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The Consistency Equation in Single Field Inflationary Models astro-ph/0603016 Andrew R. Liddle Marina Cortês Astronomy Centre University of Sussex May 06 Inflation 25 years Starobinsky,1980 Guth,1981 Sato,1981 Albrecht & Steinhardt, 1982 Hawking & Moss, 1982 Linde 1983 •period of accelerated expansion in the early universe •proposed to explain problems of the standard Hot Big Bang Model:horizon, flatness, monopole •Most interesting feature: means to explain appearance of structure •Quantum Cosmology: we are observing the effects of quantum fluctuations in the microwave sky & in the Dark-Matter Power Spectrum •DIY: scalar field: -dynamical expansion has to be switched off -self interacting field ->potential energy with small slope: field rolling down the hill with friction Perturbations:simplest form • Scale invariant equal on all scales: no divergences on small scales or large scales • Adiabatic: all particle species are perturbed in fixed ratio: perturbing the photons perturbs the baryons • Gaussian random noise • When light encompasses the wave the wave starts to oscillate Slow Roll approximation • Kinetic energy has to be small and must remain small for some time • Einstein Field Equation and Klein Gordon: 2 8 1 2 2 8 H 2 V ( ) H 3m 2 V ( ) 3mPl 2 Pl 3H V ( ) 3H V ( ) Fluctuations: lowest order expressed in terms of Hubble parameter: makes computations easier •Friedmann equation translates one into the other 4 H2 AS (k ) 5mPl H 2 V H k aH H 2 AT (k ) 5 mPl k aH Infinite Hierarchy of Slow Roll paramaters: derivatives of the Potential must be small in an order-by-order expansion mPl H 2 2 4 H mPl H 2 4 H mPl H H 2 12 2 4 H 13 mPl H 2 H ( 4) 2 H2 4 Next order (Stewart & Lyth 1993) (measuring the errors in slow roll predictions) • Compare the slow roll approximation 3H V ( ) with the exact expression 3H V ( ) • The error is H • Spectra will pick up errors of order and •Corrections in and : 2 AS (k ) 4 1 (2C 1) C H 2 5mPl H k aH AT (k ) 1 (C 1) 2 H 5 mPl k aH Next order (Stewart & Lyth 1993) (measuring the errors in slow roll predictions) • No assumptions beyond linear perturbation theory (drops slow roll) =>Next order in dynamics not perturbations • Ananda, Clarkson and Wands, ICG: - next order in perturbations - extra effect: mixing of scalars and tensors (to appear) d ln AS2 d ln AT2 nS 1 nT d ln k d ln k nS 1 4 2 (8C 8) 2 (6 10C ) 2C 2 nT 2 (6 4C ) 2 (4 4C ) dnS 8 2 10 2 2 d ln k (40 32C ) 3 (60 62C ) 2 3 (12 20C ) (8 14C ) 2C 2C 2 2 2 dnT 4 2 4 d ln k (28 16C ) 3 (40 28C ) 2 (8 8C ) (4 4C ) 2 2 Consistency Equation Smoking gun of inflation 2 A nT 2 T 2 A S • Connects the two spectra in a way unique to inflation • Correlates the relative amplitude of the two spectra with the slope of the tensor spectrum •Single field •Multifield: - isocurvature modes testing the consistency equation (prospects for detecting tensor modes) • CMB and space based experiments • Planck higher sensitivity to polarization Figures computed by A. Challinor: The Planck Consortia Clover & Planck QUIET • Should be able to map out the • Ground based electric polarization power spectrum ClE on all scales up • Large arrays of polarization to and beyond the global sensitive detectors (necessary maximum of l~1000 to detect the wavelength of primordial gravitational waves) • Will at best only be able to • distant future (NASA funding) detect tensor modes if r is greater than a few percent (already a considerable achievement) structure 2 A A 2 2 S T nT • Correlates the relative amplitude of the two spectra with the slope of the tensor spectrum • Tensors are one order higher • Differential equation for the tensors • Integral equation for the scalars => tensors fully specify the physical situation 2 AT A 2 2 S nT Taylor expand both sides: d 2 nT A2 d 2 2 nT A2 2 2 2 k 1 T 2 k 2 T ln ln dA k 1 d A k A AS 2 S ln ln 2 2 S 2 T 2 d ln k k0 2 d ln k k0 nT d ln k k0 2 d ln k k0 Order of approximation is maintained in differentation and integration: Infinite Hierarchy of Consistency Equations d 2 AT 2 (i ) d (i ) AS 2 nT i 0,1,... d ln k (i ) d ln k (i ) In terms of spectral indices d (i 1) (nS 1) d (i 1) nT d (i ) ln nT i 1,2,... d ln k (i 1) d ln k (i 1) d ln k (i ) Consistency Equation Hierarchy Next Order in Slow Roll – First Consistency Equation – Second Consistency Equation Next Order Next Order AT AT 2 2 nT 2 2 1 2 (nS 1) AS AS n dnS nT nT (nS 1) nT T (nT (nS 1)) dnT d ln k 2 d ln k Separate tensors and scalars 1 2 AS 1 (nS 1) 2 AT 1 2 nT 2 nT Tensors no longer specify the situation! First Consistency Equation Lowest Order 2 AT nT 2 2 AS First Consistency Equation Next Order 2 AT AT 2 nT 2 2 1 2 (1 nS ) AS AS Second Consistency Equation Lowest Order nT nT (nS 1) dnT d ln k Second Consistency Equation Next Order n dnS nT nT (nS 1) nT T nT (nS 1) dnT d ln k 2 d ln k The approximate Consistency Equation (models with running) Chung, Shiu & Trodden astro-ph/0305193 Chung & Romano astro-ph/0508411 • Propose similarity of two scales • Suppose n-1 changes sign at some scale • This involves a large or at least non- negligible running • Running cannot be sustained for long (must achieve 60 e-folds)=> flat bump in the potential Two Scales: nS changes sign: spectrum changes from blue to red k 1: nS 1 0 Flat bump in the potential V 2 k2 : ~ 0 nT nS 1 V •But: Consistency equation exhausts connections between spectra! (nS 1) nT nS 1 N No inflationary input dnT d ln k dnS d ln k dnS d ln k denominator is one order higher in slow roll N will be large 1 N ~ O Enforcing the relation between scalars and tensors turns the expression into N at scale k1 4 2 2 In terms of observables nT N n dnS d ln k 2 T Comparing with the non-inflationary expression at scale k1 (nS 1) nT N dnT d ln k dnS d ln k We see that their proposal is to confirm that dnT nT 2 d ln k Which is the second consistency equation at scale k1! So the proposed new consistency equation introduces no novelty on the relation between scalars and tensors Constant Running Lidsey & Tavakol 1 A2 (nS 1) 2 2 nS 1 2 ~ S exp erf c A2 T 2 dnS d ln k dnS d ln k 2 dnS d ln k Relation advertised as independent of the potential Could be used to test for inflation However this is not a consistency equation: -does not include the tensors -as it is it seeks out to measure constant c which is the same as measuring the running -typical inflation models predict some deviation from constant running => as such it is not testing for a signature of inflation Conclusions • We presented expressions for the spectral indices and their running to lowest and next order in Slow-Roll • We showed that the consistency equation can be differentiated to present a complete description between the spectra in inflationary models: Infinite Hierarchy of Consistency Equations • As this exhausts the connection between the spectra in inflationary models any other relations between these must derive from one of the consistency equations or from combinations of more of them prospects • Study the relation between scalars and tensors to next order • Consistency equation for Multi-field (the standard situation is an inequality) • Next Order Corrections in perturbation theory (relevant?)

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posted: | 4/9/2010 |

language: | English |

pages: | 25 |

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