Primordial Gravitational Waves from Cosmic Inflation

Mathematical Tripos Part III Essay #75 (colour in electronic version) Submied 5th May 2017, updated 26th August 2017

Mike S. Wang DAMTP, University of Cambridge

Abstract. Cosmic inflation generates primordial gravitational waves (PGWs) through the same physical process that seeds all structure formation in the observable universe. We will demonstrate this mechanism in detail and relate it to the distinctive signatures PGWs could leave in the observable temperature anisotropies and polarisation of the cosmic microwave background (CMB). The detection of primordial gravitational waves is of great significance to validating and understanding inflationary physics, and we shall see why CMB polarisation oers a promising path. In the end, we will remark on the unique observational challenges and prospects of probing primordial gravitational waves in future experiments.

Keywords. Inflation, primordial gravitational waves (theory), gravitational waves and CMBR polarisation, power spectrum, gravitational waves / experiments.

Contents

I A Brief Overview 1

II The Inflation Paradigm and Generation of Gravitational Waves 1 II.1 Dynamics of single-field slow-roll inflation ...... 1 II.2 antum fluctuations and the primordial power spectrum ...... 2 II.3 Scalar, vector and tensor perturbations in the FLRW background ...... 3

III CMB Signatures from Primordial Gravitational Waves 5 III.1 Temperature anisotropies from PGWs ...... 5 III.2 Polarisation from PGWs ...... 9

IV B-Mode Polarisation: A Promising Route for Detecting PGWs? 12 IV.1 Cosmological sources of B-mode polarisation ...... 12 IV.2 Statistical aspects of B-mode polarisation ...... 13

V A “Smoking Gun”: Physical Significance of PGW Discovery 14 V.1 Alternatives to inflation ...... 14 V.2 Energy scale and the inflaton field excursion ...... 15 V.3 Constraining models of inflation ...... 16

VI Future Experiments: Challenges and Prospects 17 VI.1 Challenges in detecting PGWs ...... 19 VI.2 Future prospects and concluding remarks ...... 19

Appendices 23 1

I A Brief Overview non-inationary physics, is the generation of primordial gravitational waves resulting from tensor perturbations in the geometry of the very early universe; as Technological advances in the past decade have ush- such, PGWs are often said to be a “smoking gun” for ered cosmology into an exciting era where theory and validating the ination theory [1]. observations are directly confronted. Increasing precision by orders of magnitude in measurements of the cosmological parameters enables us to further probe II.1 Dynamics of single-field slow-roll inflation conditions of the very early universe, thus deepening our understanding of fundamental physics. Currently As an entry point, we rst consider a simple model in the ination paradigm has been successful in solving which ination is driven by a single scalar eld ϕ(t,x), numerous puzzles in the standard Big Bang cosmol- known as the inaton, with an interaction potential ogy, such as the horizon problem, the atness problem V (ϕ). Its energy density and pressure and the relic problem; however, evidence for its hap- pening is yet to be found, and its energy scales to be 0 1 ˙2 ρ ≡ −T 0 = ϕ + V (ϕ), determined [1]. 2 (1) 1 1 P ≡ T i = ϕ˙2 − V (ϕ) The cosmic microwave background (CMB) is a power- 3 i 2 ful utility for discovering evidence of ination; primor- can be calculated from the energy-momentum tensor dial gravitational waves (PGWs) generated from the same inationary mechanism that seeds large struc- 1 T = ∂ ϕ∂ ϕ − g ∂λϕ∂ ϕ + V (ϕ) . (2) ture formation leave observable imprints in the CMB. µν µ ν µν "2 λ # In Sec. II and Sec. III, we will explore in detail this mechanism and the related physical observables. Sec. IV The Friedman equations demonstrates why CMB polarisation oers a promis- 1 H 2 = ρ, (3) ing route for PGW detections. The signicance of such 2 3MPl a detection for understanding fundamental physics in 2 1 the very early universe is discussed in Sec. V, before H˙ + H = − (ρ + 3P) (4) 6M2 we nally remark on the obstacles as well as positive Pl outlooks of future experimental eorts in Sec. VI. are obtained from the Einstein eld equation applied to the most general metric for an expanding universe Unless otherwise noted, the conventions adopted in assuming the cosmological principle—the Friedmann– this paper are: 1) mostly-plus Lorentzian signature Lemaître–Robertson–Walker (FLRW) metric. The un- − ~ ( ,+,+,+); 2) natural units in which√ = c = 1, and the perturbed form of the FLRW metric can be taken as reduced Planck mass MPl = 1/ 8πG; 3) Latin alphabet for spatial indices and Greek alphabet for spacetime in- ds2 = a(τ )2 − dτ 2 + dx · dx . (5) dices; 4) the Hubble parameter denoted by H ≡ a˙/a and the comoving Hubble parameter denoted by H ≡ a˙, From the Friedman equations, it is easy to see that with a being the scale factor. the condition of ination a¨ > 0 is equivalent to ϕ˙2 < V (ϕ). Further, dierentiating Eqn. (3) with respect to time and employing Eqns. (1) and (4) lead to the II The Inflation Paradigm and Genera- Klein–Gordon equation governing the scalar inaton tion of Gravitational Waves dynamics ϕ¨ + 3Hϕ˙ + V,ϕ = 0 (6) The theory of ination postulates a brief period (within where the subscript “,ϕ” denotes ϕ-derivatives. 10−34 s) of quasi-exponential accelerated expansion during which the scale factor increased by over 60 e-folds. A simple, approximate case is the slow-roll model: the The intense expansion is sourced by a negative pres- inaton rolls down a region of small gradients in the sure component in energy-momentum of the matter potential with its potential energy dominating over ˙2 contents, and drives the universe towards almost per- kinetic energy, |V | ϕ . Dierentiating this condition fect homogeneity, isotropy and atness [2]. with respect to time shows that this process is sustained if A key prediction of ination, which does not exist in ϕ¨ V,ϕ .

Two slow-roll parameters, dened for general ination- period. However, as quantum eects are important ary models, gauge this process in the early universe, by the uncertainty principle the inaton eld locally uctuates around its background d ln H H˙ ϵ B − ≡ − , value, ϕ(t,x) = ϕ¯(t) + δϕ (t,x). This means dier- a H 2 d ln (7) ent amounts of ination occur at dierent locations in d ln ϵ ϵ˙ η B ≡ . spacetime, leading to density inhomogeneities in the d ln a Hϵ universe from which structure ultimately forms. In the slow-roll model, these parameters are both 1 in magnitude We start our quantisation procedure from the inaton ˙2 2 !2 action [2] assuming the unperturbed FLRW metric (5) 1 ϕ MPl V,ϕ ϵ ≡ ≈ ϵV ≡ , 2M2 H 2 2 V Z √ Pl (8) 3 1 µν S = dτ d x −g − g ∂µϕ∂ν ϕ − V (ϕ) 2 V,ϕϕ " 2 # η ≈ 4ϵV − 2ηV , ηV ≡ M . Z (9) Pl V 1 2 = dτ d3x a2 ϕ02 − ∇ϕ − 2a2V (ϕ) Here ϵV , ηV are the potential slow-roll parameters which 2 are often more convenient to use in slow-roll scenar- where we denote the derivative of ϕ with respect to ios [2]. Their linear relations above with the slow-roll conformal time τ by ϕ0 to distinguish from the deriva- parameters follow from the Friedman equation (3) and tive ϕ˙ with respect to cosmic time t. Introducing the the Klein–Gordon equation (6) in the slow-roll approx- eld re-denition f (τ ,x) = a(τ ) δϕ(τ ,x) and ignoring imation. metric uctuations in the inationary background1, we expand the action (9) to second order II.2 antum fluctuations and the primordial power spectrum

The background inaton eld ϕ¯(t) is only time- dependent and acts as a “clock” during the inationary

Z 0 !2 !2 !2 (2) 3 1 2 f H f ∇f 2 f S = dτ d x a − − − a V,ϕϕ 2  a a a a  Z   1  2 0  = dτ d3x f02 − ∇f − H (f 2) + (H 2 − a2V )f 2 2  ,ϕϕ  Z 1 3 02 2 0 2 2 2 = dτ d x f − ∇f + H + H − a V,ϕϕ f 2 Z 00 ! 1 2 a = dτ d3x f 02 − ∇f + − a2V f 2 . 2 a ,ϕϕ       We note that in slow-roll approximations, H ≈ const. tion for the Lagrangian ≈ 00 and ρ V , so by Eqn. (3) 1 2 a L = f 02 − ∇f + f 2 (11) 00 2 " a # a 2 2 2 2 2 ≈ 2a H ≈ a V,ϕϕ a V,ϕϕ a 3ηV we arrive at the Mukhanov–Sasaki equation a00 as η 1. Therefore, f 00 − ∇2 f − f 2 = 0. (12) a Z 00 ( ) 1 02 2 a 2 S ≈ dτ d3x f − ∇f + f 2 . (10) In canonical quantisation, f (τ ,x) as well as its conju- 2 " a # . gate momentum π (τ ,x) ≡ ∂L ∂f 0 = f 0 are promoted By considering the associated Euler–Lagrange equa- to be operators obeying the equal-time canonical com- mutation relation (CCR) 1A more rigorous treatment can be found in [3], but for our 0 0 purposes the analysis below is sucient for de Sitter expansion. fˆ(τ ,x),πˆ (τ ,x ) = iδ (x − x ). (13) f g 3

ˆ 3 2 We expand f (τ ,x) and πˆ (τ ,x) in Fourier space as and read o Pf = k /2π fk (τ ) . By solution (19), Z 3 d k ∗ † − −2 fˆ(τ ,x) = f a e ik·x + f a eik·x , Pδϕ (k) = a Pf (k ) 3/2 k k k k (2π ) 3 Z (14) 2 k 1 1 d3k = (−Hτ ) 1 + πˆ (τ ,x) = f 0∗a† e−ik·x + f 0a eik·x 2π 2 2k " (kτ )2 # 3/2 k k k k (2π ) !2 H k2 † = 1 + (21) where ak, ak are the time-independent annihilation and 2π a2H 2 creation operators for each mode satisfying !2* + H a ,a† = δ (k − k0), → , - k k0 (15) 2π f g and fk satises Eqn. (12) in Fourier space on super-horizon scales k aH. f 00 + ω2 (τ )f = 0 (16) k k k Now we arrive at an important result: since H is slowly 2 2 − 00 ≡ with ωk B k a /a, k k . Now Eqns. (13) and (15) varying, we approximate the inaton power spectrum demand that the Wronskian by evaluating at horizon crossing k = aH

∗ ∗ 0 ∗0 !2 W(f , f ) ≡ f f − f f = −i. (17) H k k k k k k k P (k) ≈ k , where H ≡ . (22) δϕ 2π k a Since the expansion is quasi-de Sitter during ination, i.e. a ≈ eHt and H ≈ const., we have Z ∞ 0 Z ∞ II.3 Scalar, vector and tensor perturbations in the dt 0 −Ht 0 1 FLRW background τ (t) = − 0 ≈ − dt e = − , t a(t ) t aH and Eqn. (16) specialises to For later comparison and completeness, we will de- ! scribe briey scalar and vector perturbations as well as 00 2 2 f + k − fk = 0. (18) tensor perturbations in the FLRW background space- k τ 2 time. The general perturbed FLRW metric takes the The exact solution to this is given by form −ikτ ! ikτ ! e i e i 2 2 2 i f (τ ) = A √ 1 − + B √ 1 + , ds = a(τ ) − (1 + 2A) dτ + 2Bi dx dτ k kτ kτ 2k 2k i j but we must choose the positive-frequency solu- + (1 + 2C)δij + 2Eij dx dx (23) tion suitably normalised such that lim →−∞ f (τ ) = f g √ τ k where E is traceless, and spatial indices are raised and e−ikτ / 2k. This ensures that Eqn. (17) is satised and ij lowered using δ . the vacuum state is the ground state of the Hamilto- ij nian [2]. Hence we adopt Scalar perturbations — Scalar density inhomogeneities ! e−ikτ i from cosmic ination grow through gravitational insta- fk = √ 1 − . (19) bility, which explains large structure formation seen 2k kτ in the observable universe [4]. Gauge freedom allows We are ready now to determine the power spectrum for us to push scalar perturbations into the curvature: in a physical observable q comoving gauge where δϕ = 0, the spatial metric ˜ 2 2 2ζ D ∗ 0 E 2π 0 gij = a(t) e δij ; (24) q(k)q (k ) ≡ Pq (k)δ (k − k ) (20) k3 ζ˜ is the gauge-invariant comoving curvature perturba- in the case of the inaton eld q = δϕ = f /a. Using tion ζ evaluated in this gauge. In spatially-at gauge, Eqn. (14), we can calculate the zero-point uctuation ζ takes the form [2] D E 0 fˆ(τ ,0)fˆ†(τ ,0) 0 δϕ ζ = −H . (25) Z 3 3 0 ¯˙ d k d k ∗ † ϕ = f f 0 0 a ,a 0 3/2 3/2 k k k k0 (2π ) (2π ) f g By comparing with the inaton power spectrum (22), Z 3 k 2 we nd the scalar perturbation power spectrum = d lnk fk 2 !2 2π 1 H P = k . (26) ζ 2 π 2MPlϵ 2 4

