A STATUS REPORT ON SINGLE-FIELD INFLATION. Raquel H. Ribeiro. DAMTP, University of Cambridge. Lorentz Center, Leiden

Transcription

1 A STATUS REPORT ON SINGLE-FIELD INFLATION Raquel H. Ribeiro DAMTP, University of Cambridge Lorentz Center, Leiden July 19,

2 Message to take home Non-gaussianities are a powerful discriminator between inflationary scenarios and can lead to strong constraints to the parameters of a given model. They are important [recall Wandelt s talk]. The bispectrum is a very rich object. More than using its amplitude only, we should use the entire shape to decode info about the model in the data. Single-field inflation can be (easily) falsifiable with Planck. There are more consistency checks in addition to Maldacena s factorisation theorem (and the Suyama-Yamaguchi relation) [recall Zaldarriaga s & Tasinato s talks].

3 Outline of a short story 1 2 Primordial non-gaussianities: a (very) simplified ID Non-gaussianity technologies generic single-field models can we learn more beyond the amplitude of the bispectrum? 3 Telling models apart using the bispectrum shape

4 References alert arxivs , , , and RHR s thesis in collaboration with Clare Burrage (Nottingham) & David Seery (Sussex) preliminary work in collaboration with Donough Regan (Sussex), Sébastien Renaux-Petel (Cambridge) and David Seery (Sussex)

5 1 Primordial non-gaussianities: a (very) simplified ID Why do we like them so much?

6 Primordial Non-Gaussianities They can teach us the physics of inflation! h ( ~ k 1 ) ( ~ k 2 ) ( ~ k 3 )i =(2 ) 3 (3) ( ~ k 1 + ~ k 2 + ~ k 3 )B(k 1,k 2,k 3 ) bispectrum amplitude shape f NL 5 6 B(k 1,k 2,k 3 ) P (k 1 )P (k 2 )+P (k 2 )P (k 3 )+P (k 1 )P (k 3 )

7 Can we see it in the sky? Gaussian realisation as seen by Planck Liguori et al [astro-ph/ ] fnl=0

8 Can we see it in the sky? contact with observations Gaussian realisation as seen by Planck Liguori et al [astro-ph/ ] fnl=0 fnl=3000 fnl=-3000

9 2 Non-gaussianity technologies The amplitude and shape of the bispectrum

10 Horndeski theory [Horndeski 1974] [Kobayashi et al , Charmousis et al ] The Horndeski class contains all single-field inflation models stable from a QFT viewpoint. The full Horndeski theory is L P (X, ) X = g ab r a r b G 3 (X, ) G 4 (X, )R + G 4,X apple ( ) 2 (r µ r ) 2 G 5 (X, )G µ r µ r G 5,X 6 apple ( ) 3 3 (r µ r ) 2 + 2(r µ r ) 3

11 Comoving gauge = 0 In this gauge measures local fluctuations in the expansion history. h ij = a 2 (t)e 2 (t,~x) ij If there are no isocurvature modes, is conserved on super-horizon scales. The three-point correlators are also time independent. [Lyth 1985, Lyth & Wands astro-ph/ , separate universe approach]

12 Comoving gauge = 0 In this gauge measures local fluctuations in the expansion history. h ij = a 2 (t)e 2 (t,~x) ij If there are no isocurvature modes, is conserved on super-horizon scales. The three-point correlators are also time independent. [Lyth 1985, Lyth & Wands astro-ph/ , separate universe approach] This is *different* from the effective field theory approach a la Cheung et al [ ], in which 6= 0 and the scalar component of the metric perturbations decouples.

13 Recent developments [~1 year] S 3 = In all Horndeski models the cubic action for can be written as Z conformal time d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 [Burrage, RHR & Seery ] [See also Gao & Steer , and de Felice & Tsujikawa ]

14 Recent developments [~1 year] S 3 = In all Horndeski models the cubic action for can be written as Z conformal time d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 [Burrage, RHR & Seery ] [See also Gao & Steer , and de Felice & Tsujikawa ] This action is *exact* in slow-roll. No need to worry about boundary terms or field redefinitions.

