CMB anisotropies at all orders: the non-linear Sachs-Wolfe formula
AAccepted for publication in JCAP
CMB anisotropies at all orders: thenon-linear Sachs-Wolfe formula
Omar Roldan
Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972,Rio de Janeiro, RJ, BrazilE-mail: [email protected]
Abstract.
We obtain the non-linear generalization of the Sachs-Wolfe + integrated Sachs-Wolfe (ISW) formula describing the CMB temperature anisotropies. Our formula is valid at allorders in perturbation theory, is also valid in all gauges and includes scalar, vector and tensormodes. A direct consequence of our results is that the maps of the logarithmic temperatureanisotropies are much cleaner than the usual CMB maps, because they automatically removemany secondary anisotropies. This can for instance, facilitate the search for primordial non-Gaussianity in future works. It also disentangles the non-linear ISW from other effects.Finally, we provide a method which can iteratively be used to obtain the lensing solution atthe desired order.
Keywords:
CMB theory, non-linear CMB, Sachs-Wolfe formula, integrated Sachs-Wolfe,CMB second-order perturbations, lensing. a r X i v : . [ a s t r o - ph . C O ] A ug ontents The Cosmic Microwave Background (CMB) temperature anisotropies is one of the most im-portant observables in cosmology. The CMB data is used for instance to constrain modelsof inflation [1], the amount of primordial non-Gaussianity [2], isocurvature perturbations [1],and to extract the cosmological parameters of the Λ CDM model [3]. Because of that, manyefforts have been made to properly understand the physics of the CMB, this physics can beseparated into three stages: before, during and after the period of recombination. Regard-ing the latter case, we can formally split the CMB anisotropies into primary and secondaryanisotropies. While the primary anisotropies (those already present at the emission time) aresupposed to be known (for instance, by solving the Boltzmann and the Einstein’s equationsduring and before recombination), secondary gravitational anisotropies must be obtained bysolving the geodesic equation of photons in its way down to the observer. Note that addi-tional anisotropies can arise due to secondary scatterings of photons with hot gas during thereheating period, however, this process is not covered here.Temperature anisotropies were systematically investigated for the first time by Sachsand Wolfe [4] in 1967 by using first-order perturbation theory, and their famous formula isquite easy to understand ∆ T o ¯ T o = T e + (Φ e − Φ o ) − ( v e · n e − v o · n o ) + I , (1.1)– 1 –here, the subscripts e and o means that quantities must be evaluated at the emission andobservation event respectively. Here T , Φ and I are respectively the intrinsic temperatureanisotropies, the gravitational potential and the integrated Sachs-Wolfe effect (ISW). TheISW is an integrated term due to the time variation of the metric perturbations along thepath of the photon (it gives the accumulated redshift of photons when traveling along theevolving inhomogeneities). Finally, ( v e · n e − v o · n o ) gives the linear Doppler effect due tothe observer’s and emitter’s peculiar velocity ( v o and v e ), with n o the direction of observationand − n e the direction of emission.The CMB temperature anisotropies are so small that the previous equation gives a verygood description of the observed data, at least on large scales where secondary scatterings arenegligible. Second-order perturbation theory of the CMB is however important to describein a unified picture several important effects which are not taken into account by Eq. (1.1).These are for instance, lensing [5, 6], time delay [7], Doppler modulation and aberration[8, 9] and the Rees-Sciama effect [10, 11]. These effects although smaller than the first-order ones are very relevant for a correct understanding of the CMB physics, so second-orderperturbation theory represents an essential tool for an accurate analysis of current and futureCMB data. The full second-order generalization of the Sachs-Wolfe formula was obtainedin 1997 by Mollerach and Matarrese [12] by using a method introduced in [13, 14]. Theirsecond-order expression is somehow big and a direct interpretation of each term is difficult.Further progress in obtaining simple formulas have been given in [15, 16]In the search for non-Gaussianity, second-order perturbations is enough to study thethree-point function (or its Fourier counterpart, the bispectrum). However, if one wants to goto the four-point function (or the trispectrum) for instance, third-order perturbation theory isneeded to fully account for all the contributions. The CMB anisotropies up to third order werefirst computed in [17] by using two methods, the first one is the same used by [12] in 1997,and the second one which is simpler and closer to our method, allowed them to obtain a fullynon-linear Sachs-Wolfe formula for the specific case in which the metric is totally determinedby two scalars variables, Φ and Ψ . Additional effort to obtain the non-linear description ofthe CMB can be found in [18–20]. Choosing a particular parametrization of the metric is essential for obtaining exact solutionsin cosmology, and this was the case in this work. By writing the line element as d s = a ( η ) e d ˆ s with the conformal metric given byd ˆ s ≡ − d η + 2 β j (cid:0) e − M (cid:1) ji d x i d η + (cid:0) e − M (cid:1) ij d x i d x j , (1.2)we were able to obtain an exact expression for the observed CMB temperature, T o = ¯ T o e Θ ,where ¯ T o is the observed mean temperature and Θ ≡ ( T e − T o ) + (Φ e − Φ o ) + I ( x e , x o ) + ln (cid:18) γ e (1 − n e · v e ) γ o (1 − n o · v o ) (cid:19) . (1.3)Here T , Φ and I are respectively, the non-linear generalization of the intrinsic temperatureanisotropies, the gravitational potential and the integrated Sachs-Wolfe effect. The logarithmterm corresponds to the Doppler effect, with γ the Lorentz factor. x e = ( η, x i ) e and x o =( η, x i ) o are the spacetime coordinates of the emission and observation events, and η is theconformal time. – 2 –n terms of Θ , we can easily obtain the temperature anisotropies as ∆ T o ¯ T o = e Θ − · · · , (1.4)and in the case of perturbation theory we just need to truncate the series at the desired order.Because of the relation Θ = ln (cid:0) T o / ¯ T o (cid:1) we will call Θ the logarithmic temperatureanisotropies . Note that: • The intrinsic temperature anisotropies T e are defined through the relation (this notationwas also used in [19]) T e = (cid:104) T (cid:105) e e T e where (cid:104) T (cid:105) e is the background temperature at thetime of emission. Note also the presence of the factor T o in Eq. (1.3), which is absentin previous works in literature. This factor is important for two reasons: it makes theexpression for Θ symmetric in the emission and observation points and it ensures thegauge invariance of Θ . Without this factor, neither Θ nor ¯ T o would be gauge invariant(although the T o would). So introducing T o ensures also the gauge invariance of themean temperature ¯ T o as it should be. The definition of T o is given in the next section. • Although T o contains crossed terms involving the fields at the emission and observationpoint (for instance, at second order it contains terms of the form Φ e Φ o ), Θ does notcontain such mixed terms. That is, Θ is composed of a sum of locally defined terms. Inparticular the Doppler term is just: ln ( γ o (1 − n o · v o )) − ln ( γ e (1 − n e · v e )) . • Note also that, in Θ the ISW effect is clearly separated from other terms (althoughit is correlated with lensing §4.4) like Φ , T and v . It makes the study of the ISW (aswell as lensing) easier by directly using Θ rather than ∆ T o / ¯ T o . In previous expressionsin