Scale-dependence of the power spectrum is measured It is helpful to decompose hij in Fourier modes by the scalar spectral index, or tilt, Z X d3k = (p) , ik·x. hij / hij (η k) e (30) d ln Pζ (2π )3 2 n B 1 + (27) p=±2 s d lnk where a value of unity corresponds to scale-invariance. For k along the z-axis, we choose a set of basis tensors The power spectrum could be approximated by a ± 1 power-law with some reference scale [1] k?, m( 2) (zˆ) = (ˆx ± iˆy) ⊗ (ˆx ± iˆy) (31) 2 !ns−1 k satisfying the orthogonality and reality conditions [6] Pζ (k) = As(k?) . (28) k? (p) ˆ (q)ij ˆ ∗ pq mij (k) m (k) = δ , (32) ∗ f g Vector perturbations — Primordial vector perturbations (p) ˆ (−p) ˆ (p) ˆ mij (k) = mij (k) = mij (−k). (33) are negligible after ination since they are associated with vorticity, which by conservation of angular mo- In such a basis, we have mentum is diluted with the scale factor (see [5]). (±2) 1 (±2) ˆ (±2) hij (η,k) = √ mij (k)h (η,k). (34) [To avoid clustering of superscripts, we adopt the fol- 2 lowing convention in the context of tensor perturbations (gravitational waves): an overdot “˙” represents As in the inaton case, we start from the combined derivatives with respect to the conformal time, now de- Einstein–Hilbert action and the matter action noted by η, rather than the cosmic time t.] M2 Z √ S = Pl d4x −gR Tensor perturbations — Primordial gravitational waves 2 Z are, mathematically speaking, tensor perturbations to √ 1 + d4x −g − ∂ ϕ∂µϕ − V (ϕ) (35) the spacetime metric. In the FLRW background, we " 2 µ # can write the perturbation as where R is the Ricci scalar, and expand to second order 2 2 2 i j ds = a(η) − dη + δij + hij dx dx (29) to nd M2 Z (2) Pl 3 2 ˙ ˙ij i jk where hij is symmetric, traceless and transverse, i.e. S = dη d x a hijh − ∂ihjk ∂ h . (36) i i 8 h i = 0 and ∂ih j = 0, since we can always absorb the other parts of the tensor into scalar or vector perturba- This laborious calculation can be found in Appendix A. tions which decouple from true tensor perturbations at the linear order [1]. These conditions imply that Using Eqns. (30), (32), (33) and (34), we could rewrite terms in the second order action in the Fourier space hij has only two degrees of freedom, which we shall denote as the helicity p = ±2. as follows

Z Z 3 3 0 Z X d k d k 1 0 d3x h˙ h˙ij = h˙(p) (η,k)h˙(q) (η,k0)m(p) (kˆ)m(q)ij (kˆ 0) d3x ei(k+k )·x ij 3/2 3/2 2 ij p,q=±2 (2π ) (2π ) Z 0 1 X d3k d3k = h˙(p) (η,k)h˙(q) (η,k0)m(p) (kˆ)m(q)ij (kˆ 0)(2π )3δ (k + k0) 2 3/2 3/2 ij p,q=±2 (2π ) (2π ) Z 1 X 2 = d3k h˙(p) (η,k) 2 p=±2 f g and similarly so that Z X Z 2 Z 3 i jk 1 3 2 (p) 2 (2) MPl X 3 2 (p) 2 2 (p) 2 d x ∂ihjk ∂ h = − d k k h (η,k) S = dη d k a h˙ + k h . (37) 2 16 p=±2 f g p=±2 5

By comparing Eqn. (37) with the action (9) in Fourier II.3.2 Evolution of gravitational waves space, we see that following the√ same quantisa- (p) tion procedure with δϕ → (MPl/ 8)h for each In the absence of anisotropic stress, the traceless part independently-evolving helicity state, one can derive of the ij-component of the Einstein eld equation gives the power spectrum as dened in the two-point correlator, h¨(±2) + 2Hh˙(±2) + k2h(±2) = 0 (45) ∗ 2π 2 (±2) ±ikη h(p) (k) h(p) (k0) ≡ P (k)δ (k − k0), (38) with solutions h ∝ e /a. Details of the deriva- k3 h f g tion may be found in Appendix B. to be (at horizon crossing)

!2 8 H P (k) = k . (39) III CMB Signatures from Primordial h 2 π MPl 2 Gravitational Waves

As in the case of scalar perturbations, we can dene Observational and precision cosmology has been mak- the tensor spectral index ing remarkable leaps in recent times and since its discovery, the cosmic microwave background has been an d ln Ph (k) nt B (40) indispensable utility directly probing the very early uni- d lnk verse. Local uctuations in physical properties such as so that the tensor perturbation power spectrum (39) temperature and density were imprinted into the CMB can be approximated by a power law at the time of recombination, when photons decoupled !n k t from the primordial plasma and became essentially Ph (k) = At(k?) (41) free-streaming, presenting an almost perfect blackbody k? thermal spectrum. Angular variance in CMB radiation in analogy with Eqn. (28). thus encodes the information of perturbations generated during the hypothetical inationary era, lending us insights into the geometry and matter contents of II.3.1 A consistency condition the early universe [7].

Comparing the scalar and tensor power spectra (26), Two key observables of the CMB are the temperature (39) and their power-law approximations (28), (41), we anisotropy and polarisation. We will discuss the dis- see that the tensor-to-scalar ratio, dened below, is tinctive signatures of PGWs in these observables, and explain why the latter gives a particularly promising At r B = 16ϵ. (42) route in the detection of PGWs in the next section. As We shall see later the CMB polarisation measurements are sensitive to this value, and it contains critical infor- III.1 Temperature anisotropies from PGWs mation about inationary physics [1]. III.1.1 Concepts and notions We have, from Eqns. (7), (39) and (40), d ln P d ln a The blackbody spectrum — The Lorentz-invariant dis- n = h t d ln a d lnk tribution function of CMB photons in the phase space !−1 d ln H d lnk is isotropic and homogeneous in the rest frame, but = 2 Doppler-shifted relativistically for an observer with d ln a d ln a k=aH relative velocity v to the background −1 = −2ϵ(1 − ϵ) 1 ≈ −2ϵ (43) f¯(pµ ) ∝ exp Eγ (1 + e · v)/T¯CMB − 1 where we have used lnk = ln a + ln H at horizon cross- f g ing, so d lnk d ln a = 1 − ϵ. Therefore a consistency where e is the direction of the incoming photon and −1/2 condition is obtained for canonical single-eld slow-roll E its observed energy, γ ≡ (1 − v · v) , and T¯CMB ' ination 2.7255 K is the isotropic CMB temperature. This is a r ≈ −8nt. (44) blackbody spectrum with temperature varying with 6

direction e as T (e) ≈ T¯CMB(1 − e · v), |v| 1: we see where ϵ ≡ aE is the comoving photon energy. here a dipole anisotropy—which along with kinematic quadrupole and multipole anisotropies at order |v|2 We write or higher—must be subtracted to give the cosmological d ln f¯ f (η,e,x,ϵ) = f¯(ϵ) 1 − Θ(η,x,e) (51) anisotropies [6]. d ln ϵ   x =   Optical depth and visibility — Along a line of sight where Θ is the fractional temperature uctuation . On − −   x0 (η0 η)e between conformal times η and η0, where physical grounds the observation takes place at position x0, the optical depth is dened by df df = , Z η dη path dη scatt. 0 τ B dη an¯eσT (46) η where the LHS is a total derivative along the photon path, and the RHS describes scattering eects. This and the visibility function is dened by leads to the linearised Boltzmann equation for Θ(η,x,e) −τ g(η) B −τ˙ e . (47) ∂Θ d ln ϵ + e · ∇Θ − = −an¯eσTΘ + an¯eσTe · vb − ∂η dη Here e τ is interpreted as the probability that a photon Z 3an¯eσT 2 does not get scattered in the interval (η,η0), and g(η) + dmˆ Θ(mˆ ) 1 + (e · m) (52) π is the probability density that a photon last-scattered 16 f g . . at time η. They satisfy the integral relation [6] where at linear order d dη = ∂ ∂η + e · ∇ along a

Z η0 line of sight, n¯e is the average electron density, σT is 0 0 −τ (η) dη g(η ) = 1 − e . (48) the cross-section for Thomson scattering and vb is the η velocity of baryons and electrons tightly coupled by Coulomb scattering [6]. Rotations of a random eld — Rotational transforma- tions of a scalar random eld f (nˆ ) on the sphere can Multipole expansion and normal modes — We can ex- be performed by acting on the spherical multipole co- pand the fractional temperature uctuation in Fourier ecients f`m via the Wigner D-matrices. The relevant space in the basis of the Legendre polynomials P` (x) mathematics is found in Appendix C. X ` Θ(η,k,e) = (−i) Θ` (η,k)P` (kˆ · e), (53) Angular power spectrum — The two-point correlator `>0 of a scalar random eld f (nˆ ) is rotationally-invariant or in the basis of the spherical harmonics if [7] D ∗ E X 0 0 f`m f`0m0 = C`δ`` δmm (49) Θ(η,k,e) = Θ`m (η,k)Y`m (e). (54) `,m where C` is the angular power spectrum associated with the random eld f . Then by the addition formula [6]

X 4π ∗ P` (kˆ · e) = Y (kˆ)Y` (e) 2` + 1 `m m III.1.2 The Boltzmann equation for anisotropies |m |6` we have In what follows in this section, it is useful to express 3-vectors in terms of an orthonormal tetrad with com- ` 4π ∗ Θ` m (η,k) = (−i) Θ` (η,k)Y (kˆ). (55) µ µ ` + `m ponents (E0) , (Ei ) such that 2 1

g (E )µ (E )ν = −1, g (E )µ (E )ν = 0, µν 0 0 µν 0 i III.1.3 Linear anisotropies from gravitational waves µ ν gµν (Ei ) (Ej ) = δij .

We denote the tetrad components of the direction of We will temporarily suppress the polarisation label ± propagation vector e by eıˆ, then the 4-momentum of a (p), p = 2, for readability. It is useful to work in an photon with energy E is orthonormal tetrad, whose components are given by ! µ µ 1 µ ıˆ ϵ ıˆ µ −1 µ −1 j E,p = 1,e (50) (E0) = a δ0 , (Ei ) = a δi − hi δj . (56) a 2 7

The time-component of the geodesic equation at linear factor due to the spatial dependence of gravitational order, for the metric (29), gives then an equation [7] waves. After some algebraic manipulations with the satised by ϵ (see Appendix D) Rayleigh expansion and the Wigner 3j-symbols (see Appendix E), we have the result 1 dϵ 1 ıˆ ˆ + h˙ije e = 0. (57) ϵ dη 2 r ± π h˙( 2) (η,kzˆ)eıˆe ˆ e−ikχ cos θ = − h˙(±2) (η,kzˆ) ij 2 For tensor perturbations, there are no perturbed scalars s X √ or 3-vectors, so vb = 0 and the monopole ` (` + 2)! j` (kχ) × (−i) 2` + 1 Y` ± (e). (60) (` − 2)! (kχ)2 2 Z ` dmˆ Θ(mˆ ) = 0.

Using Eqn. (57) and the integrating factor e−τ in the We are now ready to extend this result to Fourier modes linearised Boltzmann equation (52), we obtain with k along a general direction by a rotation via the Wigner D-matrix

d −τ −τ −τ e Θ = −τ˙ e Θ − e an¯eσTΘ r dη ± ˆ π h˙( 2) (η,k)eıê ˆ e−ikχk·e = − h˙(±2) (η,k) 1 1 ij 2 − e−τ h˙ eıê ˆ = − e−τ h˙ eıê ˆ 2 ij 2 ij s X √ (` + 2)! j` (kχ) × (−i)` 2` + 1 which can be integrated to (` − 2)! (kχ)2 `,m Z 1 η0 ` −τ ıˆ ˆ × D (ϕk,θk,0)Y`m (e) (61) Θ(η0,x0,e) ≈ − dη e h˙ije e . (58) m ±2 2 0 where explicitly Note here we have 1) recognised τ˙ = −an¯eσT from denition (46); 2) evaluated τ (η ) = 0, τ (0) = ∞; and r 0 4π 3) neglected the temperature quadrupole at last-scatter- ` ∗ ˆ Dm ±2(ϕk,θk,0) = ∓2Y`m (k). ing on large scales as it has not had the time grow. 2` + 1 ˙ The physical interpretation of this is thathij is the shear In analogy with Eqns. (54) and (55), we expand the frac- ˙ ıˆ ˆ of the gravitational waves and hije e contributes to tional temperature uctuation in spherical multipoles the temperature anisotropies as a local quadrupole, as h is traceless [6]. Z 1 X d3k Θ (η ,x ) = √ (−i)`Θ(p) (η ,k) ` m 0 0 3/2 ` 0 We shall now attempt to evaluate the integral expres- 2 p=±2 (2π ) sion. First take a Fourier mode of h given in Eqn. (34) r ij 4π along the z-axis, × D` (ϕ ,θ ,0) eik·x0 , (62) 2` + 1 mp k k ˙(±2) ıˆ ˆ 1 ˙(±2) 2 hij (η,kzˆ)e e = √ h (η,kzˆ) (ˆx ± iˆy) · e 2 2 f g Z 2` + 1 η0 1 (±2) 2 ±2iϕ (p) 0 −τ ˙(p) 0 = √ h˙ (η,kzˆ) sin θ e Θ` (η0,k) = dη e h (η ,k) 2 2 4 0 r s 4π (±2) (` + 2)! j` (kχ) = h˙ (η,kzˆ)Y ± (e) (59) × (63) 15 2 2 (` − 2)! (kχ)2 where θ, ϕ are the spherical polar coordinates of e, and where inside the integral χ = η − η0. we have recalled the explicit form of the (l,m) = (2,±2) 0 spherical harmonic r III.1.4 Angular power spectrum of temperature 15 ±2iϕ 2 Y ± (e) = e sin θ. anisotropies 2 2 32π The contribution from the integrand in Eqn. (58) at a To make contact with observations, we shall now particular time η = η0 − χ, where χ is the comoving dis- compute the two-point correlator for the temperature tance from the observer now, is modulated by a phase anisotropies induced by gravitational waves. 8