15 Case Study Dirac-Born-Infeld inflation [Silverstein & Tong hep-th/ ] The DBI action provides a low-energy description of a D3-brane moving in warped space S = Z d 4 x p g apple 1 f( ) p1 f( )X 1 V ( ) warping of the throat coupling between the brane and other dof D3-brane throat This action admits a power-law inflationary solution and c s = q 1 f 2

16 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 DBI g 2 = " c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s apple 5" +(1 c 2 s) g 4 = "2 2c 4 s " 4 g 5 = "3 4c 4 s where " Ḣ H 2

17 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 DBI g 2 = " c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s apple 5" +(1 c 2 s) g 4 = "2 2c 4 s " 4 g 5 = "3 4c 4 s where " Ḣ H 2

18 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 mix leading and next-order DBI g 2 = " c 4 s g 4 = "2 2c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s " 4 g 5 = "3 4c 4 s apple 5" +(1 c 2 s) where " Ḣ H 2 The vertices mix different orders in slow-roll.

19 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 mix leading and next-order mixes next-order and next-next-order DBI g 2 = " c 4 s g 4 = "2 2c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s " 4 g 5 = "3 4c 4 s apple 5" +(1 c 2 s) where " Ḣ H 2 The vertices mix different orders in slow-roll.

20 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 mix leading and next-order mixes next-order and next-next-order DBI g 2 = " c 4 s g 4 = "2 2c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s " 4 g 5 = "3 4c 4 s apple 5" +(1 c 2 s) next-next-order where " Ḣ H 2 The vertices mix different orders in slow-roll.

21 In DBI inflation, the cubic action for the comoving curvature perturbation,, simplifies Z S 3 = d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2 mix leading and next-order mixes next-order and next-next-order DBI g 2 = " c 4 s g 4 = "2 2c 4 s apple " 3(1 c 2 s) g 3 = " c 2 s " 4 g 5 = "3 4c 4 s apple 5" +(1 c 2 s) where " Ḣ H 2 The vertices mix different orders in slow-roll. Keeping only leading order contributions in slow-roll quantities to the bispectrum and assuming the attractor solution of Silverstein & Tong f (equi) NL! 1 c 2 s using observational constraints c s O( )

22 How large can the corrections to fnl be? Previous estimates (based on gi) suggested that the next-order corrections could be as large as 70% for DBI inflation [for ε 1/20]. The uncertainty associated with f NL in DBI can be f NL DBI corrected

23 How large can the corrections to fnl be? Previous estimates (based on gi) suggested that the next-order corrections could be as large as 70% for DBI inflation [for ε 1/20]. The uncertainty associated with f NL in DBI can be f NL DBI corrected Compare this with the error bars for the equilateral template f NL f NL 10 Planck CMBPol and CoRE

24 How large can the corrections to fnl be? Previous estimates (based on gi) suggested that the next-order corrections could be as large as 70% for DBI inflation [for ε 1/20]. The uncertainty associated with f NL in DBI can be f NL DBI corrected Compare this with the error bars for the equilateral template f NL f NL 10 Planck CMBPol and CoRE We need to know how to fully compute these corrections to keep the theory error down [recall Gong s talk].

25 Collecting next-order contributions in DBI vertex k 1 external lines internal lines g i k 3 k 2

26 Collecting next-order contributions in DBI vertex k 1 external lines Taking the limit of small speed of sound: internal lines g i f (equi) NL! 1 c 2 s apple 35 1+(4 E 3)" 108 k 3 k 2 3% for 1/20 The uncertainty with our theoretical estimate was reduced to a tenth of what it was before.

