4 Cosmological Perturbation Theory

Transcription

1 4 Cosmologcal Perturbaton Theory So far, we have treated the unverse as perfectly homogeneous. To understand the formaton and evoluton of large-scale structures, we have to ntroduce nhomogenetes. As long as these perturbatons reman relatvely small, we can treat them n perturbaton theory. In partcular, we can expand the Ensten equatons order-by-order n perturbatons to the metrc and the stress tensor. Ths makes the complcated system of coupled PDEs manageable. 4.1 Newtonan Perturbaton Theory Newtonan gravty s an adequate descrpton of general relatvty on scales well nsde the Hubble radus and for non-relatvstc matter (e.g. cold dark matter and baryons after decouplng). We wll start wth Newtonan perturbaton theory because t s more ntutve than the full treatment n GR Perturbed Flud Equatons Consder a non-relatvstc flud wth mass densty, pressure P and velocty u. Denote the poston vector of a flud element by r and tme by t. The equatons of moton are gven by basc flud dynamcs. 1 Mass conservaton mples the contnuty equaton whle momentum conservaton leads to the Euler t = r r ( u), (4.1.1) (@ t + u r r ) u = r rp r r. (4.1.2) The last equaton s smply F = ma for a flud element. The gravtatonal potental determned by the Posson equaton s r 2 r =4 G. (4.1.3) Convectve dervatve. Notce that the acceleraton n (4.1.2) s not gven t u (whch measures how the velocty changes at a gven poston), but by the convectve tme dervatve D t u (@ t + u r) u whch follows the flud element as t moves. Let me remnd you how ths comes about. Consder a fxed volume n space. The total mass n the volume can only change f there s a flux of momentum through the surface. Locally, ths s what the contnuty equaton t + r j ( u j ) =. Smlarly, n the absence of any forces, the total momentum n the volume 1 See Landau and Lfshtz, Flud Mechancs. 77

2 78 4. Cosmologcal Perturbaton Theory can only change f there s a flux through the t ( u )+r j ( u u j ) =. Expandng the dervatves, we t ( u )+r j ( u u j )= [@ t + u j r j ] u + u [@ t + r j ( u j )] {z } = = [@ t + u j r j ] u. In the absence of forces t s therefore the convectve dervatve of the velocty, D t u, that vanshes, t u. Addng forces gves the Euler equaton. We wsh to see what these equaton mply for the evoluton of small perturbatons around a homogeneous background. We therefore decompose all quanttes nto background values (denoted by an overbar) and perturbatons e.g. (t, r) = (t) + (t, r), and smlarly for the pressure, the velocty and the gravtatonal potental. Assumng that the fluctuatons are small, we can lnearse eqs. (4.1.1) and (4.1.2),.e. we can drop products of fluctuatons. Statc space wthout gravty Let us frst consder statc space and gnore gravty ( ). It s easy to see that a soluton for the background s = const., P = const. and ū =. The lnearsed evoluton equatons for the fluctuatons are t (4.1.4) and r r (4.1.5), one t = r r ( u), t u = r r P. 2 t r 2 r P =. (4.1.6) For adabatc fluctuatons (see below), the pressure fluctuatons are proportonal to the densty fluctuatons, P = c 2 s, wherec s s called the speed of sound. Eq. (4.1.6) then takes the form of a wave 2 t c 2 sr 2 =. (4.1.7) Ths s solved by a plane wave, = A exp[(!t k r)], where! = c s k,wthk k. Wesee that n a statc spacetme fluctuatons oscllate wth constant ampltude f we gnore gravty. Fourer space. The more formal way to solve PDEs lke (4.1.7) s to expand n terms of ts Fourer components Z d 3 k (t, r) = (2 ) 3 e k r k (t). (4.1.8) The PDE (4.1.7) turns nto an ODE for each Fourer 2 t + c 2 sk 2 k =, (4.1.9) whch has the soluton k = A k e! kt + B k e! kt,! k c s k. (4.1.1)

3 79 4. Cosmologcal Perturbaton Theory Statc space wth gravty Now we turn on gravty. Eq. (4.1.7) then gets a source 2 t c 2 sr 2 r =4 G, (4.1.11) where we have used the perturbed Posson equaton, r 2 = A exp[(!t k r)], but now wth =4 G. Ths s stll solved by! 2 = c 2 sk 2 4 G. (4.1.12) We see that there s a crtcal wavenumber for whch the frequency of oscllatons s zero: k J p 4 G c s. (4.1.13) For small scales (.e. large wavenumber), k>k J, the pressure domnates and we fnd the same oscllatons as before. However, on large scales, k<k J, gravty domnates, the frequency! becomes magnary and the fluctuatons grow exponentally. The crossover happens at the Jeans length J = 2 r = c s k J G. (4.1.14) Expandng space In an expandng space, we have the usual relatonshp between physcal coordnates r and comovng coordnates x, r(t) =a(t)x. (4.1.15) The velocty feld s then gven by u(t) =ṙ = Hr + v, (4.1.16) where Hr s the Hubble flow and v = aẋ s the proper velocty. In a statc spacetme, the tme and space dervates defned from t and r were ndependent. In an expandng spacetme ths s not the case anymore. It s then convenent to use space dervatves defned wth respect to the comovng coordnates x, whch we denote by r x. Usng (4.1.15), we have r r = a 1 r x. (4.1.17) The relatonshp between tme dervatves at fxed r and at fxed @a 1 (t)r = + r x = + r = Hx r x. From now on, we wll drop the subscrpts x. x

