JGRG0, YITP, Kyoto 5 th September 00 Non-linear perturbations from cosmological inflation David Wands Institute of Cosmology and Gravitation University of Portsmouth
summary: non-linear perturbations offer distinctive observational signatures of physics of inflation
gravity + inflation: accelerated expansion of FLRW cosmologies horizon, entropy, flatness... relativistic perturbations quantum fluctuations of free scalar and tensor modes non-linear perturbations second-order tensor modes from scalar fluctuations second-order density perturbations and non-gaussian distributions inflation driven by modified gravity e.g., Starobinsy 980
coming of age... c Masashi Kio 00
inflation circa 990
extended inflation simple, compelling model for gravity-driven inflation... false vacuum + first-order transition + Brans-Dice gravity L R V false vacuum La & Steinhardt 989 Barrow & Maeda 990 Steinhardt & Accetta 990 solves graceful exit problem of Guth s old inflation Φ grows during inflation, Hubble rate decreases, until first-order transition completes scale-free gravity one dimensionless parameter
dynamical solution to hierarchy problem E. Weinberg 989 Liddle & Wands 99 start near Planc scale bubble nucleation rate H M M 4 GUT Pl 4 GUT exp S E M M Pl M GUT S E >> is dimensionless Euclidean action shape parameter percolation parameter P H 4 Pl 4 GUT P grows as Φ grows gravity gets weaer and H decreases 4 M M exp S E phase transition completes / inflation ends when p= M M exp 4 Pl GUT S E / MGUT
power law inflation power-law inflation a t p ; p 4 linear perturbations [conformal transform to Einstein frame Brans 96, Maeda 989] reproduces scale-invariant spectrum as ω non-linear perturbations CoBE 994 first-order transition leads to distribution of bubbles spectrum of bubbles also becomes scale-invariant as ω big bubble problem Weinberg 989; Liddle & Wands; Maeda & Saai 99
hybrid inflation Linde 99 inflaton field changes shape of false vacuum potential S E t => t~ M 4 exp[- S E t] t ends by sudden phase transition first- or second-order non-linear perturbations only on small scales inhomogeneous bubbles or tachyonic preheating Lyth; Fonseca, Sasai & Wands 00 + see poster by Gong spectrum of relic gravitational waves on characteristic scale Easther 09
Sources of primordial gravitational waves: Quantum fluctuations of gravitational field First-order phase transitions? Preheating after inflation? Cosmic string cusps? Primordial density perturbations
Second-order GW from first-order density perturbations Tomita 967; Matarrese et al 994; Hwang; K. Naamura; Ananda, Clarson & Wands 006 scalar, vector and tensor modes couple at second and higher order tensor perturbations become gauge-dependent in longitudinal gauge for general FRW cosmology w=p/, c s =dp/d where second-order source is transverse-tracefree part of Baumann, Steinhardt, Taahashi, Ichii, hep-th/07090
GW from density perturbations in radiation era Ananda, Clarson & Wands, gr-qc/060 almost scale-invariant primordial density power spectrum 4 P R for ah 9 generates almost scale-invariant gravitational wave bacground GW,0 0,0 c 4 R for ah d d ln GW e.g., 0 9 GW,0 0 for ΔR 0 ah
Constraints on primordial density perturbations GW,0 0,0 4 R Assadullahi & Wands, arxiv:0907.407 LIGO/VIRGO Δ R 0. 07, 00Hz Advanced LIGO/VIRGO Δ R 80 4 LISA Δ R 0 4, mhz BBO/DECIGO Δ R 0 7, Hz Pulsar timing data rules out intermediate mass primordial blac holes Saito & Yooyama, arxiv:08.49 Phys Rev Lett Bugaev & Klimai, arxiv:09080664
second-order density perturbations non-linear evolution lead to non-gaussian distribution non-zero bispectrum and higher-order correlators Local-type non-gaussianity super-hubble evolution of Gaussian random field from multi-field inflation Equilateral-type non-gaussianity sub-hubble interactions in -inflation/dbi inflation Topological defects cosmic strings from phase transitions templates required to develop optimal estimators matched filtering to extract small non-gaussian signal
the N formalism for primordial perturbations in radiation-dominated era curvature perturbation on uniform-density hypersurface t during inflation field perturbations I x,t i on initial spatially-flat hypersurface N final initial H dt x on large scales, neglect spatial gradients, treat as separate universes N N initial N I I I Starobinsy `85; Sasai & Stewart `96 Lyth & Rodriguez 05 wors to any order
the N formalism order by order at Hubble exit I I I... I N I... I I N I I t I, J N I I I J... sub-hubble quantum interactions super-hubble classical evolution N N N N N N Byrnes, Koyama, Sasai & DW
N is local function of single Gaussian random field, where odd factors of /5 because Komatsu & Spergel, 00, used /5 simplest local form of non-gaussianity applies to many models inflation including curvaton, modulated reheating, etc... 5......... 6 x x x x f x x x N N x x x x x N x x N N N NL
large non-gaussianity from inflation? single inflaton field adiabatic perturbations => constant on large scales during conventional slow-roll inflation for any * adiabatic model Creminelli&Zaldarriaga 004 /DBI - inflation N N multi-field models typically f NL ~ for slow-roll inflation could be much larger from sudden transition at end of inflation? modulated reheating curvaton f NL ~/ decay >>? f f local NL equil f NL c local NL s 5 n new epyrotic models f NL >> * but as Sasai-san!
