CMB: Theory and History

Leonardo Senatore (Physics Dept. and SLAC)

CMB: theory and history The Cosmic Microwave Background The Cosmic Microwave Background • It is a remarkable aspect of the theory of the Big Bang • It is also an extremely accurately predicted observable • allows us to learn about the fundamental constituents of the universe at early times • allows us to map the reconstruct the very initial conditions of the universe

• This lecture is largely based on a review by: Challinor and Peiris 0903.5158

but see also for example Hu 0802.3688 and the most-complete book Modern Cosmology by Dodelson Thermal History • The universe starts very hot. Indeed, it starts so hot that there are no nuclei. As the expansion progress and slows down, nuclear reaction fall out of equilibrium and nuclei form and their abundances freeze (Big Bang Nucleosynthesis (BBN)). The overall abundance of nuclei depends on how many photons are there, as they break the newly formed nuclei. In fact, nuclei form at a much lower temperature than their binding energy.

• Therefore, observation of current abundance of nuclei implies a prediction for a bath of thermal radiation at a certain temperature. This was the ﬁrst prediction of the Cosmic Microwave Background (CMB), by Gamov (1948), Alper & Herman (1948), but it was largely unnoticed. By observed homogeneity of the universe, it was expected to be homogenous. • After BBN the universe cools down: reactions that changes number of photons fall out of equilibrium very early on, and the distribution of photons remains in kinetic equilibrium for a long time due to Thomson scattering with electron.

• At some later time (when universe is 4000K (again, delay due to large number of photons per baryon)), hydrogen starts to recombine. Thermal History

• Quickly, photons, whose energy is typically T 0 . 1eV , much lower than the energy for exciting ⇠ Hydrogen, begin to travel freely: their mean freeE patch becomes of order Hubble distance (the (1) !1 1 k 0.45 h Mpc (2) maximum distance). In other words, the probability⇠ for a photon that we get in our eyes to have last (3) scattered at some time, it is highly peaked at )a distance of order 300 kyr, and then they travelled ⌫ (4) quite freely for 13 Gyr. dm (5) • This time is called the Last Scattering Surfaceq (6) (q) (7) 2 Pcounter,slow ⌫0s = f⌫, slowcsP11(k) (8) • Therefore, there should be a quite homogenouskfs radiation& kNL of a few Kelvins coming to us from all (9) 1 k directions. fs (10) k 2 k NL ✓ NL ◆ (1) 0 (11) di↵ ' (1) (1) (12) • This was serendipitously discovered in 1964 bydi↵ Penzias' dm and Wilson k . kfs (13) 14f P (14) ⌫ dm, dm 8f P (15) ⇠ ⌫ dm, dm k & kfs (16) (1 f ) (17) ⌫ (1) 1+f⌫ dm(a) D D(a) (1 + f⌫ log(a)) (18) (1) / ⇠ · dm (19) (1) f⌫ di↵ (20) log(a ) 8 (21) equality ⇠ cs (22)

dm@idm (23) v /v 1 (24) NL ⌧ (qi) > 1forqi >kNL (25) [(~x ,t )(~x ,t )] = (3)(~x ~x ) f (everything,...,@ @ (x) ...) 1 1 2 2 C D 1 2 very complicated i j |past light cone =atlongwavelengthswecanTaylorexpand= (26) (1) (27) q ,q k (28) 1 2 (29) Discovery Nobel Prize 1978: Penzias and Wilson • This was serendipitously discovered in 1964 by Penzias and Wilson

• Princeton physicists

• were looking for that, • and told them how to interpret

• their discovery

• There was a problem • with pigeons Thermal History

Credit Hu and White 2004 The hunt for Inhomogeneities • This was a spectacular conﬁrmation of the Big Bang picture: the universe was homogenous at very early times, with great precision. • The numbers matched the ones roughly predicted from BBN. • Cosmology was transforming in a highly quantitative discipline • A natural picture for the formation of structures in the universe emerged: – the universe started very homogenous – starts, galaxies, and clusters (large-scale structure) formed out of gravitational growth of small inhomogeneities.

• There had to be inhomogeneities in the CMB! k⌘ 1 (k⌘) constant ⌧ ) ⇠ ) 1 k⌘ 1 (k⌘) 0 (1) ) ⇠ (k⌘)2 !

C2(~k)=0 (2) 1 (k, ⌘) C (~k) cos kr (⌘)+C (~k) (3) ' 1 (1 + R)1/4 s 2 ✓ ⇣ ⌘◆ (4) R (5) !1 vb = ikˆvb (6) f = ⇥0 + (7) lm (8) { } (1 + cos(✓)2) (9)

T =ThomsonCross-section (10) ⇥ (11) d @ = + e =derivativealongﬂow (12) d⌘ @⌘ · r ˙= @/@⌘ (13) Df @f dx @f d✏ @f de @f = + + + (14) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (15) d⌘ 1 f ✏ (16) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (17) { 1 2 3} The hunt for Inhomogeneities (18) )• There had to be inhomogeneities in the CMB! • Some✓ numbers:2⇡/l the CMB has now a temperature of 2.72K. Since when it was emitted the (19) ⇠ 3 temperature was about 3000K, it means that the redshift of about z 10 ECMB =photonenergyinCMBrestframe LSS ⇠ (20) ¯ T 0.1eV (1) TCMB = T =isotropictemperature ⇠ (21) •TNow, the evolution of small inhomogeneities in the universe tells usE that, if the universe is (2) 5 !1 1 dominated3 10by dark matter, we have k 0.45 h Mpc (22) (3) T ⇠ · ⇠ a 3 (4) dm(a)=dm(ain) if dm(a0) 1 ) dm(aLSS) 10 (23) ain ) ⇠ )⌫ ⇠ (5) 3 z 10 dm T 5 (24) (6) LSS– there are various detailed corrections that tells us that the correct prediction is 3 10 ⇠ q T ⇠ · (7) T 0.1eV a (25) 3 ⇠ (q) dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (1) (8) aain ) ⇠ ) ⇠ • of course this disagreement matters a lot if you are an experimentalist!! 32 (26) Pcounter,slow ⌫ s =z f⌫, slow10c P11(k) (2) (9) 0 LSS ⇠ s k k T 0.1eV (3)(10) fs & NL ⇠ 1 k (4) fs (11) k 2 k NL ✓ NL ◆ (1) 0 (12) di↵ ' (1) (1) (13) di↵ ' dm k . kfs (14) 14f P (15) ⌫ dm, dm 8f P (16) ⇠ ⌫ dm, dm k & kfs (17) (1 f ) (18) ⌫ (1) 1+f⌫ dm(a) D D(a) (1 + f⌫ log(a)) (19) (1) / ⇠ · dm (20) (1) f⌫ di↵ (21) log(a ) 8 (22) equality ⇠ cs (23)

dm@idm (24) v /v 1 (25) NL ⌧ (qi) > 1forqi >kNL (26) [(~x ,t )(~x ,t )] = (3)(~x ~x ) f (everything,...,@ @ (x) ...) 1 1 2 2 C D 1 2 very complicated i j |past light cone =atlongwavelengthswecanTaylorexpand= (27) (1) (28) q ,q k (29) 1 2 (30) The Discovery of Inhomogeneities • In 1971 a large dipole pattern was detected (see R. Muller Scientiﬁc American 1978 for an overview) – This is interpreted as the dipole induced by the relative motion of our galaxy wrt the intergalactic medium (~600km/sec) • After many experiments falling short of the target,