4 8 4. Cosmologcal Perturbaton Theory Wth ths n mnd, let us look at the flud equatons n an expandng unverse: Contnuty equaton Substtutng (4.1.17) and (4.1.18) for r r t n the contnuty equaton (4.1.1), we get Hx r (1 + ) + a r (1 + )(Hax + v) =, (4.1.19) Here, I have ntroduced the fractonal densty perturbaton. (4.1.2) Sometmes s called the densty contrast. Let us analyse ths order-by-order n perturbaton theory: At zeroth order n fluctuatons (.e. droppng the perturbatons and v), we +3H =, where I have used r x x = 3. We recognse ths as the contnuty equaton for the homogeneous mass densty, / a 3. At frst order n fluctuatons (.e. droppng products of and v), we get Hx r + a r Hax + v =, (4.1.22) whch we can wrte as + r v =. (4.1.23) a The frst term vanshes by (4.1.21), so we fnd = 1 a r v, (4.1.24) where we have used an overdot to denote the dervatve wth respect to tme. Euler equaton Smlar manpulatons of the Euler equaton (4.1.2) lead to v + Hv = 1 a r P 1 a r. (4.1.25) In the absence of pressure and gravtatonal perturbatons, ths equaton smply says that v / a 1, whch s somethng we already dscovered n Chapter 1. Posson equaton It takes hardly any work to show that the Posson equaton (4.1.3) becomes r 2 =4 Ga 2. (4.1.26)

5 81 4. Cosmologcal Perturbaton Theory Exercse. Derve eq. (4.1.25) Jeans Instablty t (4.1.24) wthr (4.1.25) and (4.1.26), we fnd +2H c 2 s a 2 r2 =4 G. (4.1.27) Ths mples the same Jeans length as n (4.1.14), but unlke the case of a statc spacetme, t now depends on tme va (t) and c s (t). Compared to (4.1.11), the equaton of moton n the expandng spacetme ncludes a frcton term, 2H. Ths has two e ects: Below the Jeans length, the fluctuatons oscllate wth decreasng ampltude. Above the Jeans length, the fluctuatons experence power-law growth, rather than the exponental growth we found for statc space Dark Matter nsde Hubble The Newtonan framework descrbes the evoluton of matter fluctuatons. We can apply t to the evoluton dark matter on sub-hubble scales. (We wll gnore small e ects due to baryons.) Durng the matter-domnated era, eq.(4.1.27) reads m +2H m 4 G m m =, (4.1.28) where we have dropped the pressure term, snce c s = for lnearsed CDM fluctuatons. (Non-lnear e ect produce a fnte, but small, sound speed.) Snce a / t 2/3,wehave H =2/3t and hence m + 4 3t m 2 3t 2 m =, (4.1.29) where we have used 4 G m = 3 2 H2.Tryng m / t p gves the followng two solutons: 8 < t 1 / a 3/2 m /. (4.1.3) : t 2/3 / a Hence, the growng mode of dark matter fluctuatons grows lke the scale factor durng the MD era. Ths s a famous result that s worth rememberng. Durng the radaton-domnated era, eq.(4.1.27) gets modfed to m +2H m 4 G X I I I =, (4.1.31) where the sum s over matter and radaton. (It s the total densty fluctuaton = m + r whch sources!) Radaton fluctuatons on scales smaller than the Hubble radus oscllate as sound waves (supported by large radaton pressure) and ther tme-averaged densty contrast vanshes. To prove ths rgorously requres relatvstc perturbaton theory (see below). It follows that the CDM s essentally the only clustered component durng the acoustc oscllatons of the radaton, and so m + 1 t m 4 G m m. (4.1.32)

6 82 4. Cosmologcal Perturbaton Theory Snce m evolves only on cosmologcal tmescales (t has no pressure support for t to do otherwse), we have m H 2 m 8 G 3 r m 4 G m m, (4.1.33) where we have used that r m. We can therefore gnore the last term n (4.1.32) compared to the others. We then fnd 8 < const. m /. (4.1.34) : ln t / ln a We see that the rapd expanson due to the e ectvely unclustered radaton reduces the growth of m to only logarthmc. Ths s another fact worth rememberng: we need to wat untl the unverse becomes matter domnated n order for the dark matter densty fluctuatons to grow sgnfcantly. Durng the -domnated era, eq.(4.1.27) reads m +2H m 4 G X I I =, (4.1.35) where I = m,. As far as we can tell, dark energy doesn t cluster (almost by defnton), so we can wrte m +2H m 4 G m m =, (4.1.36) Notce that ths s not the same as (4.1.28), because H s d erent. Indeed, n the - domnated regme H 2 const. 4 G m. Droppng the last term n (4.1.36), we get whch has the followng solutons m / m +2H m, (4.1.37) 8 < : const. e 2Ht / a 2. (4.1.38) We see that the matter fluctuatons stop growng once dark energy comes to domnate. 4.2 Relatvstc Perturbaton Theory The Newtonan treatment of cosmologcal perturbatons s nadequate on scales larger than the Hubble radus, and for relatvstc fluds (lke photons and neutrnos). The correct descrpton requres a full general-relatvstc treatment whch we wll now develop Perturbed Spacetme The basc dea s to consder small perturbatons g µ around the FRW metrc ḡ µ, g µ =ḡ µ + g µ. (4.2.39) Through the Ensten equatons, the metrc perturbatons wll be coupled to perturbatons n the matter dstrbuton.

