6 Structure Formation – I

6 Structure formation – I 6.1 Newtonian equations of motion We have decided that perturbations will in most cases effectively be described by the Newtonian potential, Φ. Now we need to develop an equation of motion for Φ, or equivalently for the density fluctuation ρ (1 + δ)¯ρ. In the Newtonian approach, we treat dynamics of cosmological≡ matter exactly as we would in the laboratory, by finding the equations of motion induced by either pressure or gravity. We begin by casting the problem in comoving units: x(t) = a(t)r(t) (177) δv(t) = a(t)u(t), so that x has units of proper length, i.e. it is an Eulerian coordinate. First note that the comoving peculiar velocity u is just the time derivative of the comoving coordinate r: x˙ =a ˙r + ar˙ = Hx + ar˙, (178) where the rhs must be equal to the Hubble flow Hx, plus the peculiar velocity δv = au. The equation of motion follows from writing the Eulerian equation of motion as x¨ = g0 + g, where g = Φ/a is the peculiar −∇∇∇∇∇∇∇ acceleration, and g0 is the acceleration that acts∇∇∇ on a particle in a homogeneous universe (neglecting pressure forces to start with, for simplicity). Differentiating x = ar twice gives a¨ x¨ = au˙ + 2˙au + x = g0 + g. (179) a The unperturbed equation corresponds to zero peculiar velocity and zero peculiar acceleration: (ä/a) x = g0; subtracting this gives the perturbed equation of motion u˙ + 2(˙a/a)u = g/a = Φ/a. (180) −∇∇∇∇∇∇∇∇∇∇ This equation of motion for the peculiar velocity shows that u is affected by gravitational acceleration and by the Hubble drag term, 2(˙a/a)u. This arises because the peculiar velocity falls with time as a particle attempts to catch up with successively more distant (and therefore more rapidly receding) neighbours. In the absence of gravity, we get δv 1/a: momentum redshifts away, just as with photon energy. ∝ The peculiar velocity is directly related to the evolution of the density field, through conservation of mass. If we restrict ourselves to a the linear approximation where δ 1, the linearized continuity equation for conservation of momentum≪ and matter as experienced by fundamental observers moving with the Hubble flow is: δ˙ = u. (181) −∇∇∇∇∇∇∇∇∇∇∇∇·∇ (see e.g. Section 15.3 of Peacock 1999 for the details). The solutions of these two equations can be decomposed into modes either parallel to g or independent of g (these are the homogeneous and inhomogeneous solutions to the equation of motion). We are particularly interested in the growing mode. First eliminate u 52 by taking the divergence of the equation of motion for u, and then the time derivative of the continuity equation. This requires a knowledge of g, which comes via Poisson’s equation: g = 4πGaρ0δ. The resulting∇∇∇∇∇∇∇∇∇∇∇∇·∇ 2nd-order equation for δ is ∇∇∇∇∇∇∇∇∇∇∇∇·∇ − a˙ δ¨ + 2 δ˙ = 4πGρ0 δ. (182) a 2 This is easily solved for the Ωm = 1 case, where 4πGρ0 = 3H /2 = 2/3t2, and a power-law solution works: − δ(t) t2/3 or t 1. (183) ∝ The first solution, with δ(t) a(t) is the growing mode, corresponding to the gravitational instability∝ of density perturbations. Given some small initial seed fluctuations, this is the simplest way of creating a universe with any desired degree of inhomogeneity. Radiation-dominated universe The analysis so far does not apply when the universe was radiation dominated (cs = c/√3). For this period of the early Universe it is therefore common to resort to general relativity perturbation theory or use special relativity fluid mechanics and Newtonian gravity with a relativistic source term (see e.g. Section 15.2 of Peacock 1999 for the details). In the interests of brevity and completeness we simply quote the result of this analysis where the resulting evolution equation for δ has a driving term on the rhs that is a factor 8/3 higher than in the matter-dominated case a˙ 32π δ¨ + 2 δ˙ = Gρ0δ, (184) a 3 For the Ωm = 1 case a power-law solution gives: − δ(t) t or t 1. (185) ∝ The growing mode during radiation domination (a t1/2) has δ(t) a(t)2 . ∝ ∝ the general case We have solved the growth equation for the matter-dominated Ω = 1 case and seen the solution for the radiation- dominated Ω = 1 case which we can safetly use to describe the early high redshift Universe. In the more general case it is necessary to integrate the differential equation numerically. In general, we can write δ(a) a f[Ω(a)], (186) ∝ where the factor f expresses a deviation from the simple growth law. The case of matter + cosmological constant is of the most common practical interest, and a very good approximation to the answer is given