Structure formation Consider a flat, matter dominated universe Ignore the cosmological constant - it wasn’t (very) important at high z when the first structures were forming

Imagine a spherical volume of the universe which is slightly denser than the background. How will this overdense region evolve with time as the Universe expands?

Recall Birkhoff’s theorem - gravitational force inside a sphere depends only on the matter inside Overdense region behaves exactly like a small closed Universe! ASTR 3830: Spring 2004 expansion of the flat size background universe solution for a realistic overdensity with some aspherical perturbation

closed universe solution for exact spherical time symmetry Schematic evolution: • Density contrast grows as universe expands

• Perturbation `turns around’ at R = Rturn, t = tturn • If exactly spherical, collapses to a point at t = 2 tturn • Realistically, bounces and virializes at radius R = Rvirial

ASTR 3830: Spring 2004 Can use the virial theorem (textbook Section 3.1) to derive the final radius of the collapsed perturbation. Let perturbation have mass M, kinetic energy , and gravitational potential energy .

Virial theorem: + 2 = 0 (in equilibrium) Energy conservation: + = constant

GM 2 GM 2 GM 2 - = - + KE = - Rturn Rvirial 2Rvirial energy at energy when applying turnaround virialized virial theorem Note: †ignore prefactor in potential energy here 1 Conclude: R = R virial 2 turn

ASTR 3830: Spring 2004

† How dense are collapsed objects? Can apply the solution for a closed universe to calculate the final overdensity in a spherical collapse model: Friedmann equation:

2 2 A 2 2 Ê 8pGˆ 3 a˙ = - kc where constant A: A = Á ˜ r0a0 a Ë 3 ¯ Solutions:

Ê 3Aˆ 2 3 † a = Á ˜ t 2 3 k = 0 solution† derived previously Ë 2 ¯

1 A2 a = 2 (1- cosY) 2 c Parametric solution for a closed, † 1 A2 k = 1 universe (not derived here) t = 3 (Y - sinY) 2 c ASTR 3830: Spring 2004

† Calculate the turnaround time for the collapsing sphere by finding when the size of the small closed universe has a maximum: da 1 A2 = 2 sinY = 0 Æ Y = 0,p,K d Y 2 c At turnaround, y = p, which corresponds to time, 1 A2 † t = p turn 2 c 3 The scale factor of the perturbation and of the background universe are just: † 1 A2 A2 asphere = 2 (1- cosp) = 2 2 c c 2 3 2 3 Ê 3Aˆ Ê 3p ˆ A2 a = Á ˜ t 2 3 = Á ˜ background Ë 2 ¯ turn Ë 4 ¯ c 2 ASTR 3830: Spring 2004 †

† The density contrast at turnaround is therefore:

-3 Ê ˆ 2 rsphere asphere 9p = Á ˜ = rbackground Ë abackground ¯ 16

At the time when the collapsing sphere virialized, at t = 2 t : turn† • Its density has increased by a factor of 8 • Background universe’s density has decreased by a factor of (22/3)3 = 4 Final result: a collapsing object virializes when its density is greater than the mean density of the universe by a factor of 18 p2 ~ 180… First objects to form are small and dense - these later merge to form larger structures - `bottom-up’ structure formation. ASTR 3830: Spring 2004 Non-spherical collapse Real perturbations will not be spherical - can gain qualitative insight into real behavior by considering collapse of an ellipsoidal overdensity:

Fastest collapse in the shortest direction: • Perturbation first pancakes • Then forms filaments Filamentary structure is seen both in numerical simulations of structure formation and in galaxy redshift surveys. ASTR 3830: Spring 2004 ASTR 3830: Spring 2004