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Structure Formation

A typical output from a simulation of large-scale structure formation. The brightest spots in this image are clusters of galaxies. Credit: Uri Keshet et al.

We now have a flat universe with plenty of matter: how do we make galaxies (and humans!)? Discussion follows Ryden Chap 12.

Note "frothiness" of universe on large scale: voids of up to 500 Mpc

Sizes of Things

3 characteristic sizes

Clusters

\color{red}{M \sim 10^{15} M_o }

\color{red}{R \sim 2Mpc}

Galaxies

\color{red}{M \sim 10^{11} M_o }

\color{red}{R \sim 20kpc}

Globular clusters/Dwarf Galaxies

\color{red}{M \sim 10^6 M_o }

\color{red}{R \sim 1kpc}

How and when do they form?

What happens to fluctuations;
Four competing processes
Gravitation: causes inhomogeneites to become unstable and collapse
Cosmic fluid pressure: causes inhomogeneites to vanish as acoustic waves
Photon viscosity: slows down processes
expansion of universe smooths out everything: "Hubble friction"

Grav. Instab.

Average density

\color{red}{ \bar \varepsilon \left( t \right) = \frac{1}{V}\int {\varepsilon \left( {\vec r,t} \right)} d\vec r}

and can then define (dimless) density fluctuations

\color{red}{ \delta \left( {\vec r,t} \right) = \frac{{\varepsilon \left( {\vec r,t} \right) - \bar \varepsilon \left( t \right)}}{{\bar \varepsilon \left( t \right)}}}

-1 < |δ| < ∞
We worked out the free-fall time once before: i.e. how long would it take for a particle to fall from the outside of a star to the centre if there was no resistance:
\color{red}{ \tau = \left( {\frac{{R^3 }}{{GM}}} \right)^{1/2} \Rightarrow \left( {\frac{3}{{4\pi G\rho }}} \right)^{1/2} }

Suppose we have a sphere of excess density \color{red}{\rho = \bar \rho \left( {1 + \delta } \right)} : how long will it take to collapse?

Force on surface layer accelerates it inwards
\color{red}{ \ddot R = - \frac{{G\Delta m}}{{R^2 }} = - \frac{{4\pi G\bar \rho \delta \left( t \right)R\left( t \right)}}{3}}
Total mass stays constant
\color{red}{ M = \frac{{4\pi G\bar \rho \left( {1 + \delta \left( t \right)} \right)R\left( t \right)^3 }}{3}}
so
\color{red}{ R\left( t \right) = \frac{{R_0 }}{{\left( {1 + \delta \left( t \right)} \right)^{1/3} }} \approx R_0 \left( {1 - \frac{1}{3}\delta \left( t \right)} \right) \Rightarrow \ddot R = - \frac{1}{3}\ddot \delta }
(density fluctation increases as R decreases) Hence
\color{red}{ \delta \left( t \right) \approx \frac{{\delta _0 }}{2}e^{t/t_{dyn} } }
where
\color{red}{ t_{dyn} = \left( {\frac{1}{{4\pi G\rho }}} \right)^{1/2} \simeq \tau }
(Note that for t > > t_dyn, δ(t) is no longer small)

Pressure

stops the collapse.

Size R ∼ wavelength λ of acoustic wave
If wave travels with speed c_s, restoring time t_pre ∼ λ/c_s
If t_pre < t_dyn, collapse won't happen. so need speed of sound in gas.
Before decoupling, Universe is filled with photon-baryon fluid. \color{red}{ P = w\varepsilon } with w = 1/3 for photons, w = 0 for matter so 0<w_pb<1/3.
will give acoustic oscillations : speed of sound
\color{red}{ c_s = \sqrt {\frac{{dP}}{{d\rho }}} = \sqrt {\frac{{c^2 dP}}{{d\varepsilon }}} = c\sqrt {w_{pb} } }
(i.e. in a pure γ or ν gas, v = .58c)

Self-gravitating fluid: fluctuations will cause higher density, and will attract fluid:

but if it gets too dense, pressure will drive it apart.

