# Astro 321 Set 4: Inflation. Wayne Hu

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1 Astro 321 Set 4: Inflation Wayne Hu

2 Horizon Problem The horizon in a decelerating universe scales as η a (1+3w)/2, w > 1/3. For example in a matter dominated universe η a 1/2 CMB decoupled at a = 10 3 so subtends an angle on the sky η = a 1/2 η So why is the CMB sky isotropic to 10 5 in temperature if it is composed of 10 4 causally disconnected regions If smooth by fiat, why are there 10 5 fluctuations correlated on superhorizon scales

3 Flatness & Relic Problems Flatness problem: why is the radius of curvature larger than the observable universe. (Before the CMB determinations, why is it at least comparable to observable universe Ω K < Ω m ) Also phrased as a coincidence problem: since ρ K a 2 and ρ m a 3, why would they be comparable today modern version is dark energy coincidence ρ Λ = const. Relic problem why don t relics like monopoles dominate the energy density Inflation is a theory that solves all three problems at once and also supplies a source for density perturbations

4 Accelerating Expansion In a matter or radiation dominated universe, the horizon grows as a power law in a so that there is no way to establish causal contact on a scale longer than the inverse Hubble length (1/aH, comoving coordinates) at any given time: general for a decelerating universe 1 η = d ln a ah(a) H 2 ρ a 3(1+w), ah a (1+3w)/2, critical value of w = 1/3, the division between acceleration and deceleration In an accelerating universe, the Hubble length shrinks in comoving coordinates and so the horizon gets its contribution at the earliest times, e.g. in a cosmological constant universe, the horizon saturates to a constant value

5 Causal Contact Note confusion in nomenclature: the true horizon always grows meaning that one always sees more and more of the universe. The Hubble length decreases: the difference in conformal time, the distance a photon can travel between two epochs denoted by the scale factor decreases. Regions that were in causal contact, leave causal contact. Horizon problem solved if the universe was in an acceleration phase up to η i and the conformal time since then is shorter than the total conformal age η 0 η 0 η i total distance distance traveled since inflation apparent horizon

6 Flatness & Relic Comoving radius of curvature is constant and can even be small compared to the full horizon R η 0 yet still η 0 R η 0 η i In physical coordinates, the rapid expansion of the universe makes the current observable universe much smaller than the curvature scale Likewise, the number density of relics formed before the accelerating (or inflationary) epoch is diluted to make them rare in the current observable volume Common to reference time to the end of inflation η η η i. Here conformal time is negative during inflation and its value (as a difference in conformal time) reflects the comoving Hubble length - defines leaving the horizon as k η = 1

7 Exponential Expansion If the accelerating component has equation of state w = 1, ρ = const., H = H i const. so that a exp(ht) ai η = d ln a 1 a ah = 1 ai ah i a 1 (a i a) ah i In particular, the current horizon scale H 0 η 0 1 exited the horizon during inflation at η 0 H 1 0 = 1 a H H i a H = H 0 H i

8 Sufficient Inflation Current horizon scale must have exited the horizon during inflation so that the start of inflation could not be after a H. How long before the end of inflation must it have began? a H = H 0 a i H i a i H 0 ρc =, H i ρ i a i = T CMB T i ρ 1/4 c = GeV, T CMB = GeV ( ) a H = ρ 1/4 2 ( i T i a i GeV GeV ln a i a H = ln ( ρ 1/4 i GeV ) ( ln ) T i GeV )

9 Perturbation Generation Horizon scale η during inflation acts like an even horizon - things leaving causal contact Particle creation similar to Hawking radiation from a black hole with hubble length replacing the BH horizon T H H i Because H i remains roughly constant during inflation the result is a scale invariant spectrum of fluctuations due to zero-point fluctuations becoming classical Fluctuations in the field driving inflation (inflaton) carry the energy density of the universe and so their zero point fluctuations are net energy density or curvature fluctuations Any other light field (gravitational waves, etc...) will also carry scale invariant perturbations but are iso-curvature fluctuations

10 Scalar Fields Stress-energy tensor of a scalar field T µ ν = µ ϕ ν ϕ 1 2 ( α ϕ α ϕ + 2V )δ µ ν. For the background φ φ 0 (a 2 from conformal time) ρ φ = 1 2 a 2 φ2 0 + V, p φ = 1 2 a 2 φ2 0 V So for kinetic dominated w φ = p φ /ρ φ 1 And potential dominated w φ = p φ /ρ φ 1 A slowly rolling (potential dominated) scalar field can accelerate the expansion and so solve the horizon problem or act as a dark energy candidate