T 5 – Intrinsic anisotropies were detected by COBE in 1992: 3 10 T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (1) aain ) ⇠ ) ⇠ z 103 (2) LSS ⇠ T 0.1eV (3) ⇠ Nobel Prize 2006: Mather and Smooth (4)

• We understood how structure get formed! How structures gets formed

http://lambda.gsfc.nasa.gov/ Actual distribution of structures in our Universe

Movie del viaggio nelle galassie

http://astro.uchicago.edu/cosmus/projects/sloanmovie/ Measuring the Inhomogeneities • COBE discovery not only explained us the basic picture of how structure got formed, it also opened up a series of extraordinary questions. • In fact, there was the need to characterize the spectrum: – they appeared to be highly gaussian – and what was the wavelength distribution of the power in anisotropies?

• This questions needed to be answered by making measurements. T 5 • Furthermore, since 3 10 , it became pretty clear that it should be possible to make T ⇠ · extremely accurate predictions, by expandinga in the smallness of the inhomogeneities. 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (1) – The CMB was to become the instrumentaain to turn) cosmology into a⇠ high-precision) science ⇠ z 103 (2) LSS ⇠ T 0.1eV (3) ⇠ (4) Measuring the Spectrum Inhomogeneities • QMAP, Boomerang, Maxima begun to ﬁrst measure the spectrum of the inhomogeneities.

• They found a very tantalizing peak structure Measuring the Spectrum Inhomogeneities • Since then, many improvements Interpreting the measurements • The peak structure is very tantalizing. Can we understand it?

• Let us begin to study the theory of the inhomogeneities. • Let us expand in the smallness of the perturbations. • At zeroth order, we have that the one-particle distribution (number of photons per proper phase- space volume) is a Black Body:

– where ECMB =photonenergyinCMBrestframe

TCMB = T¯ =isotropictemperature (1) • Due to our motion wrt the CMB, we have T 5 3 10 (2) T ⇠ · • This is a angle-dependent black body a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (3) aain ) ⇠ ) ⇠ • with z 103 (4) LSS ⇠ • This is our distributionT if0 the.1eV universe were homogeneous. We are interested in the inhomogeneities. (5) ⇠ (6) CMB observables • It is useful to decompose the perturbation in spherical harmonics:

– each multipole encodes information on a scale ✓ 2⇡/l ⇠ • We can make predictions only on the statistical properties ofE CMBthe fluctuations=photonenergyinCMBrestframe (we do not have a (1) ¯ theory that predicts the amplitude of each ). So, we computeTCMB directly= T =isotropictemperature correlation functions of (2) T 5 – the statistical properties of the fluctuations should be compatible3 with10 the background space, (3) T ⇠ · which for FRW is rotation and translation. Since under rotation transforms under a a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (4) Wigner rotation matrix as , rotation invariance of atheain perturbations) ⇠ ) ⇠ implies z 103 (5) LSS ⇠ ». T 0.1eV (6) ⇠ ) (7) ». ✓ 2⇡/l (1) ⇠ »… ECMB =photonenergyinCMBrestframe (2) • This defines the power spectrum and the Bispectrum . Angle brakes denotes TCMB = T¯ =isotropictemperature (3) ensemble average over many universes. T 5 • If perturbations are Gaussian, all information is in 2-point3 function10 (4) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (5) aain ) ⇠ ) ⇠ z 103 (6) LSS ⇠ T 0.1eV (7) ⇠ (8) Linear Inhomogeneities • General quasi-linear perturbations around an FRW background can be classified according to their rotational properties (helicity-0,1,2…). Let us start assuming we have only helicity-0 perturbations, and let us assume the background FRW metric is flat. The most general metric with such a perturbation can be brought by a suitable choice of coordinates that fixes all the freedom in the coordinates (gauge fixing) to the following form:

• We characterize a photon by its energy (as seen from an observer at rest in our coordinates) and direction e = e ,e ,e with { 1 2 3} (1) • The photon 4-momentum) is 1 ✓ 2⇡/l f ✏ (2) • We define the perturbed⇠ distribution of photons as ' e aT¯[1+⇥(⌘,x,e)] 1 ECMB =photonenergyinCMBrestframe (3) ¯ e = e1,e2,e3 (1) TCMB = T =isotropictemperature { } (4) T 5 (2) 3 10 ) (5) • notice that T ⇠ · . Since the✓ unperturbed2⇡/l distribution (3) a 3 is time-independent.dm(a)=dm(ain) if dm(a0) 1 ⇠ dm(aLSS) 10 (6) aain ) ⇠ E)CMB =photonenergyinCMBrestframe⇠ (4) 3 zLSS 10 ¯ (7) ⇠ TCMB = T =isotropictemperature (5) T 0.1eV (8) ⇠ T 5 3 10 (9) (6) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (7) aain ) ⇠ ) ⇠ z 103 (8) LSS ⇠ T 0.1eV (9) ⇠ (10) Linear Inhomogeneities: gravity Df • The evolution of this distribution is determined by the Boltzmann equation = C[f] Df @f @x @f @✏ @f @e @f d⌘ =Df +@f dx +@f d✏ @+f de @f 1 d⌘ @⌘ = @⌘+· @x @⌘+ @✏ @⌘+ · @e – is the free-streaming fterm ✏ (1) Df d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e ' e aT¯[1+⇥(⌘,x,e)] 1 =– Df C [ f ] is the collision term (due to Thomson scattering) (1) = C[f] e = e1,e2,e3 (1) (2) d⌘ d⌘ { } – at linear order, we can deal with each one separately. (3) 1 1 ) f f ✏ ✏ (2)(2) • Let us¯ start with the free streaming term, which is determined by the motion in the✓ perturbed2⇡/l metric. (4) ' e aT [1+' ⇥e(a⌘T¯,x[1+,e)]⇥(⌘,x,e1)] 1 ⇠ • The geodesic equation for a single photon gives e = ee1,e=2,ee13,e2,e3 ECMB =photonenergyinCMBrestframe(3)(3) (5) { { } } ¯ TCMB = T =isotropictemperature(4)(4) (6) ) ) ✓ 2⇡/l T 5 (5) ✓ 2⇡/l⇠ 3 10 (5) (7) ⇠ T ⇠ · E – Ewhere=photonenergyinCMBrestframeCMB =photonenergyinCMBrestframe and d @ (6)(6) CMB ˙=¯ @/@⌘ = + e =derivativealongflow a 3 TCMB¯ = T =isotropictemperatured⌘ @⌘ · r dm(a)=dm(ain) (7) if dm(a0) 1 dm(aLSS) 10 (8) TCMB = T =isotropictemperatureDf @f dx @f d✏ @f de @f aain(7)) ⇠ ) ⇠ T 5 – comoving3 energy10 = evolves due+ to˙= gravitational@/@⌘ redshifts+ + 3 (8) (1)(1) T T ⇠ 5· zLSS 10 (9) 3 10 d⌘ @⌘a Dfd⌘ · @@xf ddx⌘ @✏@f dd✏⌘@f· @ede⇠ @f (8) T ⇠– direction· evolves due to gravitational =gradient orthogonal+ to direction+ (gravitationalT+ 0 .lensing)1eV 3 (2) (10) dm(a)=dm(aain) if dm(a0) 1 dm(aLSS⇠) 10 (9) • By plugging inDf a ain d ) ⌘ @⌘ d ⌘ ⇠ · @ x ) d ⌘, @✏ , we getd ⌘eq.⇠· for@ e 3 dm(a)=dm(a3in) = C[f] if dm(a0) 1 dm(aLSS) 10⇥ (9) (2) (11) zLSS 10d⌘ aain )Df ⇠ ) ⇠ (10) 3⇠ = C[f] d @ (3) zLSS T10 0.1eV d⌘ = (11)(10)+ e =derivativealongflow (1) ⇠ ⇠ 1 d⌘ @⌘ · r T 0.1eV f ✏ 1 (12)(11) (3) aT¯[1+⇥(⌘f,x,e)] ✏ (4) ⇠ ' e ¯ 1 ' e aT [1+⇥(⌘,x,e)] 1 ˙= @/@⌘ (12) (2) e = e1,e2,ee3 = e1,e2,e3 Df @f dx @f d✏ @f (5)de(4)@f { } { } = + + + (3) (6)(5) ) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e ) ✓ 2⇡/l (7) ✓ 2⇡/l ⇠ Df (6) ⇠ ECMB =photonenergyinCMBrestframe = C[f] (8) (4) ECMB =photonenergyinCMBrestframe d⌘ (7) TCMB = T¯ =isotropictemperature (9) T = T¯ =isotropictemperature 1 (8) CMB T 5 f ✏ (5) 3 10 ' aT¯[1+⇥(⌘,x,e)] (10) T 5T ⇠ · e 1 3 10 a 3 (9) T ⇠ · dm(a)=dm(ain) if dm(a0)e =1 e1,e2,edm3 (aLSS) 10 (11) (6) a aain ) ⇠ { ) } ⇠ 3 3 (7) dm(a)=dmz(LSSain) 10 if dm(a0) 1 ) dm(aLSS) 10 (12)(10) ⇠ aain ) ⇠ ✓) 2⇡/l ⇠ (8) 3 T 0.1eV (13) zLSS 10 ⇠ ⇠ (11) ⇠ ECMB =photonenergyinCMBrestframe(14) (9) T 0.1eV ¯ (12) ⇠ TCMB = T =isotropictemperature (10) (13) T 5 3 10 (11) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (12) aain ) ⇠ ) ⇠ z 103 (13) LSS ⇠ T 0.1eV (14) ⇠ (15) Linear Inhomogeneities: gravity • Therefore we can suggestively write:

