm{d}}{\mathrm{d}x}P(x) = xP(x) - \int\mathrm{d}x\: P(x)

\end{equation}

by integration by parts, where all integration boundaries are taken to be $-\infty\ldots+\infty$. both expressions in the final expression are not finite. where's the mistake?

## Wednesday, April 10, 2013

### thermal equilibrium and the CMB

why does the CMB have a Planckian spectrum? after all, it has been generated over a range in redshift ($\Delta z\simeq 100$) over which the temperature changed, and there has been energy exchange with the electron plasma.

bonus question: can you show that the asymptote of the integral sine, $\int_0^x\mathrm{d}\ln(t)\:\sin(t)$, is $\pi/2$ for $x\rightarrow\infty$?

bonus question: can you show that the asymptote of the integral sine, $\int_0^x\mathrm{d}\ln(t)\:\sin(t)$, is $\pi/2$ for $x\rightarrow\infty$?

## Wednesday, April 3, 2013

### limitations of the NFW-profile shape

the NFW-profile is an approximation to the density distribution inside cold dark matter structures. where does the NFW-approximation break down and for what reason? if there was annihilation of dark matter, what could you say about the central brightness?

bonus question: where is the first minimum of $x^{\sin(x)}$, $x>0$?

bonus question: where is the first minimum of $x^{\sin(x)}$, $x>0$?

## Wednesday, March 27, 2013

### decaying vorticity

can you show that in the course of linear structure formation during matter domination the vorticity $\omega=\mathrm{rot}\upsilon$ decays $\propto 1/a$ with the scale factor $a$? what's the relation during radiation domination? by how much has the vorticity decreased from the end of inflation (when vorticity modes could have been seeded) until today?

bonus question: what's the value of the integral

\begin{equation}

\int_0^\pi\mathrm{d}x\:\sin(x)^{x^x}

\end{equation}

CQW accepts numerical values and we'd be really interested in an analytical solution, because the numerical value is somewhat...suggestive.

bonus question: what's the value of the integral

\begin{equation}

\int_0^\pi\mathrm{d}x\:\sin(x)^{x^x}

\end{equation}

CQW accepts numerical values and we'd be really interested in an analytical solution, because the numerical value is somewhat...suggestive.

## Wednesday, March 20, 2013

### Higgs mechanism

what fraction of your body weight is due to the Higgs mechanism? in what way is the remaining mass being produced?

bonus question: can you show that parity transformation $\vec{r}\rightarrow -\vec{r}$ introduces a factor $(-1)^\ell$ in the spherical harmonics $Y_{\ell m}(\theta,\phi)$?

bonus question: can you show that parity transformation $\vec{r}\rightarrow -\vec{r}$ introduces a factor $(-1)^\ell$ in the spherical harmonics $Y_{\ell m}(\theta,\phi)$?

## Wednesday, March 13, 2013

### electroweak phase transition

before the electroweak phase transition of the Universe, all weak bosons are massless just like the photons, and with the temperatures well above TeV, the Higgs-particle hasn't given any mass to the charged leptons yet. would there be a way of distinguishing electrons, muons and taus?

physics bonus question: in quantum mechanics, what property of the Hamiltonian operator is responsible for the conservation of probability?

maths bonus question: can you show that $\exp(A)$ is unitary if $A$ is anti-Hermitean, $A^+=-A$, where $A^+$ is the complex-conjugated, transposed matrix $A$?

physics bonus question: in quantum mechanics, what property of the Hamiltonian operator is responsible for the conservation of probability?

maths bonus question: can you show that $\exp(A)$ is unitary if $A$ is anti-Hermitean, $A^+=-A$, where $A^+$ is the complex-conjugated, transposed matrix $A$?

## Wednesday, March 6, 2013

### thinking outside the box

we can observe the cosmos on the past light cone and naturally, we see finite amounts of objects and of fluctuations which sets statistical limits on inferences from cosmological data (aka cosmic variance). are there possibilities of looking outside the past light cone and what type of information would that provide?

bonus question: can you estimate the value of

\begin{equation}

\log\int_0^\infty\mathrm{d}t\:t^x\exp(-t)

\end{equation}

for large $x$?

bonus question: can you estimate the value of

\begin{equation}

\log\int_0^\infty\mathrm{d}t\:t^x\exp(-t)

\end{equation}

for large $x$?

## Wednesday, February 27, 2013

### back to the future

imagine there hadn't been an inflationary epoch - how long would we need to wait until hydrogen atoms were formed and the cosmic microwave background was generated?

maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

maths bonus question: where's the global maximum of $\sqrt[x]{x}$, $x>0$?

physics bonus question: in quantum mechnics, can a perfect sphere rotate?

## Wednesday, February 20, 2013

### Hubble slow motion

does time pass slower in distant galaxies? if you could observe a planetary system at high redshift and observe the orbits of planets (or any other dynamical system driven by gravity), would the observation be consistent with a lower gravity instead of a slower passage of time? what about atomic processes like chemical reactions or radiative processes?

bonus question: can you show that for a real-valued matrix $A$ the matrix exponential $\exp(A)$ transforms according to

\begin{equation}

\exp(R^{-1}AR) = R^{-1}\exp(A)R

\end{equation}

with any invertible matrix $R$?

bonus question: can you show that for a real-valued matrix $A$ the matrix exponential $\exp(A)$ transforms according to

\begin{equation}

\exp(R^{-1}AR) = R^{-1}\exp(A)R

\end{equation}

with any invertible matrix $R$?