literature (see for instance [12, 17]), many integrated terms are coupled with otherquantities making it difficult to isolate the ISW effect from the rest. So our results canbe stated in a different way: by taking the logarithm of the temperature anisotropieswe are making kind of “resummations” and removing these spurious non-linearities.This is similar to what happens in quantum field theory, in which the disconnectedFeynman diagrams are removed by taking the logarithm of the propagators. Finally,since propagators in quantum mechanics are just correlation functions, we expect thatthe correlation functions of Θ are much simpler than those of ∆ T o / ¯ T o . For instance,by considering ∆ T o / ¯ T o instead of Θ , we are considering spurious quadratic, cubic, ...,terms which could create bias in the search for primordial non-Gaussianity. • Even if we treat Θ as a first-order quantity, ∆ T o / ¯ T o will not be linear as it contains allpowers of Θ . This shows that even if Θ is a Gaussian distributed quantity (which ingeneral is not the case, but it would be a good approximation if we evolve Θ linearlyfrom single-field initial conditions during inflation), ∆ T o / ¯ T o is not Gaussian. Because ofthis, it seems better to use Θ as the variable to be studied in future CMB experiments.That is, we propose to study the maps of the logarithmic temperature anisotropies ln (cid:0) T o / ¯ T o (cid:1) . Such a map will be free of Doppler modulation (see below) and othercouplings which otherwise will be present in a normal map of ∆ T o / ¯ T o . In practice, In particular, the off-diagonal part of the two-point correlation function does not vanish. Note that the couplings induced by aberration and lensing cannot be removed by such procedure. – 3 –hat is measured in an experiment like Planck are the variations in the intensity I obs ( ν, n ) = 2 hν c (cid:16) hνk B T o ( n ) (cid:17) − , (1.5)so that Θ can be calculated directly from the variation δI obs without explicitly giving T o . • If we still want to analyze the data in terms of ∆ T / ¯ T rather than Θ , the theoreticaln-point correlation function of ∆ T / ¯ T and Θ are easily related for the specific case of aGaussian distributed Θ (see for instance [21]). • Within the linear regime, it is well known that for adiabatic perturbations, in the Poissongauge and in the large scale limit (where we can neglect v e · n e and the ISW term) wehave T = − / , so that Θ ≈ Φ e / (here without considering the contributions atthe observer). It has been shown in [21] that this relation continues to hold at thenon-linear level. So, this result in conjunction with the formula Eq. (1.3) suggest thatthe metric parametrization introduced in this work and our definition of the non-linearintrinsic perturbations T are appropriated to extend the results of the linear theory tothe non-perturbative level. • Since conformal transformations yield null geodesics into null geodesics, the photon’spath is totally determined by the conformal metric d ˆ s . Therefore the integrated Sachs-Wolfe term I , as well as the lensing terms encoded into x e and n e are totally determinedby β i and M ij . The explicit form of these quantities are given in the next sections. • In principle Φ , β i and M ij are independent quantities, but in the linear regime (andduring matter domination) general relativity predicts that M ii / . So by measuring Φ by an independent method like the use of the Poisson equation [22], and comparingwith the lensing and ISW measurements, we can test general relativity. Note that byhaving the non-linear version of the ISW and lensing effects we could make a betterinterpretation of future data, this is because even if Einstein’s gravity is correct, theinadequate use of the linear approximation to analyze the data could indicate a deviationfrom the expected relation between Φ and M ii . • An immediate consequence of Eq. (1.4) is that Doppler modulation of the temperatureanisotropies always exist regardless of the nature of the dipole. Let’s explain it a bitmore. Split the logarithmic anisotropies as
Θ = Θ d + ˜Θ , where in a multipolar expansion Θ d refers to the dipole of the logarithmic anisotropies and ˜Θ contains all the remainingmultipolar components, that is, (cid:96) ≥ . Then we see from Eq. (1.4) that the observedtemperature anisotropies up to second order are given by ∆ T o ¯ T o = Θ d + ˜Θ + ˜Θ + Θ d d ˜Θ . (1.6)The last term is what we call the Doppler modulation of the temperature anisotropies,and it leads to coupling between neighbors multipolar components ( (cid:96), (cid:96) ± ) in the two-point correlation function that are proportional to the magnitude of the CMB dipole. The appropriate name will be dipolar modulation, but in the case in which the dipole is mainly ofkinematical origin this modulation is due to the Doppler effect. We will adopt this name here because theCMB dipole is believed to be due to our peculiar velocity. – 4 –hese couplings (as well as aberration couplings) were measured by Planck in [24].The results were consistent in amplitude and direction (at the σ -level) with the wellknown measured CMB dipole, that is, they are consistent with the prediction of simpleformula Θ d ˜Θ .According to Eq. (1.6), Planck’s measurements tell us nothing about the nature of theCMB dipole. However, measuring Doppler modulation is important for the followingreason: suppose that a more precise measurement of Doppler modulation is made bya future CMB experiment like CoRE [25], suppose also that the results show a signifi-cant deviation from the simple expectation Θ d ˜Θ , then that would imply that the term ˜Θ in Eq. (1.6) contains dipolar-like modulation couplings, and they necessarily comefrom primordial non-Gaussianity terms that couple the long-mode (dipolar components)with the short-modes (the higher multipoles). Such a result will rule-out single-field-inflationary models and would require a non-negligible amplitude for the dipolar com-ponents. These facts were first noted in [26]. Although the previous results followsimmediately from Eq. (1.6), they were far from obvious by using previously existingformulas, like the one given in [12]. Finally, we want to mention that the conclusions of[26] regarding dipolar modulation were restricted to the large scale case, but here theproof holds at any scale.We want to stress that in our results we have assumed a perfect blackbody spectrumfor the CMB (Eq. (1.5)). However, it is known that spectral distortions (deviations fromthe blackbody spectrum) start been relevant at second order. A non-linear treatment oftheses spectral distortions was introduced in the nice paper of Stebbins [27]. There, Stebbinsconsidered the observed spectrum as a superposition of blackbody with different temperaturesand introduced the concept of mean logarithmic temperature which must be related to ourdefinition of Θ . On the other hand, in the same way as we are proposing the use Θ =ln (cid:0) T o / ¯ T o (cid:1) for the future CMB maps, it was also advocated in [28] the use of thelogarithmically averaged temperature moments to describe the spectral distortions. In [29] itwas also noted the importance of the used of the exponential notation, though they consideredparticular cases.Finally, note that in order to make quantitative predictions, the Sachs-Wolfe formula isnot enough as we need to specify the fields T , Φ , β i , etc. as well as the integration path x i ( η ) .In this sense further progress is needed to obtain (analytical or numerical) non-linear solutionsof the Einstein’s (or Boltzmann’s) equations. On the other hand, as perturbative solutionsare still of high importance, in §4.2 we obtain the full second-order Sachs-Wolfe formula.Perturbative solutions of the metric and fluid perturbations are known in some specific cases(e.g., by