Investigating a second consistency relation for the trispectrum

1. Investigating a second consistency relation for the trispectrum Jonathan Ganc Qualifier for PhD Candidacy October 12, 2009 Department of Physics, University of Texas

2. Overview • Cosmic Inflation • Characterizing inflation, calculating non-Gaussianity; the in-in formalism • The bispectrum consistency relation for single-field inflation • The trispectrum has at least one consistency relations • Is there another consistency relation for the trispectrum? • Conclusion and further work Verifying a second consistency relation for the trispectrum. Jonathan Ganc

3. I. Cosmic Inflation • A period of exponential expansion in the very early universe with a nearly constant Hubble parameter: a(t)=a0e∫Hdt. • Resolves many potential problems in cosmology: • the horizon problem • the flatness problem • the monopole problem • seeding large-scale perturbations • Lasted long enough for the universe to expand by a factor of about e60. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

4. What produced inflation? • Inflation took place well above the energy scale of known physics (≫1 TeV); i.e. we have no idea what caused it. • Can be simply modelled by a scalar field slowly rolling down a nearly flat potential; there are also innumerable more complicated models. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

5. Chen et al 2007 “Slow-variation” inflation • For a large class of single field inflationary models, we can write the field Lagrangian as ℒ=P(X,φ), whereX≡-1/2gμν∂μφ∂νφ. • The speed of sound cs is defined (Garriga & Mukhanov 1999): • We define three “slow variation parameters”: ; for “slow-variation” inflation, we assume them all to be small. • Note that standard “slow-roll” inflation is included in “slow-variation” inflation. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

6. Inflation seeds large-scale fluctuations through quantum fluctuations • For single field inflation, the inflaton φ is a quantum field inside the horizon:For slow-variation inflation (Chen et al 2006): (I will use ≃ to indicate equality to lowest order in slow variation). Verifying a second consistency relation for the trispectrum. Jonathan Ganc

7. Curvature perturbation ζ • Fluctuations in the inflaton δφare converted to perturbations in the spatial curvature ζ: • ζ produces anisotropy in the CMB temperature and in the matter distribution. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

8. Inflation produces large-scale fluctuations cont’d • For single field inflation, fluctuations freeze as they are stretched outside the horizon (Bardeen, Steinhardt, & Turner 1983). • Later, the horizon expands and the modes reenter the horizon.

9. II. Characterizing inflation: the power spectrum • A straightforward calculation yields the power spectrum Pζ(k) of ζ: where (originally calculated, for cs≠1, by Garriga and Mukhanov 1999) Verifying a second consistency relation for the trispectrum. Jonathan Ganc

10. Characterizing inflation: non-Gaussianity • Non-Gaussianity is determined by the connected part of three-point and higher cosmological correlation functions. • Typically, theoretical results are calculated using in-in formalism: • Weinberg 2005 • Similar to typical QFT “out-in” scattering; e.g., we ultimately let t→t(1+iε) (as tnears -∞) in order to calculate in the interacting vacuum. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

11. Bispectrum calculations • In 2003, Maldacena calculated the bispectrum for single field slow-roll inflation: • Others (notably Seery et al 2005 and Chen et al 2007) later calculated the bispectrum for more general kinetic terms (slow variation inflation). Verifying a second consistency relation for the trispectrum. Jonathan Ganc

12. III. Bispectrum consistency relation • Maldacena (2003) used his explicit result for the bispectrum (in single field slow-roll inflation) to find a bispectrum formula in the “squeezed limit” (k3≪k1≈k2): • Creminelli and Zaldarriaga (2004) found a straightforward kinematic argument that generalized this result (unchanged) to the case of any (even non-canonical) single field inflation. • This result holds regardless of kinetic term, vacuum state, or form of potential. power spectrum spectral tilt

13. Bispectrum consistency relation: the significance • The consistency relation involves measurable quantities: trispectrum <ζk1ζk2ζk3>, power spectrum Pζ(k), and spectral tilt ns. • Assuming local form for non-Gaussianity (Komatsu and Spergel 2001): ζ= ζg+3/5fNLζg2, we find fNL=5/12 (1-ns). • Observationally:ns=0.960 ± 0.013 (68% CL) (Komatsu et al 2009).fNL=38±21 (68% CL) (Smith et al 2009) • It does not look like fNL=5/12 (1-ns)=0.017. If this holds up, we have ruled out single field inflation!