2 r r Z 3 3 0 D ∗ E 1 ` −`0 4π 4π X d k d k D (p) (q)∗ 0 E Θ` Θ 0 0 = √ (−i) (−i) h (k)h (k ) m ` m 2` + 1 2`0 + 1 3/2 3/2 2 p,q=±2 (2π ) (2π ) * + (p) (q) 0 Θ (η0,k) Θ (η0,k ) 0 0 ` ` ` ` i(k−k )·x0 , - × D (ϕk,θk,0)D 0 (ϕk0 ,θk0 ,0) e h(p) (k) h(q) (k0) mp m q `−`0 Z 2 (−i) 1 X 3 3 0 2π 0 pq = d k d k Ph (k)δ (k − k )δ p 0 4π 2 k3 (2` + 1)(2` + 1) p,q=±2 (p) (q) 0 Θ (η0,k) Θ (η0,k ) 0 0 ` ` ` ` i(k−k )·x0 × D (ϕk,θk,0)D 0 (ϕk0 ,θk0 ,0) e h(p) (k) h(q) (k0) mp m q 2 `−`0 Z (p) Z − X Θ (η ,k) 0 ( i) ` 0 ˆ ` ` = p d lnk Ph (k) dk Dmp (ϕk,θk,0)Dm0 p (ϕk,θk,0) 2 (2` + 1)(2`0 + 1)  h(p) (k)  p=±2   | {z }   4π   = δ``0δmm0   2`+1  2  `−`0 Z (p) (−i) Θ` (η0,k) 4π = p d lnk Ph (k) δ``0δmm0 (2` + 1)(2`0 + 1)  h(p) (k)  2` + 1    2  Z Θ(p) (η ,k)  4π ` 0  = δ``0δmm0 d lnk Ph (k), (2` + 1)2  h(p) (k)        where formally d3k = k2 dk dkˆ, and in the third to and substitute this into Eqns. (63) and (64) the fourth lines the Wigner D-matrix orthogonality Z −1 4π η∗ condition has been used [6]. C` ∼ Ph d lnk (2` + 1)2 −1 η0 Comparing with Eqn. (49), we see that the angular s 2 2` + 1 (` + 2)! j (kη − 1) power spectrum for temperature anisotropies gener- × ` 0 − − 2 ated by gravitational waves is simply  4 (` 2)! (kη0 1)     −1   Z η∗ 2 −  π (` + 2)!  j` (kη0 1)  (p) 2 = Ph d lnk  Z − −1 − 4 4π Θ (η0,k) 4 (` 2)! η (kη0 1) C ` P 0 ` = d lnk h (k). (64) Z / − 2 (2` + 1)2  h(p) (k)  π (` + 2)! η0 η∗ 1 dx j (x)   = P `   − h 4   4 (` 2)! 0 1 + x x   Z ∞ 2   π (` + 2)! j (x) ≈ P x ` . A crude approximation — To gain an intuitive under- h d 5 4 (` − 2)! 0 x standing and make use of C , we assume that [6]: ` To arrive at the last line, we note that for 1 ` 1) the shear in gravitational waves is impulsive at η0/η∗ ≈ 60, the integral is dominated by x ≈ l. Finally, horizon entry; using the numerical formula [6] Z ∞ 2 j (x) 4 (` − 1)! 2) the visibility function is sharply peaked at the ` = , dx 5 time of recombination η∗, so τ = 0; 0 x 15 (` + 3)! η>η∗ we have 3) the primordial power spectrum Ph (k) is scale- π 1 C` ∼ Ph . (66) invariant. 15 (` − 2)(` + 2) Because of our approximation, this scale-invariant an- This means that we can write gular power spectrum is only valid for gravitational waves on scales ` ∈ (1,60) which enter the horizon (±2) −1 (±2) −1 h˙ (η,k) ∼ −δ (η − k )h (k) (65) after last-scattering (η∗ < k < η0). 9

−|s | ≡ − s ¯ |s | III.2 Polarisation from PGWs where [7] ð ( 1) ð . Numerous s Y`m properties, which are useful in subsequent calculations, are III.2.1 Concepts and notions listed in Appendix F.

Linear polarisation tensor — In the coordinate basis, we Stokes parameters — In analogy with electromagnetism can write the complex null basis vectors m± as where the correlation matrix of the electric elds for a plane wave propagating in direction zˆ are captured by a a a m± = (∂θ ) ± i cosecθ (∂ϕ ) . (72) the 4 real parameters I, Q, U and V The linear polarisation tensor can then be expressed D ∗ ED ∗ E Ex E Ex E 1 I + QU + iV as D x ED yE ≡ , (67) ∗ ∗ − − EyEx EyEy 2 U iVI Q 1 ab a b − a b .* /+ * + P = (Q + iU )m−m− + (Q iU )m+m+ . (73) 4 f g we could, also characterise- the, polarisation of- the CMB using these Stokes parameters.2 The trace is the total This relates to the symmetric traceless part of Eqn. (67): intensity I, the dierence between intensities in x- and the projection of Pab onto the complex null basis gives y-directions is Q and U the dierence when x,y-axes a b are rotated by 45°. V describes circular polarisation, Q ± iU = m±m±Pab . (74) which vanishes for Thomson scattering [6]. E- and B-modes — We can decompose the symmetric Spin — Locally the propagation direction e of the ob- traceless linear polarisation tensor as served photon and its associated spherical polar unit θˆ φˆ c vectors , form a right-handed basis. In the plane Pab = ∇ha∇biPE + ε (a ∇b)∇c PB (75) spanned by θˆ, φˆ, we dene the complex null basis for some real scalar potentials PE ,PB . Here ε is the m± B θˆ ± iφˆ (68) alternating tensor and h i denotes the symmetric traceless part of a tensor, i.e. ∇ ∇ = ∇ ∇ − g ∇2/2. with respect to which the linear Stokes parameters ha bi (a b) ab Then (see derivations in Appendix G) are determined. Under a left-handed rotation through e angle ψ about , Q + iU = ð ð(PE + iPB ), (76) ±iψ − = ð¯ ð¯ − . m± −→ e m±; Q iU (PE iPB ) As in the case for temperature anisotropies, we may a quantity on the 2-sphere sη is said to be spin s if it correspondingly transforms as expand in spherical harmonics the elds s isψ sη −→ e sη, (69) X (` − 2)! P (e) = E` Y` (e) E (` + 2)! m m e.g. Q ± iU has spin ±2. `,m s (77) X (` − 2)! Spin-weighted spherical harmonics — The spin-raising P (e) = B` Y` (e) B (` + 2)! m m and spin-lowering operators ð and ð¯ act on a spin-s `,m quantity η as s so that s −s ðsη = − sin θ (∂θ + i cosecθ ∂ϕ ) sin θsη, X (70) (Q ± iU )(e) = (E`m ± iB`m ) Y (e). (78) ð¯ η = − sin−s θ (∂ − i cosecθ ∂ ) sins θ η. ±2 `m s θ ϕ s `,m We can now dene the spin-weighted spherical har- These coecients E`m and B`m are associated with monics s what are known as E- and B-mode CMB polarisation (` − |s|)! s induced by (primordial) gravitational waves. Y = ð Y` (71) s `m (` + |s|)! m

2The average is taken over a time span longer than the wave III.2.2 The Boltzmann equation for polarisation period but shorter in comparison with amplitude variations. In analogy with Eqn. (51), these parameters are dened for the frequency independent fractional thermodynamic equivalent temper- Pre-recombination polarisation is negligible since e- atures [7]. cient Thomson scattering isotropises CMB radiation. 10

Towards recombination, the mean-free path of pho- III.2.3 Linear polarisation from gravitational waves tons increases until they can free-stream over an ap- preciable distance (compared with the wavelength We shall now consider the generation of linear polarisa- associated with perturbations) between scatterings tion from gravitational waves, and thus restore the po- such that a temperature quadrupole is generated [6]. larisation labels (p). If we neglect reionisation and treat The linear polarisation observed today is directly re- last-scattering as instantaneous [6], i.e. g(η) ≈ δ (η−η∗) lated to quadrupole anisotropies at the time of last- where we have reused η∗ to denote the conformal time scattering [8], and standard results in scattering theory at last-scattering, which takes place close to recombi- give the Boltzmann equation for linear polarisation nation, then the integral solution (80) above becomes

d(Q ± iU )(e) √ = τ˙ (Q ± iU )(e) 6 η (Q ± iU )(η0,x0,e) = − d 10 X √ 3 X 1 × Θ2m − 6E2m (η∗,x∗) ±2Y 2m (e) (81) − τ˙ E2m − √ Θ2m ±2Y 2m (e). (79) 5 |m |62 |m |62 6 * + . where x = x − χ e and χ = η − η . Now Eqn. (62) where d dη is along, the background- lightcone [7]. ∗ 0 ∗ ∗ 0 ∗ says in Fourier space, By using the integrating factor e−τ again as we did for 1 deriving Eqn. (58), we obtain a line-of-sight integral (p) − ` (p) Θ`m (η,k) = √ ( i) Θ` (η,k) solution to the polarisation observed today 2 r √ 4π ` Z η × D (ϕ ,θ ,0), 6 X 0 2` + 1 mp k k (Q ± iU )(η0,x0,e) = − dη g(η) 10 and similar expressions hold for E`m and B`m. With |m |62 0 √ the substitution of the equation above set to ` = 2, × − − Θ2m 6E2m (η,x0 χe) ±2Y 2m (e) (80) taking the Fourier transform of Eqn. (81) (with respect to x0) yields the polarisation generated by a single where χ = η0 − η. helicity-p state Note that if reionisation is absent, the integral mainly receives contributions from the time of recombination.

√ r √ 6 1 2 4π (p) (p) −ikχ kˆ·e X 2 (Q ± iU )(η ,k,e) = − √ (−i) Θ − 6E (η∗,k) e ∗ D (ϕ ,θ ,0) Y (e) 0 10 5 2 2 mp k k ±2 2m 2 |m |62 * + r √ 1 12π (p) (p) −ikχ kˆ·e X 2 = , - Θ − 6E (η∗,k) e ∗ D (ϕ ,θ ,0) Y (e), (82) 10 5 2 2 mp k k ±2 2m |m |62 where we have used the translation property of Fourier where the projection functions are transform, resulting in the modulation by a phase factor.

2 ! To determine the angular power spectra associated 1 d j` 4 dj` 2 ϵ` (x) B + + − 1 j` , with E- and B-modes, we must evaluate the polarisation 4  dx2 x dx x2  contribution above from gravitational waves. First  !  (84) 1  dj` 2  taking kˆ = zˆ, we obtain the following expression (see β` (x) B  + j` .  2  dx x  calculations in Appendix H) √ √ X ` (Q ± iU )(η0,kzˆ,e) ∝ − 5 (−i) 2` + 1 ` p By comparing results Eqns. (83) and (84) with Eqn. (78), × ±2Y`p (e) ϵ` (kχ∗) ± iβ` (kχ∗) (83) ` 2 we act with Dm ±2(ϕk,θk,0) to nd for general direc- 11 tions kˆ the matter-dominated era is derived in Appendix B