This gives Jeans length and Jeans mass
Jeans length = size of overdense region: anything bigger will collapse λ_J∼c_st_dyn
\color{red}{ \lambda _J \approx c_s t_{dyn} \approx \frac{{c_s c}}{{\left( {G\bar \varepsilon } \right)^{1/2} }}}
More accurately,
\color{red}{ \lambda _J \approx 2\pi c_s t_{dyn} }

However universe is expanding: flat universe has

\color{red}{ H^{ - 1} = \left( {\frac{{3c^2 }}{{8\pi G\bar \varepsilon }}} \right)^{1/2} = \left( {\frac{3}{2}} \right)^{1/2} t_{dyn} }

i.e. Hubble time ∼ collapse time. Hence in flat universe,

\color{red}{ \lambda _J \approx \frac{{c_s }}{H}\left( {2\pi \sqrt {\frac{2}{3}} } \right)}

Note this changes rapidly: in rad-dom universe

\color{red}{ \lambda _J \approx 3\frac{c}{H} \approx .6Mpc}

at decoupling

\color{red}{ a\left( t \right) = \left( {\frac{t}{{t_0 }}} \right)^{2/3} = \frac{1}{{1 + z}},H\left( t \right) = \frac{{\dot a}}{a} = \frac{2}{{3t}},z = 1100}

In matter-dom universe c_s = _b is speed of sound in gas, say few km/s. More accurately

\color{red}{ P = \frac{{\rho kT_m }}{{m_H }} \Rightarrow c_b = \sqrt {\frac{{dP}}{{d\rho }}} = \sqrt {\frac{{kT_m }}{{m_H }}} }

and we found T∼ 3000 K at decoupling.

but λ_J isn't very observable

Measuring the Jeans length

Jeans Mass

Critical mass of matter (not radiation) inside sphere defined by Jeans length

\color{red}{ M_J = \rho _b \left( {\frac{4}{3}\pi \lambda _J ^3 } \right)}

If actual mass M > M_J, sphere will collapse

Hence before dec.

\color{red}{ M_J = 4\pi \rho _b \left( {\frac{c}{H}} \right)^3 }

Note ρ_b ∼ 1/a³ and \color{red}{H\left( t \right) = \frac{{\dot a}}{a} = \frac{1}{{2t}}} are decreasing,

so M_J is increasing.

After dec, \color{red}{M_J \Rightarrow M_J \left( {\frac{{c_B }}{{c_s }}} \right)^3 }

and decreases thereafter

i.e. we could start collapsing material up to dec, but the lighter structures become unstable and get washed out. After dec, can only create small structs which gives us a problem

Note: ν's don't help: they freeze out but don't become non-rel: speed of sound ∼ .5 c.
Interpreted literally, this means that all structures formed in short time around dec.
but....

Add some WIMPs

These are heavy, freeze out very early and become non-rel. Repeat previous Jeans analysis, but now include expansion and ingnore pressure (since WIMPS are cold!) as before, consider a sphere, rad R with and over-density δ \color{red}{\rho = \bar \rho \left( {1 + \delta } \right)} and so

\color{red}{ \frac{{\ddot R}}{R} = - \frac{{4\pi G\bar \rho }}{3} - \frac{{4\pi G\bar \rho \delta \left( t \right)}}{3}}

But now add in expansion: ρ(t)∝ 1/a³ so

\color{red}{ R\left( t \right) = \frac{{R_0 }}{{a\left( t \right)\left( {1 + \delta \left( t \right)} \right)^{1/3} }} \Rightarrow \frac{{\ddot R}}{R} = \frac{{\ddot a}}{a} - \frac{1}{3}\ddot \delta - \frac{2}{3}\frac{{\dot a}}{a}\dot \delta }

and we end up with a Friedmann-like equation

\color{red}{ \frac{{\ddot a}}{a} - \frac{1}{3}\ddot \delta - \frac{2}{3}\frac{{\dot a}}{a}\dot \delta = - \frac{{4\pi G\bar \rho \delta }}{3} - \frac{{4\pi }}{3}G\bar \rho }