7 83 4. Cosmologcal Perturbaton Theory Perturbatons of the Metrc To avod unnecessary techncal dstractons, we wll only present the case of a flat FRW background spacetme, h ds 2 = a 2 ( ) d 2 jdx dx j. (4.2.4) The perturbed metrc can then be wrtten as h ds 2 = a 2 ( ) (1 + 2A)d 2 2B dx d ( j + h j )dx dx j, (4.2.41) where A, B and h j are functons of space and tme. We shall adopt the useful conventon that Latn ndces on spatal vectors and tensors are rased and lowered wth j, e.g. h = j h j. Scalar, Vectors and Tensors It wll be extremely useful to perform a scalar-vector-tensor (SVT) decomposton of the perturbatons. For 3-vectors, ths should be famlar. It smply means that we can splt any 3-vector nto the gradent of a scalar and a dvergenceless vector B B {z} scalar ˆB =. Smlarly, any rank-2 symmetrc tensor can be wrtten h j =2C j +2@ j E {z } scalar + {z} ˆB, (4.2.42) vector + 2@ ( Ê j) {z } vector + 2Êj, (4.2.43) {z} tensor where j 3 j r 2 E, ( Ê j) Ê j j Ê. (4.2.45) 2 As before, the hatted quanttes are Ê = Ê j =. The tensor perturbaton s traceless, Ê =. The 1 degrees of freedom of the metrc have thus been decomposed nto SVT degrees of freedom: scalars: A, B, C, E vectors: ˆB, Ê tensors: Ê j What makes the SVT-decomposton so powerful s the fact that the Ensten equatons for scalars, vectors and tensors don t mx at lnear order and can therefore be treated separately. In these lectures, we wll mostly be nterested n scalar fluctuatons and the assocated densty perturbatons. Vector perturbatons aren t produced by nflaton and even f they were, they would decay quckly wth the expanson of the unverse. Tensor perturbatons are an mportant predcton of nflaton and we wll dscuss them brefly n Chapter 6.

8 84 4. Cosmologcal Perturbaton Theory The Gauge Problem Before we contnue, we have to address an mportant subtlety. The metrc perturbatons n (4.2.41) aren t unquely defned, but depend on our choce of coordnates or the gauge choce. In partcular, when we wrote down the perturbed metrc, we mplctly chose a specfc tme slcng of the spacetme and defned specfc spatal coordnates on these tme slces. Makng a d erent choce of coordnates, can change the values of the perturbaton varables. It may even ntroduce fcttous perturbatons. These are fake perturbatons that can arse by an nconvenent choce of coordnates even f the background s perfectly homogeneous. For example, consder the homogeneous FRW spacetme (4.2.4) and make the followng change of the spatal coordnates, x 7! x = x + (,x). We assume that s small, so that t can also be treated as a perturbaton. Usng dx =d k d x k,eq.(4.2.4) becomes ds 2 = a 2 ( ) d 2 2 d x d j +2@ ( j) d x d x j, (4.2.46) where we have dropped terms that are quadratc n and We apparently have ntroduced the metrc perturbatons B = and Ê =. But these are just fcttous gauge modes that can be removed by gong back to the old coordnates. Smlar, we can change our tme slcng, 7! + (,x). The homogeneous densty of the unverse then gets perturbed, ( ) 7! ( + (,x)) = ( )+. So even n an unperturbed unverse, a change of the tme coordnate can ntroduce a fcttous densty perturbaton =. (4.2.47) Smlarly, we can remove a real perturbaton n the energy densty by choosng the hypersurface of constant tme to concde wth the hypersurface of constant energy densty. Then = although there are real nhomogenetes. These examples llustrate that we need a more physcal way to dentfy true perturbatons. One way to do ths s to defne perturbatons n such a way that they don t change under a change of coordnates. Gauge Transformatons Consder the coordnate transformaton X µ 7! X µ X µ + µ (,x), where T, L L + ˆL. (4.2.48) We have splt the spatal shft L nto a scalar, L, and a dvergenceless vector, ˆL. We wsh to know how the metrc transforms under ths change of coordnates. The trck s to explot the nvarance of the spacetme nterval, ds 2 = g µ (X)dX µ dx = g ( X)d X d X, (4.2.49) where I have used a d erent set of dummy ndces on both sdes to make the next few lnes clearer. Wrtng d X =(@ X /@X µ )dx µ (and smlarly for dx ), we fnd g µ (X) g ( X). (4.2.5) Ths relates the metrc n the old coordnates, g µ, to the metrc n the new coordnates, g.

9 85 4. Cosmologcal Perturbaton Theory Let us see what (4.2.5) mples for the transformaton of the metrc perturbatons n (4.2.41). I wll work out the -component as an example and leave the rest as an exercse. Consder µ = =n(4.2.5): g (X) @ g ( X). (4.2.51) The only term that contrbutes to the l.h.s. s the one wth = =. Consder for example = and =. The o -dagonal component of the metrc g s proportonal to B,sot s a frst-order perturbaton. X /@ s proportonal to the frst-order varable,sothe product s second order and can be neglected. A smlar argument holds for = and = j. Eq. (4.2.51) therefore reduces 2 g (X) = g X). (4.2.52) Substtutng (4.2.48) and (4.2.41), we get a 2 ( ) 1+2A = 1+T 2 a 2 ( + T ) 1+2Ã = 1+2T + a( )+a T Ã = a 2 ( ) 1+2HT +2T +2Ã +, (4.2.53) where H a /a s the Hubble parameter n conformal tme. Hence, we fnd that at frst order, the metrc perturbaton A transforms as A 7! Ã = A T HT. (4.2.54) I leave t to you to repeat the argument for the other metrc components and show that B 7! B = B T L, (4.2.55) h j 7! h j = h j 2@ ( L j) 2HT j. (4.2.56) Exercse. Derve eqs. (4.2.55) and (4.2.56). In terms of the SVT-decomposton, we get A 7! A T HT, (4.2.57) B 7! B + T L, ˆB 7! ˆB ˆL, (4.2.58) 1 C 7! C HT 3 r2 L, (4.2.59) E 7! E L, Ê 7! Ê ˆL, Ê j 7! Êj. (4.2.6) Gauge-Invarant Perturbatons One way to avod the gauge problems s to defne specal combnatons of metrc perturbatons that do not transform under a change of coordnates. These are the Bardeen varables: A + H(B E )+(B E ), ˆ Ê ˆB, Ê j, (4.2.61) C H(B E )+ 1 3 r2 E. (4.2.62)

