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- 1. A Further DiscriminatorySignature of Inflation Laila Alabidi Yukawa Institute of Theoretical Physics talk presented at KEK 19th of March 2012
- 2. Based onvarious papers with D.H. Lyth (Lancaster U), JamesLidsey and Ian Huston (Queen Mary, University ofLondon).and most recently01203.???? (very soon we hope!)withKazunori Kohri (KEK), Misao Sasaki (YITP),YuuitiSendouda (Hirosaki U.)
- 3. CMB
- 4. Some parameter definitions Scale factor distance distance dt = a(t)d0 at time t at initial time a ˙ derivative w.r.t. Hubble parameter H= a time Conformal time dτ 2 = a2 dt2 a HConformal Hubble parameter H= a = a derivative w.r.t. conformal time
- 5. 0s 98 Inflation: brief review :1 ePr “Cosmological Principle”-- the universe has to be both homogeneous and isotropic, simply because it seemed to be! Original matter/radiation perturbations which sourced the evolution of structure were put in by hand.
- 6. Inflation: a brief 80s 19 review t: sPo Paradigm: the near exponential expansion of the universe at t=10-37s Effect: the comoving Hubble Horizon decreases. d 1 dt aH 0 Result: explanation of: Universal homogeneity, causality and isotropy. Origin of structure: allows for a quantum to classical transition of vacuum fluctuations.
- 7. The basic problem
- 8. A ll ‘m ac s! a i hie el od n’ v ai e t m m he yan s!M
- 9. Which is the correct model?
- 10. OverviewInflation: the parameters.Inflation models I.Observation: the bounds.Results I: from CMB and related data sets.Going beyond the CMB constraints: theinduced gravitational waves.Inflation models II.Results II: prospects from gravitational waveexperiments.
- 11. Inflation: the requirementsInflation requires an accelerated rate of change of expansion a0 ¨ recall the acceleration equation? energy density a ¨ 4πG =− (ρ + 3P ) a 3 pressure negative ρ is nonsensical, but negative pressure is not a scalar field! potential scalar field 1 ˙2 1 ˙2 ρ = φ + V (φ) P = φ − V (φ) 2 2 Just considering canonical kinetic terms for now!
- 12. Inflation: slow roll ¨ ˙ dV (φ) φ + 3H φ + =0 dφ and we can define 2 } 1 V = 2 V V η= 1 V V V ξ= V2 a ‘ is a derivative with respect to the “inflaton” field φ
- 13. Number of e-folds Time re-parametrisation A measure of how long inflation lasts V ae N = ln ai = Hdt V,φ dφStart time is the time that scales of cosmological interest leave the horizon
- 14. A brief introduction to perturbations
- 15. background φ = φ0 + δφ perturbation quantity imprint themselves on the background (via the Einsteinequations) generating what we call the curvature perturbation ζ(φ) the characteristics and future evolution of which is determined by the model of inflation
- 16. Inflation: theobservational parameters
- 17. Spectrum and Spectral scale on whichIndex the ‘spectrum’perturbation is defined 2π 2 (3) ζ(k)ζ(k ) = k3 δ (k − k )Pζ (k) 1 V Pζ = 24π2 The ‘s pectral index’ defines the scale dependance of the spectrum P d ln ζ ns − 1 = d ln k ns = 1 + 2η − 6 The scale dependance of which is called the ‘running’ dns 2 2ns = d ln k ns = 16η − 24 − 2ξ
- 18. Gravitational wavesTraceless, transverse part of the perturbed spatial metric with a spectrum H 2 Pgrav = 8 2π The ratio to the scalar spectrum is defined as Pgrav Pζ ≡r In terms of slow roll parameters r = 16
- 19. Not really anobservable but...