27 [Franche et al ] V = 1 2 V 0 V 2 V = V 00 V = 1 3H f 0 f 3/2 sgn( f 1/2 ) potential warp factor

28 [Franche et al ] V = 1 2 V 0 V 2 V = V 00 V = 1 3H f 0 f 3/2 sgn( f 1/2 ) potential warp factor Fractional corrections to fnl can be written in the equilateral limit as f NL f NL ' " V 0.41 V f NL 1/2

29 [Franche et al ] V = 1 2 V 0 V 2 V = V 00 V = 1 3H f 0 f 3/2 sgn( f 1/2 ) potential warp factor Fractional corrections to fnl can be written in the equilateral limit as f NL f NL ' " V 0.41 V f NL 1/2 " V V 1 Taking & f NL 50, the next-order corrections in slowroll can be as large as 35%. Burrage, RHR & Seery

30 Limitations of the slow-roll regime RHR, How do we know that next-next-order effects will not produce sizeable effects? We can focus on a subset of the slow-roll approximation which is compatible with an approximately scale-invariant spectrum of perturbations. This procedure generates exact results in slow-roll (no need for perturbative expansion in, for example). "

31 Limitations of the slow-roll regime RHR, How do we know that next-next-order effects will not produce sizeable effects? We can focus on a subset of the slow-roll approximation which is compatible with an approximately scale-invariant spectrum of perturbations. This procedure generates exact results in slow-roll (no need for perturbative expansion in, for example). " The results: are compatible with the known formulae in the literature strongly scale-dependent

32 Summary I We are in the age of precision Cosmology: theoretical uncertainties need to be placed below Planck s exquisite sensitivity. Next-order corrections in slow-roll are typically large (several tens of percent). We have the technology to compute these corrections. Precision-wise we should be ready for Planck.

33 3 Telling models apart Beyond the amplitude of the bispectrum

34 A bispectrum shape It is a complicated object of momenta, so it is useful to plot it. The shape function is given by k1k 2 2k 2 3B(k 2 1,k 2,k 3 ). Fergusson & Shellard parametrization k 1 = k t 4 (1 + + ) k 2 = k t 4 (1 + ) k 3 = k t 2 (1 )

35 S 3 = Z Bispectrum shapes d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2

36 S 3 = Z Bispectrum shapes d 3 x d a 2 g1 a 03 + g g 3 (@ ) 2 + g g 2 (@ 2 0 ) 2

37 Can we use shapes to distinguish between models? orthogonally designed shape Creminelli et al found a new shape in a higher derivative galileon model, which hadn t been found before. One could think that, if ever observed, it would provide strong evidence in favour of these type of models.

38 Common orthogonal shape This shape was rediscovered in P(X, ) models and also in lower order galileon models: Creminelli et al. shape shape at leading-order RHR & Seery shape at next-order in P(X, ) models Burrage, RHR & Seery This shape, though harder to find in the data, will always be present in any Horndeski model provided sufficient fine-tuning is allowed.

39 Modal decomposition A generic bispectrum shape will be an admixture of the fundamental harmonics [a la Fergusson et. al ]: ~constant ~equilateral template S(k 1,k 2,k 3 )= any bispectrum shape

40 Modal decomposition A generic bispectrum shape will be an admixture of the fundamental harmonics [a la Fergusson et. al ]: ~constant ~equilateral template S(k 1,k 2,k 3 )= any bispectrum shape These are the largest harmonics which survive the orthogonalisation procedure.

41 DBI inflation RHR & Seery hb k?, R 0 ni n P 2 (k? ) inner product 2 copies of the power spectrum The first two consistency relations are: Lowest order consistency relations 0 < 0 : DBI predicts a negative value of f NL (whereas WMAP measurements marginally favour positive values). These are the observables in term of observables as first coined by Copeland et al. hep-ph/

42 Summary II The overall shape of the bispectrum might not serve as a sensitive discriminator of microphysics. Using the modal decomposition, five gi can be fixed with only five measurements. [also recall Senatore s talk/particle physics method] Even if there are not enough measurements available, one can verify/ invalidate consistency relations for each model. n 0 x Regan et al. astro-ph/ x x x x x n schematics of a fingerprint of some model

43 Effective Field Theories some brief notes Understanding EFTs is a great endeavour (spontaneously broken symmetry ideas, systematic study of non-gaussianities in single-field models). A la Cheung et al. [recall Senatore s talk] They provide the low-energy action for the perturbations by focusing on the leading low-energy dof (metric perturbations contribute at next-order in slow-roll only). Here we miss the info provided by next-order results. EFTs open the possibility to study new physics, where the modal decomposition fingerprint can be the picture of the playground where data and fundamental physics meet.