(±2) E (η ,k) j2(kη) `m 0 h˙(p) (η,k) = −3h(p) (k) , (89) (±2) ,k η B`m (η0 ) √ r    √ √ (p)  5 12π  ` (±2) (±2) where h (k) is the primordial value. Thus the inte- = − (−i) 2` + 1 Θ2 − 6E2 (η∗,k) 2 10 5 grand here contains the product of factors j2 (kη∗) and 2 2 ≈ ϵ (kχ ) ϵ` (kχ∗) (or β`). The rst factor peaks around kη∗ 2, × ` ∗ D` (ϕ ,θ ,0) ±β (kχ ) m ±2 k k whereas the second factor peaks around kχ∗ ≈ l. Hence  ` ∗  √   for large-angle behaviour l χ /η∗, the integral is √  √ ∗ 12π `  (±2)  (±2) dominated by modes with kχ l at the right tails of = − (−i) 2` + 1Θ − 6E (η∗,k) ∗ 10 2 2 2 2 ϵ` (or β`). ϵ` (kχ∗) ` × Dm ±2(ϕk,θk,0) (85) The explicit forms (84) of ϵ` and β` and the asymp- ±β` (kχ )  ∗  ∼   totic expression for spherical Bessel functions j` (x)   x−1 sin(x − `π/2) give where the pre-factors have been restored. ! E B 1 `π Now we can write down the - and -mode power ϵ` (x) ∼ − sin x − , spectra using the orthogonality condition of Wigner 2x 2 ! D-matrices, since it is analogous to calculating the 1 `π β` (x) ∼ cos x − . temperature anisotropy power spectrum: 2x 2 Z 2 2 CE 6π ϵ` (x) and β` (x) are rapidly oscillating and thus can ` = d lnk P (k) CB h be replaced by their averages in the integral, i.e. ` 25 2   ϵ` (x),β` (x) → 1/8x , so we are left with   √ 2   (p) (p) Θ − 6E (η ,k) 2   2 2 ∗ ϵ2(kχ ) Z ∞ ` ∗ E B π Ph lp dx 2 ×   2 (86) C , C ≈ j (x)  h(p) (k)  β (kχ ) ` ` 3 2    ` ∗  4 χ∗ 0 x         * +2   π Ph lp  (p) √ (p)  ≈ , - h(p)(k) Θ  − E /h(p) . where is primordial, 2 6 2 is 288 χ∗ independent of the polarisation state and we have * + summed over the helicity states p = ±2. We see therefore the E- and, B-mode- polarisation generated by gravitational waves has roughly equal powers Tight-coupling and large-angle behaviour — Physically, on large scales. we expect a temperature quadrupole to build up over a scattering time scale due to the shear of gravitational i j III.2.4 Statistics of the CMB polarisation waves Θ(e) ∼ −(lp/2)h˙ije e , where lp is the photon mean-free path close to recombination [7]. A more in-depth treatment using the tight-coupling approxi- Cross-correlation — The angular power spectrum en- mation and polarisation-dependent scattering shows codes the auto-correlation of an observable obeying that [6] rotational invariance, but we can also have cross- correlations between dierent observables, e.g. the √ (p) (p) 5 (p) two-point correlation between E- and B-modes Θ − 6E (η,k) ≈ √ lph˙ (η,k). (87) 2 2 3 3 D ∗ E EB E`mB`0 0 = C` δ``0δmm0 , (90) Substitution of this into Eqn. (86) gives m or between temperature anisotropies and polarisa- Z CE 2π tion [9] ` = l2P d lnk CB p h  `  9 D ∗ E ΘE   Θ`mE 0 0 = C δ``0δmm0 ,   2 ` m ` (p) 2 (91)   h˙ (η∗,k) D ∗ E ΘB ϵ` (kχ∗) 0 0   × . (88) Θ`mB`0m0 = C` δ`` δmm . h(p) (k) β2(kχ )    ` ∗          On scales outside the horizon at  matter–radiation Parity symmetry — A possible further constraint on the equality, the form of the gravitational wave shear in polarisation statistics is parity invariance. The E- and 12

B-modes possess electric parity and magnetic parity IV B-Mode Polarisation: A Promising respectively;3 that is under a parity transformation [7] Route for Detecting PGWs?

` `+1 The measurements of CMB polarisation will soon be of E`m −→ (−1) E`m, B`m −→ (−1) B`m . (92) critical importance in modern cosmology: the polari- Non-violation of statistical isotropy and parity symme- sation signal and the cross-correlations provide consis- try necessarily implies zero cross-correlation between tency checks for standard cosmology, and complement B and Θ or E. the cosmological information encoded in temperature anisotropies, which are ultimately bound by cosmic Cosmic variance — As the confrontation between the- variance; the denitive detection of B-mode would ory and observations often lies between the predictions indicate non-scalar perturbations, distinguishing dif- for the probability distribution and the measurements ferent types of primordial uctuations and imposing of the physical variables, it is impossible to avoid men- signicant constraints on cosmological models [1, 6]. tioning estimators and statistical variance. In observa- In particular, for our purposes, the measurements of tional cosmology, the intrinsic cosmic variance arises B-mode polarisation oers a promising route for de- as one attempts to estimate ensemble expectations with tecting primordial gravitational waves. only one realisation of the universe, which contributes to the overall experimental errors [6]. IV.1 Cosmological sources of B-mode polarisa- Given a general zero-mean random eld f (nˆ ) on the tion 2-sphere, we measure the spherical multipoles f`m and construct the unbiased estimator (Primordial) perturbations in background spacetime may be decomposed into scalar, vector and tensor 1 X Cˆ = f f ∗ (93) types, which crucially decouple at linear order [1]. The ` ` + `m `m 2 1 m E- and B-mode split of CMB polarisation has implications on what type of uctuations may be present for the corresponding true angular power spectrum C`. during the inationary period [9, 10]: This has a variance [6] 1) scalar perturbations create only E- not B-mode 2 2 var Cˆ` = C (94) polarisation; 2` + 1 ` 2) vector perturbations create mostly B-mode polar- due to the fact that we only have 2` + 1 independent isation; modes for any given multipole. 3) tensor perturbations create both E- and B-mode Therefore an estimator for cross-correlation may be, polarisation. for example [9], We have proved the last claim in §§ III.2.3, and as for the rst claim, one can intuitively see that scalar per- EB 1 X ∗ Cˆ = E` B ; ` 2` + 1 m `m turbations do not possess any handedness so they can- m not create any B-modes which are associated with the the cosmic variance for auto-correlations may be, for “curl” patterns in CMB temperature maps, but vector example, and tensor perturbations can [9]. 2 var CˆE = CE CE ; Although we can show this directly just as in §§ III.2.3 ` ` + ` ` 2 1 but for scalar perturbations, we will instead argue from and for cross-correlations [7], the Boltzmann equation (79) for linear polarisation by showing that (see details in Appendix I) 1 var CˆΘE = CΘE CΘE + CΘCE , ` 2` + 1 ` ` ` ` r r (` + )2 − `2 − ΘB 1 ΘB ΘB Θ B 1 4 4 var Cˆ = C C + C C . E˙` + k E`+1 − E`−1 ` 2` + 1 ` ` ` `  2` + 3 2` − 1    3   This follows from that under a parity transformation, e → −e,  3 1  but θˆ(−e) = θˆ(e) and φˆ (−e) = −φˆ (e) so (Q ± iU )(e) → (Q ∓  = τ˙ E` − δ`2 E2 − √ Θ2  , (95) 5 6 iU )(−e).    * +  , - 13

r (` + 1)2 − 4 Primordial inhomogeneous magnetic elds — In the B˙` + k B`+1  2` + 3 early universe, magnetic anisotropic stress can gen-  r erate both vorticity and gravitational waves, and leave  `2 − 4  − B`− = τ˙B`. (96) signatures in the CMB temperature anisotropy and 2` − 1 1  polarisation (including B-modes). However, these pri-  mordial elds are not well-motivated by theoretical  We see that the B-mode equation (96) does not have a models, which predict either very small eld ampli- source term from temperature quadrupoles, so scalar tudes or a blue tilt in ns. Furthermore, they can be perturbations do not produce B-mode polarisation. distinguished from primordial gravitational waves by their non-Gaussianity, or detection of the Faraday ef- For B-mode polarisation to truly vindicate the exis- fect [15] (interaction between light and magnetic elds tence of primordial gravitational waves, we must con- in a medium). sider other cosmological sources of B-modes. These include: 1) topological defects; 2) global phase tran- Gravitational lensing — This deforms the polarisation sition; 3) primordial inhomogeneous magnetic elds; pattern on the sky by shifting the positions of photons 4) gravitational lensing. relative to the last-scattering surface. Some E-mode polarisation is consequently converted to B-modes, as Topological defects — We have remarked in § II.3 that the geometric relation between polarisation direction primordial vector modes are diluted away with expan- and angular variations in the E-mode amplitude is not sion. However, topological defects such as cosmic preserved [7]. More in-depth investigations of lens- strings, which are often found in grand unication ing eects and careful de-lensing work are needed to models, actively and eciently produce vector pertur- remove its contamination of the primordial B-mode bations which in turn create B-mode polarisation [1]. signal (see further discussions in § VI.1). Nonetheless, the presence of topological defects alone poorly accounts for the polarisation signals seen in the data of the BICEP2 Collaboration4 [11]. The peaks pro- IV.2 Statistical aspects of B-mode polarisation duced by cosmic strings in the polarisation spectrum, if they are formed, are at high ` ∼ 600–1000 (gener- The observational importance of CMB polarisation also ated at last-scattering) and at low ` ∼ 10 (generated at stems from the exhaustion of information that could reionisation). ever be extracted from CMB temperature anisotropies (and E-mode polarisation). Soon the cosmic variance Global phase transitions — It has been shown on dimen- intrinsic to the latter would fundamentally limit our sional grounds and by simulations that the symmetry- ability to achieve much greater accuracies [16]; in par- breaking phase transition of a self-ordering scalar eld ticular, we see that the weighting factor (2` + 1)−1 in could causally produce a scale-invariant spectrum of Eqn. (94) attributes greater variances at lower `, i.e. on gravitational waves [12]. However, similar to topologi- large scales of our interests where gravitational waves cal defects, global phase transitions of self-order scalar (GWs) contribute signicantly to anisotropies6. eld do not reproduce the BICEP2 data [13]. To demonstrate this point, we turn to the tensor-to- The key physical point is that in these two alter- scalar ratio r introduced in §§ II.3.1 as an example: native cosmological models of B-mode polarisation, imagine that r is the only unknown variable and our the causally-produced uctuations are on sub-horizon measurements of large-scale temperature anisotropies scales; only the inationary model alters the causal are noise-free. We saw in §§ III.1.4 that C` ∼ Ph ∼ At structure of the very early universe and accounts for for gravitational waves, and similarly for curvature correlations on super-horizon scales [13, 14]. The dis- perturbations C` ∼ As, so we can estimate r as the tinctive signature of an anti-correlation between CMB excess power over C` temperature and polarisation, imprinted by adiabatic uctuations at recombination and seen on large scales Cˆ` − C` rˆ = (97) ` ∼ 5 ` CGW 50–150 in WMAP data, is convincing evidence ` (r = 1) for the ination theory [7]. using a set of angular power spectrum estimators [6]

4Acronym for Background Imaging of Cosmic Extragalactic 6In Appendix B, we show that as the universe expands gravi- Polarisation. tational waves are damped by the scale factor within the Hubble 5Acronym for the Wilkinson Microwave Anisotropy Probe. horizon. 14

Cˆ` given in §§ III.2.4. Here C` is the true angular power standard Big Bang cosmology; quantum uctuations in CGW spectrum from curvature perturbations, and ` (r = the very early universe are amplied and subsequently 1) is the true angular power spectrum from GWs if r frozen in on super-horizon scales, seeding large-scale were equal to 1. structure formation. Detecting PGWs provides strong evidence for the existence of ination, and in addition Amongst all weighted averages, the inverse-variance it will reveal to us weighting gives the least variance, so one can make a prediction on the error σ (r ) in the null hypothesis the energy scale of ination and thus that of the H0 : r = 0, very early universe; 2 the amplitude of the inaton eld excursion which 1 X 2` + 1 CGW(r = 1) = ` . (98) constrains models of ination; σ 2(r ) 2 C `  `    any violation of the various consistency condi-   CGW C ≈  tions for testing ination models; Approximating ` (r = 1)/ ` 0.4 as constant for ` < 60, we obtain a rough estimate clues about modied gravity and particle physics beyond the Standard Model. 1 0.42 ≈ × (60 + 1)2 =⇒ σ (r ) ≈ 0.06. σ 2(r ) 2 The last point is anticipated as the validation of ina- Using the actual spectra observed gives σ (r ) = 0.08 tion theory inevitably involves the testing of all fun- which is not far from our rough estimate [6]. The latest damental theories upon which it is built. The links Planck data puts an upper bound r < 0.10 − 0.11 at between primordial gravitational waves and modied 95% condence level (CL) [17], consistent with r = 0. gravity and eective eld theory are explored in some We thus see that the scope of detecting PGWs within recent literature [19]. We shall here consider the other temperature anisotropies alone is slim. points in turn.