14. Maldacena 2003 Bispectrum consistency relation: the argument • Expand • We want to find the correlation <ζk1ζk2> as k3≪k1≈k2. In comoving gauge, the metric is: ds2= -dt2 +e2ζ(x)a2(t)dx2. • For small distances (i.e. corresponding to the length scales of the k1, k2 modes), ζk3 is approximately constant; thus, we can consider the effect of ζk3 as a rescaling of the scale factor: aeff(t)=eζk3(x) a(t). Verifying a second consistency relation for the trispectrum. Jonathan Ganc

15. Creminelli and Zaldarriaga 2004, Cheung et al 2008 Bispectrum consistency relation: the argument cont’d • Any measurable quantity f can ultimately only be a function of physical (not comoving) distance, so: Remember: aeff(t)=eζk3(x) a(t) Figure adapted from a talk by Komatsu 2009.

16. Creminelli and Zaldarriaga 2004, Cheung et al 2008 Bispectrum consistency relation: the argument cont’d • Expanding in terms of the background field ζk3 |Δx|≈ 1/k1,1/k2 Fourier Transform Verifying a second consistency relation for the trispectrum. Jonathan Ganc

17. Creminelli and Zaldarriaga 2004, Cheung et al 2008 Bispectrum consistency relation: the argument cont’d • Finally, we correlate the result with ζk3: as desired. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

18. Creminelli and Zaldarriaga 2004, Cheung et al 2008 Bispectrum consistency relation: summary • The important thing to note is that we made no assumptions except that we could expand <ζk1ζk2> in terms of a singlebackground field ζk3. • Thus, the relation holds for any single field inflation model. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

19. IV. The trispectrum • The connected part of the four-point correlation function: • With respect to the bispectrum, provides independent information about inflation • Single field calculations include Seery & Lidsey 2007 and Seery, Sloth, Vernizzi 2009 (canonical slow-roll inflation), Chen et al 2009 and Arroja et al 2009 (non-canonical slow-variation inflation). Verifying a second consistency relation for the trispectrum. Jonathan Ganc

20. Trispectrum shapes • We only have a non-zero trispectrum when Σiki=0. • Thus, the wavenumbers form a closed quadrilateral. • We name certain configurations based on the relative length of sides:

21. Maldacena-like trispectrum consistency relation • An argument like that for the bispectrum determines the trispectrum in the squeezed limit (Seery, Lidsey, & Sloth 2007): • Again, these are measurable quantities and the relationship can be tested, potentially ruling out single field inflation. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

22. V. Is there another trispectrum consistency relation? • There are three tree graphs that contribute to the trispectrum: • Seery, Sloth, and Vernizzi 2009 found kinematic argument for scalar exchange and graviton exchange terms in the folded limit. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

23. Seery et al 2009 argument: scalar exchange • Expand <ζk1ζk2> in terms of ζk12: • Note that ζk34 =ζk12. Thus, we can correlate <ζk1ζk2>ζk12, <ζk3ζk4>ζk12 over ζk12: • Thus, <ζ4>SE=O(Pζ3ε2) Diagram: ns-1=O(ε); ε≈10-2 Verifying a second consistency relation for the trispectrum. Jonathan Ganc

24. Seery et al 2009 argument: graviton exchange • An essentially identical argument for graviton exchange yields:This term goes as O(Pζ3ε), so it’s dominant over the scalar exchange term (O(Pζ3ε2)). • (χ12,34≡ φ1 - φ3 is the angle between the projections of k1 & k3 on the plane orthogonal to k12) Diagram: r=scalar-tensor ratio =O(ε) Verifying a second consistency relation for the trispectrum. Jonathan Ganc

25. Seery et al 2009 summary • Seery, Sloth,and Vernizzi 2009 determined that, in the folded limit, the scalar exchange (SE) and graviton exchange (GE) terms mustgive: • Thus <ζ4>SE+GE∝ Pζ(k12) ∝ k12-3. • For local form: • ζ= ζg+3/5fNLζg2+9/25gNLζg3 • we find τNL=36/25fNL2. If the contact interaction is sufficiently small, then fNL2=25/64r cos2χ12,34. =O(ε) =O(ε2) Verifying a second consistency relation for the trispectrum. Jonathan Ganc

26. What about the contact interaction (CI) • For canonical slow-roll inflation Seery et al 2009 used the explicit form for the contact interaction as calculated in Seery, Lidsey, & Sloth 2007. • They verified that the contact interaction is small in the folded limit; i.e. <ζ4>CI ∝ k120. • However, they don’t claim that CI term will be negligible in more general models. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

27. I noticed that... • ...in later papers, which calculate the bispectrum for more general (slow-variation) single-field inflation models (e.g. Chen et al 2009 and Arroja et al 2009), contact interaction terms also don’t blow up in the folded limit. • Let’s see why... Verifying a second consistency relation for the trispectrum. Jonathan Ganc

28. Reviewing in-in formalism • Whether kinematic or explicit, our calculations are done within the framework of the in-in formalism: where HI is the interaction Hamiltonian in the interaction picture and ζI is ζ in the interaction picture. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

29. Applying the in-in formalism • the 3 connected tree diagrams correspond to terms from the in-in formalism: Verifying a second consistency relation for the trispectrum. Jonathan Ganc

30. How does a k12-3 factor arise in exchange terms? Scalar exchange • Look at SE term: • The bracketed term equals the sum of all fully contracted terms, where (Chen et al 2009): Note that the time variable tis uniquely given by the momentum variable (e.g. p’⇒(p’,t’) or k⇒(k,t)) Verifying a second consistency relation for the trispectrum. Jonathan Ganc

31. How does a k12-3 factor arise in exchange terms? (cont’d) Scalar exchange • All connected terms have the following (or equivalent) contractions: 1: 2: 3: 4: Then, . • But, u(k12)∝k12-3/2, and we see each term has a factoru(k12,t’)u*(k12,t’’)∝k12-3. • Thus, <ζ4>SE∝k12-3. 2 1 3 4

32. How does a k12-3 factor arise in exchange terms? (cont‘d) • In the derivation, the essential point is having two connected vertices. • Since the situation is identical with GE terms, <ζ4>GE∝k12-3. • Graphically, this effect is equivalent to the fact that the exchange terms have a propagator. GE SE Verifying a second consistency relation for the trispectrum. Jonathan Ganc

33. But, contact interaction has no propagator • For connected terms, every ζ’picontracts with ζki , giving: . • This time, there is no propagator to give a k12 term. • So far, it looks like CI terms have no k12 factors. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

34. But, can the time integral blow up? • Remember that in-in formalism also has a time integral: • We still have to consider if this time integral can blow up in the folded limit, because then the contact interaction will contribute. h(η) = some scalar function of η Verifying a second consistency relation for the trispectrum. Jonathan Ganc

35. For slow-variation inflation, the time integral can’t blow up. From earlier: • There may also be terms with u’, but the effect is identical. • Being in folded limit (k2→k1, k4→k3) has no effect on the convergence of the integral.(Remember to calculate in the interacting vacuum: let η→η(1+iε).) Verifying a second consistency relation for the trispectrum. Jonathan Ganc

36. For slow-variation inflation, the time integral can’t blow up. • Thus (as I observed), we can’t get large CI terms for slow-variation models. Unfortunately, it’s not clear this will be true for more exotic models. • Generally speaking, it will probably hold in approximately De Sitter universes because then u∝e-ikη(Maldacena 2003). Verifying a second consistency relation for the trispectrum. Jonathan Ganc

37. Non Bunch-Davies vacuums • As another consideration, does our conclusion about the time integral still hold if inflation takes place in a non Bunch-Davies vacuum? • To represent a non Bunch-Davies vacuum, include negative frequency modes in u(k) (Chen et al 2009) : ; otherwise, the calculation is identical. • Normally, C+=1, C-=0. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

38. Non Bunch-Davies vacuum (cont’d) • Even for canonical single field inflation, there is a term (Seery, Lidsey, & Sloth 2007): • This yields a time integral: • This term diverges (actually, there will be some cutoff time for the integral so the term will be finite but it can still be very large). • So, CI terms can blow up for non Bunch-Davies vacuums. folded limit Verifying a second consistency relation for the trispectrum. Jonathan Ganc

39. VI. Summary of results for trispectrum • Squeezed limit: True consistency relation: will always hold. • Folded limit: Will hold for slow-variation inflation and a Bunch-Davies vacuum. Verifying a second consistency relation for the trispectrum. Jonathan Ganc

40. Further work • Try to generalize my result for the folded limit beyond slow-variation inflation. • Resolve a question about potential contamination of the trispectrum in the squeezed limit for the case of a non-standard kinetic term. • Further explore the implications of the trispectrum consistency relations for observation of gNL and τNL; can they be large for single-field inflation and, if so, when? Verifying a second consistency relation for the trispectrum. Jonathan Ganc

41. Questions? Verifying a second consistency relation for the trispectrum. Jonathan Ganc