1. Jonathan Ganc Qualifier for PhD Candidacy October 12, 2009 Department of Physics, University of Texas

2. I. Cosmic Inflation II. Characterizing inflation, calculating non- Gaussianity; the in-in formalism III. The bispectrum consistency relation for single-field inflation IV. The trispectrum has at least one consistency relations V. Is there another consistency relation for the trispectrum? VI. Conclusion and further work 2Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

3.  A period of exponential expansion in the very early universe with a nearly constant Hubble parameter: a(t)=a 0 e ∫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 e 60. 3Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

4.  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. 4Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

5.  For a large class of single field inflationary models, we can write the field Lagrangian as ℒ=P(X,φ), where X≡-1/2g μν ∂ μ φ∂ ν φ.  The speed of sound c s 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. 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc5 Chen et al 2007

6.  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). 6Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

7.  Fluctuations in the inflaton δφ are converted to perturbations in the spatial curvature ζ:  ζ produces anisotropy in the CMB temperature and in the matter distribution. 7Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

8.  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. 8

9.  A straightforward calculation yields the power spectrum P ζ (k) of ζ: where (originally calculated, for c s ≠1, by Garriga and Mukhanov 1999) 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc9

10.  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 t nears -∞) in order to calculate in the interacting vacuum. 10 Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

11.  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). 11Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

12.  Maldacena (2003) used his explicit result for the bispectrum (in single field slow-roll inflation) to find a bispectrum formula in the “squeezed limit” (k 3 ≪k 1 ≈k 2 ):  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. 12 10/12/2009 power spectrum spectral tilt

13.  The consistency relation involves measurable quantities: trispectrum, power spectrum P ζ (k), and spectral tilt n s.  Assuming local form for non-Gaussianity (Komatsu and Spergel 2001): ζ= ζ g +3/5f NL ζ g 2, we find f NL =5/12 (1-n s ).  Observationally: n s =0.960 ± 0.013 (68% CL) (Komatsu et al 2009). f NL =38±21 (68% CL) (Smith et al 2009)  It does not look like f NL =5/12 (1-n s )=0.017. If this holds up, we have ruled out single field inflation! 13

14.  Expand  We want to find the correlation as k 3 ≪k 1 ≈k 2. In comoving gauge, the metric is: ds 2 = -dt 2 +e 2ζ(x) a 2 (t)dx 2.  For small distances (i.e. corresponding to the length scales of the k 1, k 2 modes), ζ k3 is approximately constant; thus, we can consider the effect of ζ k3 as a rescaling of the scale factor: a eff (t)=e ζ k3 (x) a(t). 14 10/12/2009 Maldacena 2003 Verifying a second consistency relation for the trispectrum. Jonathan Ganc

15.  Any measurable quantity f can ultimately only be a function of physical (not comoving) distance, so: Figure adapted from a talk by Komatsu 2009. Remember: a eff (t)=e ζ k3 (x) a(t) Creminelli and Zaldarriaga 2004, Cheung et al 2008

16.  Expanding in terms of the background field ζ k3 16Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 Creminelli and Zaldarriaga 2004, Cheung et al 2008 Fourier Transform |Δx| ≈ 1/k 1,1/k 2

17.  Finally, we correlate the result with ζ k3 : as desired. 17Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 Creminelli and Zaldarriaga 2004, Cheung et al 2008

18.  The important thing to note is that we made no assumptions except that we could expand in terms of a single background field ζ k3.  Thus, the relation holds for any single field inflation model. 18Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 Creminelli and Zaldarriaga 2004, Cheung et al 2008

19.  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). 19Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

20.  We only have a non-zero trispectrum when Σ i k i =0.  Thus, the wavenumbers form a closed quadrilateral.  We name certain configurations based on the relative length of sides: 20 10/12/2009

21.  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. 21Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

22.  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. 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc22

23.  Expand in terms of ζ k12 :  Note that ζ k34 =ζ k12. Thus, we can correlate ζk12, ζk12 over ζ k12 :  Thus, SE =O(P ζ 3 ε 2 ) 23Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 n s -1=O(ε); ε≈10 -2 Diagram:

24.  An essentially identical argument for graviton exchange yields: This term goes as O(P ζ 3 ε), so it’s dominant over the scalar exchange term (O(P ζ 3 ε 2 )).  (χ 12,34 ≡ φ 1 - φ 3 is the angle between the projections of k 1 & k 3 on the plane orthogonal to k 12 ) 24Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 Diagram: r=scalar-tensor ratio =O(ε)

25. 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc25  Seery, Sloth, and Vernizzi 2009 determined that, in the folded limit, the scalar exchange (SE) and graviton exchange (GE) terms must give:  Thus SE+GE ∝ P ζ (k 12 ) ∝ k 12 -3.  For local form: ζ= ζ g +3/5f NL ζ g 2 +9/25g NL ζ g 3 we find τ NL =36/25f NL 2. If the contact interaction is sufficiently small, then f NL 2 =25/64r cos2χ 12,34. =O(ε 2 ) =O(ε)

26.  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. CI ∝ k 12 0.  However, they don’t claim that CI term will be negligible in more general models. 26Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

27. ...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... 27Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

28.  Whether kinematic or explicit, our calculations are done within the framework of the in-in formalism: where H I is the interaction Hamiltonian in the interaction picture and ζ I is ζ in the interaction picture. 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc28

29. scalar exchange graviton exchange contact interaction  the 3 connected tree diagrams correspond to terms from the in-in formalism: 10/12/2009 Verifying a second consistency relation for the trispectrum. Jonathan Ganc29

30.  Look at SE term:  The bracketed term equals the sum of all fully contracted terms, where (Chen et al 2009): 30Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 Note that the time variable t is uniquely given by the momentum variable (e.g. p’⇒(p’,t’) or k⇒(k,t)) Scalar exchange

31.  All connected terms have the following (or equivalent) contractions: 1: 2: 3: 4: Then,.  But, u(k 12 )∝k 12 -3/2, and we see each term has a factor u(k 12,t’)u * (k 12,t’’)∝k 12 -3.  Thus, SE ∝k 12 -3. Scalar exchange 1 2 3 4

32.  In the derivation, the essential point is having two connected vertices.  Since the situation is identical with GE terms, GE ∝k 12 -3.  Graphically, this effect is equivalent to the fact that the exchange terms have a propagator. 32Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 GE SE

33.  For connected terms, every ζ’ pi contracts with ζ ki, giving:.  This time, there is no propagator to give a k 12 term.  So far, it looks like CI terms have no k 12 factors. 33Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

34.  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. 34Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 h(η) = some scalar function of η

35.  There may also be terms with u’, but the effect is identical.  Being in folded limit (k 2 →k 1, k 4 →k 3 ) has no effect on the convergence of the integral. (Remember to calculate in the interacting vacuum: let η →η (1+iε).) 35Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 From earlier:

36.  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). 36Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

37.  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. 37Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

38.  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. 38Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009 folded limit

39.  Squeezed limit: True consistency relation: will always hold.  Folded limit: Will hold for slow-variation inflation and a Bunch-Davies vacuum. 39Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

40.  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 g NL and τ NL ; can they be large for single-field inflation and, if so, when? 40Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

41. Verifying a second consistency relation for the trispectrum. Jonathan Ganc41