AST4320: LECTURE 10 M. DIJKSTRA

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1 AST4320: LECTURE 10 M. DIJKSTRA 1. The Mass Power Spectrum P (k) 1.1. Introduction: the Power Spectrum & Transfer Function. The power spectrum P (k) emerged in several of our previous lectures: It fully characterised the properties of a Gaussian random (density) field. It determined σ(m) in Press-Schechter theory. We constrain(ed) the slope d log P (k)/d log k from the observed galaxy two-point correlation function in assignment 5. Now we will discuss some key properties of P (k). Linear theory allowed us to describe the time evolution of a density perturbation δ with wavenumber k. This time evolution was (1) δ(k, t) = δ(k, t = 0)D + (t), where we derived that D + (t) t 2/3 a in the Einstein-de Sitter Universe (Ω m = 1.0, Ω Λ = 0.0). Therefore, (2) P (k, t) = P (k, t = 0)D 2 +(t) P 0 (k)d 2 +(t). One of the goals of modern cosmology is to calculate P 0 (k). There are no preferred length scales in the very early Universe, and the only functional form for P 0 (k) with no scale length is a power law: (3) P 0 (k) = Ak n. We may think of this P 0 (k) as the primordial power spectrum as it described density fluctuations at t = 0. As we will see next, perturbations with different wave numbers evolved differently in the very early Universe. This modifies the matter power spectrum from the power law form given above. These modifications are encoded in the so-called transfer function T (k). This transfer function T (k) encodes the information on the evolution of some density perturbation δ(k), and therefore affects the power spectrum as (4) P 0 (k) = Ak n T 2 (k). Our goal is now to obtain some intuition for T (k), and expressions for some limiting behaviour of T (k). Before proceeding it is useful to discuss the concept of a horizon, as it plays a key role in shaping the transfer function. 1

2 2 M. DIJKSTRA 1.2. The Horizon. There are many horizons in cosmology. We focus on the so called particle horizon, which corresponds to the maximum proper distance over which there can be causal communication at time t (Quote lifted from M. Longair s book). The particle horizon thus corresponds to the maximum proper distance a photon could have travelled between t = 0 and t = t. This distance corresponds to (5) r H (t) = a(t) t 0 cdt a(t ). The integral gives the total comoving distance traversed by the photon. The term a(t) converts that into a proper distance at time t. During radiation domination the scale factor a(t) t 1/2 Ct 1/2, where C is a constant. We can solve the integral to give r H = 2ct. This is a factor of two larger than the maximum distance a photon could travel in a static medium. The factor of 2 accounts for the fact that space itself is expanding during the photon s flight. Now consider the evolution of a perturbation of some proper length/wavenumber L. The time evolution of the wavelength of this perturbation is given by L = L 0 (a/a 0 ) = L 0 (t/t 0 ) 1/2. Consider a perturbation for which L > r H : because L > r H there is no way to know what lies outside of L. It is therefore impossible to determine what the mean density of the background should be, and therefore whether the perturbation L is overdense or under dense. We need general relativity to describe the time evolution of perturbations larger than the horizon scale. I will comment on this later. While we currently cannot say much about the expected time evolution of the perturbation L, we can predict that L will become smaller than the horizon in the future: If L > r H at some time t 1, then L 2 = L(t 2 /t 1 ) 1/2 at some later time t 2 while the horizon at this time r H,2 = r H (t 2 /t 1 ). The perturbation size equals the horizon scale when (6) r H (t 2 /t 1 ) = L(t 2 /t 1 ) 1/2 r H (t 2 /t 1 ) 1/2 = L t 2 = t 1 (L/r H ) 2 > t 1. After the perturbation enters the horizon we can apply our classical (non-relativistic) perturbation theory The Transfer Function T (k). Figure 1 shows the time evolution of a perturbation δ(k) (with corresponding wavelength or size of the perturbation L = 2π/k) that enters the horizon during the radiation dominated era at scale factor a enter ). There are three key events during the evolution of this perturbation: When the matter density starts to dominate the Universal energy density - this happens at the redshift of matter-radiation equality z eq the dark matter perturbation grows as δ a. This is what we derived in previous lectures. At redshifts z > z eq - i.e. a < a eq - the Universal energy density is dominated by radiation. During radiation domination the scale factor grows as a a 1/2 (as opposed to a t 2/3 that we found during matter dominance). This different time

3 AST4320: LECTURE 10 3 Figure 1. This Figure shows (schematically) the time evolution of an overdensity δ on some scale L that enters the horizon at a enter a eq. The time evolution goes through three different phases: (i) δ a 2 before horizon entry which follows from general relativistic perturbation theory; (ii) δ =constant after horizon entry, and up until a eq. This stalling of the growth of the perturbation is known as the Meszaros effect; (iii) when matter starts to dominate the Universal energy density δ a as we derived in previous lecture. This Figure illustrates that the Meszaros effect suppress the growth of this perturbation by a factor of (a enter /a eq ) 2 compared to uninhibited growth. dependence of the scale factor - combined with the fact that radiation dominates the Universal (mass-)energy density - gives rise to a drastically different predicted time evolution for δ. As we will see next, δ barely grows at all during radiation dominance. This stalling of the growth of density perturbation in the radiationdominated era is known as the Meszaros effect. The fluctuations are said to be frozen in the background. Mathematically, the Meszaros effect is easy to understand. Recall that the density evolution of a perturbation δ was given by the following differential equation: (7) δm + 2ȧ a δ m = 4πGρ m δ m. Divide both sides by H 2 = 8πGρtot 3, where ρ tot = ρ rad + ρ m. Using H = ȧ a we find