10 86 4. Cosmologcal Perturbaton Theory Exercse. Show that, and ˆ don t change under a coordnate transformaton. These gauge-nvarant varables can be consdered as the real spacetme perturbatons snce they cannot be removed by a gauge transformaton. Gauge Fxng An alternatve (but related) soluton to the gauge problem s to fx the gauge and keep track of all perturbatons (metrc and matter). For example, we can use the freedom n the gauge functons T and L n (4.2.48) to set two of the four scalar metrc perturbatons to zero: Newtonan gauge. The choce B = E =, (4.2.63) gves the metrc ds 2 = a 2 ( ) (1 + 2 )d 2 (1 2 ) j dx dx j. (4.2.64) Here, we have renamed the remanng two metrc perturbatons, A and C, n order to make contact wth the Bardeen potentals n (4.2.61) and (4.2.62). For perturbatons that decay at spatal nfnty, the Newtonan gauge s unque (.e. the gauge s fxed completely). 2 In ths gauge, the physcs appears rather smple snce the hypersurfaces of constant tme are orthogonal to the worldlnes of observers at rest n the coordnates (snce B = ) and the nduced geometry of the constant-tme hypersurfaces s sotropc (snce E = ). In the absence of ansotropc stress, =. Note the smlarty of the metrc to the usual weak-feld lmt of GR about Mnkowsk space; we shall see that plays the role of the gravtatonal potental. Newtonan gauge wll be our preferred gauge for studyng the formaton of large-scale structures (Chapter 5) and CMB ansotropes (Chapter??). Spatally-flat qauge. A convenent gauge for computng nflatonary perturbatons s C = E =. (4.2.65) In ths gauge, we wll be able to focus most drectly on the fluctuatons n the nflaton feld (see Chapter 6) Perturbed Matter In Chapter 1, we showed that the matter n a homogeneous and sotropc unverse has to take the form of a perfect flud T µ =( + P )Ū µ Ū P µ, (4.2.66) where Ūµ = a µ, Ū µ = a 1 µ the stress-energy tensor for a comovng observer. Now, we consder small perturbatons of T µ = T µ + T µ. (4.2.67) 2 More generally, a gauge transformaton that corresponds to a small, tme-dependent but spatally constant boost.e. L ( ) and a compensatng tme translaton T = L ( ) to keep the constant-tme hypersurfaces orthogonal wll preserve E j =andb = and hence the form of the metrc n eq. ( ). However, such a transformaton would not preserve the decay of the perturbatons at nfnty.

11 87 4. Cosmologcal Perturbaton Theory Perturbatons of the Stress-Energy Tensor In a perturbed unverse, the energy densty, thepressurep and the four-velocty U µ can be functons of poston. Moreover, the stress-energy tensor can now have a contrbuton from ansotropc stress, µ. The perturbaton of the stress-energy tensor s T µ =( + P)Ū µ Ū +( + P )( U µ Ū + Ū µ U ) P µ µ. (4.2.68) The spatal part of the ansotropc stress tensor can be chosen to be traceless, =, snce ts trace can always be absorbed nto a redefnton of the sotropc pressure, P. The ansotropc stress tensor can also be chosen to be orthogonal to U µ,.e. U µ µ =. Wthout loss of generalty, we can then set = =. In practce, the ansotropc stress wll always be neglgble n these lectures. We wll keep t for now, but at some pont we wll drop t. Perturbatons n the four-velocty can nduce non-vanshng energy flux, T j, and momentum densty, T. To fnd these, let us compute the perturbed four-velocty n the perturbed metrc (4.2.41). Snce g µ U µ U = 1 and ḡ µ Ū µ Ū = 1, we have, at lnear order, g µ Ū µ Ū +2Ūµ U µ =. (4.2.69) Usng Ū µ = a 1 µ and g =2a 2 A,wefnd U = Aa 1. We then wrte U v /a, where v dx /d s the coordnate velocty, so that From ths, we derve O(2) U µ = a 1 [1 A, v ]. (4.2.7) z } { U = g U + g U = a 2 (1 + 2A)a 1 (1 A) =a(1 + A), (4.2.71) U = g U + g j U j = a 2 B a 1 a 2 ja 1 v j = a(b + v ), (4.2.72).e. U µ = a[1 + A, (v + B )]. (4.2.73) Usng (4.2.7) and (4.2.73) n(4.2.68), we fnd T =, (4.2.74) T =( + P )v, (4.2.75) T j = ( + P )(v j + B j ), (4.2.76) T j = P j j. (4.2.77) We wll use q for the momentum densty ( + P )v. If there are several contrbutons to the stress-energy tensor (e.g. photons, baryons, dark matter, etc.), they are added: T µ = P I T I µ. Ths mples = X I I, P = X I P I, q = X I q I, j = X I j I. (4.2.78) We see that the perturbatons n the densty, pressure and ansotropc stress smply add. The veloctes do not add, but the momentum denstes do.