First and last terms give us usual expanding universe (put δ = 0) so put \color{red}{\frac{{\ddot a}}{a} = - \frac{{4\pi }}{3}G\bar \rho } gives eqn. for growth of fluct:

\color{red}{ \ddot \delta + 2\frac{{\dot a}}{a}\dot \delta = 4\pi G\bar \rho \delta }

ρ and δ only refer to matter. Can be rewritten

\color{red}{ \ddot \delta + 2H\dot \delta = \frac{3}{2}H^2 \Omega _m \delta }

Means if Ω_m<<1 (e.g if universe is rad or Λ dominated) eqn simplifies to

\color{red}{ \ddot \delta + 2H\dot \delta \approx 0}

whihc has solutions whihc alwyas stay small, so universe remains smooth. In matter dom,

\color{red}{ \ddot \delta + \frac{4}{{3t}}\dot \delta = \frac{2}{{3t^2 }}\delta }

whihc has power-law solutions

\color{red}{ \delta = Dt^n ,n = \frac{2}{3}, - 1}

n = -1 is not interesting (decaying solution) so

\color{red}{ \delta \propto t^{2/3} \propto a \propto \frac{1}{{1 + z}}}

i.e. if we had to wait for matter alone, would only get collapse (growth of fluctuations) after t_ls or z =1100. However, CDM decouples earlier and dominates earlier

Hence Scenario

CDM decouples
CDM dominates and clumps
Baryons decouple
Baryons clump onto CDM

Up to now we have idealised density fluctuations into spheres: in reality, they would be far more complicated. Just as we decompose a sound wave into Fourier, write δ as Fourier integral over co-moving volume V

\color{red}{ \delta \left( {\vec r} \right) = \frac{V}{{\left( {2\pi } \right)^3 }}\int {e^{ - i\vec k.\vec r} \delta _k d\vec k} }

and then each F component obeys

\color{red}{ \ddot \delta _k + 2H\dot \delta _k - \frac{3}{2}\Omega _m H^2 \delta _k = 0}

Wavelength assoc with this mode

\color{red}{ \lambda _k = \frac{{2\pi }}{k}a\left( t \right)}

so need \color{red}{\lambda _J < \lambda _k < \frac{c}{H}} Then power spectrum \color{red}{P\left( k \right) = \left\langle {\left| {\delta _{\vec k} } \right|^2 } \right\rangle } Best guess is that spectrum will be Gaussian: \color{red}{p\left( \delta \right) \sim e^{ - \frac{{\delta ^2 }}{{2\sigma ^2 }}} }

Power Spectrum

Pink noise has P(k)∼k, so higher frequncies are emphasised. Expected form

\color{red}{ P\left( k \right) \propto k^n ,n = 1}

If n= 0, would have "white-noise" spectrum (In acoustics "white noise" has flat power spectrum), means points would be randomly distributed. Larger n will give more "power at short distances": i.e. fluctuations will more likely to produce smaller objects. consider spheres, rad L, average mass

\color{red}{ \left\langle M \right\rangle = \frac{{4\pi L^3 \varepsilon _{m,0} }}{{3c^2 }}}

Then rms fluctuation
\color{red}{ \frac{{\delta M}}{M} = \left\langle {\left( {\frac{{M - \left\langle M \right\rangle }}{{\left\langle M \right\rangle }}} \right)^2 } \right\rangle ^{1/2} \propto \left( {k^3 P\left( k \right)} \right)^{1/2} \propto L^{ - \frac{{3 + n}}{2}} }
For random (Poisson distrib.) of masses m, number in sphere rad. L will be N = M/ m,

so fluctuation will be √N, so
\color{red}{ \frac{{\delta M}}{M} \sim \frac{{\sqrt N }}{N} \sim \frac{1}{{\sqrt N }}}
so n = 0
however we can write \color{red}{\frac{{\delta M}}{M} \approx M^{ - \left( {3 + n} \right)/6} } Must not diverge at large M (otherwise would make super-large structures) so
For n < -3, fluctuations will diverge at large L or M, so universe would not be homogeneous.

