El principio del fin: 1975

In the two previous sections we have considered anisotropies in the cos-mic cos-microwave background that arise from effects in the recent universe:

the motion of the earth relative to the cosmic microwave background, and the scattering of light by intergalactic electrons in clusters of galaxies along the line of sight. Now we turn to general anisotropies, including the highly revealing primary anisotropies that have their origin in the early universe.¹

It is convenient to expand the difference T (ˆn) between the microwave radiation temperature observed in a direction given by the unit vector ˆn and the present mean value T₀of the temperature in spherical harmonics Y^m

 (ˆn):

T (ˆn) ≡ T (ˆn) − T0 =

m

a_mY^m

 (ˆn) , T₀ ≡ 1 4π

d²ˆn T (ˆn) , (2.6.1)

6E. D. Reese et al., Astrophys. J. 533, 38 (2000).

7Y. Rephaeli, Astrophys. J. 245, 351 (1981); S. Cole and N. Kaiser, Mon. Not. Roy. Astron. Soc.

233, 637 (1988).

8E. Komatsu and T. Kitayam, Astrophys. J. 526, L1 (1999).

9B. S. Mason et al., Astrophys. J. 591, 540 (2003).

10E. Komatsu and U. Seljak, Mon. Not. Roy. Astron. Soc. 336, 1256 (2002); J. R. Bond et al., Astrophys. J. 626, 12 (2005).

1P. J. E. Peebles and J. T. Yu, Astrophys. J. 162, 815 (1970); R. A. Sunyaev and Ya. B. Zel’dovich, Astrophys. Space Sci. 7, 20 (1970); Ya. B. Zel’dovich, Mon. Not. Roy. Astron. Soc. 160, 1 (1972).

the sum over  running over all positive-definite integers, and the sum over m running over integers from − to . Since T (ˆn) is real, the coefficients a_m must satisfy the reality condition

a^∗

m= a_{ −m}. (2.6.2)

(We are defining the spherical harmonics so that Y^m

 (ˆn)^∗= Y_^−m(ˆn).) As we saw in Section 2.4, the earth’s motion contributes to T (ˆn) an anisotropy that to a good approximation is proportional to P₁(cos θ ) ∝ Y₁⁰(θ , φ) (with the z-axis taken in the direction of the earth’s motion), so the main a_m produced by this effect is that with  = 1 and m = 0.

The coefficients a_m reflect not only what was happening at the time of last scattering, but also the particular position of the earth in the universe.

No cosmological theory can tell us this. The quantities of greatest cosmolog-ical interest are averages, which may be regarded either as averages over the possible positions from which the radiation could be observed, or averages over the historical accidents that produced a particular pattern of fluctu-ations. The ergodic theorem described in Appendix D shows that, under reasonable assumptions, these two kinds of average are the same. These aver-ages will be denoted · · · . As discussed in Chapter 10, for anisotropies that arise from quantum fluctuations during inflation, it is these averages over historical accidents that are related to quantum mechanical expectation val-ues. We will return shortly to the question of how to use observations from one position in a universe produced by one specific sequence of accidents to learn about these averages, but first we must introduce some notation.

We assume that the universe is rotationally invariant on the average, so all averages T (ˆn1)T (ˆn2)T (ˆn3) · · ·  are rotationally invariant functions of the directions ˆn1, ˆn2, etc. In particular, T (ˆn) is independent of ˆn, Since T (ˆn) is defined as the departure of the temperature from its angular average, its angular average

T (ˆn)d²ˆn/4π vanishes. Averaging over the position of the observer, we have

T (ˆn)d²ˆn = 0, so since T (ˆn) is independent of ˆn, it too vanishes.