- 20. Primordial Black HolesThe spectrum on small scales has not been ‘well’ measured If spectrum very large (0.03) then PBHs will form. Have an ‘upper’ bound due to astrophysical constraints Pζend 10 −2
- 21. PBH pre-requisitesFor an enhanced spectrum towards the end ofinflation: Pζe → 10−2Then a decreasing slow roll parameter is required: →0and a running of the spectral index: n 0 s
- 22. Classification of single field models “Small” field vs. “Large” field models ∆φ = φend − φ∗ “Observable” gravitational waves generated up to 4-efolds after horizon exit ∆φ ∼ 0.5Mpl r ∼ 0.1Gravitational waves on the order of 0.1 will be detectable soon!
- 23. Small field models
- 24. Tree level potential p p0, inflaton rolls away from the origin φ V = V0 1− µ Taylor expand about the vacuum, then assume one of the p’s dominates. Logarithmic potential 2 gsφ V = V0 1+ ln 2π Q p0 logarithmic and exponentialtowards the p0, logarithmic, towards the origin p0, inflaton rolls inflaton rolls , inflaton Dvali, Shafi Schaefer (1994) origin rolls towards the origin Exponential potential V = V0 1 − e−qφ/MplDimopoulos, Lazarides, Lyth Ruiz de Austri (2003) Stewart (1995); Lazarides Panagiotakopoulos (1995) 2 p−1 1 − ns = *not so if μmpl N p−2
- 25. Hilltop-type inflation models p q V = V0 (1 + ηp φ − ηq φ ) Kohri et al (2007) The Running Mass Model End of 2 µ0 + A0 2 Inflation V = V0 1 − φ Scales 2 leave A0 + 2 φ 2 horizon 2(1 + α ln φ) Stewart (1996) These models have bet ween 3 and 4 independent parameters so they can be ‘fit’ to data these are the only 2 models which can lead to PBH Kohri et al (2007) and Drees et al (2011) formation.
- 26. Large field models
- 27. Monomial potential α V ∝φ 4α 2+α r= , 1 − ns = N 2N Linde 1983 Silverstein Westphal 2008 α positive integer: chaotic α 1: monodromySinusoidal potential Hilltop regime V0 2 |η0 |V = 1 + cos φ 2 Mpl 16|η0 | Chaotic regime r = 2N |η | e 0 − 1 2|η0 |N Freese, Frieman Olinto 1990; Adams, Bond, Freese, Frieman Olinto 1993 e +1 1 − ns = 2|η0 | 2|η |N e 0 −1
- 28. Observational bounds
- 29. 95Observational Constraints % c. lSpectral index (r and ns’=0) 0.939 ns 0.987Tensor fraction (ns’=0) r 0.24Running −0.084 ns 0.017
- 30. Results
- 31. +p 2 gs φ −p φ φ V = V0 1 −V = V0 1 + ln V = V0 1 − 2π Q µ µ p=0 p-∞ p0 V = V0 1 − e −qφ/mpl p0
- 32. α V ∝φ N=30 N=47 N=61 Axion Monodromy McAllister, Silverstein Westphal (2008) η=0Planck sensitivity
- 33. 3 APMW 0 Multifield/Chaotic Inflation 1 Natural Inflation Brane Inflation 2 tensor desert log(r) 3 Original Hybrid Model 4 Modular Inflation (p=2) 5 Modular Inflation (p3) Mutated Hybrid Inflation 6 0.9 0.95 1 1.05 n
- 34. Degenaracies stillexist, so what next?
- 35. Future data may alleviate some of thedegeneracy, e.g. PLANCK or CMBPOL. Butnot fully.Further signatures of inflation models.Be more philosophical and ask whatmakes a model ‘natural’.
- 36. The Data Fantasy•Space based detectors of gravitational waves, DECIGO andLISA.• DECIGO and LISA cover a frequency range of approx. 10-3 to101.5 Hz.•This corresponds to scales which leave the horizon at theend of inflation.•Funding has not been approved yet but ...•Proposals and white papers for these projects haveappeared on the arXiv!