• If the observer is at , a photon will have last scattered from a location given by, if , then

• We can integrate this equation. The ﬁrst term gives the difference in gravitational energy between the initial and the ﬁnal location (overall gravitational redshift). The second contribution gives the so-called Integrated Sachs-Wolfe effect – where the integral in taken along trajectory (line-of-sight)

– Due to evolution of perturbation, this integral has support only at radiation epoch (early ISW) and at late time when dark energy kicks in and potential decays (leading to a blue shift). Df @f @x @f @✏ @f @e @f = + + + d⌘ @⌘ @⌘ · @x @⌘ @✏ @⌘ · @e Linear Inhomogeneities:Df scattering • We now take into account the collision term= C [ f ] . The evolution due to Thomson scattering gives (1) d⌘ 1 f ✏ (2) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (3) { 1 2 3} – where T =ThomsonCross-section (4) ) ✓ 2⇡/l (5) ⇥ ⇠ (1) d @ ECMB =photonenergyinCMBrestframe (6) ¯ = + e TCMB=derivativealongﬂow= T =isotropictemperature (7) (2) d⌘ @⌘ · r T 5 3 10 (8) ˙= @/@⌘ T ⇠ · (3) a 3 Df @f ddmx(a)=@fdm(adin✏) @f de if@fdm(a0) 1 dm(aLSS) 10 (9) = + + aain +) ⇠ ) ⇠ (4) d⌘ @⌘ dz ⌘ · @10x3 d⌘ @✏ d⌘ · @e (10) LSS ⇠ Df T 0.1eV (11) = C[f] ⇠ (5) d⌘ (12) 1 f ✏ (6) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (7) { 1 2 3} (8) ) ✓ 2⇡/l (9) ⇠ ECMB =photonenergyinCMBrestframe (10)

TCMB = T¯ =isotropictemperature (11)

T 5 3 10 (12) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (13) aain ) ⇠ ) ⇠ z 103 (14) LSS ⇠ T 0.1eV (15) ⇠ (16) Df @f @x @f @✏ @f @e @f = + + + d⌘ @⌘ @⌘ · @x @⌘ @✏ @⌘ · @e Linear Inhomogeneities:Df scattering • We now take into account the collision term= C [ f ] . The evolution due to Thomson scattering gives (1) d⌘ 1 f ✏ (2) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (3) { 1 2 3} – where T =ThomsonCross-section (4) ) ✓ 2⇡/l (5) ⇥ ⇠ (1) Scattering into of beam. ECMB =photonenergyinCMBrestframe (6) d @ 2 Notice (1 + cos( ¯ ✓ ) ) dependence = + e TCMB=derivativealongflow= T =isotropictemperature (7) (2) d⌘ @⌘ · rT =ThomsonCross-section (1) T 5 3 10 (8) ˙= @/@⌘ T⇥ ⇠ · (2) (3) d @ a 3 Df @f ddmx(a=)=@f+dme(adin✏) =derivativealongflow@f de if@fdm(a0) 1 dm(aLSS) 10 (3) (9) = + d⌘ @⌘ + · r aain +) ⇠ ) ⇠ (4) ˙= @/@⌘ 3 (4) d⌘ @⌘ dzLSS⌘ · @10x d⌘ @✏ d⌘ · @e (10) Df⇠ @f dx @f d✏ @f de @f Df T 0=.1eV + + + (5)(11) = C[f] d⇠⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e (5) Df (12) d⌘ = C[f] (6) 1 d⌘ f ✏ 1 (6) f ✏ (7) aT¯[1+⇥(⌘,x,e)] ' e ' e a1T¯[1+⇥(⌘,x,e)] 1 e =e ,e ,e (8) e = e1,e2,e3 { 1 2 3} (7) { } (9) ) ✓ 2⇡/l (10) (8) ) ⇠ ✓ 2⇡/l ECMB =photonenergyinCMBrestframe (11) (9) ⇠ TCMB = T¯ =isotropictemperature (12) ECMB =photonenergyinCMBrestframe (10) T 5 3 10 (13) T TCMB = T¯ =isotropictemperature⇠ · (11) a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (14) T 5 aain ) ⇠ ) ⇠ 3 10 z 103 (15) (12) T ⇠ · LSS ⇠ T 0.1eV (16) ⇠ a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (17) (13) aain ) ⇠ ) ⇠ z 103 (14) LSS ⇠ T 0.1eV (15) ⇠ (16) Df @f @x @f @✏ @f @e @f = + + + d⌘ @⌘ @⌘ · @x @⌘ @✏ @⌘ · @e Linear Inhomogeneities:Df scattering • We now take into account the collision term= C [ f ] . The evolution due to Thomson scattering gives (1) d⌘ 1 f ✏ (2) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (3) { 1 2 3} – where T =ThomsonCross-section (4) ) ✓ 2⇡/l (5) ⇥ ⇠ (1) E =photonenergyinCMBrestframe (6) d @ CMB Doppler due to velocity of electrons ¯ = + e TCMB=derivativealongflow= T =isotropictemperature (7) (2) d⌘ @⌘ · r T 5 3 10 (8) ˙= @/@⌘ T ⇠ · (3) a 3 Df @f ddmx(a)=@fdm(adin✏) @f de if@fdm(a0) 1 dm(aLSS) 10 (9) = + + aain +) ⇠ ) ⇠ (4) d⌘ @⌘ dz ⌘ · @10x3 d⌘ @✏ d⌘ · @e (10) LSS ⇠ Df T 0.1eV (11) = C[f] ⇠ (5) d⌘ (12) 1 f ✏ (6) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (7) { 1 2 3} (8) ) ✓ 2⇡/l (9) ⇠ ECMB =photonenergyinCMBrestframe (10)