The simplest non-trivial quantity characterizing the anisotropies in the microwave background is the average of a product of two T s. Rotational invariance requires that the product of two as takes the form

a_ma_m^ = δ_^δ_{m −m}^C_, (2.6.3) for in this case the average of the product of two T s is rotationally invariant:

T (ˆn)T (ˆn^) =

m

C_Y^m

 (ˆn)Y_^−m(ˆn^) =



C_

2 + 1 4π



P_(ˆn · ˆn^) , (2.6.4)

2.6 Primary fluctuations in the microwave background: A first look

where P_ is the usual Legendre polynomial. Given the left-hand side, we can find C_by inverting the Legendre transformation

C_= 1 4π

d²ˆn d²ˆn^P_(ˆn · ˆn^)T (ˆn)T (ˆn^) . (2.6.5) Instead of Eq. (2.6.3), we could equivalently define the multipole coefficients C_by

a_ma^∗

^m^ = δ_^δm m^C_,

which shows that the C_are real and positive. For perturbations T that are Gaussian in the sense described in Appendix E, a knowledge of the C_ tells us all we need to know about averages of all products of T s.

Of course, we cannot average over positions from which to view the microwave background. What is actually observed is a quantity averaged over m but not position:

C^obs The fractional difference between the cosmologically interesting C_and the observed C^obs

 is known as the cosmic variance. Fortunately, for Gaussian perturbations, the mean square cosmic variance decreases with . The mean square fractional difference is If T is governed by a Gaussian distribution, then so are its multipole coefficients a_{ m} (but not quantities quadratic in the a_{ m}, such as C_.) It

2Non-Gaussian terms in the probability distribution of anisotropies would show up as non-vanishing averages of products of odd numbers of the a_{ m}, as well as corrections to formulas like Eq. (2.6.8) for the averages of products of even numbers of the a_{ m}. Such non-Gaussian terms are produced both in the early universe and at relatively late times. For a review, see N. Bartolo, E. Komatsu, S. Matarrese, and A. Riotto, Phys. Rep. 402, 103 (2004). Non-Gaussian terms produced by quantum fluctuations during inflation were calculated in the tree graph approximation by J. Maldacena, J. High Energy Phys.

Using Eq. (2.6.3), we find that the first term on the right-hand side of Eq. (2.6.8) contributes (2 + 1)²C²

 to the sum in Eq. (2.6.7), while the second and third terms each contribute (2 + 1)C_²to the sum, so that

C_− C_^obs C_

²

= 2

2 + 1 . (2.6.9)

This sets a limit on the accuracy with which we can measure C_ for small values of . On the other hand, the same analysis shows that for  = ^,

C_− C_^obs C_

 C_− C_^obs

C_



= 0 , (2.6.10)

so the fluctuations of C^obs

 away from the smoothly varying quantity C_ are uncorrelated for different values of . This means that when C_^obs is measured for all  in some range  in which C_ actually varies little, the uncertainty due to cosmic variance in the value of C_obtained in this range is reduced by a factor 1/√

. Even so, measurements of C_ for  < 5 probably tell us little about cosmology. Also, measurements for  > 2, 000 are corrupted by foreground effects, such as the Sunyaev–Zel’dovich effect discussed in the previous section. Fortunately there is lots of structure in C_at values of  between 5 and 2,000 that provides invaluable cosmological information.

The primary anisotropies in the cosmic microwave background arise from several sources:

1. Intrinsic temperature fluctuations in the electron–nucleon–photon plasma at the time of last scattering,³ at a redshift of about 1,090.

05 (2003) 013. The effect of loop graphs is considered by S. Weinberg, Phys. Rev. D 72, 043514 (2005) [hep-ph/0506236]; Phys. Rev. D. 74, 023508 (2006) [hep-ph/0605244]; K. Chaicherdsakul, Phys.

Rev. D 75, 063522 (2007) [hep-th/0611352]. For late-time contributions, see M. Liguori, F. K. Hansen, E. Komatsu, S. Matarrese, and A. Riotto, Phys. Rev. D 73, 043505 (2006) [astro-ph/0509098]. The weakness of microwave background anisotropies indicates that any non-Gaussian terms are likely to be quite small. So far, there is no observational evidence of such terms.