12 88 4. Cosmologcal Perturbaton Theory Fnally, we note that the SVT decomposton can also be appled to the perturbatons of the stress-energy tensor: and P have scalar parts only, q has scalar and vector parts, and j has scalar, vector and tensor parts, q q +ˆq, (4.2.79) j j +@ ( ˆ j) + ˆ j. (4.2.8) Gauge Transformatons Under the coordnate transformaton (4.2.48), the stress-energy tensor transform as Evaluatng ths for the d erent components, we fnd T µ T ( X). (4.2.81) 7! T, (4.2.82) P 7! P T P, (4.2.83) q 7! q +( + P )L, (4.2.84) v 7! v + L, (4.2.85) j 7! j. (4.2.86) Exercse. Confrm eqs. (4.2.82) (4.2.86). Gauge-Invarant Perturbatons There are varous gauge-nvarant quanttes that can be formed from metrc and matter varables. One useful combnaton s + (v + B), (4.2.87) where v v. The quantty s called the comovng-gauge densty perturbaton. Exercse. Show that s gauge-nvarant. Gauge Fxng Above we used our gauge freedom to set two of the metrc perturbatons to zero. Alternatvely, we can defne the gauge n the matter sector: Unform densty gauge. We can use the freedom n the tme-slcng to set the total densty perturbaton to zero =. (4.2.88)

13 89 4. Cosmologcal Perturbaton Theory Comovng gauge. Smlarly, we can ask for the scalar momentum densty to vansh, q =. (4.2.89) Fluctuatons n comovng gauge are most naturally connected to the nflatonary ntal condtons. Ths wll be explaned n and Chapter 6. There are d erent versons of unform densty and comovng gauge dependng on whch of the metrc fluctuatons s set to zero. In these lectures, we wll choose B =. Adabatc Fluctuatons Smple nflaton models predct ntal fluctuatons that are adabatc (see Chapter 6). Adabatc perturbatons have the property that the local state of matter (determned, for example, by the energy densty and the pressure P ) at some spacetme pont (,x) of the perturbed unverse s the same as n the background unverse at some slghtly d erent tme + (x). (Notce that the tme shft vares wth locaton x!) We can thus vew adabatc perturbatons as some parts of the unverse beng ahead and others behnd n the evoluton. If the unverse s flled wth multple fluds, adabatc perturbatons correspond to perturbatons nduced by a common, local shft n tme of all background quanttes; e.g. adabatc densty perturbatons are defned as I (,x) I ( + (x)) I ( ) = I (x), (4.2.9) where s the same for all speces I. Thsmples = I I = J J for all speces I and J. (4.2.91) Usng 3 I = 3H(1 + w I) I, we can wrte ths as I 1+w I = where we have defned the fractonal densty contrast J 1+w J for all speces I and J, (4.2.92) I I I. (4.2.93) Thus, for adabatc perturbatons, all matter components (w m ) have the same fractonal perturbaton, whle all radaton perturbatons (w r = 1 3 ) obey r = 4 3 m. (4.2.94) It follows that for adabatc fluctuatons, the total densty perturbaton, tot = tot tot = X I I I, (4.2.95) s domnated by the speces that s domnant n the background snce all the I are comparable. We wll have more to say about adabatc ntal condtons n If there s no energy transfer between the flud components at the background level, the energy contnuty equaton s satsfed by them separately.

14 9 4. Cosmologcal Perturbaton Theory Isocurvature Fluctuatons The complement of adabatc perturbatons are socurvature perturbatons. Whle adabatc perturbatons correspond to a change n the total energy densty, socurvature perturbatons only correspond to perturbatons between the d erent components. Eq. (4.2.92) suggests the followng defnton of socurvature fluctuatons S IJ I J. (4.2.96) 1+w I 1+w J Sngle-feld nflaton predcts that the prmordal perturbatons are purely adabatc,.e. S IJ =, for all speces I and J. Moreover, all present observatonal data s consstent wth ths expectaton. We therefore won t consder socurvature fluctuatons further n these lectures Lnearsed Evoluton Equatons Our next task s to derve the perturbed Ensten equatons, G µ =8 G T µ, from the perturbed metrc and the perturbed stress-energy tensor. We wll work n Newtonan gauge wth! g µ = a (4.2.97) (1 2 ) j In these lectures, we wll never encounter stuatons where ansotropc stress plays a sgnfcant role. From now on, we wll therefore set ansotropc stress to zero, j =. As we wll see, ths enforces =. Perturbed Connecton Coe cents To derve the feld equatons, we frst requre the perturbed connecton coe cents. Recall that µ = 1 2 gµ (@ g g ). (4.2.98) Snce the metrc (4.2.97) s dagonal, t s smple to nvert g µ = 1 a (1 + 2 ) j!. (4.2.99) Substtutng (4.2.97) and (4.2.99) nto(4.2.98), gves = H +, (4.2.1) (4.2.11) = j, (4.2.12) j = H j +2H( + ) j, (4.2.13) j = H j j, (4.2.14) jk = 2 k) + jk l. (4.2.15) I wll work out as an example and leave the remanng terms as an exercse.

15 91 4. Cosmologcal Perturbaton Theory Example. From the defnton of the Chrsto el symbol we have Substtutng the metrc components, we fnd at lnear order n. = 1 2 g (2@ g ) = 1 2 g. (4.2.16) = 1 2a 2 (1 2 )@ [a 2 (1 + 2 )] = H +, (4.2.17) Exercse. Derve eqs. (4.2.11) (4.2.15). Perturbed Stress-Energy Conservaton Equpped wth the perturbed connecton, we can mmedately derve the perturbed conservaton equatons from r µ T µ = µ T µ + µ µ T µ T µ. (4.2.18) Contnuty Equaton Consder frst the = T T + µ µ T + µ µ T {z } O(2) T T {z } O(2) T {z } O(2) jt j =. (4.2.19) Substtutng the perturbed stress-energy tensor and the connecton coe cents ( + )+@ q +(H + +3H 3 )( + ) (H + )( + ) (H ) j ( P + P) j =, (4.2.11) and hence + q +3H( + ) 3 +3H( P + P) 3 P =. ( ) Wrtng the zeroth-order and frst-order parts separately, we get = 3H( + P ), ( ) = 3H( + P) + 3 ( + P ) r q. ( ) The zeroth-order part ( ) smply s the conservaton of energy n the homogeneous background. Eq. ( ) descrbes the evoluton of the densty perturbaton. The frst term on the rght-hand sde s just the dluton due to the background expanson (as n the background