assuming matter domination or the large scale limit), see for instance [30, 31]. Forother useful results at second order, see [32–35]. In §3.1 we solve the time-component of the geodesic equation, which allow us to relate theobserved temperature T o with the emission temperature T e by a simple relation. The results Two independent works realized that such effect could be observed by the Planck satellite, [9, 23]. I am very grateful to Cyril Pitrou for let me know about the works I cite in this paragraph. – 5 –re given in Eq. (3.17), and can be expressed as T o = T e a e a o e Φ e − Φ o + I γ e (1 − n e · v e ) γ o (1 − n o · v o ) , (1.7)where v o (and v e ) is the peculiar velocity of the observer (and the emitter). Note that, givenan observer with four-velocity u , its peculiar velocity is defined according to u com = γ ( u − v ) , γ = − u · u com = 1 √ − v · v , (1.8)where, u · v = 0 and u com is the four-velocity of comoving observers. In pp.5 of [27] a similarresult to Eq. (1.7) was obtained but without including vector and tensor perturbations, andwithout considering the velocity of the emitter and observer.We now define the logarithmic intrinsic temperature anisotropies and clarify some con-cepts about the mean values. After that, we will get the final form of the generalized Sachs-Wolfe formula. (Logarithmic) Intrinsic temperature anisotropies Before the epoch of recombination, the Universe was so hot and dense that photons frequentlyinteracted with the free electrons via Thomson scattering, while the electrons frequently in-teracted with protons via Coulomb scattering, thus forming the so called photon-baryon fluid.As a result, the fluid reached a state of thermal equilibrium and the photons are well describedby a blackbody distribution function . However, because of the inhomogeneities the thermalequilibrium is just local, meaning that different local observers in the rest frame of the fluidwill measure different values for the temperature T , that is, T = T ( x ) . During recombinationthe Compton scattering rate decreases and anisotropies in the photon’s distribution functionwill appear, that is, T = T ( x, n ) .We will write the temperature of the photon’s fluid as T ( x, n ) = (cid:104) T (cid:105) e T , where T = T ( x, n ) , (1.9)and (cid:104) T (cid:105) ∝ /a ( η ) is the background temperature. We will call T the logarithm perturba-tions. This expression is meaningful for η ≤ η e , when the photons and baryons are still inequilibrium. That is, T is not defined for η > η e . Below we will provide an extension of T for η > η e , so that T is a field defined in the whole spacetime.Note that η = const is defined by the physical argument that (cid:104) T (cid:105) = const as in Mirbabayi& Zaldarriaga [16]. In that sense, when transforming the CMB temperature, it is better touse gauge transformations (active transformations, acting on the fields) rather than passivetransformations (transformations on the coordinates), because in the latter case the transfor-mation of the time-coordinate becomes intricate. This issue will be discussed in details in afuture work. See also [16] for an specific example. We stress that the mean (cid:104)(cid:105) is taken on thespace-like 3D-hypersurfaces of constant η . However, what is important for the CMB is themean taken on the last scattering surface S e,o . Here we define S e,o as the 2D-surface (usually The explicit form of I is given in Eq. (3.19). Note that v is related to v com , the velocity of comoving-observers w.r.t u by the relation v com = − v , see§2.2. See sections 8.7.1 and 11.3.1 of [36] for further details on this. Here, an observer can be for instance an electron. Again, (cid:104) T (cid:105) e is relevant for defining the time of emission η e . – 6 – igure 1 . Different observers define different hypersurfaces S e,o each one with its own mean valuetemperature ¯ T e . The deviation of ¯ T e from the mean temperature at the hypersurface η = η e willtherefore depend on the observer’s position x o , and that information is encoded into T o . thought as a deformed spherical shell) formed at the intersection between the hypersurfaceof η = η e and the past light-cone of the observer. It follows from Eq. (1.7) that the observedmean temperature is ¯ T o ( x o ; η e ) = a e a o (cid:104) T (cid:105) e e T o , (1.10)where we have defined the intrinsic temperature anisotropies at the observer’s spacetimeposition x o as a mean value on the last scattering surface S e,o e T o ≡ exp ( T e + Φ e − Φ o + I + ln γ o (1 − n o · v o ) − ln γ e (1 − n e · v e )) . (1.11)From Eq. (1.10) it follows that T o = T o ( x o ; η e ) transforms under gauge transformations in thesame way as the logarithmic anisotropies T e but evaluated at the observer’s position. Sincethis definition is valid for any observer with η o > η e , it provides a natural extension for thefield T to the whole spacetime. Note however that by construction T o depends only on thespacetime position x o not on the direction of observation n o . This is in contrast with intrinsictemperature anisotropies T e which according to the discussion at the beginning of this section,could depend on n e . It follows from Eqs. (1.7) and (1.10) that the observed temperature canbe written as T o = ¯ T o e Θ , with Θ ≡ ( T e − T o ) + (Φ e − Φ o ) + I + ln (cid:18) γ e (1 − n e · v e ) γ o (1 − n o · v o ) (cid:19) . (1.12)Finally, we define the mean temperature ¯ T e of the last scattering surface as ¯ T e ( x o ; η e ) ≡ (cid:104) T (cid:105) e e T o . (1.13)Because in general ¯ T e (cid:54) = (cid:104) T (cid:105) e , then through the previous equation, T o tell us how anisotropicthe last scattering surface is (see figure 1). From Eq. (1.10) it follows that ¯ T o = ¯ T e a e a o . (1.14) Mean values on S e,o represent integrations w.r.t the direction of observation n o . In it does, as the intrinsic temperature anisotropies have a quadrupole component which act as a sourcefor the CMB polarization [37, 38]. – 7 –he previous relation is simply the statement that the mean temperature evolves only throughthe cosmological expansion. Additionally, the quantity Θ is what we call the logarithmicCMB temperature anisotropies, Eq. (1.12) is the non-linear generalization of the Sachs-Wolfeformula, and I (given in Eq. (3.19)) is the non-linear generalization of the integrated Sachs-Wolfe effect. As we will show in §3.2, the presence of the factor T o will guarantee the gaugeinvariance of Θ and ¯ T o . Eq. (1.12) is the main result of this paper.In the remaining sections we do the explicit calculations and consider particular cases.So for instance, in §2 give a quick review of fundamental concepts and introduce the notation.In §3 we introduce a tetrad basis which facilitates the resolution of the geodesic equation andallow us to interpret the metric components β i as the tetrad components of the four-velocityof comoving observers. Then we compare our results with the previous ones in literature.Firstly, we consider the first-order case in §3.2 and discuss the gauge invariance. In §4.4 weshow how to obtain the lensing term up to the desired order, and then in §4.2 we obtainthe second-order Sachs-Wolfe formula which is simpler than the previous ones in literatureand then we give the conclusions. In a companion paper [39] we discuss the subtle issue ofsecond-order gauge transformations on the CMB, prove the gauge invariance of our second-order formula and introduce the concept of a cosmological river-frame. Further applicationsof our results and comparison with existing ones will appear elsewhere [40]. In this section we quickly review some concepts which will be important to find the exactsolution for the Sachs-Wolfe formula and at the same time give us a clear geometrical meaningof each terms in that formula.
