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Primordial Gravitational Waves from Cosmic Inflation

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Primordial Gravitational Waves from Cosmic Inflation Mathematical Tripos Part III Essay #75 (colour in electronic version) Submied 5th May 2017, updated 26th August 2017 Primordial Gravitational Waves from Cosmic Inflation 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 primor- dial gravitational waves resulting from tensor pertur- bations 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 preci- sion 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 mech- anism 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 whichp = 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 dur- ing 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, jV j 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 : 2 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 p 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 − rϕ − 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 rf 2 f S = dτ d x a − − − a V;ϕϕ 2 2 a a a a 3 Z 6 7 1 6 2 0 7 = dτ d3x f602 − rf − H (f 2) + (H 2 − a2V )7f 2 2 4 ;ϕϕ 5 Z 1 3 02 2 0 2 2 2 = dτ d x f − rf + H + H − a V;ϕϕ f 2 Z 00 ! 1 2 a = dτ d3x f 02 − rf + − a2V f 2 : 2 a ;ϕϕ 62 73 6 7 46 57 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 − rf + 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 − r2 f − f 2 = 0: (12) a Z 00 ( ) 1 02 2 a 2 S ≈ dτ d3x f − rf + 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 ∗ y − −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∗ay e−ik·x + f 0a eik·x 2π 2 2k " (kτ )2 # 3=2 k k k k (2π ) !2 H k2 y = 1 + (21) where ak, ak are the time-independent annihilation and 2π a2H 2 creation operators for each mode satisfying !2* + H a ;ay = δ (k − k0); ! , - k k0 (15) 2π f g and fk satises Eqn.
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