On the other hand, the CMB B-mode polarisation may circumvent this problem: it receives no scalar contribu- V.1 Alternatives to inflation tions, and is unlike the temperature anisotropies and E-mode polarisation to which the gravitation wave con- To validate the ination theory we must consider tribution is sub-dominant [6]. The peak location and competing cosmological models, which chiey in- the peak height of the polarisation power spectrum are clude [1]: 1) ekpyrotic cosmology; 2) string gas cosmol- sensitive to the epoch of last-scattering when pertur- ogy; 3) pre-Big Bang cosmology. bation theory is still in the linear regime: they depend, Ekpyrotic cosmology respectively, on the horizon size at last-scattering and — In this model, the universe its duration; this signature is not limited by cosmic starts from a cold beginning, followed by slow con- variance until late reionisation [8]. In contrast, temper- traction and then a bounce returning to the standard ature uctuations can alter between last-scattering and FLRW cosmology. Its cyclic extension presents a sce- today, e.g. through the integrated Sachs–Wolfe eect nario where the ekpyrotic phase recurs indenitely. It of an evolving gravitational potential. Therefore B- has been shown that in this model, not only quantum mode polarisation complements information extracted uctuations but inhomogeneities are also exponentially from temperature anisotropies and already-detected amplied without ne-tuning [20]; during the bounc- E-mode polarisation [18], and oers a promising route ing phase, the null energy condition is violated—a sign for primordial gravitational wave detection. usually associated with instabilities [1]; furthermore, taking gravitational back-reaction into account, the curvature spectrum is strongly scale-dependent [21]. V A “Smoking Gun”: Physical Signifi- In the new ekpyrotic models, some of these issues cance of PGW Discovery are resolved; however, a substantial amount of non- Gaussianity is now predicted [22] and more impor- tantly, the absence of detectable levels of PGWs makes Having discussed the causal mechanism, the cosmolog- it very distinguishable from ination. ical imprints and the practical detection of primordial gravitational waves, we now turn to the signicance of String gas cosmology — This model assumes a hot Big a PGW discovery. As we mentioned earlier in this pa- Bang start of the universe with energies at the string per, the theory of ination solves a number of puzzles in scale and with compact dimensions. The dynamics 15 of interacting strings means three spatial dimensions at such enormous energies are the apparent unica- are expected to de-compactify but this requires ne- tion of gauge couplings and the lower bounds on the tuning. Also, a smooth transition from the string gas proton lifetime—such energy scales are forever beyond phase to the standard radiation phase violates the the reach of human-made ground particle colliders [7]; null energy condition, believed to be important in in comparison, the Large Hadron Collider currently ultraviolet-complete (UV-complete) theories, or the operates at up to 13 TeV [27], lower by O 1013 . scale-inversion symmetry, believed to be fundamental to string theory; in addition, the production of a nearly Lyth bound — We can relate the inaton eld excursion scale-invariant power spectrum requires a blue-tilted ∆ϕ to the tensor-to-scalar ratio r. Recall in §§ II.3.1 we 2 ≡ 2 ˙ scalar spectral index [1, 23, 24]. have derived that r = 16ϵ 8/MPl ϕ/H , which can be written using the number of e-folds dN = H dt Pre-Big Bang cosmology — Motivated by string gas cos- as !2 mology, this scenario describes a cold, empty, at initial r 1 dϕ = . state of the universe. Dilatons drive a period of “super- 8 MPl dN ination” until the string scale is reached, after which Integration gives the radiation-dominated era initiates [1, 25]. However, r Z N? r the issue of string phase exit to the radiation phase is ∆ϕ r (N ) r? poorly understood and some literature claims that the = dN ≡ Ne (100) MPl 0 8 8 horizon, atness and isotropy problems in standard cosmology are not explained [26]. where N? is the number of e-folds between the end of ination and the horizon exit of the CMB pivot scale, To summarise, the key obstacles of current alternative and the eective number of e-folds theories to competing with ination include: 1) failure s of a smooth transition to the standard Big Bang evolu- Z N? r (N ) tion; 2) absence of a signicant amplitude of primordial Ne B dN (101) 0 r? gravitational waves. The latter means primordial gravitational waves, and thus B-mode polarisation, are a is model-dependent as r evolves [19]. In slow-roll ina- “smoking gun” of ination [1]. tion, we can treat ϵ as approximately constant so r also is, giving a lower bound on the inaton eld excursion r r V.2 Energy scale and the inflaton field excursion ∆ϕ r? N? r? & N? ≈ . (102) MPl 8 60 0.002 Energy scale of the early universe — Recall that for slow- Hence we see that if at least 60 e-folds are required to roll ination, ϕ˙2 V so Eqns. (1), (3), (39) and (42) solve the atness and the horizon problems [1], emis- together lead to a relation between the energy scale of sion of a substantial amount of gravitational waves ination V 1/4 and the tensor-to-scalar ration r, would mean a super-Planckian eld excursion. (A N = 1/4 more conservative√ bound of e 30 gives instead 3π 2 ∆ϕ/M & 1.06 r /0.01, but the conclusion is unal- V 1/4 ≈ r P M . (99) Pl ? 2 s Pl tered.) * + −1 Super-Planckian eld variation has consequences for For ducial values at r,? = 0.01,- k? = 0.05 Mpc and 10 ln 10 As = 3.089 (the value given in [17]), we have ination model-building. For instance, in the context / of supergravity, one may expect the inaton potential calculated that V 1 4 ≈ 1.03 × 1016 GeV using M = ? Pl to have an innite power series, say 2.43 × 1018 GeV, i.e. 2 1/4 m 2 4 λ6 6 λ8 8 / r V = V0 + ϕ + λ4ϕ + ϕ + ϕ + ··· . V 1 4 = 1.06 × 1016 GeV. 2 M2 M4 0.01 Pl Pl Standard eld theories would truncate such series af- We see that the energy scale during ination reaches ter the rst few terms, as is the case in the Standard that of Grand Unication theories, just a few orders of Model, its minimal supersymmetric extensions and magnitude below the Planck scale. It is dicult to over- many others [28]. However, for more generic eld theo- state the huge implications this has for high-energy ries, the coupling coecients λ arising from integrated- particle physics. To date the only hints about physics out elds at higher energies may be large, e.g. at 16

O(1), in which case the series diverges. A detectable r remains “at”, i.e. ϵV 1 and ηV 1. In addition to could place us in an uncharted territory of high-energy ϵV and ηV , the family of potential slow-roll parameters physics, propelling advances in beyond-the-Standard- include [17] [cf. Eqn. (8)] Model UV-complete theories. 1/2 2 V,ϕV,ϕϕϕ ξV B M etc. (106) Pl V 2 V.3 Constraining models of inflation

which encode derivatives of the inaton potential at Broadly speaking, ination models are classied as: increasing orders, so they control the shape of the po- 1) single-eld, of which single-eld slow-roll ination tential V (ϕ). Often in literature (e.g. [1, 7]) the Hubble is a simple case, but including some apparently multi– slow-roll parameters dened analogously to the above eld models such as hybrid ination; or 2) multi-eld, are adopted, with the potential variable V (ϕ) replaced where more than one scalar eld is invoked [1]. by the Hubble parameter H (ϕ).

Generic single-eld ination — Single-eld ination Deviation from scale-invariance — A perfectly scale- may be large or small according to the inaton eld invariant spectrum, i.e. the Harrison–Zel’dovich– Peebles spectrum, has ns = 1. Deviations from scale- excursion, and blue (ns > 1) or red (ns < 1) depending invariance may be captured by the scalar spectral index on the tilt ns. Non-canonical kinetic eects may appear in general single-eld models given by an action of the and its running form (M = 1 here) Pl dns Z α B , (107) 1 √ s d lnk S = d4x −g R + 2P (X,ϕ) (103) 2 f g which parametrise the scalar power spectrum as [cf. µν where X ≡ −g ∂µϕ∂ν ϕ/2, P is the pressure of the Eqn. (28)] scalar uid and ρ = 2XP − P its energy density. For ,X 1 k instance, slow-roll ination has P (X,ϕ) = X − V (ϕ). !ns−1+ αs ln k 2 k? These models are characterised by a speed of sound Ps = As . (108) k s ? P,X cs B (104) αs may be small even for strong scale-dependence as ρ,X it only arises at second order in slow-roll. A signi- where cs = 1 for a canonical kinetic term, and cs 1 cant value of αs would mean that the third potential signals a signicant departure from that. In addition, slow-roll parameter ξV plays an important role in in- a time-varying speed of sound cs(t) would alter the ationary dynamics [1,7]. prediction for the scalar spectral index as ns = −2ϵ − − η s, where V.3.2 Significance of the tensor power spectrum c˙s s B (105) Hcs measures its time-dependence [1]. To link the model features above to the detection of PGWs, we observe that the spectral indices are related Multi-eld ination — Multi-eld models produce to the potential slow-roll parameters (in the case of novel features such as large non-Gaussianities with canonical single-eld ination P = X − V ) by dierent shapes and amplitudes, and isocurvature − = − , perturbations which could leave imprints on CMB ns 1 2ηV 6ϵV anisotropies. However, the extension to ordinary nt = −2ϵV single-eld models to include more scalar degrees of at rst order. The second equation is Eqn. (43), and the freedom also diminishes the predictive power of ina- rst comes from Eqns. (8) and (26) tion. Diagnostics beyond the B-mode polarisation may be needed for such models [1]. d ln H d ln ϵ n − 1 = 2 − s d lnk d lnk 2 d ln H 1 d ln ϵ V.3.1 Model features ≈ − H dt H d ln t = −2ϵ − η Shape of the ination potential — For single-eld slow- roll ination, we merely say the inaton potential V ≈ 2ηV − 6ϵV 17 where we have replaced d lnk = (1 − ϵ)H dt ≈ H dt VI Future Experiments: Challenges and at horizon crossing k = aH. The runnings αs and Prospects αt (dened analogously for tensor perturbations) can also be linked to potential slow-roll parameters, so the 70 years ago the cosmic microwave background was measurements of r, spectral indices n , n and runnings s t predicted by Alpher and Hermann; 53 years ago the α , α can break the degeneracy in (potential) slow- s t CMB was “accidentally” discovered by Penzias and roll parameters and control the shape of the inaton Wilson; 25 years ago the CMB anisotropy was rst potential, thus constraining inationary models. observed by the COBE DMR7. What marked the gaps Consistency conditions — The measurements of PGW of many years in between was not our ignorance of signals can be used to test consistency of dierent ina- the signatures encoded in the CMB, but the lack of tion models (see Tab. 1), and therefore lter out single- necessary technology to make precise measurements eld ination through the sound speed, and on multi- [16]. eld ination through cos ∆, which is the directly mea- Now that has all changed within the last decade or surable correlation between adiabatic and isocurvature so: observational cosmology has beneted from leaps perturbations for two-eld ination; or more generally in precision technology, and many may refer to the through sin2 ∆, which parametrises the ratio between present time as the “golden age of cosmology” in im- the adiabatic power spectrum at horizon exit during plicit parallelism with the golden age of exploration, ination and the observed power spectrum [1]. when new continents were discovered and mapped out [1].

Bounds from current CMB observations — Various cos- Table 1. Consistency conditions for inflation models. mological parameters related to ination [17] have been measured, including but not limited to: model consistency conditions

single-eld slow-roll r = −8nt 1) a red tilt of ns = 0.9645–0.9677 at 68% CL de- generic single-eld r = −8ntcs pending on the types of data included. The scale- 2 invariant Harrison–Zel’dovich–Peebles spectrum multi-eld r = −8nt sin ∆ is 5.6σ away; 2) a value of running for scalar perturbations consistent with zero, αs = −0.0033 ± 0.0074, with the Planck 2015 full mission temperature data, high- Symmetries in fundamental physics — Sensitivity of in- ` polarisation and lensing ; ation to the UV-completion of gravity is a crux in its model-building, but also creates excitement over exper- 3) an upper bound at 95% CL for the tensor-to-scalar imental probes of fundamental physics as the very early ratio r0.002 < 0.10–0.11, or r0.002 < 0.18 assuming universe is the ultimate laboratory for high energy phe- nonzero αs and αt. The subscript denotes the pivot −1 nomena. Many inationary models are motivated by scale k? in units of Mpc . string theory, which is by far the best-studied theory of Therefore our current observations disfavour models quantum gravity, and supersymmetry is a fundamental with a blue tilt, e.g. hybrid ination, and suggests a spacetime symmetry of that (and others) [1]. negative curvature of the potential V,ϕϕ < 0. Models Controlling the shape of the inaton potential over predicting large tensor amplitudes are also virtually super-Planckian ranges requires an approximate shift- ruled out, e.g. the cubic and quartic power law poten- symmetry ϕ → ϕ + const. in the ultraviolet limit of tials [7]. For the testing of select inationary models the underlying particle theory for ination. The con- and the reconstruction of a smooth inaton potential, struction of controlled large-eld ination models with see [17]. approximate shift symmetries in string theory has been Graphic presentation of current CMB observations — realised recently. Therefore, the inaton eld excur- Constraints on the tensor-to-scalar ratio r . in sion as inferred from PGW signals could probe the 0 002 the ΛCDM model with B-mode polarisation results symmetries in fundamental physics, and serve as a se- lection principle in string-theoretical ination models 7Acronyms for the Cosmic Background Explorer and Dieren- [1,7]. tial Microwave Radiometers. 18 added to existing Planck 2015 results from the BI- CEP2/Keck Array+Planck default conguration are shown in Fig. 1 (see [29]). The Planck 2015 results have conrmed measurements of temperature anisotropies and E-mode polarisation. The (re-dened) EE angular power spectra C` and `(` + 1) `(` + 1) DTT B CTT , DTE B CTE ` 2π ` ` 2π ` are shown in Fig. 2 (see [30]). A full-sky map of CMB polarisation ltered at around 5° is shown in Fig. 3 (see [31]).

Figure 1. Constraints (68% and 95% CL) on the tensor- to-scalar ratio r0.002 in the ΛCDM model with B-mode polarisation results added to existing Planck data from the BICEP2/Keck Array+Planck (BKP) default likelihood configuration. Zero running and consistency relations are assumed. Solid lines show the approximate ns-r relation for quadratic and linear potentials, to first order in Figure 2. Planck 2015 CMB spectra with the base slow roll; the laer separates concave and convex poten- ΛCDM model fit to full temperature-only and low-` tials. Doed lines show loci of approximately constant polarisation data. The upper panels show the spectra numbers of e-folds N (from the horizon exit to the end and the lower panels the residuals. The horizontal scale of inflation) for power-law single-field inflation. Credit changes from logarithmic to linear at the “hybridisa- © ESA/Planck Collaboration. tion” scale ` = 29. Also note the change in vertical scales in lower panels on either side of ` = 29. Credit © ESA/Planck Collaboration. As it currently stands, the observations of r are not signicant enough for the null hypothesis H0 : r = 0 to be rejected. The constraints placed by CMB temperature data on the tensor perturbation amplitude, although improved by the inclusion of Type IA supernova lumi- nosities and baryon acoustic oscillations (BAO)8 [7], waves and thus ination, experimental eorts directed are close to the cosmic variance limit. at detecting B-mode signals more sensitively have been driving more stringent constraints on tensor perturba- Since B-mode polarisation, as we have explained ear- tions recently. We shall now present an overview of lier, is a powerful route for vindicating gravitational the experimental challenges and prospects lying ahead 8 Through more precise values of the matter density and ns. in detecting PGWs. 19

cently gravitational lensing in B-mode polarisation has been detected by multiple experiments, in particular in the range 30 < ` < 150 which includes the recombination peak [33, 34]. Therefore contamination from lensing could be especially signicant for small r . 0.01, and accurate de-lensing work is needed for very high sensitivity measurements in the future.