An orthonormal dual tetrad e a ( x ) , is a set of dual vectors e a ≡ (cid:8) e , e , e , e (cid:9) attached toeach point x µ of the spacetime in which the line element looks Minkowskian ds = η ab e a e b , (2.1)and so the tetrad axes form (at each point) a locally inertial orthonormal frame. We cantransform between the tetrad frame and the coordinate frame by using the matrix e ab and itsinverse e ab , e a = e ab d x b , d x a = e ab e b . (2.2)Now, the orthonormal tetrads e a (that is, the duals of e a ) are related to the coordinatevectors ∂ a (the duals of d x a ) by e a = e ba ∂ b , ∂ a = e ba e b . (2.3)Since any vector (or tensor) can be expressed in any base, we can write for instance (for avector v and co-vector k ) v = v a ∂ a = v b e b , k = k a d x a = k b e b , (2.4) For an introduction to the tetrads we refer the reader to [41, 42]. Here, I am using the very nice notationused [43]. – 8 –nd by using the change of basis matrices, we can obtain the transformation rules for thecomponents v a = v b e ab , k a = k b e ba ,v b = v a e ba , k b = k a e ab . (2.5)The same analysis can be made for tensors. In particular, for the metric tensor we have thatthe components transform as: g ab = e µa e νb g µν , but we defined the tetrads to be orthonormal,in the sense that the metric looks Minkowskian (Eq. (2.2)), therefore g ab = η ab , and we get η ab = e µa e νb g µν , g µν = e aµ e bν η ab , (2.6)with similar expressions for the inverse matrices g ab and η ab . Finally, since the metric g isused to rise and lower spacetime indexes, we can easily see that the metric η is used to riseand lower tetrad indexes, that is: v a = η ab v b and v a = η ab v b . In this paper we use the signature − for the metric. So, the four-velocity of a given observersatisfies u · u = − , where a “ · ” represents the scalar product between four-vectors, that is, u · u = u a u a = u a u a .Two observers u and u are related by u = γ (1) , (cid:0) u + v (1) , (cid:1) , where u · v (1) , = 0 , γ (1) , = − u · u = 1 (cid:112) − v (1) , · v (1) , , (2.7)and v (1) , is the relative velocity of u w.r.t u . For an observer u a , the four-momentum p a of given a photon can be written as p = E ( u − n ) , with u · n = 0 , E = − p · u , (2.8)where E and n a are the observed energy and direction of arrival. Note that n · n = 1 andthat d a ≡ − n a is the direction of propagation of the photon. In the following, it will be usefulto introduce the concepts of comoving-observers u com and tetrad-comoving observers ˜ u , theyare defined by the relations u i com = 0 , comoving-observers , (2.9) ˜ u i = 0 , tetrad-comoving-observers . (2.10)Note that in general, a comoving observer do not coincide with a tetrad-comoving observer.In fact, for the former the tetrads components of the four-velocity are u a com = e a u , showingthat in general u i do not vanish. For tetrad-comoving-observers the energy and direction ofincoming photons has a simple form ˜ E = − p , ˜ n a = (cid:0) , p i /p (cid:1) , (2.11) Eqs. (2.7)-(2.8) are given in a series of articles that follow the so-called 1 + 3 covariant approach to generalrelativity. See for instance [44–46]. Note that we are writing scalars in capital letters and vectors and tensors in small letters. – 9 –dditionally the decomposition of the four-velocity u = ˜ γ (˜ u + ˜ v ) is quite simple ˜ γ = u = (cid:113) u i u i , ˜ v a = (cid:0) , u i /u (cid:1) . (2.12)Using these results we can relate the energy E and direction n as observed by u , with theenergy ˜ E and direction ˜ n as seen by tetrad-comoving-observers simply by E = ˜ E (cid:0) u + ˜ n · u (cid:1) , (2.13) ˜ n i = n i − u i u − n . (2.14)For comparison with other works in the literature, let’s now relate the observed energy E tothe energy seen by comoving observers E com . We can obtain two equivalent expressions: thefirst one is obtained by applying Eq. (2.13) two times E = E com (cid:0) u + ˜ n · u (cid:1)(cid:16) u com + ˜ n · u com (cid:17) , (2.15)and the other one follows by applying the boost directly from the comoving observer to theu-observer E = E com γ (1 + n · v com ) = E com γ (1 − n · v ) , (2.16)where v com is the velocity of u com w.r.t u , and we have introduced the peculiar velocity v ≡ − v com , that is, u com = γ ( u − v ) . Although we will call v the peculiar velocity of theobserver, it is clear that this is not the velocity of u with respect to u com . We have introducedthis concept in order to be closer to the notation used in many other works, see for instanceEq.(1) of [24]. It follows from the previous equations that γ (1 − n · v ) = (cid:0) u + ˜ n · u (cid:1)(cid:16) u com + ˜ n · u com (cid:17) . (2.17)Although the photons’s energy has a simply form in the comoving frame, E com = − p u com = − p √ g , (2.18)most of the time we prefer to work with tetrad-comoving-observers because of the nice prop-erties given in this frame (see Eqs. (2.11)-(2.12)). In particular, for the direction ˜ n we have n a = n a making it safe to use bold-notation (see below). By contrast, in the comovingframe we have ( n com ) = 0 but in general n com (cid:54) = 0 . Additionally, the physics becomes moretransparent when using locally orthonormal basis (tetrads) instead of coordinates basis. That is, we apply two boost: one from the comoving frame to the tetrad-comoving one, and then oneadditional boost to the u-observer frame. – 10 – .3 Observed CMB temperature
It is well known (see for instance pp.588 of [47]) that in absence of secondary scatterings theCMB temperature at the point of observation T o is related with the temperature at emission T e by (this is a consequence of the Liouville theorem) T o = E o E e T e , (2.19)where E e ( E o ) is the energy of photons at the emission (observation) point. Note that ingeneral, the temperature is a function of both: the spacetime position x and direction nT o = T ( x o , n o ) , T e = T ( x e , n e ) . (2.20)The direction of emission (as seen by a local observer) is d e = − n e . Note that before theperiod of recombination it is expected that the temperature of the photon fluid is isotropic,in that sense it will not depend on the direction of emission. However, during the period ofrecombination a small quadrupole anisotropy arise in the photon distribution function [48, 49],that is why we kept the n e dependence in the emission temperature.By using Eqs. (2.11) and (2.13), the observed temperature can be written as T o = T e p ( x o ) p ( x e ) (cid:0) u + ˜ n · u (cid:1) o ( u + ˜ n · u ) e , (2.21)This can also be written in bold notation as T o = T e p ( x o ) p ( x e ) (cid:112) u o + ˜ n o · u o (cid:112) u e + ˜ n e · u e , (2.22)where the bold notation is used as a shorthand to express the spatial components in thetetrad basis, that is ˜ n = (˜ n i ) , u = ( u i ) , and u · ˜ n = u i ˜ n i . We stress that ˜ n is the observeddirection of incoming photons as seen by the tetrad-comoving-observers, which is related tothe direction of observation n by Eq. (2.14). The previous equation is equivalent to that givenin appendix A of [16], although there the authors were only interested in obtaining the CMBtemperature up to second order in the Poisson gauge, and by neglecting primordial vectorand tensor perturbations. In this paper however, we will not use the bold-notation. In this section we introduce the metric and tetrads which will allow us to obtain the Sachs-Wolfe formula. Note that two common notations for the metric ared s = a ( η ) (cid:2) − (1 + 2 φ ) d η + 2 ω i d x i d η + { (1 − ψ ) δ ij + 2 γ ij } d x i d x j (cid:3) , (3.1) = a ( η ) (cid:2) − e d η + 2 ω i d x i d η + (cid:8) e − δ ij + 2 γ ij (cid:9) d x i d x j (cid:3) , (3.2)where δ ij is the delta Kronecker tensor, x µ = ( η, x i ) , η the conformal time, a is the scalefactor and γ ij is defined as traceless in order to make the separation of the spatial part of themetric unambiguous. Usually each quantity is expanded perturbatively into first, second, or– 11 –hird order perturbations. Here however, we will treat each quantity non-perturbatively. Wepropose to use the following parametrization of the metricd s = a ( η ) (cid:20) − e d η + 2 β j (cid:0) e Φ − Ψ − Γ (cid:1) ji d x i d η + (cid:16) e − (cid:17) ij d x i d x j (cid:21) = a ( η ) e d ˆ s , (3.3)where Γ is a symmetric and traceless matrix, and the notation Ψ + Γ really means