TCMB = T¯ =isotropictemperature (11)

T 5 3 10 (12) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (13) aain ) ⇠ ) ⇠ z 103 (14) LSS ⇠ T 0.1eV (15) ⇠ (16) Linear Inhomogeneities: full Boltzmann • Combining the two terms, we can conveniently write the Boltzmann equation as

– where is the optical depth to the last scattering surface; – = visibility function, peaked at LSS surface.

• A formal solution is found by integrating the RHS up to some very early time where . We get, at the reception event:

– where is the source (the RHS of the top equation) . • There are by now numerical codes (CMBFAST, CAMB, CLASS) that solve this equation numerically. (1 + cos(✓)2)

T =ThomsonCross-section (1) ⇥ (2) d @ = + e =derivativealongﬂow (3) d⌘ @⌘ · r ˙= @/@⌘ (4) Df @f dx @f d✏ @f de @f = + + + (5) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (6) d⌘ 1 f ✏ (7) ' e aT¯[1+⇥(⌘,x,e)] 1 Linear Inhomogeneities:e = full eBoltzmann1,e2,e3 (8) • We can get some insight by approximating the LSS surface as{ instantaneous. We then have } (9) • . ) ✓ 2⇡/l (10) ⇠ ECMB =photonenergyinCMBrestframe (11)

TCMB = T¯ =isotropictemperature (12)

T 5 3 10 (13) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (14) aain ) ⇠ ) ⇠ z 103 (15) LSS ⇠ T 0.1eV (16) ⇠ (17) (1 + cos(✓)2)

T 5 3 10 (13) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (14) aain ) ⇠ ) ⇠ z 103 (15) LSS ⇠ T 0.1eV (16) ⇠ (17) Projection on Sphere • At this point, if we have the solution of the fields in space and time, we can compute the effect at the reception event. In FRW universe, translation invariance makes the use of Fourier space very useful It is therefore useful to write the relevant formulas when the fields are in Fourier space. We have that for a field of the form

• its contribution to the multipole lm on the LSS sphere (whose radius is ) : { } (1 + cos(✓)2) (1)

T =ThomsonCross-section (2) ⇥ (3) d @ = + e =derivativealongﬂow (4) d⌘ @⌘ · r ˙= @/@⌘ (5) Df @f dx @f d✏ @f de @f = + + + (6) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (7) d⌘ 1 f ✏ (8) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (9) { 1 2 3} (10) ) ✓ 2⇡/l (11) ⇠ ECMB =photonenergyinCMBrestframe (12)

TCMB = T¯ =isotropictemperature (13)

T 5 3 10 (14) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (15) aain ) ⇠ ) ⇠ z 103 (16) LSS ⇠ T 0.1eV (17) ⇠ (18) Projection on Sphere • At this point, if we have the solution of the fields in space and time, we can compute the effect at the reception event. In FRW universe, translation invariance makes the use of Fourier space very useful It is therefore useful to write the relevant formulas when the fields are in Fourier space. We have that for a field of the form

• its contribution to the multipole lm on the LSS sphere (whose radius is ) : { } (1 + cos(✓)2) (1)

T =ThomsonCross-section (2) ⇥ (3) d @ = + e =derivativealongﬂow (4) d⌘ @⌘ · r ˙= @/@⌘ (5) EachDf Fouerir @modef contributesdx @f d✏ @f de @f at some level= + + + (6) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (7) d⌘ 1 f ✏ (8) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (9) { 1 2 3} (10) ) ✓ 2⇡/l (11) ⇠ ECMB =photonenergyinCMBrestframe (12)

TCMB = T¯ =isotropictemperature (13)

• its contribution to the multipole lm on the LSS sphere (whose radius is ) : { } (1 + cos(✓)2) (1)

T =ThomsonCross-section (2) ⇥ (3) d @ = + e =derivativealongﬂow (4) d⌘ @⌘ · r ˙= @/@⌘ (5) SphericalDf Bessel@f function,dx peaked@f atd✏ @f de @f = + + + (6) Most dof⌘ contribution@⌘ dcomes⌘ · @ fromx modesd⌘ @✏ of orderd⌘ · @e Also Dfshorter modes contribute, if they are not perpendicular = C[f] (7) d⌘ 1 f ✏ (8) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (9) { 1 2 3} (10) ) ✓ 2⇡/l (11) ⇠ ECMB =photonenergyinCMBrestframe (12)

TCMB = T¯ =isotropictemperature (13)

• its contribution to the multipole lm on the LSS sphere (whose radius is ) : { } (1 + cos(✓)2) (1)

T =ThomsonCross-section (2) ⇥ (3) d @ = + e =derivativealongﬂow (4) d⌘ @⌘ · r ˙= @/@⌘ (5) Df @f dx @f d✏ @f de @f Angular dependence= + we wish to+ have + (6) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (7) d⌘ 1 f ✏ (8) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (9) { 1 2 3} (10) ) ✓ 2⇡/l (11) ⇠ ECMB =photonenergyinCMBrestframe (12)

TCMB = T¯ =isotropictemperature (13)

T 5 3 10 (14) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (15) aain ) ⇠ ) ⇠ z 103 (16) LSS ⇠ T 0.1eV (17) ⇠ (18) Projection on Sphere • If we have