3Strictly speaking, in the approximation of a sudden drop in opacity at a fixed temperature T_L
3, 000 K, it is not the intrinsic fluctuation in temperature we observe, but the consequent fluctuation in the redshift z_Lof last scattering. Since the unperturbed temperature ¯T (t) goes as 1/a(t), the value aLof a(t) at which the total temperature ¯T (t)+δT (t) reaches a fixed value TLis shifted by an amount δaLsuch that −(TL/aL)δaL+ δT (tL) = 0. The observed temperature (leaving aside other effects) is TLa(tL)/a0, so to first order this is shifted by a fractional amount T_LδaL/a0T₀= δT (tL)aL/a0T₀= δT (tL)/TL, just as if it were the intrinsic temperature fluctuation that we observe.

2.6 Primary fluctuations in the microwave background: A first look

2. The Doppler effect due to velocity fluctuations in the plasma at last scattering.

3. The gravitational redshift or blueshift due to fluctuations in the gravitational potential at last scattering. This is known as the Sachs–

Wolfe effect.⁴

4. Gravitational redshifts or blueshifts due to time-dependent fluctuations in the gravitational potential between the time of last scattering and the present. (It is necessary that the fluctuations be time-dependent;

a photon falling into a time-independent potential well will lose the energy it gains when it climbs out of it.) This is known, somewhat confusingly, as the integrated Sachs–Wolfe effect.⁴

A proper treatment of these effects requires the use of general relativity. This will be the subject of Chapters 5–7. On the other hand, from the time the temperature dropped below 10⁴K until vacuum energy became important at a redshift of order unity, the gravitational field of the universe was dominated to a fair approximation by cold dark matter, which can be treated by the methods of Newtonian physics. Therefore in this introductory section we will concentrate on the Sachs–Wolfe and integrated Sachs–Wolfe effect, which turn out to dominate the multipole coefficients C_for relatively small

, less than about 40. We will make only a few tentative remarks in this section about the contribution of intrinsic temperature fluctuations and of the Doppler effect.

In considering the Sachs–Wolfe and integrated Sachs–Wolfe effects, we return to the Newtonian approach to cosmology outlined in Eqs. (1.5.21)–

(1.5.27). The treatment of perturbations to a homogeneous isotropic cosmology in this approach is presented in Appendix F. For the moment, we need only one result of this analysis, that the perturbation to the gravitational potential, when expressed as a function of the co-moving coordinate x, is a time-independent function δφ(x). This perturbation has two effects. First, there is the usual gravitational redshift: A photon emitted at a point x at the time of last scattering will have its frequency and hence its energy shifted by a fractional amount δφ(x), so the temperature seen when we look in a direction ˆn will be shifted from the average over the whole sky by an amount

T (ˆn) T₀



1

= δφ (ˆnrL) . (2.6.11)

4R. K. Sachs and A. M. Wolfe, Astrophys. J. 147, 73 (1967).

Here r_L is the radial coordinate of the surface of last scattering, given by

and t₀is the present. The perturbation to the gravitational potential also has the effect of changing the rate at which the universe expands by a fractional amount δφ(x), and since the temperature in a matter-dominated universe is falling like a⁻¹ ∝ t^−2/3, this shifts the value of the redshift at which the universe reaches the temperature
3, 000 K of last scattering in direction

ˆn by a fractional amount

 δz

With all other effects neglected, the temperature observed in direction ˆn would be 3,000 K divided by 1 + z, so this fractional shift in 1 + z changes the observed temperature by a fractional amount

T (ˆn) The sum of the fractional shifts (2.6.11) and (2.6.13) gives the Sachs–Wolfe

effect:  The factor 1/3 will be obtained in Chapter 7 as a result of a better-grounded relativistic treatment.