Ψ1 + Γ where is the identity matrix, that is, (Ψ + Γ) ij = Ψ δ ij + Γ ij . The conformal metric isd ˆ s ≡ − d η + 2 β j (cid:0) e − M (cid:1) ji d x i d η + (cid:0) e − M (cid:1) ij d x i d x j , (3.4)with M = Φ + Ψ + Γ . Note that indexes in β i and M ij are raised and lowered with δ ij .Hereafter, we will mainly work with the conformal metric Eq. (3.4), and whenever weneed to express quantities in the physical metric we just multiply by the appropriated con-formal factor, as given for instance in Eq. (3.3) (more details below).We will still rewrite the conformal metric in a different way that will allow us to give aninteresting interpretation of β i and to easily express the metric in terms of tetrads, d ˆ s = − (cid:0) β d η (cid:1) + (cid:104)(cid:0) e − M (cid:1) ji d x i + β j d η (cid:105) (cid:104)(cid:0) e − M (cid:1) jk d x k + β j d η (cid:105) , (3.5)where we have introduced β ≡ (cid:112) β i β i . It is interesting to note that null paths in theconformal s-t are also null paths in the physical s-t, that implies that the path of photonsis totally determined by just two quantities: β i and M ij (and its derivatives, which enter thegeodesic equation). This is important for effects like lensing, time-delay and the integratedSachs-Wolfe (ISW). The conformal metric in the form given in Eq. (3.5) provides a naturalbasis of orthonormal dual vectors e a = e aµ d x µ , whose tetrads components are e µ = β δ µ , e i = β i , e ij = (cid:0) e − M (cid:1) ij . (3.6)The tetrads for the physical s-t are obtained from the above ones, simply multiplying by theconformal factor ae Φ . We now note that β a are the tetrad components of the four-velocity ofcomoving-observers. In fact, for a comoving observer ( u i com = 0 ) we have u a com = ( ae Φ ) e a u com = e a = β a , (3.7)where we have multiplied by the conformal factor ae Φ in order to get quantities in the physicals-t. Additionally, we used the normalization condition to obtain ae Φ u com = 1 .Below, we provide some relations which will be useful in the next section. They are theinverse tetrads, e a = 1 β δ a , e i = − β (cid:0) e M (cid:1) ij β j , e ij = (cid:0) e M (cid:1) ij , (3.8)and the derivative of the exponential matrix, the Baker-Campbell-Hausdorff formula (or theZassenhaus formula) [50] (cid:0) ∂ µ e − M (cid:1) = − A µ e − M , A µ ≡ (cid:90) d s e − sM ( ∂ µ M ) e sM . (3.9) This is basically the ADM decomposition of the metric. Hereafter we will use “s-t” as a short-hand for spacetime. – 12 – .1 The geodesic equation
In order to obtain the explicit form of ˜ n and p needed to obtain the observed CMB tempera-ture in Eq. (2.21), we need to solve the geodesic equation. Since photons follow null paths, wecan use the conformal s-t instead of the physical s-t, this will make calculations easier. Notethat, if p µ is the photon four-momentum in physical s-t, then ˆ p µ = ( ae Φ ) p µ is the photonfour-momentum in conformal s-t (Appendix D of [51]), consequently, ˆ p a = ( ae Φ ) p a . Note alsothat according to Eq. (2.11), the direction of observation as seen in the tetrad-comoving-frameis ˜ n i = − p i /p = − ˆ p i / ˆ p . With those considerations in mind, we can now proceed to obtainthe observed CMB temperature. We start with the geodesic equation in the conformal s-t[42] d ˆ p µ dλ = 12 ( ∂ µ ˆ g αβ ) ˆ p α ˆ p β = ( ∂ µ e aν ) e b ν ˆ p a ˆ p b , (3.10)where λ is an affine parameter. Using ddλ = ˆ p ddη , and after dividing on each size by (cid:0) ˆ p (cid:1) weget − β ˙ˆ p µ ˆ p = ( ∂ µ e aν ) e b ν ˆ p a ˆ p ˆ p b ˆ p , (3.11)where we used ˆ p = − β ˆ p , and a “dot” over a variable means total derivative w.r.t conformaltime. By noting that ˆ p = β ˆ p + β i ˆ p i , we can write p = 1ˆ p (cid:18) β + β i ˆ p i ˆ p (cid:19) , (3.12)and therefore the geodesic equation takes the form ˙ˆ p µ ˆ p = 1 β + β i ˜ n i (cid:110)(cid:0) ∂ µ β (cid:1) + ˜ n i (cid:104) ( ∂ µ β i ) + ( A µ ) ij β j (cid:105) + β ˜ n i ( A µ ) ij ˜ n j (cid:111) , (3.13)where we have used Eq. (3.9) for A µ . The equation above can be integrated for µ = 0 , yielding ˆ p ( x o ) = ˆ p ( x e ) e I ,I ≡ (cid:90) η o η e d ηβ + β i ˜ n i (cid:110)(cid:0) ∂ β (cid:1) + ˜ n i (cid:104) ( ∂ β i ) + ( A ) ij β j (cid:105) + β ˜ n i ( A ) ij ˜ n j (cid:111) , (3.14)which after substituting into Eq. (2.21) yields (multiplying by the conformal factor) T o = T e (cid:0) ae Φ (cid:1) e ( ae Φ ) o e I (cid:0) β + ˜ n i β i (cid:1) e ( β + ˜ n i β i ) o (cid:0) u + ˜ n · u (cid:1) o ( u + ˜ n · u ) e , (3.15)where we have used Eq. (3.12). Note that Eq. (3.12) is nothing else that the relation betweenthe energy in the comoving frame E com = − p , and the energy in the tetrad-comoving-frame ˜ E = − p , that is, E com = ˜ E (cid:0) u com + ˜ n · u com (cid:1) , (3.16)which follows from Eq. (2.13). We see that β + ˜ n i β i = u com + ˜ n · u com represents aDoppler boost. This is however, a point-to-point (along the photon’s path) boost which– 13 –akes the observed temperature by tetrad-comoving-observers into the observed temperatureby comoving observers. On the other hand, since this boost is determined by β i , which isdirectly related to the − i components of the metric, we will call this “a metric-Dopplereffect”.By using Eq. (3.15) together with Eq. (2.17) we can equivalently write T o = T e a e a o e Φ e − Φ o + I γ e (1 − n · v ) e γ o (1 − n · v ) o . (3.17)This is the equation we used in §1.2 to obtain the generalized Sachs-Wolfe formula Eq. (1.12).To complete the solution for the observed temperature, we need to obtain both ˜ n i andthe coordinates x µ of the photon’s path. We will address this problem in §4.4. Finally, wecan write the ISW in a covariant way by noting that ˜ β a ≡ (cid:0) , β i /β (cid:1) , (3.18)is the velocity of comoving observers w.r.t tetrad-comoving-observers (it follows from Eq. (2.12)).In terms of ˜ β we have I ≡ (cid:90) η o η e d η ˜ β · ˜ β (cid:48) − ˜ β + ˜ n · ˜ β (cid:48) + ˜ n · A · (cid:16) ˜ n + ˜ β (cid:17) β · ˜ n . (3.19)Here, we are treat ( A ) ij as the non-vanishing components of a (space-like) tensor A in thetetrad-frame, that is: ( A ) ij ≡ ( A ) ij and ( A ) a = 0 . In this section we use the generalized Sachs-Wolfe formula Eq. (1.12) to obtain the well knownresults at first order. Then we will discuss the gauge invariance of our result, emphasizingthe importance of the factor T o .From the definition of A µ , Eq. (3.9), we have up to first order A µ = ∂ µ M . Additionally,since we are interested in writing the observed temperature up to first order, we can take ˜ n at zero-order as it is always multiplying first-order