16 92 4. Cosmologcal Perturbaton Theory equaton), the r q term accounts for the local flud flow due to pecular velocty, and the term s a purely relatvstc e ect correspondng to the densty changes caused by perturbatons to the local expanson rate [(1 )a s the local scale factor n the spatal part of the metrc n Newtonan gauge]. It s convenent to wrte the equaton n terms of the fractonal overdensty and the 3-velocty, and v = q + P. ( ) Eq. ( ) then becomes + 1+ P P r v 3 +3H P =. ( ) Ths s the relatvstc verson of the contnuty equaton. In the lmt P, we recover the Newtonan contnuty equaton n conformal tme, + r v 3 =, but wth a generalrelatvstc correcton due to the perturbaton to the rate of exanson of space. Ths correcton s small on sub-horzon scales (k H) we wll prove ths rgorously n Chapter 5. Euler Equaton Next, consder the = component of eq. µ T µ + µ µ T µ T µ =, ( ) and T j T j + µ µ T + µ µj T j T jt j j T j j k T k j =. ( ) Usng eqs. (4.2.74) (4.2.77), wth T = q n Newtonan gauge, eq. ( ) becomes h q j ( P + P) j 4Hq (@ j 3@ j ) P H j q j + H j q j + 2 j k) + jl l P k j =, ( ) {z } 3@ P or Usng eqs. ( ) and ( ), we get P 4Hq ( + P )@ =. ( ) v + Hv 3H P v = r P + P r. (4.2.12) Ths s the relatvstc verson of the Euler equaton for a vscous flud. Pressure gradents (r P) and gravtatonal nfall (r ) drve v. The equaton captures the redshftng of pecular veloctes (Hv) and ncludes a small correcton for relatvstc fluds ( P / ). Adabatc fluctuatons satsfy P / = c 2 s. Non-relatvstc matter fluctuatons have a very small sound speed, so the relatvstc correcton n the Euler equaton (4.2.12) s much smaller than the redshftng

17 93 4. Cosmologcal Perturbaton Theory term. The lmt P then reproduces the Euler equaton (4.1.25) of the lnearsed Newtonan treatment. Eqs. ( ) and (4.2.12) apply for the total matter and velocty, and also separately for any non-nteractng components so that the ndvdual stress-energy tensors are separately conserved. Once an equaton of state of the matter (and other consttutve relatons) are specfed, we just need the gravtatonal potentals and to close the system of equatons. Equatons for and follow from the perturbed Ensten equatons. Perturbed Ensten Equatons Let us now compute the lnearsed Ensten equaton n Newtonan gauge. We requre the 1 perturbaton to the Ensten tensor, G µ R µ 2 Rg µ, so we frst need to calculate the perturbed Rcc tensor R µ and scalar R. Rcc tensor. We recall that the Rcc tensor can be expressed n terms of the connecton as R µ µ + µ µ. ( ) Substtutng the perturbed connecton coe cents (4.2.1) (4.2.15), we fnd R = 3H + r 2 +3H( + ) + 3, ( ) R =2@ +2H@, ( ) R j = H +2H 2 + r 2 2(H +2H 2 )( + ) j ( ). IwllderveR here and leave the others as an exercse. 5H j ( ) Example. The component of the Rcc tensor s +. ( ) When we sum over, thetermswth = cancel so we need only consder summng over =1, 2, 3,.e j j {z } O(2) = r 2 3@ (H ) + 3(H + = 3H + r 2 +3H( + )(H {z } O(2) j ) (H j ) 2 j ) + 3. ( ) j Exercse. Derve eqs. ( ) and ( ). Rcc scalar. It s now relatvely straghtforward to compute the Rcc scalar R = g R +2g R {z +g j R } j. ( )

18 94 4. Cosmologcal Perturbaton Theory It follows that a 2 R =(1 2 )R (1 + 2 ) j R j =(1 2 ) 3H + r 2 +3H( + ) + 3 3(1 + 2 ) H +2H 2 + r 2 2(H +2H 2 )( + ) H 5H (1 + 2 )r 2 ( ). ( ) Droppng non-lnear terms, we fnd a 2 R = 6(H + H 2 )+2r 2 4r (H + H 2 ) H( + 3 ). ( ) Ensten tensor. Computng the Ensten tensor s now just a matter of collectng our prevous results. The component s 1 G = R 2 g R = 3H + r 2 +3H( + ) (1 + 2 )(H + H 2 ) 1 2r 2 4r (H + H 2 ) H( + 3 ). (4.2.13) 2 Most of the terms cancel leavng the smple result G =3H 2 +2r 2 6H. ( ) The component of the Ensten tensor s smply R snce g = n Newtonan gauge: G =2@ ( + H ). ( ) The remanng components are G j = R j 1 2 g jr = H +2H 2 + r 2 2(H +2H 2 )( + ) H 3(1 2 )(H + H 2 ) j 5H j ( ) r 2 4r (H + H 2 ) H( + 3 ) j. ( ) Ths neatens up (only a lttle!) to gve G j = (2H + H 2 ) j + r 2 ( ) (2H + H 2 )( + ) + 2H Ensten Equatons +4H j ( ). ( ) Substtutng the perturbed Ensten tensor, metrc and stress-energy tensor nto the Ensten equaton gves the equatons of moton for the metrc perturbatons and the zeroth-order Fredmann equatons: Let us start wth the trace-free part of the j equaton, G j =8 GT j. Snce we have dropped ansotropc stress there s no source on the rght-hand sde. From eq. ( ), we j ( ) =. ( )