It is convenient to write δφ(x) as a Fourier transform δφ (x) =

d³q e^iq·xδφq. (2.6.15) We make use of the well-known Legendre expansion of the exponential

e^iq·x=



(2 + 1)i^P_( ˆq · ˆn) j_(qr) , (2.6.16) where j_ is the spherical Bessel function, defined in terms of the usual Bessel function J_ν(z) by j_(z) ≡ (π/2z)^1/2J_+1/2(z). Eq. (2.6.14) then

2.6 Primary fluctuations in the microwave background: A first look

Now we must consider how to calculate the average of a product of two of these fractional temperature shifts in two different directions. Although δφ (x) depends on position, the probability distribution of δφ (x) as seen by observers in different parts of the universe is invariant under spatial rotations and translations. This implies among other things that

δφqδφq^ = P_φ(q) δ³(q + q^) . (2.6.18) where P_φ(q) is a function only of the magnitude of q. (The delta function is needed so that δφ(x)δφ(y) should be only a function of x − y.) Because δφ (x) is real, its Fourier transform satisfies the reality condition δφq^∗= δφ_−q, which together with Eq. (2.6.18) tells us that P_φ(q) is real and positive.

Now, using Eqs. (2.6.17) and (2.6.18), together with the reflection prop-erty P_(−z) = (−1)^P_(z) and the orthogonality property of Legendre or, comparing with Eq. (2.6.4),

C_{, SW} = 16π²T₀² 9

∞

0

q²dq P_φ(q)j_²(qrL) (2.6.21) To the extent that the gravitational potential is produced by pressureless cold dark matter, the differential equation for δφ does not involve gradients, and so the differential equation for its Fourier transform does not involve the wave vector q. (See Eqs. (F.12) and (F.18).) The dependence of δφq

on q then can arise only from the initial conditions for these differential equations.⁵ It is therefore natural to try the hypothesis that the function

5This is not true even for the Sachs–Wolfe effect if q is so large that q/a became greater than H before the density of the universe became dominated by cold matter. As we will see in Chapter 7, this qualification affects C_{ SW}only for  > 100.

P_φ(q) has a simple form, such as a power of q. This power is conventionally written as n − 4:

P_φ(q) = N_φ²qⁿ⁻⁴ (2.6.22) where N²

φ is a positive constant. Then we can use a standard formula:

∞ and find that for  < 100, Eq. (2.6.21) gives

C_{ SW}→

In particular, even before the discovery of primary fluctuations in the cos-mic cos-microwave background there was a general expectation about the values of n and N , based not on the microwave background, but on the large scale structure of matter observed relatively close to the present. The perturba-tion δρ in the total mass density is related to the Sachs–Wolfe effect through the Poisson equation, which gives

a⁻²∇²δφ = 4π Gδρ , (2.6.25) with the factor a⁻²(t) inserted because it is X = a(t)x that measures proper distances. (See Eq. (F.12).) Expressing the Fourier transform of δρ in terms of the Fourier transform of δφ, we find the correlation function of the density fluctuations to be

δρ(x, t)δρ(x^, t^) = (4π Ga(t)a(t^))⁻²

d³q q⁴ P_φ(q) e^iq·(x−x). (2.6.26) The use of this formula to measure P_φ is discussed in Chapter 8. For the present, it is enough to note that observations of the density correla-tion funccorrela-tion long ago led to the expectacorrela-tion that P_φ takes the so-called Harrison–Zel’dovich form⁶with n = 1, and that N_φ ≈ 10⁻⁵. As we will see in Chapter 10, inflationary theories generally predict that n is close but not

6E. R. Harrison, Phys. Rev. D1, 2726 (1970); Ya. B. Zel’dovich, Mon. Not. Roy. Astron. Soc. 160, 1P (1972).

2.6 Primary fluctuations in the microwave background: A first look

precisely equal to unity. For n = 1, we obtain a result that is scale-invariant, in the sense of being independent of r_L:

C_{, SW}→ 8π²N²

φT₀²

9( + 1) . (2.6.27)

This is why experimental data on the cosmic microwave background anisotropies is usually presented as a plot of ( + 1)C_versus .