quantities. That has several consequences.i) We can drop the “tilde” in the direction of observation as it is the same (at zero-order) forall observers, that is, ˜ n = n com = n, ii) all quantities are evaluated along the unperturbedpath for which we can set n io = n ie = n i . This is called the Born approximation, and theunperturbed path has coordinates x i ( η ) = x io + ( η o − η ) n io . Under these considerations theSachs-Wolfe formula up to first order is Θ = ( T e − T o ) + (Φ e − Φ o ) + I + ln (cid:18) − n e · v e − n o · v o (cid:19) , (3.20) I = (cid:90) η o η e (cid:8) n i β (cid:48) i + n i M ij n j (cid:9) , (3.21)where a “prime” means partial derivative w.r.t conformal time, and we have used β = 1 = γ valid up to first order.Although the components of n o are equal to the components of n e , that is, n io = n ie , wehave written n e · v e instead of n o · v e because in general, the quantity n o · v e is not well defined– 14 –s it represents the scalar product of two four-vectors which are defined at different points inthe s-t.Remembering that M ij = (Φ + Ψ) δ ij + Γ ij (see after Eq. (3.4)), and expanding thelogarithm up to first order we obtain Θ = ( T e − T o ) + (Φ e − Φ o ) − ( v e · n e − v o · n o ) + I , (3.22) I = (cid:90) η o η e (cid:8) n i β (cid:48) i + Φ (cid:48) + Ψ (cid:48) + n i Γ (cid:48) ij n j (cid:9) . (3.23)This is (apart from the factor T o ) the very well known first-order Sachs-Wolfe formula givenin Eq. (1.1). We now discuss the gauge invariance of Eq. (3.22). The gauge invariance of our results upto second order are discussed in a companion paper [39]. There we will provide the full setof transformation rules for the metric components and additional relevant quantities. Wewill use the following notation: under a gauge transformation a geometrical object T (scalar,vector, tensor, connections, etc.) will transform as T → T + ∆ T . Here we just need thefirst-order gauge transformations induced by the gauge generator ξ µ = ( α, ξ i ) , so we have ∆ T = −H α , ∆Φ = α (cid:48) + H α , ∆ v i = − ξ (cid:48) i , (3.24) ∆ β i = ξ (cid:48) i − α ,i , ∆ M ij = α (cid:48) δ ij − ξ ( i,j ) , (3.25)where H ≡ a (cid:48) /a is the Hubble’s expansion rate, a “comma” means derivative, so that α ,i = ∂ i α .The parenthesis in the expression ξ ( i,j ) means symmetrization, so ξ ( i,j ) = ( ξ i,j + ξ j,i ) / .With these expressions, it is easy to show the gauge invariance of Θ , that is ∆Θ = 0 .Indeed, for the integrated Sachs-Wolfe term, we get ∆ I = (cid:90) η o η e (cid:8)(cid:0) α (cid:48)(cid:48) − n i ∂ i α (cid:48) (cid:1) + n j (cid:0) ξ (cid:48)(cid:48) j − n i ∂ i ξ (cid:48) j (cid:1)(cid:9) = (cid:0) α (cid:48) + n j ξ (cid:48) j (cid:1) (cid:12)(cid:12)(cid:12) oe , (3.26)where we made used of the fact that along the unperturbed path, the following relations holds ∂ − n i ∂ i = d / d η . Additionally, we have ∆ { ( T e − T o ) + (Φ e − Φ o ) − ( v e · n e − v o · n o )) = (cid:0) α (cid:48) + n j ξ (cid:48) j (cid:9) (cid:12)(cid:12)(cid:12) eo , (3.27)showing explicitly that Θ is gauge invariant. Since the full temperature T o = ¯ T o e Θ is anobservable, it has also to be gauge invariant, as a consequence the mean value ¯ T o also is.Note that this result was possible thanks to the presence of T o inside Θ . Without it, eachtime we perform a gauge transformation, the temperature anisotropies would acquire anadditional monopole term. These transformation rules can also be obtained easily from the rules given in [52, 53]. – 15 –
The lensing term
To complete our analysis we need to obtain ˜ n i and the coordinates x µ along the photon’spath. These quantities are needed for a fully computation of the ISW effect. Additionally,they provide the so called lensing and time-delay terms (see §4.3). In this section we arriveat an expression which can by solved easily by iteration, allowing us to obtain the solutionperturbatively up to the desired order.By manipulating Eq. (3.13), we can obtain a differential equation for n i . We howeverchoose to follow a different way which yields a compact expression and can be used to easilyobtain the coordinates of the photon’s path. We start by defining q a (which is not a four-vector) by the relation q a ≡ ˆ p a ˆ p = p a p , ⇒ q a = d x a d η = (cid:0) , ˙ x i (cid:1) , (4.1)and we remind the reader that a “hat” means that quantities belong to the conformal s-t.Now, by using ˜ n i = − p i /p = − e ia p a / ( β p ) we get ˜ n i = − β (cid:16) β i + (cid:0) e − M (cid:1) ij q j (cid:17) . (4.2)Then if we manage to obtain q i , we automatically get both n i and x i = (cid:82) d η q i . Therefore,we now focus on q i . Before proceeding, we stress that the previous relation is nothing elsethat transformation of the direction vector, from the comoving-frame to the tetrad-comoving-observers, Eq. (2.14). That is, the previous relation can be written as ˜ n i = n i com − u i com u com − n com . (4.3)Consider now the geodesic equation in conformal s-t ˆ p ˙ˆ p a + ˆΓ ab c q b q c = 0 , (4.4)then by using ˙ˆ p a / ˆ p = ˙ q a + q a ˙ˆ p / ˆ p we get − ˙ q a = q b ˆΓ ab c q c − q a (cid:16) q b ˆΓ b c q c (cid:17) = q · Γ a · q − q a (cid:0) q · Γ · q (cid:1) , (4.5)where for simplicity of notation we have written on the second line q b ˆΓ ab c q c = q · (Γ a ) · q ,that is, we treat ˆΓ ab c as the components of a matrix Γ a . The relevant part of Eq. (4.5) is thatfor the spatial indices a = i and the a = 0 component is automatically satisfied, with q = 1 .Eq. (4.5) is a autonomous cubic equation in q , without an obvious analytic solution. Thiscan however easily be solved perturbatively, so for instance, if we call q a ( n ) the solution up ton-order, we can immediately obtained (n + 1)-solution as − q a ( n +1) (cid:12)(cid:12)(cid:12) η o η = (cid:90) η o η d η q ( n ) · (cid:16) Γ a − q a ( n ) Γ (cid:17) · q ( n ) . (4.6)We now detail the first-order solution which is needed to obtain the second-order logarithmictemperature anisotropies. I thank Yves Daoust user from stackexchange.com for useful comments on this point. See: https://math.stackexchange.com/questions/2205149/non-linear-matrix-differential-equation – 16 – .1 Lensing term at first order
As described before, we can just use Eq. (4.6) to easily obtain the first-order solution for q a .Before doing the integration, however, let’s write down the integrand on the r.h.s of Eq. (4.5)in a suitable way. Let’s start with ˆΓ ab c q b q c = 12 ˆ g aµ ( − ˆ g bc,µ + ˆ g µb,c + ˆ g cµ,b ) q b q c = −
12 (ˆ g aµ ˆ g bc,µ ) q b q c + ˆ g aµ q b ˙ˆ g bµ , (4.7)where we have used the fact that q c ∂ c = d / d η . Now, since ˆ g ab,c is already first order we canset ˆ g ab = η ab on the previous equation, so we got from Eq. (4.5) − ˙ q j = − q b q c (cid:2) ˆ g bc,j + ˆ g bc, q j (cid:3) − q b (cid:16) ˙ˆ g jb + ˙ˆ g b q j (cid:17) , (4.8)then noting that ∂ = d / d η − q i ∂ i , and defining the transverse gradient as ∂ ⊥ i = ∂ i − ˜ n i (˜ n · ∂ ) , where ˜ n · ∂ ≡ ˜ n i ∂ i , (4.9)we arrive, after integration, to − q i (cid:12)(cid:12)(cid:12) η o η = (cid:26) ˜ β i + 2 ( M · ˜ n ) i − ˜ n i (˜ n · M · ˜ n ) + (cid:90) d η ∂ ⊥ i (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17)(cid:27) (cid:12)(cid:12)(cid:12) η o η . (4.10)Here we have used that up to second order β i = ˜ β i (see Eq. (3.18)), and we treat M ij as thenon-vanishing components of a (space-like) tensor M in the tetrad-frame, that is: M ij ≡ M ij and M a = 0 . This is pretty much the same as we did in Eq. (3.19) for A .It is clear that on the r.h.s of the previous equation we should keep ˜ n at zero order, thisfact was taken into account in passing from Eq. (4.7) to Eq. (4.10) by setting q i = − ˜ n i validat zero-order. Additionally at zero-order we have ˜ n ie = ˜ n io = ˜ n i and we can also remove thetilde from ˜ n , so that ˜ n i = n i .Now that we are in possession of q a , we can immediately obtain ˜ n and x a up to firstorder. Direction vector ˜ n up to first order In order to obtain ˜ n , we see from Eq. (4.2) thatup to first order ˜ n i = − (cid:16) ˜ β i + q i + M ij ˜ n j (cid:17) , so we get ˜ n i = ˜ n io − (cid:104) ( M · ˜ n ) i − ˜ n i (˜ n · M · ˜ n ) (cid:105) (cid:12)(cid:12)(cid:12) η o η − (cid:90) η o η d η ∂ ⊥ i (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17) . (4.11) Coordinates of the photon’s path