19 95 4. Cosmologcal Perturbaton Theory Had we kept ansotropc stress, the rght-hand sde would be 8 Ga 2 j. In the absence of ansotropc stress 4 (and assumng approprate decay at nfnty), we get 5 =. ( ) There s then only one gauge-nvarant degree of freedom n the metrc. In the followng, we wll wrte all equatons n terms of. Next, we consder the equaton, G =8 GT. Usng eq. ( ), we get 3H 2 +2r 2 6H =8 Gg µ T µ =8 G g T + g T =8 Ga 2 (1 + 2 )( + ) =8 Ga 2 ( ). ( ) The zeroth-order part gves H 2 = 8 G 3 a2, ( ) whch s just the Fredmann equaton. The frst-order part of eq. ( ) gves whch, on usng eq. ( ), reduces to r 2 =4 Ga 2 +8 Ga 2 +3H. ( ) r 2 =4 Ga 2 +3H( + H ). (4.2.14) Movng on to equaton, G =8 GT,wth T = g µ T µ = g T =ḡ T = a 2 q. ( ) It follows ( + H ) = 4 Ga 2 q. ( ) If we wrte q =( + P )@ v and assume the perturbatons decay at nfnty, we can ntegrate eq. ( ) to get + H = 4 Ga 2 ( + P )v. ( ) Substtutng eq. ( ) nto the Ensten equaton (4.2.14) gves r 2 =4 Ga 2, where 3H( + P )v. ( ) 4 In realty, neutrnos develop ansotropc stress after neutrno decouplng (.e. they do not behave lke a perfect flud). Therefore, and actually d er from each other by about 1% n the tme between neutrno decouplng and matter-radaton equalty. After the unverse becomes matter-domnated, the neutrnos become unmportant, and and rapdly approach each other. The same thng happens to photons after photon decouplng, but the unverse s then already matter-domnated, so they do not cause a sgnfcant d erence. 5 In Fourer space, eq. ( ) becomes k k j 1 3 jk2 ( ) =. For fnte k, we therefore must have =. For k =, =const. would be a soluton. However, the constant must be zero, snce the mean of the perturbatons vanshes.

20 96 4. Cosmologcal Perturbaton Theory Ths s of the form of a Posson equaton, but wth source densty gven by the gaugenvarant varable of eq. (4.2.87) snceb = n the Newtonan gauge. Let us ntroduce comovng hypersurfaces as those that are orthogonal to the worldlnes of a set of observers comovng wth the total matter (.e. they see q = ) and are the constant-tme hypersurfaces n the comovng gauge for whch q = and B =. It follows that s the fractonal overdensty n the comovng gauge and we see from eq. ( ) that ths s the source term for the gravtatonal potental. Fnally, we consder the trace-part of the j equaton,.e. G =8 GT. We compute the left-hand sde from eq. ( ) (wth = ), G = g µ G µ = g k G k = a 2 (1 + 2 ) k (2H + H 2 ) k H + 4(2H + H 2 ) k = 3a 2 (2H + H 2 )+2 +3H +(2H + H 2 ). ( ) We combne ths wth T = 3( P + P). At zeroth order, we fnd 2H + H 2 = 8 Ga 2 P, ( ) whch s just the second Fredmann equaton. At frst order, we get +3H +(2H + H 2 ) = 4 Ga 2 P. ( ) Of course, the Ensten equatons and the energy and momentum conservaton equatons form a redundant (but consstent!) set of equatons because of the Banch dentty. We can use whchever subsets are most convenent for the partcular problem at hand. 4.3 Conserved Curvature Perturbaton There s an mportant quantty that s conserved on super-hubble scales for adabatc fluctuatons rrespectve of the equaton of state of the matter: the comovng curvature perturbaton. As we wll see below, the comovng curvature perturbaton provdes the essental lnk between the fluctuatons that we observe n the late-tme unverse (Chapter 5) and the prmordal seed fluctuatons created by nflaton (Chapter 6) Comovng Curvature Perturbaton In some arbtrary gauge, let us work out the ntrnsc curvature of surfaces of constant tme. The nduced metrc, j, on these surfaces s just the spatal part of eq. (4.2.41),.e. j a 2 [(1 + 2C) j +2E j ]. ( ) where E j E for scalar perturbatons. In a tedous, but straghtforward computaton, we derve the three-dmensonal Rcc scalar assocated wth j, a 2 R (3) = 4r C r2 E. ( ) In the followng nsert I show all the steps.