What about the other contributions to C_? Pressure gradients are important in the dynamics of the photon–nucleon–electron plasma, so in estimating the contributions of the Doppler effect and intrinsic temperature fluctuations we must deal with differential equations in which the wave num-ber q enters in an important way. Whatever sort of perturbation we are considering, we can always write it as a Fourier integral like Eq. (2.6.15) and use Eq. (2.6.16) to express e^iq·ˆnrL as a series of Legendre polynomials P_( ˆq · ˆn) with coefficients proportional to j_(qrL). The integral over q is then dominated by values of q of order /rL in the case   1. (This is the most interesting case because Eq. (2.6.9) shows that it is only for   1 that cosmic variance can be neglected in measurements of C_.) This is because for   1, the spherical Bessel function j_(z) is peaked at z
. Specifically, for ν ≡  + 1/2 → ∞ and z → ∞ with ν/z fixed at a value other than unity, we have

j_(z) →

0 z < ν

z^−1/2(z²− ν²)^−1/4cos√

z²− ν²− ν arccos(ν/z) −^π₄

z > ν.

(2.6.28) Hence C_for large  chiefly reflects the behavior of the Fourier components of perturbations for q ≈ /rL. To put this another way, the physical wave number at time t_Lis k_L ≡ q/a(tL) (because it is a(tL)x that measures proper distances at this time) so C_for large  reflects the behavior of the perturba-tions for k_L ≈ /dA, where d_Ais the angular diameter distance of the surface of last scattering

d_A ≡ rLa(tL) = 1

^1/2_K H₀(1 + zL)

× sinh



^1/2_K 1

1 1+zL

 dx

_x⁴+
Kx²+
Mx +
R

 , (2.6.29) with 1 + zL ≡ a(t0)/a(tL).

For physical wave numbers q/a that are much less than the expansion rate H the differential equation governing any one perturbation is essentially independent of the wave number q, so that the whole dependence of the perturbation on q comes from the initial conditions. (Ratios of different perturbations, such as δφq and δρq, may of course depend on q.) Such perturbations are said to be “outside the horizon” because the wavelength 2π a/q is much larger than the horizon distance (strictly speaking, the “par-ticle horizon” distance), which we saw in Section 1.13 is roughly of order 1/H . During the radiation- or matter-dominated eras q/a decreased like t^−1/2or t^−2/3, while H decreased faster, like 1/t, so perturbations that were outside the horizon at early times subsequently came within the horizon, those with high wave number entering the horizon earlier than those with lower wave numbers.

We need to be a little more precise about the horizon distance. At the time of last scattering the universe was largely matter dominated, so as shown in Section 1.13 the horizon distance at that time was approximately 2/H (tL).

But this is the maximum proper distance that light could have traveled since the beginning of the present phase of the expansion of the universe. As we will see in Chapters 5 and 6, the dominant perturbations to the plasma of nucleons, electrons, and photons that are relevant to the anisotropies in the cosmic microwave background are sound waves, so we need to take into account the smaller speed of sound. During the era before recombination, when radiation and matter were in thermal equilibrium, the speed of sound was vs = (δp/δρ)^1/2, where δp and δρ are infinitesimal variations in the pressure and density in an adiabatic fluctuation. In such a perturbation, the entropy per baryon remains unperturbed:

0 = δσ ∝ δ

where  is the thermal energy density, nBis the baryon number density, and p is the pressure. As we saw in Section 2.2, there are so many more photons than baryons that both  and p are dominated by radiation:

 = aBT⁴ , p = 1 3a_BT⁴ , and therefore for adiabatic perturbations

3δT

T = δnB

n_B = δρB

ρB

, (2.6.30)

where ρB is the baryonic mass density. This gives a sound speed

vs=

2.6 Primary fluctuations in the microwave background: A first look

where R ≡ 3ρB/4aBT⁴. Since ρB ∝ a⁻³and T ∝ a⁻¹, we have R ∝ a, and hence dt = dR/HR, or in more detail,

dt = dR

radiation equality. The acoustic horizon distance is then d_H ≡ aL become important when k_L ≈ 1/dH, and since in C_the integral over wave number is dominated by k_L ≈ /dA, gradients become important when  reaches the value H, where H ≡ dA/dH. Both d_Aand d_H are proportional to 1/H0, so H₀cancels in the ratio.