- 37. InducedGravitational Waves
- 38. Anatomy of Induced Gravitational WavesThese are induced by scalar perturbations entering the horizon after inflation (2) ds2 = 2 (1) a (τ ) − 1 + 2Φ + 2Φ (2) dτ + 2Vi dτ dxi + · · · 2 (1) (2) 1 + 1 − 2Φ − 2Φ δij + hij dxi dxj 2 contains 1st and 2nd order terms Einstein Eqns
- 39. Get the equation of motion: 2 hij + 2Hhij − ∇ hij = −4Sij source term just 2nd order Sij = 4Φ∂i ∂j Φ + 2∂i Φ∂j Φ + · · · 4 + ∂i (Φ + HΦ)∂j (Φ + HΦ) 3H2 (1 + w) Where the Φ is a first order quantity given by 6(1 + w) 1 2 Φk + Φk + wk Φk = 0 1 + 3w τ Note: the source term depends only on the (1) scale and (2) epoch of re-entry (w, the equation of state).
- 40. Fourier transform calculate spectra the equation 2 2π hk hk = 3 δ(k + k )Ph (k, τ ) k Spectrum of induced gravitational waves scalar spectra integral over all k-space from inflation Ph (k, τ ) ∼ ˜ dk ˜ ˜ dµPζ (k)Pζ (|k − k|)I1 (τ )I2 (τ ) cosine of angle integrals over between modes conformal time independent of inflation model The time integrals are model independent, and require only Φ and thus the equation of state.Ananda et al (2007), Baumann et al (2007), Acquaviva et al (2003), Saito et al (2009)
- 41. ˜ k Time integrals v= y = 1 + v 2 − 2vµ k Matches results found on Maple x = kτ and previous Authors results asymptote to zero!
- 42. Which Models are we looking at?
- 43. ScalesV model picture leave End of horizon Inflation ε decreases and the scalar spectrum increases ﬁeld
- 44. Selection Criteria: Hilltop-Model V = V0 (1 + ηp φp − ηq φq )Pre-set couplings to {0,1}.Reject values that violate WMAP.Reject values which require super-planckian evolution.Unique values then selected whichmaximise the spectrum after N e-folds.
- 45. Selection criteria: Running Mass µ +A 2 A 0 0 0 2V = V0 1 − φ + 2 φ2 2 2(1 + α ln φ)Less intense criteria: pick values whichsatisfy WMAP.Evolve until just before the spectrumhits the PBH bound.Terminate inflation and evaluate N.
- 46. Spectra
- 47. PBH Bound PBH Bound fractional powers p=2, q=2.2,2.3,2.5,2.7,2.9 55 60 N= integral powers N={p,q}={2,3},{2,4},{3,4},{4,5}Note: integral powers are already strongly constrained at the pivot scale
- 48. Running Mass ModelPBH Bound ns’=0.005 ns’=0.002 0.007ns’0.012 lead to PBHs with 20 28 10 g MBH 10 compatible with Dark Matter
- 49. Results: the Induced Gravitational
- 50. Present Day SpectrumThe Energy density (per logarithmicinterval) is: 1 dρGW energy density of GW ΩGW (k, τ ) = ρc d log f critical energy densityThis is related to the primordial spectrum as: 2 transfer function (2) a(τ )k 2 ΩGW = t (k, τ )Ph (k) aeq keqFor scales kkeq this gives us the relation: (2) 1 ΩGW = Ph (k) 1 + ze Baumann et al (2007)
- 51. The Running Mass Model PBH Bound ns’=0.002 ns’=0.005 0.007ns’0.012 Models are terminated at N=64DM N=43 29N39
- 52. For comparison ... DE BBO CIG O The spectra of primordial gravitational waves look very different! this image is taken from Kuroyanagi et al (2010)
- 53. Hilltop models lines=fractional q, stars and circles = integral q N= 55 60N=
- 54. ConclusionsHilltop model: generates potentially obser vable inducedGWS reasonable e-folds. also p=2,q=3 within the reach of BBO/DECIGO.Running Mass: within the reach of BBO/DECIGO for N50. DM production measurable by LISA.
- 55. Thank Youありがとうございます！

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