Instrumentation and astrophysical foregrounds — Cur- Figure 3. Planck 2015 CMB polarisation filtered at rent and future CMB experiments should reach a sensi- around 5°. Credit © ESA/Planck Collaboration. tive level that gravitational lensing noise is comparable to instrumental noise, but eorts for improving sensitivity is only sensible if foreground contamination VI.1 Challenges in detecting PGWs can be removed coherently [7]. The main astrophysical foregrounds include the following [8] The main challenge in detecting PGWs is that the primordial B-mode polarisation signal is faint. The CMB 1) free-free: although intrinsically unpolarised, free- polarisation anisotropy is only a few percent of the tem- free emission from ionised hydrogen clouds can perature anisotropy in the standard thermal history; be partially polarised due to Thomson scattering. and B-mode polarisation is at least an order of mag- The eect is small and does not dominate at any nitude lower than the E-mode polarisation [4]. What frequencies; complicates the picture are reionisation, (weak) gravi- 2) dust: this may be the dominant foreground at high tational lensing and foreground contamination such as frequencies. Interstellar dust causes microwave polarised galactic emission (in particular thermal dust thermal emission, which generates E,B-mode po- emission). larisation if galactic magnetic elds are present [19]. Recently, the much-reported detection of B- Reionisation — When intergalactic medium begins to mode polarisation from BICEP2/Keck Array was reionise, the emitted free electrons re-scatter CMB pho- due to bright dust emission at 353 GHz [35]; tons so polarisation signals from last-scattering are suppressed but there is instead a slight increase in po- 3) point sources: these are likely to be negligible larisation power on large angular scales.9 This eect except possibly for satellite missions; controlled by the epoch of reionisation is small and can 4) synchrotron: this is a major concern at low fre- be analysed by determining the corresponding optical quencies as it is highly polarised. depth τreion. [7]. There is some scope in exploiting the E,B-mode po- Gravitational lensing10 — We have mentioned gravi- larisation nature of foregrounds for their removal, but tational lensing as a cosmological source of B-mode the technique may be compromised if the foreground polarisation in § IV.1. As polarisation receives contri- produce the two modes unequally. However, multi- butions from gradients of the baryon velocity but tem- frequency coverage in CMB measurements may be perature uctuations from both velocity and density able to remove these polarised foregrounds [8]. of the photon-baryon plasma which are out of phase, acoustic oscillations are narrower for polarisation spec- Looking ahead, the power of any future experiment tra than for the temperature anisotropy, and therefore endeavouring to detect PGWs lies upon three crucial el- gravitational lensing eects are more signicant for ements [36]: 1) instrumental sensitivity; 2) foreground the former. removal; 3) size of the survey (e.g. angular range of view, number of detectors). A careful analysis reveals that the converted, extrin- sic B-mode from intrinsic E-mode is small in generic models with a peak (∼ 10% relative change to the undis- VI.2 Future prospects and concluding remarks torted spectra) at angular scales around ` ∼ 1000. Re- B 9 Although the primordial -mode polarisation signals Reionisation also screens temperature anisotropies—uniform are weak, there are no fundamental technological or screening is important on virtually all scales. 10The following discussions on gravitational lensing eects are cosmological barriers to achieving great levels sensi- mostly based on [32]. tivities. The Planck mission has already obtained a 20 sensitivity level of r ∼ 0.1 (not for B-modes though). amplitudes on dierent scales. Currently, a value of Next generation of ground-based and balloon experi- nt determined from a full-sky polarisation map, when ments surveying smaller regions of the sky with known extrapolated to much smaller wavelengths, has a signif- low-foreground contamination may achieve r ∼ 0.01, icant error. Thus to provide even stronger constraints and a dedicated polarisation satellite surveying the full on tensor modes, one must look for other observational sky, such as CMBPol, may bring this down to r ∼ 0.001. channels such as direct detection, e.g. small scale mea- Levels of such sensitivity would mark a qualitative shift surements could be performed in laser interferometer in our capability to test the ination theory [7, 37]. experiments. Information from direct detection and from B-mode polarisation can complement each other. Satellite missions — Forecasts for a speculative future satellite mission using the Fisher methodology have There have already been proposals and concept stud- been made in a concept study, where the theoretical ies of space-based interferometers such as the Laser angular power spectra are split into a primordial con- Interferometer Space Antenna (LISA) and the Big Bang tribution, a residual foreground contribution and an Observer (BBO). Although the tensor energy density instrumental noise contribution. For a ducial model could be very low, BBO may still be able detect its with r = 0.01 without foreground contamination, the signal at certain frequencies with a high signicance −4 forecast errors are ∆r = 5.4 × 10 and ∆nt = 0.076 for level. To separate stochastic PGWs from other cosmo- a low-cost mission aimed at detecting B-mode polarisa- logical contributions such as global phase transitions, tion on scales above about 2°. A graphic representation these sources must either be precisely modelled, or ex- of the forecast constraints in the ns-r plane with a pes- periments with even greater sensitivity (e.g. Ultimate simistic foreground assumption (contamination with DECIGO12) are called for; similarly, astrophysical fore- residual amplitude 10% in C`) is shown in Fig. 4. More grounds would also need to be better studied and their forecast details can be found in the concept study for contamination must be carefully removed. the CMBPol mission [1].

Cosmic ination was hypothesised to solve a series of puzzles in standard Big Bang cosmology, but it also provides the causal mechanism for large structure formation. The same mechanism generates a background of stochastic gravitational waves with a nearly scale- invariant spectrum, which leave relatively clean imprints in CMB observables when cosmological perturbations were still in the linear regime.

As it currently stands, the ination paradigm is still incomplete without a denitive proof of its existence. However, the arrival of precision cosmology has en- Figure 4. Forecasts of future constraints in the ns-r hanced detection sensitivity to B-mode polarisation plane for a low-cost mission with pessimistic foreground. by many orders of magnitude; provided we can utilise The contours shown are for 68.3% (1σ) and 95.4% (2σ) CL. Results for WMAP (5-year analysis), Planck and CMB probes with direct detection and remove fore- CMBPol are compared (coloured). Large-field and small- ground contamination with in-depth study of galactic field regimes are distinguished at r = 0.01. Credit © emissions, there is a hopeful prospect that we may ob- CMBPol Study Team. serve primordial gravitational waves in the foreseeable future. The conrmed observation would not only vindicate the ination theory, but also open up a path to 11 Direct detection — The CMB has provided a powerful uncharted territories in fundamental physics at ultra- window into the very early universe, yet it alone is still high energies. The features of the signal can tell us fur- not sucient. One obstacle arises from the tensor spec- ther about non-Gaussianities or even parity-violation tral index nt, which is cosmic-variance limited (e.g. it of Nature. On the other hand, non-detection could receives residual signals from gravitational lensing); its still constrain tensor perturbations, ruling out many determination also requires a measurement of the GW 12Acronym for the Deci-Hertz Interferometer Gravitational 11The following discussions are mostly based on [19, 38]. Wave Observatory. 21 large-eld inationary models. [10] U. Seljak and M. Zaldarriaga, “Signature of gravity waves in polarization of the microwave background,” History is often viewed in retrospect; we believe the Phys. Rev. Lett. 78 (1997) 2054–2057, Golden Age of Cosmology is still ahead of us. arXiv:astro-ph/9609169 [astro-ph]. [11] J. Lizarraga, J. Urrestilla, D. Daverio, M. Hindmarsh, M. Kunz, and A. R. Liddle, “Constraining topological Acknowledgement defects with temperature and polarization anisotropies,” Phys. Rev. D90 no. 10, (2014) 103504, arXiv:1408.4126 [astro-ph.CO]. The author thanks Anthony Challinor at the Depart- [12] K. Jones-Smith, L. M. Krauss, and H. Mathur, “A ment of Applied Mathematics and Theoretical Physics Nearly Scale Invariant Spectrum of Gravitational and the Institute of Astronomy, University of Cam- Radiation from Global Phase Transitions,” Phys. Rev. bridge for insightful discussions and valuable guid- Lett. 100 (2008) 131302, arXiv:0712.0778 ance. [astro-ph]. [13] R. Durrer, D. G. Figueroa, and M. 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Appendices

We are reminded that in these appendices the convention adopted in § II.3 is in place: an overdot denotes a conformal time η-derivative.

A. Second Order Action for Gravitational Waves A.1 Statement of the problem and set-ups Problem — The action governing gravitational waves is the second order expansion of the full action

S = SEH + Sϕ (109) where the Einstein–Hilbert action is

M2 Z √ S = Pl d4x −gR (110) EH 2 and the matter action is Z 4 √ Sϕ = d x −gLϕ (111) with the scalar-eld Lagrangian 1 L = − ∂ ϕ∂µϕ − V (ϕ). (112) ϕ 2 µ

Set-up — The perturbed at FLRW metric gµν is conformally equivalent to the perturbed Minkowski metric

2 2 gµν = a(η) g¯µν = a(η) ηµν + hµν (113)

µν ¯ µν where the pure tensor perturbation hµν is spatial h0µ ≡ 0, traceless η hµν = 0 and transverse ∂µ h = 0. We will temporarily set MPl = 1. The pre-superscript denotes the order of the quantity. Barred quantities are associated 0 with the perturbed Minkowski metric, and unbarred associated with the perturbed FLRW metric, e.g. ∂¯ = −∂η 0 −2 but ∂ = −a ∂η.

A.2 Preliminary results

In deriving these results, we bear in mind that hµν is symmetric, purely-spatial, traceless and transverse; its indices are raised with the unperturbed Minkowski metric.

Result 1. The perturbed inverse Minkowski metric is g¯ab = ηab − hab + O h2 , (114) and the perturbed inverse FLRW metric is gab = a−2g¯ab .

2 Result 2. For conformally equivalent metrices g˜ab = Ω gab , the associated Ricci scalars are related by ˜ −2 2 a R = Ω R − 6∇ ln Ω − 6∇a ln Ω∇ ln Ω . (115)

Proof. Since g˜ab = Ω−2gab , direct computation gives

1 Γ˜ a = g˜ad ∂ g˜ + ∂ g˜ − ∂ g˜ bc 2 b cd c bd d bc 1 = Γa + Ω−2gad g˜ ∂ Ω + g˜ ∂ Ω − g˜ Ω∂ bc 2 cd b bd c bc d ˜ a a a a = Γ bc + δc ∂b ln Ω + δb ∂c ln Ω − gbc ∇ ln Ω. 24

Hence

˜ ˜c ˜c ˜c ˜d ˜c ˜d Rab = ∂c Γ ab + ∂b Γ ac + Γ ab Γ cd − Γ ad Γ bc c = Rab − ∂c gab ∇ ln Ω − 2∂a ∂b ln Ω + 2∂a ln Ω∂b ln Ω c c + 2Γ ab ∂c ln Ω − 2gab ∂c ln Ω∂ ln Ω d c c d c d − gab Γ cd ∇ ln Ω + gbc Γ ad ∇ ln Ω + gac Γ bd ∇ ln Ω.

But

c c d c d c c d ∂c gab ∇ ln Ω = ∇c gab ∇ ln Ω + gbd Γ ac ∇ ln Ω + gad Γ bc ∇ ln Ω − gab Γ cd ∇ ln Ω c c ∂a ∂b ln Ω = ∇a∇b ln Ω + Γ ab ∂ ln Ω so we have ˜ c 2 Rab = Rab − 2∇a∇b ln Ω + 2∇a ln Ω∇b ln Ω − 2gab ∇c ln Ω∇ ln Ω − gab ∇ ln Ω. Therefore

˜ −2 ab ˜ R = Ω g Rab −2 2 a = Ω R − 6∇ ln Ω − 6∇a ln Ω∇ ln Ω .

Result 3. Using the formula

ϵ2 det(X + ϵA) = det X det(I + ϵB) = det X 1 + ϵ tr B + (tr B)2 − tr B2 + O ϵ3 , B ≡ X −1A, 2 * + we have the perturbed FLRW metric determinant, up to second order -

(0)g = −a8, (1) (0) (0) µν 2 (0) −2 µν 2 g = g tr g a hν ρ = g tr a η a hν ρ = 0, 8 (2) 1 (0) 2 (0) µν 2 (0) ρσ 2 a µν g = g 0 − tr g a hν ρ g a hσ λ = hµν h . 2 2 p √ p √ − − −1 − 1 −1 − 1 −2 2 O 3 Thus using binomial expansion (g + δg) = g 1 + g δg = g 1 + 2g δg 8g δg + δg , √ (0) −g = a4, (116) √ 1 √ (0) (1) −g = (0) −g g−1 (1)g = 0, (117) 2 4 √ 1 √ (0) a (2) −g = (0) −g g−1 (2)g = − h hµν . (118) 2 4 µν

Result 4. The perturbed Minkowski metric Christoel symbols up to second order are

1 Γ¯0 = h˙ , ij 2 ij 1 Γ¯i = h˙i − hikh˙ + O h3 , (119) j0 2 j k j 1 Γ¯i = ∂¯ hi + ∂¯ hi − ∂¯ih − hil ∂¯ h − hil ∂¯ h + hil ∂¯ h + O h3 jk 2 j k k j jk j lk k l j l jk and all others are identically zero to all orders. 25

Result 5. We compute the following quantities up to second order 1 1 g¯µν ∂¯ Γ¯ρ = − hijh¨ + hjk ∂¯i ∂¯ h , ρ µν 2 ij 2 i jk 1 1 1 −g¯µν ∂¯ Γ¯ρ = − ∂ hij ∂ h + h¨ + ∂¯k hij ∂¯ h , ν µρ 2 η η ij 2 ij 2 k ij µν ¯ρ ¯σ g¯ Γ µν Γ ρσ = 0, 1 1 −g¯µν Γ¯σ Γ¯ρ = − h˙ h˙ij + ∂¯ h ∂¯ihjk µρ ν σ 4 ij 4 i jk which add up to give the perturbed Minkowski metric Ricci scalar up to second order 1 3 R¯ = −hijh¨ − ∂¯ h ∂¯ihjk − h˙ h˙ij (120) ij 4 i jk 4 ij Result 6. Combining Result 2 using the conformal factor Ω = a, Result 4 and Result 5, we extract the perturbed FLRW metric Ricci scalar up to second order 6 6 (0)R = − −∂2 ln a − a−2a˙2 = (H 0 + H ), (121) a2 η a2 (1)R = 0, (122) ! 1 3 (2) (2)R = −a−2 hijh¨ + ∂¯ h ∂¯ihjk + h˙ h˙ij − 6a−2 Γ¯i ∂¯0 ln a ij 4 i jk 4 ij i0 ! 1 3 = −a−2 hijh¨ + ∂¯ h ∂¯ihjk + h˙ h˙ij − 3a−2Hhijh˙ . (123) ij 4 i jk 4 ij ij

A.3 Full calculations Einstein–Hilbert action — We need to calculate the second order quantity which by Eqn. (117) is just

(2) √ √ √ −gR = (0) −g (2)R + (2) −g (0)R .