• then

– mapping from Fourier space power spectrum to spherical-harmonic space

• In the CMB, the main contribution comes from f = ⇥ 0 + , so we need to have this one. lm (1) ˆ • There is also the contribution from . For{ scalar} perturbations,2 we have v b = i k v b , and so (1 + cos(✓) ) (2) f = ⇥0 + (1) T =ThomsonCross-section (3) lm (2) ⇥ { } 2 (4) d @ (1 + cos(✓) ) (3) = + e =derivativealongﬂow (5) d⌘ @⌘ · r T =ThomsonCross-section (4) ˙= @/@⌘ ⇥ (6) (5) Df @f dx @fd d✏@@f de @f = + += + +e =derivativealongﬂow (7) (6) d⌘ @⌘ d⌘ · @dx⌘ d@⌘⌘ @✏ ·dr⌘ · @e Df ˙= @/@⌘ (7) = C[f] Df @f dx @f d✏ @f de @f (8) d⌘ = + + + (8) 1 d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e f ✏ Df (9) ' e aT¯[1+⇥(⌘,x,e)] 1 = C[f] (9) e = e ,e ,e d⌘ (10) { 1 2 3} 1 f ✏ (11) (10) ) ' e aT¯[1+⇥(⌘,x,e)] 1 ✓ 2⇡/l e = e ,e ,e (12) (11) ⇠ { 1 2 3} ECMB =photonenergyinCMBrestframe (13) (12) ¯ ) TCMB = T =isotropictemperature✓ 2⇡/l (14) (13) ⇠ T 5 3 10 ECMB =photonenergyinCMBrestframe (15) (14) T ⇠ · T T¯ aCMB = =isotropictemperature 3 (15) dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (16) aT ) 5 ⇠ ) ⇠ ain 3 10 (16) 3 zLSS 10 T ⇠ · (17) ⇠ a 3 T 0.1eV dm(a)=dm(ain) if dm(a0) 1 dm(aLSS(18)) 10 (17) ⇠ aain ) ⇠ ) ⇠ z 103 (19) (18) LSS ⇠ T 0.1eV (19) ⇠ (20) Projection on Sphere • If we have

• then

– mapping from Fourier space power spectrum to spherical-harmonic space

• In the CMB, the main contribution comes from f = ⇥ 0 + , so we need to have this one. lm (1) ˆ • There is also the contribution from . For{ scalar} perturbations,2 we have v b = i k v b , and so (1 + cos(✓) ) (2) f = ⇥0 + (1) T =ThomsonCross-section (3) lm (2) ⇥ { } 2 (4) d @ (1 + cos(✓) ) (3) = + e =derivativealongflowT =ThomsonCross-section (5) (4) Much less sharp contributiond⌘ @⌘ · r ˙= @/@⌘ ⇥ (6) (5) Df @f dx @fd d✏@@f de @f = + += + +e =derivativealongflow (7) (6) d⌘ @⌘ d⌘ · @dx⌘ d@⌘⌘ @✏ ·dr⌘ · @e Df ˙= @/@⌘ (7) = C[f] Df @f dx @f d✏ @f de @f (8) d⌘ = + + + (8) 1 d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e f ✏ Df (9) ' e aT¯[1+⇥(⌘,x,e)] 1 = C[f] (9) e = e ,e ,e d⌘ (10) { 1 2 3} 1 f ✏ (11) (10) ) ' e aT¯[1+⇥(⌘,x,e)] 1 ✓ 2⇡/l e = e ,e ,e (12) (11) ⇠ { 1 2 3} ECMB =photonenergyinCMBrestframe (13) (12) ¯ ) TCMB = T =isotropictemperature✓ 2⇡/l (14) (13) ⇠ T 5 3 10 ECMB =photonenergyinCMBrestframe (15) (14) T ⇠ · T T¯ aCMB = =isotropictemperature 3 (15) dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (16) aT ) 5 ⇠ ) ⇠ ain 3 10 (16) 3 zLSS 10 T ⇠ · (17) ⇠ a 3 T 0.1eV dm(a)=dm(ain) if dm(a0) 1 dm(aLSS(18)) 10 (17) ⇠ aain ) ⇠ ) ⇠ z 103 (19) (18) LSS ⇠ T 0.1eV (19) ⇠ (20) Acoustic Physics • The two fields we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– therefore, Thomson scattering keeps the photon distribution isotropic and the photons-baryon system can be approximately modeled with a tightly coupled ﬂuid with velocity . • The evolution equation for is given by:

– where and Acoustic Physics • The two ﬁelds we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– where and

Drag term to account for normal redshift of velocity in expanding universe Acoustic Physics • The two ﬁelds we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– where and

Force due to gradient in photons density (and so more scattering). It reduces as R !1 vb = ikˆvb (1)

f = ⇥0 + (2) lm (3) { } (1 + cos(✓)2) (4)

T =ThomsonCross-section (5) ⇥ (6) d @ = + e =derivativealongﬂow (7) d⌘ @⌘ · r ˙= @/@⌘ (8) Df @f dx @f d✏ @f de @f = + + + (9) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (10) d⌘ 1 f ✏ (11) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (12) { 1 2 3} (13) ) ✓ 2⇡/l (14) ⇠ ECMB =photonenergyinCMBrestframe (15)

TCMB = T¯ =isotropictemperature (16)

T 5 3 10 (17) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (18) aain ) ⇠ ) ⇠ z 103 (19) LSS ⇠ T 0.1eV (20) ⇠ (21) Acoustic Physics • The two ﬁelds we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– where and

Gravitational force Acoustic Physics • The two ﬁelds we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– where and

• The equation for is given by R (1) !1 vb = ikˆvb (2)

f = ⇥0 + (3) lm (4) { } (1 + cos(✓)2) (5)

T =ThomsonCross-section (6) ⇥ (7) d @ = + e =derivativealongﬂow (8) d⌘ @⌘ · r ˙= @/@⌘ (9) Df @f dx @f d✏ @f de @f = + + + (10) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (11) d⌘ 1 f ✏ (12) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e ,e ,e (13) { 1 2 3} (14) ) ✓ 2⇡/l (15) ⇠ ECMB =photonenergyinCMBrestframe (16)

TCMB = T¯ =isotropictemperature (17)

T 5 3 10 (18) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (19) aain ) ⇠ ) ⇠ z 103 (20) LSS ⇠ T 0.1eV (21) ⇠ (22) Acoustic Physics • The two ﬁelds we need to compute, and , depend on the plasma physics and GR at the time pre-recombination. • Before recombination, the mean free path of photons is very short:

– where and

• The equation for is given by R (1) !1 vb = ikˆvb (2) Volume Increase due to ﬂow f = ⇥0 + (3) lm (4) { } (1 + cos(✓)2) (5)

TCMB = T¯ =isotropictemperature (17)

– where and

• The equation for is given by R (1) !1 vb = ikˆvb (2)

f = ⇥0 + Volume Increase due to gravity (3) lm (4) { } (1 + cos(✓)2) (5)

TCMB = T¯ =isotropictemperature (17)

T 5 3 10 (18) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (19) aain ) ⇠ ) ⇠ z 103 (20) LSS ⇠ T 0.1eV (21) ⇠ (22) Acoustic Physics • Combiningk⌘ these1 two equations,( wek⌘ obtain) theconstant equation for an Damped-Harmonic Oscillator driven by gravity:⌧ ) ⇠ ) 1 k⌘ 1 (k⌘) 0 (1) ) ⇠ (k⌘)2 ! ~ • ForC 2 ( k )=0 , let us for a moment ignore the RHS (which turns out to be small). We obtain (2) 1 (k, ⌘) C (~k) cos kr (⌘)+C (~k) (3) ' 1 (1 + R)1/4 s 2 ✓ ⇣ ⌘◆ (4) – where is the sound horizon R (5) !1ˆ • Wev bobtain= ioscillatingkvb solutions. They are the signature of the plasma physics at the LSS time. (6)

f = ⇥0 + (7) • Doeslm this means that we obtain accosting peaks in the CMB? (8) {– well,} this still depends on the initial conditions. (1 + cos(✓)2) (9)