For random (Poisson distrib.) of masses m, number in sphere rad. L will be N = M/ m,
so fluctuation will be √N, so
\color{red}{ \frac{{\delta M}}{M} \sim \frac{{\sqrt N }}{N} \sim \frac{1}{{\sqrt N }}}
so n = 0
However also want to keep potential eenrgy finite:
\color{red}{ \delta \varphi \sim \frac{{G\delta M}}{L} \propto \frac{{\delta M}}{{M^{1/3} }} \propto M^{\left( {1 - n} \right)/6} }
so only simple power spectrum that works on all scales is n=1

HDM/CDM

need to be defined:

Hot dark Matter was still rel. when it decoupled: kT_dec >> m_xh²
CDM had kT_dec << m_hc²
More exactly, if rel. at time of creation of galaxies, would not have caused fluctuation: galaxies (or at least seeds) created close to matter-radiation equality (say z = 3700) when T ∼ 5000.
HDM would wipe out small scale fluctuations (particles free-stream)
CDM amplifies fluctuations.
if you aren't part of the solution you are part of the problem!
particle with mass m_hc² (energy in eV) become non-rel when temp is
\color{red}{ T_h \approx \frac{{m_h c^2 }}{{3k}} = 4000\left( {m_h c^2 } \right)K}
this happens at
\color{red}{ t_h \approx \frac{{5 \times 10^{11} }}{{\left( {m_h c^2 } \right)^2 }}}
note just low mass isn't enough: e.g axions (required to make theory of strong interactions CP invariant) have masses 10^-12 < m_Ac² < 10^-4 eV. HBowever, they are created at rest and never interact with any other particle, so never couple!

Implication: the H-Z spctrum is the starting point.

The acual size of the fluctuations gets modified.
free-streaming HDM wipes out all small structures
\color{red}{ \lambda _{\min } \sim ct_h \Rightarrow M_{\min } \sim \frac{{4\pi }}{3}\frac{{\lambda _{\min } }}{{a\left( {t_h } \right)}}\Omega _{m0} \rho _{c0} \sim \frac{{4 \times 10^{17} }}{{\left( {m_h c^2 } \right)}}M_O }
Hence if HDM consists of (say) 1 eV ν's which would almost close universe, would have no stable structure smaller than 10¹⁷M₀, which is too large.
CDM alows small structures to form immediately, and large structs form from them

Can simulate cluster formation with (massive) n-body codes

and on scale of galaxies.

{Performed at the National Center for Supercomputer Applications by Andrey Kravtsov (The University of Chicago) and Anatoly Klypin (New Mexico State University).

Conclusion

We can understand structure formation;

run simulations,
compare statistical properties with observations
CDM dominant, but need some HDM mix
Ω_CDM∼ .26
Ω_HDM∼ .03:
means either ν's have some mass, or have some new unknown HDM component

Re-Ionisation

A topic we have not discussed at all:
at t_ls the universe is matter-dom and the matter is neutral hydrogen.
What happens to it?
As the galaxies form, the stars emit vast quantities of UV, which re-ionises the H gas.
Universe was dark at z ∼ 1100: end of Dark Ages came by z ∼ 30.

Again Simulations are the only good way to go: need fully nonlinear, 3D treatment that includes hydrodynamics and Friedmann solutions
(Nick Gnedin, U Chicago)
Finally we will look at some (unrelated) speculations; the hairy edge of cosmology