Origin of Structure Formation of Structure Projected slice of 200,000 galaxies, with thickness of a few degrees.
Redshift Surveys Modern survey: Sloan Digital Sky Survey, probes out to nearly 1000 Mpc. 800 Mpc 400 Mpc Earth Structures limited to linear dimensions <~ 100 h -1 Mpc. Universe is homogeneous if averaged over scales >~ 200 Mpc. On smaller scales, Universe is very inhomogeneous. There is structure. Courtesy of Michael Blanton.
Origin of Structure Global Projection of the Earth s Map
COBE/DIRBE Satellite Black Body T=2.725 K
COBE/DIRBE Satellite Thermal component subtracted, ΔT=3.353 mk
COBE/DIRBE Satellite Dipole component subtracted, ΔT = 18 μk
Origin of Structure WMAP image
Fluctuations are ΔT/T ~ 10-5. Universe had inhomogeneities at z~1000. Origin of Structure WMAP image All components removed but background fluctuations
Dark Matter evolves under influence of gravitational physics millennium simulation of structure formation courtesy V. Springel
Dark Matter evolves under influence of gravitational physics millennium simulation of structure formation courtesy V. Springel
Dark Matter evolves under influence of galaxy clusters gravitational physics clustering amplitude ΛCDM prediction galaxy groups galaxies object number density Papovich (2008)
At z~1000, fluctuations were ~10-5. Gravity will cause positive fluctuations to grow with time. Today, massive clusters are more than 200x the critical density. (Galaxies, planets are even larger overdensities). Define Density Fluctuation as δ(r, t) = ρ(r, t) ρ(t) ρ(t) where ρ(t) is mean cosmic matter density at time t. From this definition, δ > -1 because ρ > 0. At z ~ 1000, δ << 1 because, ΔT/T~ 10-5. Dynamics of cosmic expansion is dictated by ρ, whereas the density fluctuations, δ, generate an additional gravitational field that affects local dynamics.
Consider region with δ > 0, so that gravity is stronger than average. Extra gravity means this region expands slowly than cosmic mean. Hence the contrast in this region increases. As relative density increases, region expands less slowly, and so on. This causes a Gravitational Instability.
Equations of Motion. Continuity Equation ρ t + (ρv) =0 Euler Equation v t +(v )v = P ρ Φ Here the potential gradient is the gravitational field, which satisfies the Poisson equation: 2 Φ=4πGρ And again we assume pressureless dust, so P = 0. For δ << 1 these 3 equations can be solved analytically (for this approximation). Otherwise we resort to N- body simulations.
Equations of Motion. Consider the problem in comoving coordinates, where r = a(t) x. v(r,t)=ȧ a r + u( r a,t) First term is cosmic expansion, 2nd term is peculiar velocity, which describes deviations from homogeneous expansion. Transform other equations from partial derivatives in time with fixed r to partial derivations in time with fixed x. Define ρx(x,t) = ρ(ax,t), etc.
Continuity Equation Now use notation, ρx=ρ, and δx=δ, and note that partial derivatives w.r.t. time imply fixed x. ρ = ρ(1 + δ) ρ a 3 Yields: Gravitational Potential is 1st term is potential for homogeneous density field, and 2nd term satisfies
In the homogeneous case, δ=0, u=0, ϕ=0, ρ=ρ Equations above imply ρ +3H ρ =0 ( Which we derived earlier from the 1st law of Thermodynamics for a pressureless expansion. ) Now consider approximate solutions from small deviations in density. 1st order approximation : keep 1st order terms in δ and u (disregard higher order terms uδ or u 2
Euler Equation in Comoving coordinates becomes Combining this with the time derivative of for P=0 leads to this equation contains only derivatives w.r.t. time. Therefore, the solutions must have the form:
D(t) is the Growth Factor, and satisfies the equation: General solutions are D(t) ~ t 2/3 and D(t) ~ t -1. Growing solution will dominate at late times, and decaying solution will be irrelevant. Normalizing D(t0) = 1 forces density contrast to be δ(x,t)=d(t) δ 0 (x) Linear Perturbation Theory yields: - Spatial Shape of density fluctuations is frozen in comoving coordinates. Only Amplitude increases as described by Growth Factor. - Growth factor D(t) follows simple differential equation, which depends on cosmological model.
Linear Perturbation Theory yields: - Spatial Shape of density fluctuations is frozen in comoving coordinates. Only Amplitude increases as described by Growth Factor. - Growth factor D(t) follows simple differential equation, which depends on cosmological model. D(a) H(a) H 0 a 0 [Ω m a 1 +Ω Λ a 2 (Ω m +Ω Λ 1)] 3/2 with the boundary condition D(t0) = D(a=1) = 1. Note!! δ0(x) would be the density fluctuations today if evolution was linear and δ0(x) << 1. This is clearly not going to be the case for virialized structures. Therefore, we refer to δ0(x) as the linearly extrapolated density fluctuation field.