4 4 M. DIJKSTRA (8) δ m H H δ m = 3ρ mδ m 2[ρ m + ρ rad ]. If we now use that deep in the radiation dominated era ρ rad ρ m, then the term on the RHS can be ignored. This is because we are multiplying a small number δ with another small number, and can see this term effectively as a second order term. The differential equation then simplifies to (9) δ m H + 2 δ m = 0. If we further use that H = ȧ/a = 1/[2t] then we are left with (10) δm + δ m t = 0 δ m = A + B log t = A + C log a. The perturbation thus only grows logarithmically with the scale factor. This growth is represented by the horizontal line in Figure 1. Before the perturbation enters the horizon, at a < a enter, it grows δ a 2. As I mentioned in the lecture, this follows from general relativistic perturbation theory. This is beyond the scope of this lecture. Because the result from GR is not intuitive, we might as well have replaced the words general relativistic perturbation theory with magic 1. Figure 1 shows that for the perturbation that entered the horizon during radiation ( ) dominance at a enter the growth was suppressed by a factor of T (k) = aenter 2. a eq This suggests that T (k) 1 if a enter a eq. Indeed, for perturbations that enter the horizon during matter domination we do not have any inhibition of growth: δ a 2 during radiation dominance, and δ a during matter dominance (see footnote). This discussion clearly suggest that there is a particular scale of interest, namely that for which a enter = a eq. This corresponds to the smallest scale for which there is no suppression of growth by the Meszaros effect. The size of this perturbation is therefore equal to the horizon scale at matter-radiation equality: (11) L 0 = r H (a eq ) = a eq teq 0 cdt a(t) =... = assignment 5 c 1 80 cmpc, H 0 2Ωm,0 z eq 1 I found a prescription later today that might provide some more insight than requiring magic: density perturbation δ k generate perturbations in the gravitational potential Φ k which correspond to metric perturbation in general relativity (matter warps space time). The equation that described the time evolution of this metric perturbation corresponds to the perturbed Poisson equation from lecture 2, which in Fourier space reads k 2 Φ k = 4πGa 2 ρδ k. If we require that the metric perturbation cannot evolve for perturbations outside the horizon, then we must have that a 2 ρδ k is independent of time. We therefore must have δ k a 2 during radiation domination (ρ rad a 4 ), and δ k a during matter domination (ρ rad a 3 ).

5 AST4320: LECTURE 10 5 where Ω m,0 denotes the present-day mass density parameter, H 0 is the present-day Hubble constant, and cmpc denotes comoving Mpc (just to emphasis that L 0 is a comoving quantity). The corresponding wavenumber is k 0 = 0.1 cmpc 1. Finally, we would like to express T (k) as a function of k. We found that the suppression ( ) T (k) for perturbations entering at a < a eq was given by T (k) = aenter 2. A perturbation of length L L 0 enters the horizon at a enter (L) a eq (L/L 0 ) = a eq (k 0 /k), where in the first approximation we assumed that the horizon scale evolves as r H a out to a eq. ( ) 2 Under this approximation we therefore have T (k) = k0 k for k k0. One of the slides shows the T (k) obtained from a more precise calculation (Bardeen et al. 1986). I have also explicitly plotted the slope d log T/d log k. The approximate calculation given above captures the limiting behaviour of T (k) and identifies the physical reason for the turn-over in the transfer function. a eq 1.4. The Power Spectrum P (k). The power spectrum is given by P 0 (k) = Ak n T 2 (k). We have specified T (k). The slope of the primordial power spectrum has been inferred (from the Cosmic Microwave Background) to be close to 1. This value is predicted naturally by inflation theories. The power spectrum therefore scales as { k k k0 (12) P (k) k 3 k k 0, with a turn-over at k = k 0. Several additional comments on P (k) The normalisation constant A is obtained by matching to observations of the Cosmic Microwave Background. Baryons affect precise shape of P (k) (see paper by Eisenstein & Hu 1998). One example of how baryons affect the mass power spectrum was given in our discussion of the acoustic peak in the two-point correlation function (the single acoustic peak in ξ(r) corresponds to a series of oscillations in the power spectrum). The case n = 1 corresponds to a special case which yields scale-invariant fluctuations. This is discussed next Why n = 1 corresponds to Scale Invariance. We derived the relation between P (k) and the variance in the mass density field averaged over some mass-scale M in previous lectures. The RMS (root mean square) amplitude of fluctuations smoothed over mass-scale M is given by (13) For a < a enter δ a 2, so σ 2 (M) (M) M (n+3/6) (14) (M) a 2 M 2/3. M 2/3. n=1