By Eqns. (116) and (123) ! √ 1 3 (0) −g (2)R = −a2 hijh¨ + ∂¯ h ∂¯ihjk + h˙ h˙ij − 3a2Hhijh˙ . ij 4 i jk 4 ij ij

Integrating the rst term by parts, Z Z Z 4 2 ij ¨ 4 2 ij ˙ 4 2 ˙ij ˙ − d x a h hij = d x 2a Hh hij + d x a h hij , we nd the integral Z Z Z Z √ 1 1 d4x (0) −g (2)R = d4x a2h˙ijh˙ − d4x a2∂¯ h ∂¯ihjk − d4x a2Hhijh˙ . 4 ij 4 i jk ij Integrate the last term by parts Z Z 1 − d4x a2Hhijh˙ = d4x ∂ a2H hijh ij 2 η ij Z 1 = d4x a2 H˙ + 2H 2 . 2 Next we have from Eqns. (118) and (121) Z √ 3 (2) −g (0)R = − d4x a2(H˙ + H 2)h hij . 2 ij 26

We now arrive at Z Z Z 1 1 1 (2)S = d4x a2h˙ h˙ij − d4x a2∂¯ h ∂¯ihjk − d4x a2 H 2 + 2H˙ h hij . (124) EH 8 ij 8 i jk 4 ij

Matter action — We need to calculate the second order quantity which by Eqn. (117) is just

(2) √ (2) √ (0) (0) √ (2) ( −gLϕ ) = −g Lϕ + −g Lϕ .

Since ϕ ≡ ϕ(t),

(2) 1 1 a−4 (2) L ≡ − gµν ∂ ϕ∂ ϕ − V (ϕ) = − (2)gµν ∂ ϕ∂ ϕ = − hikhj ∂ ϕ∂ ϕ = 0 ϕ " 2 µ ν # 2 µ ν 2 k i j and we have from Eqn. (118)

√ √ a4 1 (2) ( −gL ) = (2) −g (0) L = − h hij a−2ϕ˙2 − V (ϕ) . ϕ ϕ 4 ij "2 #

But by the Friedman equations (3) written in terms of conformal time

3H 2 = a2ρ, −6H˙ = a2(ρ + 3P) where ρ = ϕ˙2/(2a2) + V and P = ϕ˙2/(2a2) − V , we have

√ √ a4 a2 (2) ( −gL ) = (2) −g (0) L = − h hij P = h hij H 2 + 2H˙ . ϕ ϕ 4 ij 4 ij

Therefore we arrive at Z 1 (2)S = d4x a2 H 2 + 2H˙ h hij . (125) ϕ 4 ij

Finally, adding the two sectors [Eqns. (124) and (125)] together and restoring MPl, we obtain

M2 Z (2)S = Pl dη d3x a2 h˙ h˙ij − ∂¯ h ∂¯ihjk . (126) 8 ij i jk

This is precisely Eqn. (36) where the derivatives are taken with respect to the Minkowski metric.

A.4 Extension In general, the energy-momentum tensor for a single scalar eld is

µν µ ν µν T = ∂ ϕ∂ ϕ + g Lϕ (127)

(0) so Eqn. (1) gives P = Lϕ as we saw above. Equivalently the matter action (111) can be recast as [1] Z 4 √ Lϕ = d x −gP (X,ϕ) (128)

µν where pressure P is a function of both the inaton eld and the kinetic term X ≡ −g ∂µ ϕ∂ν ϕ/2. Proceeding as above with the substitution of the Friedman equations yields the same second order action for gravitational waves in FLRW background spacetime, but now valid for more general single-eld ination minimally coupled to gravity where the kinetic terms include only rst derivatives. 27

B. Evolution of Gravitational Waves

The non-zero Christoel symbols for the perturbed metric (29) are, to rst order,

0 Γ 00 = H , 1 Γ0 = H δ + h + h˙ , ij ij ij 2 ij 1 (129) Γi = Hδ i + h˙i , j0 j 2 j 1 Γi = ∂ hi + ∂ hi − ∂ih . jk 2 j k k j jk µν Observing that the Ricci scalar R = g Rµν does not receive linear-order contributions from tensor perturbations as hij is transverse, we have 6 R = H˙ + H 2 a2 equal to its value for the unperturbed FLRW metric. For convenience, we rst compute the following results

ρ ρ Γ 0ρ = 4H , Γ iρ = 0, so that the Ricci tensor is

0 k ρ 0 ρ 0 k k 0 k l Rij = ∂0Γ ij + ∂k Γ ij − ∂j Γ iρ + Γ ij Γ 0ρ − Γ ik Γ j0 − Γ i0 Γ jk − Γ il Γ jk 1 = H˙ + 2H 2 δ + h + h¨ − ∇2h + 2Hh˙ + (H 0 + 2H 2)h ij ij 2 ij ij ij ij and the Einstein tensor is 1 1 G ≡ R − g R = − 2H˙ + H 2 δ + h¨ − ∇2h + 2Hh˙ − 2H˙ + H 2 h (130) ij ij 2 ij ij 2 ij ij ij ij | {z } =δGi j all at the linear order. On the other hand, we must also calculate the tensor perturbation to the energy-momentum contribution, which for a perfect uid is 2 Tµν = (ρ + P)UµUν + Pgµν + a Πij µ where ρ,P, and Πij are the density, pressure and anisotropic stress of the uid, and U its 4-velocity (a rst order quantity). Substitution of the perturbations ρ = ρ¯ + δρ, P = P¯ + δP into the energy-momentum tensor above yields the rst order perturbation 2 2 δTij = a Ph¯ ij + a Πij . (131) Applying the Einstein eld equation to quantities (130) and (131), employing the background Friedman equation H˙ H 2 − −2 2 ¯ [see Eqns. (3) and (4)] 2 + = MPl a P and ignoring anisotropic stress Πij , we obtain the ij-component of the Einstein equation for the evolution of gravitational waves

2 h¨ij + 2Hh˙ij − ∇ hij = 0 which in Fourier space is precisely Eqn. (45) for each polarisation state. Example 1 (Evolution in the matter-dominated era). During matter domination, H = 2/η so the equation above becomes 4 h¨(p) + h˙(p) + k2h(p) = 0 (132) η which we recognise as a (spherical) Bessel equation after making the change of variables x ≡ kη, h ≡ f (x)/x,

d2 f df x2 + 2x + (x2 − 2)f = 0 dx2 dx 28

so the solution is simply the spherical Bessel function f (x) = j1(x). Thus the solution is j (kη) h(p) (η,k) = 3h(p) (k) 1 . (133) kη Either by dierentiating Eqn. (132) above to obtain a new Bessel equation, or using the Bessel function property d j (x) = −(−x)n (−x)−n j (x), n+1 dx n we obtain the gravitational wave shear j (kη) h˙(p) (η,k) = −3h(p) (k) 2 . η Example 2 (Asymptotic features for general a(η)). Outside the Hubble horizon k H , it is straight-forward to see h(±2) = const. is the growing solution for Eqn. (45). Since we can rewrite Eqn. (45) as ! a¨ ∂2 ah(±2) + k2 − ah(±2) = 0, (134) η a for modes well inside the Hubble horizon k H , we can neglect a¨/a k2, thus obtaining eikη h(±2) ∝ . a This is interpreted as the usual gravitational waves in at Minkowski spacetime, but with a damping factor a: the −2 D ij E −4 radiation nature of gravitons means the energy density averaged over many oscillations a h¨ijh¨ ∝ a , so that −1 we have adiabatic decay hij ∝ a .

C. Rotations via the Wigner D-Matrix

A 3-dimensional rotation can be completely specied by the Euler angles (α,β,γ ). For a general rotation of the spherical harmonics, we act with the operator ˆ ˆ ˆ Dˆ (α,β,γ ) = e−iα Lz e−iβLy e−iγ Lz

ˆ ˆ P ` 0 where Li are the angular momentum operators (generators of the rotation), so that DY`m = m0 Dm0mY`m is ` an expansion in the basis of the spherical harmonics. The coecients Dm0m are the components of the Wigner D-matrices. The rotation operators are unitary, † X `∗ ` ˆ ˆ 0 D D = I =⇒ DnmDnm0 = δmm , n and under an active rotation, the spherical multipole coecients of a random eld on a sphere transform as Z ∗ X ˆ 0 0 0 0 f`m −→ dnˆ Y`mD f` m Y` m `0,m0 Z ∗ X `0 0 0 0 0 = dnˆ Y`m f` m Dn0m0Y` n `0,m0,n0 X ` = Dmm0 f`m0 m0 by the orthogonality relation of the spherical harmonics Z ∗ 0 0 dnˆ Y`mY`0m0 = δ`` δmm . An explicit result for the Wigner D-matrix is that r 4π D` (ϕ,θ,0) = Y ∗ (nˆ ) m 0 2` + 1 `m p which could be derived from Y`m (zˆ) = (2` + 1)/(4π )δm0 [6]. 29

D. The Geodesic Equation for Tensor Perturbations

The rst order metric connections can be found in Eqn. (129) in Appendix B. With the tetrad given in Eqn. (56), we have ! dx µ ϵ 1 = pµ = 1,eıˆ − hi e ˆ dλ a2 2 j where λ is the ane parameter, so that

dη ϵ dxi 1 = , = eıˆ − hi e ˆ dλ a2 dη 2 j to linear order. The time-component of the geodesic equation

ϵ dpµ + Γµ pνpρ = 0 a2 dη ν ρ is then 2 ! ! ! ϵ d ϵ ϵ 1 ıˆ 1 i kˆ ˆ 1 j lˆ + H + H δij + hij + h˙ij e − h e e − h e = 0 a2 dη a2 a2  2 2 k 2 l    from which the linear-order terms can be extracted    ! d ln ϵ 1 + Hh + h˙ eıˆe ˆ − Hh eıˆe ˆ = 0 dη ij 2 ij ij which is precisely Eqn. (57).

E. Calculation of the Gravitational Shear Contribution to Temperature Anisotropies

The calculations in this appendix primarily follows that in [6]. To derive Eqn. (60) from Eqn. (59) modulated by a plane wave due to the evolution of gravitational waves, we rst use the Rayleigh plane-wave expansion

−iχk·e X L e = (−i) (2L + 1)jL (kχ)PL (cosθ ) (135) L>0 p and PL (cosθ ) = 4π/(2L + 1)YL 0(e) to rewrite it as r √ 4π (±2) −ikχ cos θ 4π (±2) X L h˙ (η,kzˆ)Y ± (e) e = √ h˙ (η,kzˆ) (−i) 2L + 1j (kχ)Y ± (e)Y (e). 15 2 2 L 2 2 L 0 15 L>0

The product of two spherical harmonics can be expanded in the basis of spherical harmonics with Wigner 3j-symbol valued coecients r X 5(2L + 1)(2` + 1) 2 L ` 2 L ` ∗ Y ± (e)Y (e) = Y (e) 2 2 L 0 4π ±2 0 m 0 0 0 `m `,m * + * + but for temperature anisotropies, m = ±2 and thus ` > 2, so , - , - r 4π (±2) −ikχ cos θ h˙ (η,kzˆ)Y ± (e) e = 15 2 2 r √ 4π (±2) X L X 2 L ` 2 L ` h˙ (η,kzˆ) (−i) (2L + 1)j (kχ) 2` + 1 Y` ± (e) 3 L ∓2 0 ±2 0 0 0 2 L>0 `>2 * + * + ∗ − ±2 where we have used the reality condition Y` ±2 = ( 1) Y` ∓2. , - , - 30