V curvaton scenario: Linde & Muhanov 997; Enqvist & Sloth, Lyth & Wands, Moroi & Taahashi 00 curvaton = a wealy-coupled, late-decaying scalar field - light during inflation m<<h hence acquires an almost scaleinvariant, Gaussian distribution of field fluctuations on large scales - energy density for massive field, =m / - spectrum of initially isocurvature density perturbations - transferred to radiation when curvaton decays with some efficiency,decay, decay G 4, decay G f NL 5 4, decay
Liguori, Matarrese and Moscardini 00
f NL =+000 Liguori, Matarrese and Moscardini 00
f NL =-000 Liguori, Matarrese and Moscardini 00
remember: f NL < 00 implies Gaussian to better than 0.%
evidence for local non-gaussianity? T/T -/, so positive f NL more cold spots in CMB various groups have attempted to measure this with the WMAP CMB data using estimators based on matched filtering all 95% CL : 7 < f NL < 47 Yadav & Wandelt WMAP data -9 < f NL < Komatsu et al WMAP5-4 < f NL < 80 Smith et al. Optimal WMAP5-0 < f NL < 74 Komatsu et al WMAP7 Large scale structure observations have recently given independent indications due to non-local bias on large scales Dalal et al 007: -9 < f NL < 70 95% CL Slosar et al 008 7 < f NL < 7 95% CL Xia et al 00 [NVSS survey of AGNs]
inflation non-gaussianity from inflation Source of non- Gaussianity Initial vacuum Excited state Folded? Sub-Hubble evolution Hubble-exit Higher-derivative interactions e.g. -inflation, DBI, ghost Features in potential Super-Hubble evolution Self-interactions+gravity Local End of inflation Tachyonic instability Local preheating Modulated preheating Local After inflation Curvaton decay Local primordial non-gaussianity Radiation + matter + last-scattering Primary anisotropies Bispectrum type Equilateral +orthogonal? Local+equilateral ISW/lensing Secondary anisotropies Local+equilateral 8//008 David Wands 6
templates for primordial bispectra local type Komatsu&Spergel 00 local in real space fnl=constant max for squeezed triangles: <<, equilateral type Creminelli et al 005 peas for ~~ orthogonal type Senatore et al 009 8//008 David Wands 7,, 6/5,,, / P P P P P P f B P NL P 6/ 5,, f B local NL P 6/5,, f B equil NL P 8 6 / 5,, f B orthog NL P
the N formalism order by order at Hubble exit I I I... I N I... I I N I I t I, J N I I I J... sub-hubble quantum interactions super-hubble classical evolution N N N N N N Byrnes, Koyama, Sasai & DW
sub-hubble interactions: requires non-minimal inetic Lagrangian -inflation Armendariz-Picon & Muhanov 999 super-luminal c s > DBI brane inflation Alishahiha, Silverstein & Tong 004 probe brane in AdS bul c s <, f NL ~ /c s L f Galileon fields Nicolis, Ratazzi & Trincherini 009 f ghost-free DGP lie scalar field with second-order equations of motion c s <, f NL ~ /c s see posters by Kobayashi & Mizuno 00 L P, f Galileon and DBI reunited de Rham & Tolley 00 brane + most general second-order gravity in AdS bul L M 4 R M5 KGH M 5 K GB
summary: non-linear perturbations offer distinctive observational signatures of gravitational physics of inflation Happy Birthday!