TCMB = T¯ =isotropictemperature (21)

T 5 3 10 (22) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (23) aain ) ⇠ ) ⇠ z 103 (24) LSS ⇠ T 0.1eV (25) ⇠ (26) 1 (k, ⌘) C (k) cos (kr (⌘)+C (k)) ' 1 (1 + R)1/4 s 2 ✓ ◆ (1) R (2) !1 vb = ikˆvb (3)

f = ⇥0 + (4) lm (5) { } (1 + cos(✓)2) (6)

T =ThomsonCross-section (7) ⇥ (8) d @ = + e =derivativealongflow (9) d⌘ @⌘ · r ˙= @/@⌘ (10) Df @f dx @f d✏ @f de @f = + + + (11) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df k=⌘ C[f1] (k⌘) constant (12) d⌘ ⌧ ) ⇠ ) 1 k⌘ 1 1 Acoustic(k⌘) Physics 0 (1) • Eq:f ✏ ) ⇠ (k⌘)2 ! (13) ' e aT¯[1+⇥(⌘,x,e)] 1 ~ e =C2e(1k,e)=02,e3 (14) (2) • . { } ~ k⌘ 11 (k⌘) constant~ (15) ) (k, ⌘) C1(k) ⌧ )cos krs(⇠⌘)+C2(k) ) (3) ' (1 + R)1/4 1 ✓ 2⇡/l k⌘ 1 (k⌘) 0 (16) (1) ⇠ ✓ ⇣ 2 ⌘◆ • Acoustic oscillations are obtained if all modes have the same) initial phase.⇠ It turns(k⌘ out) that! if all (4) ECMB =photonenergyinCMBrestframe (17) modes are super-Hubble, we have that C ( ~k )=0 , which leads to this. (2) T R = T¯ =isotropictemperature2 (18) (5) CMB !1 1 v = ikˆv (k, ⌘) C1(k) cos (krs(⌘)+C2(k)) (6) (3) T b b5 ' (1 + R)1/4 3 10 ✓ ◆ (19) T f⇠= ⇥· 0 + (7) (4) a 3 dm(alm)=dm(ain) R if dm(a0) 1 dm(aLSS) 10 (20) (8) (5) a !1) ⇠ ) ⇠ { } 2 ain v = ikˆv (6) (1 + cos(3 ✓) ) b b (9) zLSS 10 (21) ⇠ f = ⇥0 + (7) T =ThomsonCross-section (10) T 0.1eV lm (22) (8) ⇠ { } ⇥ (1 + cos(✓)2) (23)(11) (9) d @ = + e =derivativealongflowT =ThomsonCross-section (12) (10) d⌘ @⌘ · r ⇥ (11) ˙= @/@⌘ d @ (13) = + e =derivativealongflow (12) Df @f dx @fd⌘ d@⌘✏ @f · rde @f = + + + (14) d⌘ @⌘ d⌘ · @˙=x @/@⌘d⌘ @✏ d⌘ · @e (13) Df @f dx @f d✏ @f de @f Df = + + + (14) = C[f] d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e (15) d⌘ Df = C[f] (15) 1 d⌘ f ✏ (16) ' aT¯[1+⇥(⌘,x,e)] 1 e f1 ✏ (16) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e1,e2,e3 (17) { } e = e ,e ,e (17) { 1 2 3} (18) (18) ) ) ✓ 2⇡/l ✓ 2⇡/l (19) (19) ⇠ ⇠ ECMB =photonenergyinCMBrestframeECMB =photonenergyinCMBrestframe (20) (20) ¯ TCMB = T¯ =isotropictemperatureTCMB = T =isotropictemperature (21) (21) T 5 T 5 3 10 (22) 3 10 T ⇠ · (22) T ⇠ · a 3 (a)= (a ) if (a ) 1 (a ) 10 (23) adm dm in a ) dm 0 ⇠ ) dm LSS3 ⇠ dm(a)=dm(ain) if dmain(a0) 1 dm(aLSS) 10 (23) 3 azainLSS )10 ⇠ ) ⇠ (24) 3 ⇠ zLSS 10 T 0.1eV (24) (25) ⇠ ⇠ T 0.1eV (25) (26) ⇠ (26) Phase-coherence and CMB Peaks • Phase coherence of the oscillations leads to CMB peaks Dodelson hep-ph/0309057 Phase-coherence and CMB Peaks • Phase coherence of the oscillations leads to CMB peaks Dodelson hep-ph/0309057

Phase coherence: all directions have max amplitude at LSS Phase-coherence and CMB Peaks • Phase coherence of the oscillations leads to CMB peaks Dodelson hep-ph/0309057

Phase incoherence: all amplitudes are realized Phase-coherence and CMB Peaks • Phase coherence of the oscillations leads to CMB peaks Dodelson hep-ph/0309057

Phase coherence: all directions have zero amplitude at LSS Phase-coherence and CMB Peaks • Phase coherence of the oscillations leads to CMB peaks Dodelson hep-ph/0309057

Phase incoherence: all amplitudes are realized C2(k)=0 1 (k, ⌘) C (k) cos (kr (⌘)+C (k)) (1) ' 1 (1 + R)1/4 s 2 ✓ ◆ (2) R (3) !1 vb = ikˆvb (4)

f = ⇥0 + (5) lm (6) { } (1 + cos(✓)2) (7)

TCMB = T¯ =isotropictemperature (21)

T 5 3 10 (22) T ⇠ · a 3 dm(a)=dm(ain) if dm(a0) 1 dm(aLSS) 10 (23) aain ) ⇠ ) ⇠ z 103 (24) LSS ⇠ T 0.1eV (25) ⇠ (26) C2(k)=0 1 (k, ⌘) C (k) cos (kr (⌘)+C (k)) (1) ' 1 (1 + R)1/4 s 2 ✓ ◆ (2) R (3) !1 vb = ikˆvb (4)

f = ⇥0 + (5) lm (6) { } (1 + cos(✓)2) (7)