11 Equation of Motion Can use general equations of motion of dictated by stress energy conservation ρ φ = 3(ρ φ + p φ )ȧ a, to obtain the equation of motion of the background field φ φ 0 + 2ȧ a φ 0 + a 2 V = 0, In terms of time instead of conformal time d 2 φ 0 dt + 3H dφ 0 2 dt + V = 0 Field rolls down potential hill but experiences Hubble friction to create slow roll. In slow roll 3Hdφ 0 /dt V and so kinetic energy is determined by field position adiabatic both kinetic and potential energy determined by single degree of freedom φ 0

12 Slow Roll Inflation Alternately can derive directly from the Klein-Gordon equation for scalar field Scalar field equation of motion V dv/dφ µ µ φ + V (φ) = 0 so that in the background φ = φ 0 and φ 0 + 2ȧ a φ 0 + a 2 V = 0 d 2 φ 0 dt + 3H dφ 0 2 dt + V = 0 Simply the continuity equation with the associations ρ φ = 1 2 a 2 φ2 0 + V p φ = 1 2 a 2 φ2 0 V

13 Slow Roll Parameters Net energy is dominated by potential energy and so acts like a cosmological constant w 1 First slow roll parameter ɛ = 3 1 (1 + w) = 2 16πG ( V V ) 2 Second slow roll parameter d 2 φ 0 /dt 2 0, or φ 0 (ȧ/a) φ 0 δ = φ (ȧ ) = ɛ 1 φ 0 a 8πG Slow roll condition ɛ, δ 1 corresponds to a very flat potential V V

14 Perturbations Linearize perturbation φ = φ 0 + φ 1 then φ 1 + 2ȧ a φ 1 + k 2 φ 1 + a 2 V φ 1 = 0 in slow roll inflation V term negligible Implicitly assume that the spatial metric fluctuations (curvature R) vanishes, otherwise covariant derivatives pick these up GR: work in the spatially flat slicing and transform back to comoving slicing once done. Curvature is local scale factor a (1 + R)a or δa/a = R R = δa a = ȧ a δη = ȧ a a change in the field value φ 1 defines a change in the epoch that inflation ends, imprinting a curvature fluctuation φ 1 φ 0

15 Slow-Roll Evolution Rewrite in u aφ to remove expansion damping (ȧ ) 2 ü + [k 2 2 ]u = 0 a or for conformal time measured from the end of inflation η = η η end η = a a end Compact, slow-roll equation: da Ha 1 2 ah ü + [k 2 2 η 2 ]u = 0

16 Slow Roll Limit Slow roll equation has the exact solution: u = A(k ± ĩ η )e ik η For k η 1 (early times, inside Hubble length) behaves as free oscillator lim u = Ake ik η k η Normalization A will be set by origin in quantum fluctuations of free field

17 Slow Roll Limit For k η 1 (late times, Hubble length) fluctuation freezes in Slow roll replacement (ȧ ) 2 1 = 8πGa2 V a 3 φ 2 0 lim u = ± ĩ k η 0 η A = ±ihaa φ 1 = ±iha (ȧ R = iha a) 1 φ 0 Comoving curvature power spectrum 3 2a 2 V ɛ = 4πG ɛ = 1 2ɛM 2 pl 2 R k3 R 2 2π 2 = k3 H 2 A 2 4π 2 ɛmpl 2

18 Quantum Fluctuations Simple harmonic oscillator Hubble length ü + k 2 u = 0 Quantize the simple harmonic oscillator û = u(k, η)â + u (k, η)â where u(k, η) satisfies classical equation of motion and the creation and annihilation operators satisfy [a, a ] = 1, a 0 = 0 Normalize wavefunction [û, dû/dη] = i u(k, η) = 1 2k e ik η

19 Quantum Fluctuations Zero point fluctuations of ground state u 2 = 0 u u 0 = 0 (u â + uâ)(uâ + u â ) 0 = 0 ââ 0 u(k, η) 2 = 0 [â, â ] + â â 0 u(k, η) 2 = u(k, η) 2 = 1 2k Classical equation of motion take this quantum fluctuation outside horizon where it freezes in. Slow roll equation So A = (2k 3 ) 1/2 and curvature power spectrum 2 R 1 8π 2 H 2 ɛm 2 pl

