## 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):

1. physics bonus question: the plasma conductivity is proportional to the time between collisions, so the conductivity formally diverges for thin plasmas, effectively short-circuiting any charges which could build up electric fields. electrical fields are responsible, however, for the particle acceleration in pulsars, where the rapidly changing magnetic field generates a strong electric field, due to induction, accelerating charged particles.

2. physics bonus question: I forgot to write that the basic consideration concerning collision time and conductivity still holds for magnetised plasmas.

3. maths bonus question: the derivative of the function is given by

\frac{\mathrm{d}}{\mathrm{d}x}\exp\left(-\frac{1}{1-x^\alpha}\right)
=\alpha\exp\left(-\frac{1}{1-x^\alpha}\right)\frac{x^{\alpha-1}}{(1-x^\alpha)^2}

which has, according to the particular value of $\alpha$, the limits $0$ for $\alpha>1$, $\exp(-1)$ for $\alpha=1$ and $-\infty$ for $\alpha<1$ for $x\rightarrow 0$, and the limit $0$ for $x\rightarrow 1$ irrespective of $\alpha$.

4. let's derive the relation between conformal time $\eta$ and cosmic time $t$ for a FLRW-cosmology where a single fluid with a constant equation of state parameter $w$ is present up to the critical density. then, the Hubble function $H(a)$ is given by

H(a) = \exp\left(3\int_a^1\mathrm{d}\ln a\:(1+w)\right) = a^{-3(1+w)}

and let's set $H_0=1$ for simplicity. with the definition of the Hubble function $aH(a)=\mathrm{d}a/\mathrm{d}t$ one can then obtain the cosmic time $t$,

t = \int\frac{\mathrm{d}a}{aH}
\rightarrow
t = \frac{1}{3+3w}a^{3+3w}

and the conformal time $\eta$ with $\mathrm{d}\eta=\mathrm{d}t/a$,

\eta = \int\frac{\mathrm{d}a}{a^2H}
\rightarrow
\eta = \frac{1}{2+3w}a^{2+3w}

and therefore one obtains for the relation between $\eta$ and $t$:

\eta =\frac{1}{2+3w}\left[(3+3w)t\right]^{\frac{2+3w}{3+3w}}

which is the desired power law. one can see that it is only defined if $w\neq-1$ because the exponent would become infinite otherwise.

1. equations are not correct:
The first equation gives H(a) squared, not H(a).

5. thanks for pointing out the missing squares, R.H.Kail!: let's rederive the relation between conformal time $\eta$ and cosmic time $t$ for a FLRW-cosmology where a single fluid with a constant equation of state parameter $w$ is present up to the critical density. then, the Hubble function $H(a)$ is given by

H^2(a) = \exp\left(3\int_a^1\mathrm{d}\ln a\:(1+w)\right) = a^{-3(1+w)}

and let's set $H_0=1$ for simplicity. with the definition of the Hubble function $aH(a)=\mathrm{d}a/\mathrm{d}t$ one can then obtain the cosmic time $t$,

t = \int\frac{\mathrm{d}a}{aH}
\rightarrow
t \propto a^{(3+3w)/2}

and the conformal time $\eta$ with $\mathrm{d}\eta=\mathrm{d}t/a$,

\eta = \int\frac{\mathrm{d}a}{a^2H}
\rightarrow
\eta \propto a^{(1+3w)/2}

and therefore one obtains for the relation between $\eta$ and $t$:

\eta \propto t^{\frac{1+3w}{3+3w}}

which is the desired power law. one can see that it not defined for $w=-1$.

6. results of the poll concerning the Maya apocalypse were as follows: most of the people taking part were either afraid of the neutrino cosmology that provides fuel for an accelerated expansion or the classic mini black hole generated at CERN.

7. we at CQW just wanted to add that of course in an exponentially expanding universe due to domination of $\Lambda$ the relation between cosmic and conformal time is this one: setting again $H_0=1$ allows a direct computation of $t=-\ln(a)$ and $\eta = 1/a-1$. combining the two results gives $t = \ln(1+\eta)$ and $\eta = \exp(t)-1$.