T =ThomsonCross-section (8) ⇥ (9) d @ = + e =derivativealongflow (10) d⌘ @⌘ · r ˙= @/@⌘ (11) Df @f dx @f d✏ @f de @f = + + + (12) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df Initial Conditions= C[f] (13) • We therefore need to explore the initiald conditions.⌘ • Deep in the radiation era, we have that we can neglect all perturbations but the ones of photons (for adiabatic initial conditions). 1 • The ij and 00 Einstein equation read:f ✏ (14) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e1,e2,e3 (15) • Combining, we get: { } (16) Growing mode ) k⌘ 1 (k⌘) constant ⌧ ) ⇠ ) • For k⌘ 1 (k⌘)✓ constant2⇡/l 1 (17) ⌧ ) ⇠ k⌘ )1 (k⌘) 0 (1) k⌘ 1 (k⌘) ⇠constant ) ⇠ (k⌘)2 ! (1) ⌧ ) ⇠ ~ C2(k)=0 ECMBConstant!!=photonenergyinCMBrestframe C2(k)=0 (2) (2) (18) 1 1 (k, ⌘) C (k) cos (kr(k,(⌘⌘))+CC ((kk))) cos (kr (⌘)+C (k(3))) (3) ' 1 T(1 + R)1/4= T¯s =isotropictemperature' 12 (1 + R)1/4 s 2 (19) ✓ CMB ✓◆ ◆ (4) (4) R T R 5 (5) (5) !1 !1ˆ 3 10ˆ (20) vb = ikvb T ⇠ v·b = ikvb (6) (6) f = ⇥0 + f = ⇥0 + (7) (7) a 3 lm (a)=lm (a ) if (8) (a ) 1 (8) (a ) 10 (21) { } 2 dm { } dm 2 in dm 0 dm LSS (1 + cos(✓) ) (1 + cos(✓) ) aain ) (9) ⇠ )(9) ⇠ T =ThomsonCross-section T =ThomsonCross-section3 (10) (10) ⇥ zLSS 10⇥ (11) (11) (22) d @ ⇠ d @ = + e =derivativealongflow= + e =derivativealongflow (12) (12) d⌘ @⌘ · r T 0.1eVd⌘ @⌘ · r (23) ˙= @/@⌘ ⇠ ˙= @/@⌘ (13) (13) Df @f dx @f d✏ @f Dfde @f@f dx @f d✏ @f de @f = + + + = + + + (14) (14) (24) d⌘ @⌘ d⌘ · @x d⌘ @✏ dd⌘⌘ · @e@⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df Df = C[f] = C[f] (15) (15) d⌘ d⌘ 1 1 f ✏ f ✏ (16) (16) ¯ ' e aT¯[1+⇥(⌘,x,e)] 1 ' e aT [1+⇥(⌘,x,e)] 1 e = e ,e ,e e = e1,e2,e3 (17) (17) { 1 2 3} { } (18) (18) ) ) ✓ 2⇡/l ✓ 2⇡/l (19) (19) ⇠ ⇠ ECMB =photonenergyinCMBrestframeECMB =photonenergyinCMBrestframe (20) (20) ¯ TCMB = T¯ =isotropictemperatureTCMB = T =isotropictemperature (21) (21) T T 5 5 3 10 3 10 (22) (22) T ⇠ · T ⇠ · a a 3 3 dm(a)=dm(ain) ifdmdm((aa)=0) dm1 (ain) dm(aLSS)if 10dm(a0) 1 (23) dm(aLSS) 10 (23) aain ) ⇠ ) aain ) ⇠ ⇠ ) ⇠ 3 z 103 zLSS 10 (24) (24) LSS ⇠ ⇠ T 0.1eV T 0.1eV (25) (25) ⇠ ⇠ (26) (26) C2(k)=0 1 (k, ⌘) C (k) cos (kr (⌘)+C (k)) (1) ' 1 (1 + R)1/4 s 2 ✓ ◆ (2) R (3) !1 vb = ikˆvb (4)

f = ⇥0 + (5) lm (6) { } (1 + cos(✓)2) (7)

T =ThomsonCross-section (8) ⇥ (9) d @ = + e =derivativealongflow (10) d⌘ @⌘ · r ˙= @/@⌘ (11) Df @f dx @f d✏ @f de @f = + + + (12) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df Initial Conditions= C[f] (13) • We therefore need to explore the initiald conditions.⌘ • Deep in the radiation era, we have that we can neglect all perturbations but the ones of photons (for adiabatic initial conditions). 1 • The ij and 00 Einstein equation read:f ✏ (14) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e1,e2,e3 (15) • Combining, we get: { } (16) ) Growing mode • For kk⌘⌘ 11 ((kk⌘⌘))✓ constantconstant2⇡/l (17) ⌧⌧ )) ⇠⇠ )) k⌘ 1 (k⌘) ⇠constant1 (1) • For k⌘ ⌧ 1 ) (k⌘) ⇠ 2 0 (1) C2(k)=0) ECMB⇠ (k⌘) =photonenergyinCMBrestframe! (2) (18) C2(k)=0 1 (2) (k, ⌘) C1(k) cos (kr¯s(⌘)+C2(k)) (3) ' T(1CMB +1R)1/4= T =isotropictemperature (19) (k, ⌘) C1(k) ✓ cos (krs(⌘)+C2(k))◆ (3) ' (1 + R)1/4 ✓ ◆ (4) R T 5 (4)(5) R !1ˆ 3 10 (5) (20) vb!1= ikvb (6) v = ikˆv T ⇠ · (6) fb= ⇥0 +b (7) f = ⇥ + a (7) 3 lm 0 (a)= (a ) if (8) (a ) 1 (a ) 10 (21) {lm} 2 dm dm in (8)dm 0 dm LSS {(1 +} cos(✓) ) aain ) (9) ⇠ ) ⇠ (1 + cos(✓)2) (9) T =ThomsonCross-section 3 (10) ⇥T =ThomsonCross-sectionzLSS 10 (10)(11) (22) ⇥d @ ⇠ (11) d = @ + e =derivativealongflow (12) d⌘ = @⌘ + e · r T=derivativealongflow0.1eV (12) (23) ˙=d⌘ @/@⌘@⌘ · r ⇠ (13) ˙=Df@/@⌘@f dx @f d✏ @f de @f (13) (24) Df = @f + dx @f + d✏ @f + de @f (14) d⌘ = @⌘ + d⌘ · @x + d⌘ @✏ + d⌘ · @e (14) Dfd⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (15) d⌘ = C[f] (15) d⌘ 1 f ✏ 1 (16) aT¯[1+⇥(⌘,x,e)] f ' e ✏ 1 (16) ' e aT¯[1+⇥(⌘,x,e)] e = e1,e2,e3 1 (17) e = {e ,e ,e } (17) { 1 2 3} (18) ) (18) )✓ 2⇡/l (19) ✓ ⇠ 2⇡/l (19) ECMB⇠ =photonenergyinCMBrestframe (20) ECMB =photonenergyinCMBrestframe (20) TCMB = T¯ =isotropictemperature (21) TCMB = T¯ =isotropictemperature (21) T 5 T 3 105 (22) T ⇠ 3 · 10 (22) T ⇠ · a 3 dm(a)=dm(ain) a if dm(a0) 1 dm(aLSS) 103 (23) dm(a)=dm(ain) aain ) if dm(a0) ⇠ 1 ) dm(aLSS) ⇠ 10 (23) 3 aain ) ⇠ ) ⇠ zLSS 10 (24) z ⇠ 103 (24) TLSS ⇠0.1eV (25) T ⇠ 0.1eV (25) ⇠ (26) (26) C2(k)=0 1 (k, ⌘) C (k) cos (kr (⌘)+C (k)) (1) ' 1 (1 + R)1/4 s 2 ✓ ◆ (2) R (3) !1 vb = ikˆvb (4)

f = ⇥0 + (5) lm (6) { } (1 + cos(✓)2) (7)