## Wednesday, February 13, 2013

### scale-invariance of perturbations

can you find an intuitive argument why the variance of density fluctuations seeded by inflation is proportional to the wave number? what would be the corresponding relation if we lived in a universe with 4 spatial dimensions instead of 3?

bonus question: can you show that if a square real-valued matrix $A$ is skew-symmetric, $A^t=-A$, then the matrix exponential $\exp(A)$ of that matrix is orthogonal?

bonus question: can you show that if a square real-valued matrix $A$ is skew-symmetric, $A^t=-A$, then the matrix exponential $\exp(A)$ of that matrix is orthogonal?

## Wednesday, February 6, 2013

### cosmic conspiracy for rotation curves

what's the density distribution inside a dark matter halo that allows flat rotation curves? how does that correspond to the NFW-profile shape and what's the reason that the size of the galactic disk is roughly equal to the NFW scale radius?

bonus question: can you show using a power series that

\begin{equation}

\exp(A)\exp(B)=\exp(A+B)

\end{equation}

for two square, real-valued, commuting matrices $A$ and $B$, i.e. $\left[A,B\right]=0$. what changes if the two matrices are non-commuting?

bonus question: can you show using a power series that

\begin{equation}

\exp(A)\exp(B)=\exp(A+B)

\end{equation}

for two square, real-valued, commuting matrices $A$ and $B$, i.e. $\left[A,B\right]=0$. what changes if the two matrices are non-commuting?

## Wednesday, January 30, 2013

### gravity in 2d and 3d

why can't gravity rotate galaxy images in gravitational lensing, but introduce physical rotation of galaxies?

bonus question: where's the global minimum of $x^x$, $x>0$?

bonus question: where's the global minimum of $x^x$, $x>0$?

## Wednesday, January 23, 2013

### homogeneous and anisotropic cosmology

can a cosmological model homogeneous but anisotropic? what would that construction imply? are there observable consequences, and which ones would that be?

(question courtesy of R. Schmidt, Centre for Astronomy, Heidelberg)

bonus question: what's the angle between an edge and the space diagonal in an $n$-dimensional unit hypercube? what's the limit of this angle in the case $n\rightarrow\infty$?

for the solution CQW shows here the angle in question (in units of $\pi$) as a function of dimension $n$:

(question courtesy of R. Schmidt, Centre for Astronomy, Heidelberg)

bonus question: what's the angle between an edge and the space diagonal in an $n$-dimensional unit hypercube? what's the limit of this angle in the case $n\rightarrow\infty$?

for the solution CQW shows here the angle in question (in units of $\pi$) as a function of dimension $n$:

## Wednesday, January 16, 2013

### destruction of gravitationally bound systems

is it possible that gravitationally bound systems (clusters, galaxies, globular clusters, solar systems, planets with moons) are taken apart by the Hubble-expansion? if yes, under which circumstances and on what time scale?

bonus question: can you show that the integral over spherical harmonics almost always vanishes,

\begin{equation}

\int\mathrm{d}\Omega\:Y_{\ell m}(\theta,\phi) = 0

\end{equation}

except of course for the monopole $\ell=m=0$?

bonus question: can you show that the integral over spherical harmonics almost always vanishes,

\begin{equation}

\int\mathrm{d}\Omega\:Y_{\ell m}(\theta,\phi) = 0

\end{equation}

except of course for the monopole $\ell=m=0$?

## Wednesday, January 9, 2013

### conformal time in $\Lambda$-driven expansion

CQW wishes all readers a successful new year 2013, in particular with PLANCK's upcoming data release!

what's the relation between conformal time and cosmic time (or equivalently between comoving and proper distance) during epochs of exponential expansion? can you write the relation as a power-law?

physics bonus question: what's the reason why magnetic fields play such an important role in astrophysics, and electrical fields so little?

maths bonus question: can you determine the derivative of $\exp(-1/(1-x^\alpha))$ both at $x=0$ and $x=1$ for arbitrary $\alpha>0$?

i've added a plot of that function to CQW: $\alpha=2,3,4,5,6$ (green solid lines), $\alpha=1$ (red solid line), $\alpha>1$ in steps of $0.1$ (red dashed lines), $\alpha<1$ in steps of $0.1$ (blue dashed lines):

what's the relation between conformal time and cosmic time (or equivalently between comoving and proper distance) during epochs of exponential expansion? can you write the relation as a power-law?

physics bonus question: what's the reason why magnetic fields play such an important role in astrophysics, and electrical fields so little?

maths bonus question: can you determine the derivative of $\exp(-1/(1-x^\alpha))$ both at $x=0$ and $x=1$ for arbitrary $\alpha>0$?

i've added a plot of that function to CQW: $\alpha=2,3,4,5,6$ (green solid lines), $\alpha=1$ (red solid line), $\alpha>1$ in steps of $0.1$ (red dashed lines), $\alpha<1$ in steps of $0.1$ (blue dashed lines):

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