6 6 M. DIJKSTRA This equation shows that M is smaller for larger M. However, higher masses correspond to larger scales. These higher mass fluctuations therefore enter the horizon at a later time, and the Meszaros effect limits their growth by a smaller factor. We can relate the mass M of a perturbation to the scale factor at which it entered the horizon as (15) M = M h (t) ρ m r 3 H a 3 t 3 t 3/2 t 3 t 3/2, where we used that the mass density ρ m a 3 and that the proper horizon scale scales as r H t. We have derived the time-dependence of horizon entry of fluctuations of mass M. Substituting this into Eq 14 we have (16) (M) a 2 M 2/3 a 2 t 1 = constant. Fluctuations that enter the horizon 2 at a < a enter therefore have the same RMS amplitude at horizon entry. The subsequent growth of these perturbations is stalled until a eq, after which they all grow as δ a. This identical growth ensures that the RMS amplitude of these fluctuations remains independent of scale at all times. Although I have not shown it in the lecture, you can do the same analysis for perturbations that enter the horizon at a > a enter and get the same result: namely that (M)=constant! This shows that all fluctuations enter the horizon with the same RMS amplitude. This remarkable scale invariance is a special property of the power spectrum with n = Some Concluding Remarks. A very brief & broad summary of the processes that are relevant for the formation of structure in our Universe. At time t 0 some process (inflation) generates the primordial power spectrum P (k) = Ak n with n 1. Processes like horizon entry of a perturbation combined with the Meszaros effect then modify the shape of the power spectrum into P (k) = Ak n T 2 (k) at a a eq where T (k) = 1 for k k cmpc 1, and T (k) = (k 0 /k) 2 for k k 0. Baryons provide further smaller modifications of this power spectrum. At a eq < a < a rec dark matter perturbations grow as δ a, while radiation pressure prevent baryons from collapsing on all scales smaller than the Jeans length (which is very large during this epoch). Acoustic waves generated by density perturbations permeate the primordial photon-baryon plasma, which introduce further smaller corrections to the power spectrum P (k). At a > a rec the sound speed drops by five orders of magnitude, and the consequently, the Jeans mass drops by 10 orders of magnitude. Baryons are now free to collapse into the potential wells generated by the dark matter (i.e δ b δ DM ). This is important: the RMS amplitude of the density fluctuations in baryons at a rec is only δ b 10 5, which would not be enough to form non-linear objects (recall that δ a and that a increases only by a factor of 10 3 ). Press-Schechter theory allows us to take a Gaussian random density field - which describes the density field post recombination still extremely well - and transform 2 Convince yourself that this corresponds to fluctuations on mass-scales that are relevant for astrophysical objects (galaxies, groups of galaxies, clusters of galaxies).

7 AST4320: LECTURE 10 7 this into predictions for the abundance (number density) and clustering of (nonlinear) collapsed objects. While this theory has many flaws, it predicts the number density of collapsed objects as a function of mass M and redshift z remarkably well. It also highlights the important hierarchical aspect of structure formation: namely that small (i.e. low mass) objects collapse first, and that larger (i.e. more massive) objects collapse later Shortcomings/Caveats. Our discussion of structure formation does not explain the most apparent visual appearance of structures in the Universe, namely the walls, filaments, and nodes that are apparent in observed galaxy distributions and numerical simulations (see slides). The main reason is that in our discussion of the non-linear growth of structure we focussed on spherical top-hat model. In this model the evolution of the perturbation is determined entirely by its radius R(t) and the overdensity inside of it (δ(t)). Most R1 R2 Figure 2. Geometry for a simple ellipsoidal perturbation. perturbations in Gaussian random fields are not spherical. In fact, Gaussian random field theory can be used to study shapes of peaks in the density field. Consider the simple case in which an overdensity is ellipsoidal instead of spherical, and that there are two characteristic axes denoted with R 1 and R 2. Outside of this ellipsoid the overdensity is δ = 0. Clearly, the mean overdensity inside the sphere of radius R 1 is larger than that inside the sphere of radius R 2. This implies that less growth in δ is required for the perturbation reach the critical linear overdensity for collapse δ crit = The structure therefore collapses along its R 1 axes while it still has a finite size in its R 2 direction. Gravity therefore amplifies these deviations from spherical symmetry into flattened objects like walls (collapse along 1 axis), filaments (collapse along two axes), and halos (collapse along all three axes). So

8 8 M. DIJKSTRA while our discussion of structure formation in previous lectures does not produce these observed & simulated features in the mass distribution, the theory is easily adjusted to be able to explain these structures Useful Reading. Useful reading material includes For the discussion of Transfer functions I used the book by Peter Schneider Extragalactic Astronomy and Cosmology: An Introduction. Useful discussions on horizon scales in cosmology, and the difficulty of modelling super horizon scales are given in Longair Galaxy Formation. Specifically Chapter The lectures by Frank van den Bosch have some very nice & clear slides. See for a very brief discussion why δ a 2 before horizon entry. His site also has references to his book (Mo, Van den Bosch & White) which provides much more details to these lectures. Can be very useful.