T =ThomsonCross-section (8) ⇥ (9) d @ = + e =derivativealongflow (10) d⌘ @⌘ · r ˙= @/@⌘ (11) Df @f dx @f d✏ @f de @f = + + + (12) d⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df Initial Conditions= C[f] (13) • We therefore need to explore the initiald conditions.⌘ • Deep in the radiation era, we have that we can neglect all perturbations but the ones of photons (for adiabatic initial conditions). 1 • The ij and 00 Einstein equation read:f ✏ (14) ' e aT¯[1+⇥(⌘,x,e)] 1 e = e1,e2,e3 (15) • Combining, we get: { } (16) ) Growing mode • For kk⌘⌘ 11 ((kk⌘⌘))✓ constantconstant2⇡/l (17) ⌧⌧ )) ⇠⇠ )) k⌘ 1 (k⌘) ⇠constant1 Justifies us (1) • For neglecting k⌘ ⌧ 1 ) (k⌘) ⇠ 2 0 (1) C2(k)=0) ECMB⇠ (k⌘) =photonenergyinCMBrestframe! gravity source (2) (18) term C2(k)=0 1 (2) (k, ⌘) C1(k) cos (kr¯s(⌘)+C2(k)) (3) ' T(1CMB +1R)1/4= T =isotropictemperature (19) (k, ⌘) C1(k) ✓ cos (krs(⌘)+C2(k))◆ (3) ' (1 + R)1/4 ✓ ◆ (4) R T 5 (4)(5) R !1ˆ 3 10 (5) (20) vb!1= ikvb (6) v = ikˆv T ⇠ · (6) fb= ⇥0 +b (7) f = ⇥ + a (7) 3 lm 0 (a)= (a ) if (8) (a ) 1 (a ) 10 (21) {lm} 2 dm dm in (8)dm 0 dm LSS {(1 +} cos(✓) ) aain ) (9) ⇠ ) ⇠ (1 + cos(✓)2) (9) T =ThomsonCross-section 3 (10) ⇥T =ThomsonCross-sectionzLSS 10 (10)(11) (22) ⇥d @ ⇠ (11) d = @ + e =derivativealongflow (12) d⌘ = @⌘ + e · r T=derivativealongflow0.1eV (12) (23) ˙=d⌘ @/@⌘@⌘ · r ⇠ (13) ˙=Df@/@⌘@f dx @f d✏ @f de @f (13) (24) Df = @f + dx @f + d✏ @f + de @f (14) d⌘ = @⌘ + d⌘ · @x + d⌘ @✏ + d⌘ · @e (14) Dfd⌘ @⌘ d⌘ · @x d⌘ @✏ d⌘ · @e Df = C[f] (15) d⌘ = C[f] (15) d⌘ 1 f ✏ 1 (16) aT¯[1+⇥(⌘,x,e)] f ' e ✏ 1 (16) ' e aT¯[1+⇥(⌘,x,e)] e = e1,e2,e3 1 (17) e = {e ,e ,e } (17) { 1 2 3} (18) ) (18) )✓ 2⇡/l (19) ✓ ⇠ 2⇡/l (19) ECMB⇠ =photonenergyinCMBrestframe (20) ECMB =photonenergyinCMBrestframe (20) TCMB = T¯ =isotropictemperature (21) TCMB = T¯ =isotropictemperature (21) T 5 T 3 105 (22) T ⇠ 3 · 10 (22) T ⇠ · a 3 dm(a)=dm(ain) a if dm(a0) 1 dm(aLSS) 103 (23) dm(a)=dm(ain) aain ) if dm(a0) ⇠ 1 ) dm(aLSS) ⇠ 10 (23) 3 aain ) ⇠ ) ⇠ zLSS 10 (24) z ⇠ 103 (24) TLSS ⇠0.1eV (25) T ⇠ 0.1eV (25) ⇠ (26) (26) The three contributions • The sum of the three contributions gives this (with adiabatic, superHubble initial conditions):

credit Challinor and Peyris

– Notice the peaks – the doppler is meakred – the ISW peaks at late-times (long distances), signature of dark energy. CMB Observations • We already anticipate that CMB peaks were observed. This is the result of more accurate observations.

credit Planck CMB Observations • We already anticipate that CMB peaks were observed. This is the result of more accurate observations.

credit A. Hamilton Ruling out Cosmic Strings • The presence of acoustic oscillations is a very non-trivial result. It tells us that the primordial perturbations were already non-zero when the modes were super-Hubble (as this ﬁxes the initial conditions). • An alternative theory was that there were cosmic strings that are produced by some early-universe phase transition. credit Martins and Shellard Ruling out Cosmic Strings • In this case perturbations are continuously produced when modes are sub-Hubble. There is therefore no phase coherence. We obtain something like this.

– This is hugely ruled out. Huge discovery made by the peaks.

credit Martins and Shellard

credit Ringeval Ruling out Cosmic Strings • In this case perturbations are continuously produced when modes are sub-Hubble. There is therefore no phase coherence. We obtain something like this.

– This is hugely ruled out. Huge discovery made by the peaks.

credit Martins and Shellard

credit Ringeval

Ruled out A tool for precision physics • Given the smallness of the perturbations, the CMB is a perfect playground to make accurate predictions that allows us to learn with certainty many aspects of the universe. We have a very small parameter around which to expand!

• Since we can measure the CMB very well, and since we have very accurate predictions, we can be sensitive to very small effects. And almost everything affects the CMB!

• For example, we can measure very well the cosmological parameters. They effect either: – plasma physics – geometric projections A tool for precision physics • Given the smallness of the perturbations, the CMB is a perfect playground to make accurate predictions that allows us to learn with certainty many aspects of the universe. We have a very small parameter around which to expand!

• Since we can measure the CMB very well, and since we have very accurate predictions, we can be sensitive to very small effects. And almost everything affects the CMB!

• For example, we can measure very well the cosmological parameters. They effect either: – plasma physics – geometric projections – there are however degeneracies that need to be broken by other observations • Let us see some examples Examples • Effect of Curvature: change the projection of the sound horizon at LSS

credit Challinor and Peyris

credit Challinor and Peyris • Effect of Dark Energy: enhances ISW Examples • Effect of Baryon density

credit Challinor and Peyris

• Effect of Matter density

credit Challinor and Peyris CMB is Polarized • Thomson scattering is anisotropic. Quadrupole anisotropy at LSS induces a polarization in what we see in the sky:

credit W. Hu CMB is Polarized • Thomson scattering is anisotropic. Quadrupole anisotropy at LSS induces a polarization in what we see in the sky. • The field representing the polarization can be decomposed – in the sum of two fields, a gradient (E) and a curl (B) field. Conclusions • CMB has revealed us that the universe was very simple early on, confirming the Big Bang picture. • It has told us how structure formed, by gravitational collapse of small initial density perturbation • and what, qualitatively, is the seed of these fluctuations: they were super Hubble

• The smallness of the perturbation allowed us to make extremely accurate predictions of the observations, so that we have learnt so much about the cosmological parameters and about the initial conditions.

• And so much more to come.