We shall concentrate on the summation over L, i.e. the expression

X 2 L ` 2 L ` (−i)L (2L + 1)j (kχ) . L ∓2 0 ±2 0 0 0 L>0 * + * + 2 L ` , - , - The Wigner 3j-symbol is non-vanishing only for L + ` ∈ 2Z by the time-reversal formula 0 0 0 * + a b c a b c , - = (−1)a+b+c , (136) −d −e −f d e f * + * + `,` ± − hence the summation in L is only over,L = 2 by- triangle inequality, a -b 6 c 6 a +b of the Wigner 3j-symbol a b c . We receive the following contributions d e f * + r s , - `+2 2 ` + 2 ` 2 ` + 2 ` ` 3 (` + 2)! 1 A B (−i) (2` + 5)j`+ (x) = −(−i) j`+ (x) 2 ∓2 0 ±2 0 0 0 8 (` − 2)! (2` + 3)(2` + 1) 2 * + * + and similarly , - , - r s r s ` 3 (` + 2)! 1 ` 3 (` + 2)! 2 B B −(−i) j`− (x), C B −(−i) j` (x) 8 (` − 2)! (2` + 1)(2` − 1) 2 8 (` − 2)! (2` + 3)(2` − 1) where x ≡ kχ and we have used the following results r s 2 ` + 2 ` 2 ` + 2 ` 3 (` + 2)! 1 = , ±2 0 ∓2 0 0 0 8 (` − 2)! (2` + 5)(2` + 3)(2` + 1) * + * + r s 2 ` − 2 ` 2 ` − 2 ` 3 (` + 2)! 1 , - , - = , ±2 0 ∓2 0 0 0 8 (` − 2)! (2` + 1)(2` − 1)(2` − 3) * + * + r s 2 ` ` 2 ` ` 3 (` + 2)! 1 , - , - = − . ±2 0 ∓2 0 0 0 2 (` − 2)! (2` + 3)(2` + 1)(2` − 1) * + * + j (x) We utilise the recursion, relation for- , ` , - x j` = j` + j`− 2` + 1 +1 1 x x x = j` + j` + j` + j`− 2` + 1 2` + 3 +2 2` − 1 2 ! x2 x2 x2 1 1 = j` + j`− + + j` (137) (2` + 1)(2` + 3) +2 (2` − 1)(2` + 1) 2 2` + 1 2` + 3 2` − 1

2 so (2` + 3)(2` + 1)(2` − 1)j` = x (2` − 1)j`+2 + (2` + 3)j`−2 + (4` + 2)j` . Therefore the total contribution of A, B and C gives s r √ (±2) ıˆ ˆ −ikχ cos θ π (±2) X ` (` + 2)! j` (kχ) h˙ (η,kzˆ)e e e = − h˙ (η,kzˆ) (−i) 2` + 1 Y` ± (e). ij 2 (` − 2)! (kχ)2 2 `>2

F. Properties of Spin Spherical Harmonics

We list below a number of properties of the spin spherical harmonics: Orthogonality relation Z ∗ 0 0 dnˆ s Y`m s Y`0m0 = δ`` δmm ; (138) 31

Conjugation relation ∗ s+m s Y`m = (−1) −s Y` −m ; (139)

Parity relation ` s Y`m (−nˆ ) = (−1) −s Y`m (nˆ ); (140)

Wigner D-matrix relation r 4π D` (ϕ,θ,0) = (−1)m Y (θ,ϕ); (141) −m s 2` + 1 s `m

Product formula r X ( ` + )( ` + )( L + ) `1+`2+L 2 1 1 2 2 1 2 1 `1 `2 L `1 `2 L s Y` m s Y` m = (−1) SLMY . (142) 1 1 1 2 2 2 4π m1 m2 M s1 s2 S L,M,S * + * + , - , - G. Tensor Calculus on the 2-Sphere

This mathematical introduction to tensor calculus on the 2-sphere may be found in literatures such as [6]. In a basis of complex null vectors on the 2-sphere, we decompose the metric tensor and the alternating tensor as

1 i g = (m m− + m− m ), ε = (m m− − m− m ). (143) ab 2 +a b a +b ab 2 +a b a +b

The action of the alternating tensor on the null basis vectors is a rotation by π/2:

b εa m±b = ±im±a . (144)

a We can express m± in the spherical polar coordinate basis as Eqn. (72). They satisfy the parallel transport equations a b b a b b m+∇am± = ± cotθm±, m−∇am± = ∓ cotθm±. (145)

To derive Eqn. (76) from Eqns. (74) and (75), we write

a b c Q ± iU = m±m± ∇ha∇biPE + ε (a ∇b)∇c PB a b = m±m±∇a∇b (PE ± iPB ) a 2 a b = m±∇a − m± ∇am± ∇b (PE ± iPB ) | {z } b =cot θm± 2 = (∂θ ± i cosecθ∂ϕ ) − cotθ (∂θ ± i cosecθ∂ϕ ) (PE ± iPB ) f −1 g = sinθ (∂θ ± i cosecθ∂ϕ ) (sinθ ) (∂θ ± i cosecθ∂ϕ )(PE ± iPB ) f g which we recognise by Eqn. (70) as

Q + iU = ð ð(PE + iPB ), Q − iU = ð¯ ð¯(PE − iPB ). 32

H. Calculation of the Gravitational Shear Contribution to Temperature Anisotropies

The calculations in this appendix primarily follows that in [6]. To derive Eqn. (83) from Eqn. (82), we note only m = p summands survive in Eqn. (82). Employing the Rayleigh plane-wave expansion [cf. Eqn. (135)]

(Q ± iU )(η0,kzˆ,e)

−ikχ∗ cos θ ∝ e ±2Y 2p (e) X p L = ±2Y 2p (e) 4π (2L + 1)(−i) jL (kχ∗)YL 0(e) L r X p X 5(2L + 1)(2` + 1) 2 L ` 2 L ` = 4π (2L + 1)(−i)Lj (kχ ) (−1)2+L+` Y ∗ (e) L ∗ 4π p 0 m ±2 0 s s `m L `,m,s * + * + √ X X √ 2 L ` 2 L ` = 5 iL (2L + 1)j (kχ ) (−1)` 2` + 1(−1)2+L+` , - , (−1)−s−-m Y (e) L ∗ −p 0 m ±2 0 −s s `m L `,m,s * + * + √ X X √ 2 L ` 2 L ` = 5 (−i)L (2L + 1)j (kχ ) 2` + 1 , - , Y (e) - L ∗ −p 0 p ±2 0 ∓2 ±2 `p L ` * + * + where we have used in the second line the product formula, (142)-and, in the third- line the conjugation relation (139) for spin-weighted spherical harmonics (see Appendix F), as well as relabelled m,s → −m,−s and employed in the third line the time-reversal formula (136) for the Wigner 3j-symbols (see Appendix E). Furthermore, in the last line the only terms that survive in the m,s summations are m = p and s = ±2 summands (Q ± iU is spin ±2 and m = p as argued earlier). We follow a similar procedure to the one in Appendix E. The summation is over ` − 2 6 L 6 ` + 2 by the triangle inequality of the Wigner 3j-symbols, and we have the results below

2 ` + 2 ` 2 ` + 2 ` 1 `(` − 1) = , ∓2 0 ±2 ±2 0 ∓2 4 (2` + 1)(2` + 3)(2` + 5) * + * + 2 ` + 1 ` 2 ` + 1 ` ` − 1 , - , - = ∓ , ∓2 0 ±2 ±2 0 ∓2 2(2` + 1)(2` + 3) * + * + 2 ` ` 2 ` ` 3 (` − 1)(` + 2) , - , - = , ∓2 0 ±2 ±2 0 ∓2 2 (2` − 1)(2` + 1)(2` + 3) * + * + 2 ` − 1 ` 2 ` − 1 ` ` + 2 , - , - = ∓ , ∓2 0 ±2 ±2 0 ∓2 2(2` − 1)(2` + 1) * + * + 2 ` − 2 ` 2 ` − 2 ` 1 (` + 1)(` + 2) , - , - = . ∓2 0 ±2 ±2 0 ∓2 4 (2` − 3)(2` − 1)(2` + 1) * + * + 2 L ` 2 L ` Hence the sum P (−i)L (,2L + 1)j (x) - , - becomes L L −p 0 p ±2 0 ∓2 * + * + ` 1 `(` − 1) , 6(` −-1,)(` + 2) - (` + 1)(` + 2) (−i) − j + j − j − ` + ` + `+2 ` − ` + ` ` − ` + ` 2  4 " (2 1)(2 3) (2 1)(2 3) (2 1)(2 1) #  !  i ` − 1 ` + 2  + j − j .  ` `+1 ` `−1 2 2 + 1 2 + 1   Now the result (83) can be derived using the following recursion relations for the spherical Bessel functions:  j` (x) 1 0 = j`− (x) + j` (x) , (2` + 1)j (x) = `j`− (x) − (` + 1)j` (x) x 2` + 1 1 +1 ` 1 +1 from which we identify 1 2β` (x) = (` + 2)j`− − (` − 1)j` 2` + 1 1 +1 33 as well as 0 j` ` ` + 1 = (j`− + j` ) − (j` + j` ), x (2` + 1)(2` − 1) 2 (2` + 1)(2` + 3) +2 ! 00 ` ` − 1 ` ` + 1 j = j`− − j` − (` + 1)j` − (` + 2)j` ` 2` + 1 2` − 1 2 2` − 1 (2` + 1)(2` + 3) +2 and Eqn. (137), so that

`(` − 1) 6(` − 1)(` + 2) (` + 1)(` + 2) 4ϵ` (x) = j` − j` + j`− . (2` + 1)(2` + 3) +2 (2` − 1)(2` + 3) (2` − 1)(2` + 1) 2

I. Polarisation from Scalar Perturbations

We prove here that scalar perturbations do not generate B-mode polarisation. Expanding the Fourier transform of Q ± iU in normal modes,

X ` 4π ∗ (Q ± iU )(η,k,e) = (−i) (E` ± iB` )(η,k)Y (kˆ) Y (e), (146) 2` + 1 `m ±2 `m `,m we substitute this into the Boltzmann equation (79) for linear polarisation. If we take k = kzˆ then the result is a + b = c where the quantities to be examined separately are r X ` 4π ∗ X ` 4π a ≡ (−i) (E˙` ± iB˙` )Y (kˆ) Y (e) = (−i) (E˙` ± iB˙` )Y` (ˆe), 2` + 1 `m ±2 2m 2` + 1 0 `,m `

X ` 4π ∗ b ≡ (−i) ie · k(E` ± iB` )Y (kˆ) Y (e) 2` + 1 `m ±2 `m `,m s 2 2 2 X `+1 4π ∗ 1 [(` + 1) − m ][(` + 1) − 4] = − (−i) k(E` ± iB` )Y (kˆ) Y (e) 2` + 1 `m ` + 1 4(` + 1)2 − 1 ±2 `+1m `,m   r  2 2 2  2m 1 (` − m )(` − 4)  ∓ ±2Y`m (e) + ±2Y`−1m (e) `(` + 1) ` 4`2 − 1  r  2 2 2  X ` 1 ∗ 1 (` − m )(` − 4)  = −4πk (−i) (E`− ± iB`− )Y (kˆ) Y (e)  2` − 1 1 1 `−1m ` 4`2 − 1 ±2 `m `,m    i ∗ 2m  ± (E` ± iB` )Y (kˆ) Y (e)  2` + 1 `m `(` + 1) ±2 `m s 2 2 2 1 ∗ 1 [(` + 1) − m ][(` + 1) − 4] − (E` ± iB` )Y (kˆ) Y (e) 2` + 3 +1 +1 `+1m ` + 1 4(` + 1)2 − 1 ±2 `m  r r  √ 2 2  X ` 1 ` − 4 1 [(` + 1) − 4]  = − 4πk (−i) (E`− ± iB`− ) − (E` ± iB` ) Y (e)  2` − 1 2` + 1 1 1 2` + 3 2` + 1 +1 +1  ±2 ` 0 `     and  

X ` 4π ∗ 3 X 1 c ≡ (−i) τ˙ (E` ± iB` )Y (kˆ) Y (e) − τ˙ E − √ Θ Y (e) 2` + 1 `m ±2 `m 5 2m 2m ±2 2m `,m |m |62 6 r * + X ` 4π 3 X X 1 = (−i) τ˙ (E` ± iB` ) Y (e) − τ˙ δ` , E` − √ Θ`- Y (e). 2` + 1 ±2 ` 0 5 2 m m ±2 `m ` ` |m |62 6 * + , - 34

∗ p The following results have been used in the derivation above: 1) Y`m (zˆ) = (2` + 1)/(4π )δm0; 2) the recursion relation s r 1 [(` + 1)2 − m2][(` + 1)2 − s2] sm 1 (`2 − m2)(`2 − s2) cosθ Y = Y − Y + Y . s `m ` + 1 4(` + 1)2 − 1 s `+1m `(` + 1) s `m ` 4`2 − 1 s `−1m

We have now arrived at r r r X ` 2 − `2 − ` 4π ˙ ˙ ( + 1) 4 4 (−i) (E` ± iB` ) + k (E`+1 ± iB`+1) − (E`−1 ± iB`−1) ±2Y` 0 (e) 2` + 1  2` + 3 2` − 1  `     r   X ` 4π 3 X X 1  = (−i)  τ˙ (E` ± iB` ) Y (e) − τ˙ δ` E` − √ Θ` Y (e). (147) 2` + 1 ±2 ` 0 5 2 m m ±2 `m ` ` |m |62 6 * + Adding and subtracting this equation with dierent ± signs, we obtain , - r r (` + 1)2 − 4 `2 − 4 3 1 E˙ + k E − E = τ˙ E − δ E − √ Θ , ` `+1 − `−1 ` `2 2 2  2` + 3 2` 1   5 6     * + r r     (` + 1)2 − 4 `2 − 4   , - B˙` + k  B` − B`−  = τ˙B`.  2` + 3 +1 2` − 1 1      