21 97 4. Cosmologcal Perturbaton Theory Dervaton. The connecton correspondng to j s (3) jk = 1 2 l (@ j kl k l jk ), (4.3.15) where j s the nverse of the nduced metrc, j = a 2 (1 2C) j 2E j = a 2 j + O(1). ( ) In order to compute the connecton to frst order, we actually only need the nverse metrc to zeroth order, snce the spatal dervatves of the j are all frst order n the perturbatons. We have (3) jk = j (C kl + E kl )+ k (C jl + E jl ) l (C jk + E jk ) =2 k)c l l C +2@ (j E k) l E jk. ( ) The ntrnsc curvature s the assocated Rcc scalar, gven by R (3) = l (3) l k k (3) l l + k (3) l k (3) m lm k (3) m l (3) l km. ( ) To frst order, ths reduces to a 2 R (3) = l (3) l k k (3) l l. ( ) Ths nvolves two contractons of the connecton. The frst s k(3) l k = k 2 ( jl k)c j C + k l 2@ ( E jl j E k The second s Eq. ( ) therefore becomes =2 k C 3 j C +2@ E l j ( k E k ) {z } = k C +2@ k E kl. ( ) (3) l l = l l@ C + l C l E l E l l E l =3@ C. ( ) a 2 R (3) l k C +2@ k E kl 3 C = r 2 C j E j 3r 2 C = 4r 2 C j E j. ( ) Note that ths vanshes for vector and tensor perturbatons (as do all perturbed scalars) snce then C j E j =. For scalar perturbatons, E j j E so Fnally, we get j E j = m E 1 3 lm r 2 E = r 2 r 2 E 1 3 r2 r 2 E = 2 3 r4 E. ( ) We defne the curvature perturbaton as C 1 3 r2 E.Thecomovng curvature perturbaton R

22 98 4. Cosmologcal Perturbaton Theory s the curvature perturbaton evaluated n the comovng gauge (B ==q ). It wll prove convenent to have a gauge-nvarant expresson for R, so that we can evaluate t from the perturbatons n any gauge (for example, n Newtonan gauge). Snce B and v vansh n the comovng gauge, we can always add lnear combnatons of these to C 1 3 r2 E to form a gaugenvarant combnaton that equals R. Usng eqs. (4.2.58) (4.2.6) and (4.2.85), we see that the correct gauge-nvarant expresson for the comovng curvature perturbaton s R = C 1 3 r2 E + H(B + v). ( ) Exercse. Show that R s gauge-nvarant A Conservaton Law We now want to prove that the comovng curvature perturbaton R s ndeed conserved on large scales and for adabatc perturbatons. We shall do so by workng n the Newtonan gauge, n whch case R = +Hv, (4.3.16) snce B = E = and C. We can use the Ensten equaton ( ) to elmnate the pecular velocty n favour of the gravtatonal potental and ts tme dervatve: R = H( + H ) 4 Ga 2 ( + P ). ( ) Takng a tme dervatve of ( ) and usng the evoluton equatons of the prevous secton, we fnd 4 Ga 2 ( + P ) R = 4 Ga 2 H P nad + H P r2, ( ) where we have defned the non-adabatc pressure perturbaton P nad P P. ( ) Dervaton. We d erentate eq. ( ) tofnd 4 Ga 2 ( + P ) R = 4 Ga 2 ( + P ) + H ( + H ) + H( + H +H ) + H 2 ( + H ) + 3H P 2 ( + H ), ( ) where we used = 3H( + P ). Ths needs to be cleaned up a bt. In the frst term on the rght, we use the Fredmann equaton to wrte 4 Ga 2 ( + P ) as H 2 H. In the last term, we use the Posson equaton (4.2.14) towrte3h( + H ) as (r 2 4 Ga 2 ). We then fnd 4 Ga 2 ( + P ) R = (H 2 H ) + H ( + H ) + H( + H +H ) + H 2 ( + H ) + H P r 2 4 Ga 2. ( )

23 99 4. Cosmologcal Perturbaton Theory Addng and subtractng 4 Ga 2 H P on the rght-hand sde and smplfyng gves 4 Ga 2 ( + P ) R = H +3H +(2H + H 2 ) 4 Ga 2 P +4 Ga 2 H P nad + H P r2, ( ) where P nad was defned n ( ). The frst term on the rght-hand sde vanshes by eq. ( ), so we obtan eq. ( ). Exercse. Show that P nad s gauge-nvarant. The non-adabatc pressure P nad vanshes for a barotropc equaton of state, P = P ( ) (and, more generally, for adabatc fluctuatons n a mxture of barotropc fluds). In that case, the rght-hand sde of eq. ( ) scales as Hk 2 Hk 2 R, so that d ln R k 2 d ln a. ( ) H Hence, we fnd that R doesn t evolve on super-hubble scales, k H. Ths means that the value of R that we wll compute at horzon crossng durng nflaton (Chapter 6) survves unaltered untl later tmes. 4.4 Summary We have derved the lnearsed evoluton equatons for scalar perturbatons n Newtonan gauge, where the metrc has the followng form ds 2 = a 2 ( ) (1 + 2 )d 2 (1 2 ) j dx dx j. ( ) In these lectures, we won t encounter stuatons where ansotropc stress plays a sgnfcant role, so we wll always be able to set =. The Ensten equatons then are r 2 3H( + H ) = 4 Ga 2, ( ) + H = 4 Ga 2 ( + P )v, (4.4.17) +3H +(2H + H 2 ) = 4 Ga 2 P. ( ) The source terms on the rght-hand sde should be nterpreted as the sum over all relevant matter components (e.g. photons, dark matter, baryons, etc.). The Posson equaton takes a partcularly smple form f we ntroduce the comovng gauge densty contrast r 2 =4 Ga 2. ( ) From the conservaton of the stress-tensor, we derved the relatvstc generalsatons of the contnuty equaton and the Euler equaton P P +3H = 1+ P r v 3, ( ) 1 P v r P +3H 3 v = + P r. ( )

24 1 4. Cosmologcal Perturbaton Theory These equatons apply for the total matter and velocty, and also separately for any nonnteractng components so that the ndvdual stress-energy tensors are separately conserved. A very mportant quantty s the comovng curvature perturbaton R = H( + H ) 4 Ga 2 ( + P ). ( ) We have shown that R doesn t evolve on super-hubble scales, k H, unless non-adabatc pressure s sgnfcant. Ths fact s crucal for relatng late-tme observables, such as the dstrbutons of galaxes (Chapter 5), to the ntal condtons from nflaton (Chapter 6).