For a crude estimate of H, we note that d_Ais proportional to (1 + zL)⁻¹, while d_His proportional to (1+zL)^−3/2, so His of the order of (1+zL)^1/2 = 33. To get a closer estimate, we will take cosmological parameters to have sample values suggested by supernova observations and cosmologi-cal nucleosynthesis (discussed in Section 3.2):
M = 0.26,
_ = 0.74,

B = 0.043. This gives RL = 0.62, REQ= 0.21, dA= 3.38 H₀⁻¹(1 + zL)⁻¹ and d_H = 1.16 H₀⁻¹(1 + zL)^−3/2, and hence H = 2.9√

1 + zL = 96.

We can now estimate the relative magnitude of the contributions to C_ other than the Sachs–Wolfe effect:

• Doppler effect: Like any vector field, the plasma velocity can be decomposed into a term given by the gradient of a scalar, plus a

“vector” term whose divergence vanishes. Appendix F shows that the vector term decays as 1/a, so the dominant perturbations are compressional modes, for which the velocity is the gradient of a scalar.

We can therefore expect that in the integral over wave numbers q for

T /T0, the contribution of the Doppler effect will be suppressed for small wave numbers by a factor of order k_Ld_H. We will see in Chapter 7 that, because it is proportional to the vector k_L, the Doppler effect contribution does not interfere with the contribution of the Sachs–

Wolfe effect, so for a multipole order   H, the contribution of the Doppler effect will be less than that of the Sachs–Wolfe effect by a factor of order [(/dA)dH]² = ²/²_H.

• Intrinsic temperature fluctuations: As we have seen in Eq. (2.6.30) the intrinsic fractional temperature fluctuation at the time of last scattering will be just one-third the intrinsic fractional perturbation in the plasma density. As discussed in Chapters 5 and 6, the particular perturbations that are believed to dominate outside the horizon are adiabatic in the further sense that the fractional perturbation in the plasma density is equal to the fractional perturbation δρ/ ¯ρ in the total matter density.

But the perturbation to the total matter density is related to the per-turbation to the gravitational potential by Poisson’s equation (2.6.25), which if evaluated at the time of last scattering gives for the Fourier transform:

δρq(tL) = − q²

4π Ga²(tL)δφq= − k²

L

4π Gδφq ,

where, as before, k = q/a(tL). Also, the mean total mass density ¯ρ(tL) at last scattering is related to the horizon size d_H by

¯ ρ(tL)

3H²(tL) 8π G ≈

1 2π Gd_H²

,

so the order of magnitude of the intrinsic fractional temperature pertu-rbation is related to the pertupertu-rbation to the gravitational potential by

δT (tL)

T (t¯ L) = δρ(tL)

3 ¯ρ(tL) ≈ k_L²d_H²δφq . (2.6.33) Thus we expect that in the integral over wave numbers q for T /T0, the contribution of intrinsic temperature fluctuations will be suppressed for small wave numbers by a factor k²

Ld²

H. The interference of this contribution with the Sachs–Wolfe term then makes a contribution to C_ for   horizon that, like the contribution of the Doppler effect, is smaller than the Sachs–Wolfe contribution by a factor of order

H. The interference of this contribution with the Sachs–Wolfe term then makes a contribution to C_ for   horizon that, like the contribution of the Doppler effect, is smaller than the Sachs–Wolfe contribution by a factor of order

In document TESIS DOCTORAL (página 120-124)