by Carroll et al. (1992): −1 5 4/7 1 1 f(Ω) 2 Ωm Ωm Ωv +(1+ 2 Ωm)(1 + 70 Ωv) . (187) ≃ h − i This is accurate, but still hard to remember. For flat models with 0.23 Ωm + Ωv = 1, a simpler approximation is f Ωm , which is less marked than f Ω0.65 in the Λ = 0 case. This≃ reflects the more rapid ≃ m variation of Ωv with redshift; if the cosmological constant is important dynamically, this only became so very recently, and the universe spent more of its history in a nearly Einstein–de Sitter state by comparison with an open universe of the same Ωm. 53 6.2 The Jeans Scale So far, we have considered the collisionless component. For the photon- baryon gas, all that changes is that the peculiar acceleration gains a term from the pressure gradients: g = Φ/a p/(aρ). (188) −∇∇∇∇∇∇∇∇∇∇ − ∇∇∇∇∇∇∇∇∇∇ The pressure fluctuations are related to the density perturbations via the sound speed ∂p c2 . (189) s ≡ ∂ρ Now think of a plane-wave disturbance δ e−ik·r, where k and r are in comoving units. All time dependence is∝ carried by the amplitude of the wave. The linearized equation of motion for δ then gains an extra term on the rhs from the pressure gradient: a˙ 2 2 2 δ¨ + 2 δ˙ = δ 4πGρ0 c k /a . (190) a − s This shows that there is a critical proper wavelength, the Jeans length, at which we switch from the possibility of gravity-driven growth for long-wavelength modes to standing sound waves at short wavelengths. This critical length is proper 2π π λJ = proper = cs . (191) kJ rGρ It is interesting to work out the value of this critical length. Consider a universe with a coupled photon-baryon fluid and ignore dark matter (which we can do at high redshifts, near matter- 2 radiation equality). The sound speed, cS = ∂p/∂ρ, may be found by thinking about the response of matter and radiation to small adiabatic compressions: 2 δp = (4/9)ρrc (δV/V ), δρ = [ρm + (4/3)ρr](δV/V ), (192) implying −1 −1 2 2 9 ρm 2 9 1 + zrad cS = c 3 + = c 3 + . (193) 4 ρr 4 1 + z Here, zrad is the redshift of equality between matter and photons; 1 + zrad = 1.68(1 + zeq) because of the neutrino contribution. At z 3 2 ≪ zrad, we therefore have cS √1 + z. Since ρ = (1+ z) 3ΩBH0 /(8πG), the comoving Jeans length∝ is constant at 2 1/2 c 32π 2 −1 λJ = = 50 (ΩBh ) Mpc. (194) H0 27ΩB(1 + zrad) This is of order the horizon size at matter-radiation equality. Smaller- scale fluctuations in the photon-baryon fluid will not have undergone steady gravitational growth, but will have oscillated with time as standing waves. We will see later that the imprint of these oscillations is visible in the microwave background. 54 6.3 Perturbation growth in the early Universe Figure 14 shows a schematic of how a density fluctuation grows in the early Universe. For scales greater than the horizon, perturbations in matter and radiation can grow together, so fluctuations at early times grow at the same rate, independent of wavenumber. But this growth ceases once the perturbations ‘enter the horizon’ – i.e. when the horizon grows sufficiently to exceed the perturbation wavelength. At this point, growth ceases. For fluids (baryons) it is the radiation pressure that prevents the perturbations from collapsing further. For collisionless matter the rapid radiation driven expansion prevents the perturbation from growing again until matter radiation equality. This effect (called the Mészáros effect) is critical in shaping the late-time power spectrum (as we will show) as the universe preserves a ‘snapshot’ of the amplitude of the mode at horizon crossing. aenter aeq log(δ ) α a α a 2 log(a) Figure 14. A schematic of the suppression of fluctuation growth during the radiation dominated phase when the density perturbation enters the horizon at aenter < aeq. 6.4 The Shape of the Matter Power Spectrum The fluctuation or matter power spectrum Pδ(k) emcompasses the power or amplitude of each mode of density fluctuations (refer back to section 5.1). From inflation we predict a scale invariant initial Zeldovich spectrum where Pi(k) k. How does the Mészáros effect modify the shape of this initial power∝ spectrum? Figure 15 shows that the smallest physical scales (largest k scales) will be affected first and experience the strongest suppression 55 to their amplitude. The largest physical scale fluctuations (smallest k scales) will be unaffected as they will enter the horizon after matter- radiation equality. We can therefore see that there will be a turnover in the power spectrum at a characteristic scale given by the horizon size at matter-radiation equality.

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