Since q a = d x a / d η , the coordinates of the photon’strajectory are simply given by x a = (cid:82) d η q a . There is one important point we want to stresshere. Since q i depends on the fields β i and M ij , the coordinates of the photon’s path willdepend on these quantities. That means for instance that under a gauge transformationthe coordinates x i will necessarily change. The same happens if we consider two differentrealizations of the Universe, each one with the same background evolution but with differentfield perturbations (that is, different β i and M ij ). On the other hand, by construction q = 1 ,so the coordinate x = η is independent on these fields, and so x = η is insensible to any– 17 –auge transformation. By construction, the value of x = η is totally determined by thebackground evolution of the Universe (or the FLRW spacetime), in particular it is defined bythe hypersurface of constant (cid:104) T (cid:105) (see §1.2, and [16]).To obtain x i , we will use (cid:90) η o η e d η (cid:90) η o η d η (cid:48) f ( η (cid:48) ) = (cid:90) η o η e d η ( η − η e ) f ( η ) , (4.12)and the relation ˜ n io = − (cid:16) ˜ β i + q i + M ij ˜ n j (cid:17) (cid:12)(cid:12)(cid:12) η o which is valid up to first order. So fromEq. (4.10) we get x i = x io + (cid:104) ˜ n i − ( M · ˜ n ) i + ˜ n i (˜ n · M · ˜ n ) (cid:105) (cid:12)(cid:12)(cid:12) η o ( η o − η ) (4.13) + (cid:90) η o η d ¯ η (cid:104) ˜ β i + 2 ( M · ˜ n ) i − ˜ n i (˜ n · M · ˜ n ) (cid:105) − (cid:90) η o η d ¯ η (¯ η − η ) ∂ ⊥ i (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17) . In Eqs. (4.10)-(4.13) all the integrations are along the unperturbed path. Note that we haveparametrized q i , ˜ n i and x i in terms of the conformal time η . This is in contrast with severalother works, in which the coordinates x µ and the four-momentum p µ are obtained in termsof the affine parameter. See for instance Eqs. (2.20)-(2.24) of [12]. For comparison, note thatEq. (4.10) can be obtained by properly (i.e. by taking into account our Eq. (4.1)) dividingEqs. (2.22) by Eqs. (2.20) of [12].We have now all the elements to compute the logarithmic anisotropies up to secondorder. In this section we expand the logarithmic anisotropies up to second order. We will keepquantities evaluated along the photons’s curved path. In the next subsection, we expresseach quantity along the unperturbed path (the Born approximation). Let’s start with theISW. Firstly, it follows from Eq. (3.9) that up to second order A µ = ∂ µ M + [ ∂ µ M, M ] / , thenfrom Eq. (3.19) we get I = (cid:90) η o η e d η (cid:26) ˜ β · ˜ β (cid:48) + ˜ n · M (cid:48) · ˜ β + (cid:20) ˜ n · ˜ β (cid:48) + ˜ n · (cid:18) M (cid:48) + [ M (cid:48) , M ]2 (cid:19) · ˜ n (cid:21) (cid:16) − ˜ β · ˜ n (cid:17)(cid:27) = (cid:90) η o η e d η (cid:104)(cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) · ˜ n + (cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) · ˜ β ⊥ (cid:105) , (4.14)where ˜ β ⊥ is the orthogonal projection of ˜ β on ˜ n , that is, ˜ β ⊥ = ˜ β − ˜ n (˜ n · ˜ β ) . We have alsoused the fact that ˜ n · [ M (cid:48) , M ] · ˜ n = ˜ n i (cid:16) M (cid:48) ik M kj − M ik M (cid:48) kj (cid:17) ˜ n j = 0 .From Eq. (4.14) we see that there are two kind of contributions to the ISW. The termthat is explicitly linear in the fields is projected along the direction ˜ n , while the one whichis quadratic in the fields is projected in an orthogonal direction to ˜ n . Note also that, Though it is sensible to the introduction of new physical field perturbations, or different Universe realiza-tions. This is so, because field perturbations will affect the energy-momentum tensor which determines thetime-evolution via the Einstein’s equations. Even if the perturbations are small, they give a back reaction onthe background [54, 55]. Of course the term ˜ n · M (cid:48) is a vector formed by the projection of M onto ˜ n . In that sense, the full term ˜ n · M (cid:48) · ˜ β ⊥ represents a “double” projection of M , one along ˜ n and other along ˜ β ⊥ which is perpendiculardirection to ˜ n . – 18 – ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) · ˜ β ⊥ = (cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) ⊥ · ˜ β = (cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) ⊥ · ˜ β ⊥ , where (cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) ⊥ isdefined in the same manner as ˜ β ⊥ .To compute the logarithmic anisotropies, it remains to expand the Doppler effect up tosecond order. It can be written as ln (cid:18) γ e (1 − n e · v e ) γ o (1 − n o · v o ) (cid:19) = (cid:34) − v · n + v − ( v · n ) (cid:35) (cid:12)(cid:12)(cid:12) η e η o = (cid:104) − v · n + v · v ⊥ (cid:105) (cid:12)(cid:12)(cid:12) η e η o . (4.15)Again we see the same behavior as for the ISW effect. That is, terms that are linear in thefields (here v ) are projected along n , while the quadratic terms only receive contributionfrom the orthogonal direction to n (here, v ⊥ ).Finally, by using Eq. (2.17) (see also Eq. (3.15)) we can also write ln (cid:18) γ e (1 − n e · v e ) γ o (1 − n o · v o ) (cid:19) = (cid:34) ( β i − u i ) ˜ n i + β i β i ⊥ − u i u i ⊥ (cid:35) (cid:12)(cid:12)(cid:12) η e η o . (4.16)Depending on the situation, one can find it more convenient to use either the first or thesecond version of the Doppler effect (Eq. (4.15) or Eq. (4.16)). We will take the latter, asit involves ˜ n . Before going further, we write the previous equation in a covariant mannerby using the velocity of comoving observer w.r.t the tetra-frame ˜ β and the velocity of theobserver u w.r.t the tetrad-frame, that is, ˜ v a F ≡ (cid:0) , u i /u (cid:1) . Up to second order we have ˜ β a = (cid:0) , β i (cid:1) and ˜ v a F = (cid:0) , u i (cid:1) . (4.17)Here, the subscript F is because we can think of the observer u as being a fish moving througha river (the tetrad frame). This idea is explored in a companion paper [39]. Joining all theprevious results we have up to second order Θ = (cid:16) T + Φ + ( ˜ β − ˜ v F ) · ˜ n (cid:17)(cid:12)(cid:12)(cid:12) η e η o + (cid:90) η o η e d η (cid:16) ˜ n · ˜ β (cid:48) + ˜ n · M (cid:48) · ˜ n (cid:17) + ˜ β · ˜ β ⊥ − ˜ v F · ˜ v ⊥ F (cid:12)(cid:12)(cid:12) η e η o + (cid:90) η o η e d η (cid:16) ˜ β (cid:48) + ˜ n · M (cid:48) (cid:17) · ˜ β ⊥ . (4.18)We have written the logarithmic anisotropies in this way to stress that the first line is for-mally equal to the first-order logarithmic anisotropies. So (formally), the difference comesonly from the second part. These two lines are different in nature, so they could be mea-sured independently. Note that − ˜ n is the