44 Thank you for listening

45 D3-brane throat Speed of the D3-brane along the throat is related to the speed of sound of the scalar fluctuations: c s = q 1 f 2 Reduced speed of sound is related to large equilateral non-gaussianities: c s 1 f (equi) NL <f (equi) NL < 266 [95% CL, WMAP 7 astro-ph/ ] this is the regime of phenomenological interest

46 D3-brane throat Speed of the D3-brane along the throat is related to the speed of sound of the scalar fluctuations: c s = q 1 f 2 Reduced speed of sound is related to large equilateral non-gaussianities: c s 1 f (equi) NL <f (equi) NL < 266 [95% CL, WMAP 7 astro-ph/ ] this is the regime of phenomenological interest

47 Modal decomposition at a glance Traditional method of matching with templates bispectrum in the sky compare with templates extract constraints for f NL bispectrum of the theory evaluate at specific momenta configuration need access to overlapping cosine between shapes

48 Modal decomposition at a glance Traditional method of matching with templates bispectrum in the sky compare with templates extract constraints for f NL bispectrum of the theory evaluate at specific momenta configuration need access to overlapping cosine between shapes Modal decomposition bispectrum in the sky bispectrum of the theory obs n theory n place constraints on parameters of the theory + useful when judging consistency relations + makes use of the entire bispectrum

49 Where do next-order contributions come from? [first systematically identified by Chen et al. hep-th/ ] As in any Feynman diagram, corrections will arise from vertex external lines internal lines k 3 k 1 k 2

50 Discussion board Is inflation the right framework? If so, what is the fundamental physics behind inflation? Simplest scenario: one single scalar field is responsible for the quasi de Sitter expansion, while seeding the cosmological perturbations. Of all the higher derivative single-field models, Dirac Born Infeld and galileon inflation are the only established examples where nonrenormalization theorems apply. Discussing a generic single-field model can still reveal interesting observational signatures.

51 Discussion board Is inflation the right framework? If so, what is the fundamental physics behind inflation? Simplest scenario: one single scalar field is responsible for the quasi de Sitter expansion, while seeding the cosmological perturbations. Of all the higher derivative single-field models, Dirac Born Infeld and galileon inflation are the only established examples where nonrenormalization theorems apply. de Rham et al Silverstein & Tong hep-th/ Discussing a generic single-field model can still reveal interesting observational signatures.

52 Calculating the bispectrum Cosmologists usually write fnl evaluated at a specific momenta configuration: equilateral, squeezed and enfolded limits. On the other hand, there are also templates.

53 Catalogue of slow-roll parameters These parameters are defined independently of the potential Ḣ H 2 H s c s Hc s Recipe: systematically expand every background quantity around the time of horizon crossing for a reference scale, eg apple H(t) ' H(t? ) 1 "? N? + where 4N? (t) =N(t) N? (t) = ln( k? ) reference scale

54 Modal decomposition Fergusson et. al suggested the decomposition of the bispectrum shapes into an orthonormal basis: S(k 1,k 2,k 3 )= X n n R 0 n(k 1,k 2,k 3 ) any bispectrum shape The basis decomposition defines the bispectrum shape space available.

55 What to extract from the data? With only five measurements, we can fix the five vertices gi for an arbitrary Horndeski theory. Any extra measurements can be directly compared with the predictions of the theory. Also, we can construct consistency relations between the amplitudes of each harmonic, obtaining observables in terms of observables. Copeland et al. hep-ph/

56 What to extract from the data? With only five measurements, we can fix the five vertices gi for an arbitrary Horndeski theory. Any extra measurements can be directly compared with the predictions of the theory. Also, we can construct consistency relations between the amplitudes of each harmonic, obtaining observables in terms of observables. Copeland et al. hep-ph/ Instead of using shapes as an estimator to distinguish between models, we can compare the amplitudes of each harmonic.

57 What now? If the shape is recurrent, how can we find compelling evidence in favour of some inflationary model? The exotic shape arises from two delicate requirements: orthogonality with the common templates fine tuning of the couplings to suppress the leading order shapes So how can the same exotic shape appear keep appearing?