20 Tilt Curvature power spectrum is scale invariant to the extent that H is constant Scalar spectral index d ln 2 R d ln k n S 1 = 2 d ln H d ln k d ln ɛ d ln k Evaluate at horizon crossing where fluctuation freezes d ln H d ln k k η=1 = k H where ah = 1/ η = k dh d η d η k η=1 dk k η=1 = k H ( ah2 ɛ) k η=1 1 k 2 = ɛ

21 Tilt Evolution of ɛ d ln ɛ d ln k = d ln ɛ d ln η Tilt in the slow-roll approximation = 2(δ + ɛ)ȧ η = 2(δ + ɛ) a n S = 1 4ɛ 2δ

22 Gravitational Waves Gravitational wave amplitude satisfies Klein-Gordon equation (K = 0), same as scalar field ḧ +, + 2ȧ aḣ+, + k 2 h +, = 0. Acquires quantum fluctuations in same manner as φ. Lagrangian sets the normalization Scale-invariant gravitational wave amplitude 2 +, = 16πG 2 φ 1 = 16πG H2 (2π) 2 = H2 2π 2 M 2 pl

23 Gravitational Waves Gravitational wave power H 2 V E 4 i where E i is the energy scale of inflation Tensor-scalar ratio - various definitions - WMAP standard is Tensor tilt: r R = 16ɛ d ln 2 + d ln k n T = 2 d ln H d ln k = 2ɛ

24 Gravitational Waves Consistency relation between tensor-scalar ratio and tensor tilt r = 16ɛ = 8n T Measurement of scalar tilt and gravitational wave amplitude constrains inflationary model in the slow roll context Comparision of tensor-scalar ratio and tensor tilt tests the idea of slow roll itself

25 Gravitational Wave Phenomenology A gravitational wave makes a quadrupolar (transverse-traceless) distortion to metric Just like the scale factor or spatial curvature, a temporal variation in its amplitude leaves a residual temperature variation in CMB photons here anisotropic Before recombination, anisotropic variation is eliminated by scattering Gravitational wave temperature effect drops sharply at the horizon scale at recombination

26 Large Field Models For detectable gravitational waves, scalar field must roll by order M pl = (8πG) 1/2 dφ 0 dn = dφ 0 d ln a = dφ 0 1 dt H The larger ɛ is the more the field rolls in an e-fold ɛ = r 16 = 3 ( H dφ ) 2 0 = 8πG ( ) 2 dφ0 2V dn 2 dn Observable scales span N 5 so φ 0 5 dφ dn = 5(r/8)1/2 M pl 0.6(r/0.1) 1/2 M pl Does this make sense as an effective field theory? Lyth (1997)

27 Large Field Models Large field models include monomial potentials V (φ) = Aφ n ɛ δ ɛ n2 16πGφ 2 n(n 1) 8πGφ 2 Slow roll requires large field values of φ > M pl Thus ɛ δ and a finite tilt indicates finite ɛ Given WMAP tilt, potentially observable gravitational waves

28 Small Field Models If the field is near an maximum of the potential V (φ) = V µ2 φ 2 Inflation occurs if the V 0 term dominates ɛ 1 16πG µ 4 φ 2 V 2 0 δ ɛ + 1 8πG µ 2 V 0 δ ɛ = V 0 µ 2 φ 2 1 Tilt reflects δ: n S 1 2δ and ɛ is much smaller The field does not roll significantly during inflation and gravitational waves are negligible

29 Hybrid Models If the field is rolling toward a minimum of the potential V (φ) = V m2 φ 2 Slow roll parameters similar to small field but a real m 2 ɛ 1 16πG δ ɛ 1 8πG m 4 φ 2 V 2 0 m 2 V 0 Then V 0 domination ɛ, δ < 0 and n S > 1 - blue tilt For m 2 domination, monomial-like. Intermediate cases with intermediate predictions - can have observable gravity waves but does not require it.

30 Hybrid Models But how does inflation end? V 0 remains as field settles to minimum Implemented as multiple field model with V 0 supplied by second field Inflation ends when rolling triggers motion in the second field to the true joint minimum

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