direction of propagation of photons as seem bythe tetrad-comoving-observers, so the plane perpendicular to ˜ n is the plane of the photon’spolarization. We conclude that only the projection on the plane of polarization of the fieldperturbations, contribute to the explicitly quadratic terms of Θ (second line of Eq. (4.18)). In this section the second-order logarithmic anisotropies are given by using the Born approx-imation, that is, we express each quantity along the path x i = x io + ˜ n io ( η o − η ) , which is the We remind the reader that n is the direction of observation in the u -frame, while ˜ n is the direction ofobservation in the tetrad-comoving-frame. I said formally, because here each quantity is considered up to second order. Additionally they evaluatedalong the photon’s curved path, while at first order, Θ is computed using the background trajectory. – 19 –ath inferred by the observer ignoring perturbations. This can be useful for numeric compu-tations and also because the notation of previous results in literature (e.g., those in [26]) iscloser to the one we use below.We will define the deviation δx i from the Born approximation by the relation x i = (cid:104) x io + ˜ n io ( η o − η ) (cid:105) + δx i . Analogously, we will write ˜ n i = ˜ n io + δ ˜ n i . The explicit expression of δx i and δ ˜ n i , follow directly from Eqs. (4.11) and (4.13). With theses definitions we can Taylorexpand the logarithmic temperature anisotropies around the Born approximation’s path, as Θ = Θ
Born + (cid:104) δ T + δ Φ + ˜ n · δ ( ˜ β − ˜ v F ) + ( ˜ β − ˜ v F ) · δ ˜ n (cid:105) e + (cid:90) η o η e d η (cid:104)(cid:16) ˜ n · δ ˜ β (cid:48) + ˜ n · δM (cid:48) · ˜ n (cid:17) + (cid:16) δ ˜ n · ˜ β (cid:48) + 2 δ ˜ n · M (cid:48) · ˜ n (cid:17)(cid:105) , (4.19)where Θ Born is the same as Eq. (4.18) but with everything evaluated in the Born approxi-mation. Here the notation must be intuitive. For instance, ( δM (cid:48) ) ij = δx k ∂ k M (cid:48) ij with sim-ilar expressions for the other fields. The only difference is with the intrinsic logarithmicanisotropies T e which in general will depend not only on the position x e but also on thedirection of emission − ˜ n e , that is, T e = T ( x e , − ˜ n e ) . So we must use δ T e = (cid:18) δx ie ∂ i + δ ˜ n ie ∂∂ ˜ n ie (cid:19) T e . (4.20) To end this section, we remind the reader about the concepts of lensing and time-delay, whichare encoded into δx i and are correlated with the ISW. To obtain the time-delay, we project δx i along the radial direction (see Eq. (4.13)) δx i ˜ n i = (cid:90) η o η d ¯ η (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17) , (4.21)this quantity tells us that photons are not coming from a spherical shell of radius r but froma distorted surface whose “radius” in direction ˜ n is distorted by ( δx i ˜ n i ) e . There are two typesof lensing terms: the first one is given by the transverse component of δx i , δx i ⊥ = − ( M o · ˜ n o ) i ⊥ ( η o − η ) + (cid:90) η o η d ¯ η (cid:104) ˜ β i ⊥ + 2 ( M · ˜ n ) i ⊥ (cid:105) − (cid:90) η o η d ¯ η (¯ η − η ) ∂ ⊥ i (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17) , and the second one is just the local deflection angle δ ˜ n , which from Eq. (4.11) is δ ˜ n i = − (cid:104) ( M · ˜ n ) i ⊥ (cid:105) (cid:12)(cid:12)(cid:12) η o η − (cid:90) η o η d η ∂ ⊥ i (cid:16) ˜ n · ˜ β + ˜ n · M · ˜ n (cid:17) . (4.22)Lensing and time-delay are correlated with the ISW effect due to the second line of Eq. (4.19).Note that, regarding the logarithmic anisotropies Θ , these are the only quantities that arecorrelated with the ISW. This is not true however for ∆ T o / ¯ T o which involves powers of Θ and therefore will automatically correlate the ISW with other terms like T and Φ . Becauseof that, making maps of the logarithmic anisotropies will provide an optimal tool for studythe ISW.Below we briefly comment on the comparison with other works and also briefly cite someof the results that will publish in a companion paper.– 20 – Future work and conclusions
The results of appendix A of [26] (and therefore, the results of [12]) are equivalent to the onesgiven in the previous section. In particular compare Eq. (4.19) with Eqs. A.32-A.35 of [26].In comparing the results of [26], we must take into account the following relationship δ e i = δ ˜ n i + (cid:16) ˜ β + ˜ n · M (cid:17) i ⊥ , where the quantity δ e was defined in Eq.A.29 of [26]. The previous relation follows directlyfrom Eq. (4.22) and Eq.A.28 of [26]. This shows that the interpretation given in [26] for thequantity δ e as the local deflection angle is wrong, because the true local deflection angle isgiven by δ ˜ n i . Apart from this fact, the results of [26] are correct. A more detailed comparisonof our results with those already present in literature will be discussed in a future paper [40].On the other hand, since we have introduced several new concepts: a new parametriza-tion of the metric, the logarithmic intrinsic temperature anisotropies T , the direction of ob-servation by tetrad-comoving-observers ˜ n , etc, the gauge transformations of theses quantitieshave not been discussed before in literature. In addition, gauge transformations when appliedto the CMB anisotropies involves several subtle issues as it was firstly discussed in [16]. Ina companion paper [39], we will discuss the gauge transformations of the relevant quantitiesintroduced in this paper and explicitly show the gauge invariance of our second-order formulaEq. (4.19). Special emphasis is put on the subtle issues of gauge transformations on the CMB. We have obtained the non-linear generalization of the Sachs-Wolfe + integrated Sachs-Wolfeformula describing the CMB temperature anisotropies Eq. (1.12). Our result is valid at allorders in perturbation theory, includes scalar, vector and tensor perturbations, and is valid inany gauge. Direct observational consequences of our result have been discussed, in particularthe fact that the logarithmic temperature anisotropies
Θ = ln (cid:0) T o / ¯ T o (cid:1) is more suitablefor data analysis than the usual temperature anisotropies ∆ T o / ¯ T o . The reason is that bytaking the logarithm we automatically remove many secondary effects which otherwise wouldbias the analysis of the data. This will be of particular importance for the search of primordialnon-Gaussianity and for analysis of the ISW effect and lensing.Then we expanded our exact expression up to second order and got results which are verysimple and intuitive, see Eqs. (4.18) and (4.19) for two different versions. Finally, severalconcepts have been introduced as the logarithmic intrinsic anisotropies T and T o (see §1.2),the tetrad-comoving-observers in §2.2 and an useful parametrization of the metric §3 whichexpresses the − i metric components in terms of the four-velocity of comoving observers. Acknowledgments
I thank Thiago Pereira and Elvis Soares for useful discussions and suggestions. I also thankMauricio Calvão for introducing me to the 1 + 3 - covariant formalism, and for useful dis-cussions on aberration. I thank Cyril Pitrou for drawing my attention on the importance ofthe logarithmic transform on spectral distortions. Finally, I thank the anonymous referee ofpaper [26] because his (her) useful comments somehow influenced the style of this article. Although in [26] we took the perturbations to vanish at the observer position. – 21 – eferences [1]
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