EXAMPLE: Consider the dynamics of an unperturbed region with the mean density of mass m = 4πρ0 a0 3 /3, expanding with the Universe. (Taken from Peebles 1980, and Lacey & Cole 1993). Dynamical equations are the following. (Assume no mass shells cross, so m is a constant here): a Mass m d 2 a dt 2 = Gm r 2 2 da = 2Gm dt a + C Dust For C > 0 (positive total energy), solution to these equations takes the form: a = A(cosh η 1), t = B(sinh η η) A where the constants are related by 3 B 2 =4πGρ 0a 3 0/3
EXAMPLE: Consider the dynamics of an unperturbed region with the mean density of mass m = 4πρ0 a0 3 /3, expanding with the Universe. a = A(cosh η 1), t = B(sinh η η) A 3 where the constants are related by B 2 =4πGρ 0a 3 0x 3 /3 The values of A and B are determined by present-day Hubble constant, H0 = (da/dt) / a and the density parameter Ω0=8πGρ0 / (3H0 2 ), and by choosing the present value of the radius r0 = a0 x. These give together: η 0 = cosh 1 (2Ω 1 0 1) B =(1/2)H 1 0 Ω 0 (1 Ω 0 ) 3/2
Now consider dynamics of the expansion and eventual recollapse of a perturbed overdense region with the same mass m as the region of the background universe with which it is being compared. r Mass m Parametric equations take the form: a p = A p (1 cos θ), t = B p (θ sin θ) where the constants are related to those for the unperturbed case by A 3 p B 2 p = A3 B 2 =4πGρ 0a 3 0x 3 /3
As time progresses, the perturbed region expands less rapidly than the background universe, and a density contrast develops, given by 1+δ = a 3 /a 3 p where δ = Δρ/ρ = (ρp - ρ)/ρ. The region collapses to a singularity (aka virialized, bound structure) when ap = 0 when θ = 2π. For t=bp(θ - sinθ) this yields, tcoll = 2πBp. We must now match solutions for perturbed region and the unperturbed expanding universe by considering their evolution when θ 0.
a = A η 2 2! + η4 4! +... t = B η 3 3! + η5 5! +... θ 2 a p = A p 2! θ4 4! +... θ 3 t = B p 3! θ5 5! +... Eliminating the parametric variables (with some algebra...) we get 2/3 t 6 2/3 2/3 a = A 1+ 62/3 t +... B 20 20 B a p = A p t B p 2/3 6 2/3 20 1 62/3 20 t B p 2/3 +...
a = A Gravitational instabilities t B a p = A p t B p 2/3 6 2/3 20 2/3 6 2/3 20 1+ 62/3 Combined with 1+δ = (a/ap) 3 and keeping only terms to leading order gives δ = 3 2/3 2/3 62/3 1 1 + t 2/3 20 B p B 2/3 2/3 δ = 3(12π)2/3 1 1 + t 2/3 20 where we had defined t coll 20 1 62/3 20 2/3 t +... B 2/3 t +... B p t Ω =2πB = πh 1 0 Ω 0(1 Ω 0 ) 3/2 Therefore, we have shown that the perturbation grows as δ~t 2/3, the linear growing-mode solution. t Ω
δ = 3(12π)2/3 20 1 t coll 2/3 + 1 t Ω 2/3 t 2/3 For Ω0=1, tω, and we have δ = 3(12π)2/3 20 t t coll 2/3 For spheres with tcoll = t0 we have at t=t0 that δ 1.686. At later times t > tω the linear perturbation behavior departs from t 2/3, and the exact behavior follows (Peebles 1980): D(t) = And it can be shown that: 3 sinh η(sinh η η) (cosh η 1) 2 2 δ = 3 2 D(t) 1+ tω t coll 2/3
When the spherical perturbation collapses, we assume it reaches its virial equilibrium at then time tcoll when the expansion halts (ap=0) at a radius which is half of its radius at maximum expansion (θ=π). Therefore, 2 ρ 2π = (cosh η coll 1) 3 ρ vir sinh η coll η coll Compared to the critical density ρc = 3H(t) 2 / (8 πg) ρ =8π 2 (cosh η coll 1) 2 2 sinh η coll (sinh η coll η coll ) ρ c vir ηcoll << 1 (tcoll << tω) : ρvir = 18 π 2 178 ρc ηcoll >> 1 (tcoll >> tω) : ρvir = 8 π 2 80 ρc
Kitayama & Suto (1996, ApJ, 469, 480) give other equations for ρvir as a function of cosmological parameters: