Investigating a second consistency relation for the trispectrum

Investigating a second consistency relation for the

trispectrumJonathan Ganc

Qualifier for PhD CandidacyOctober 12, 2009

Department of Physics, University of Texas

I. Cosmic InflationII. Characterizing inflation, calculating non-

Gaussianity; the in-in formalismIII. The bispectrum consistency relation for

single-field inflationIV. The trispectrum has at least one

consistency relationsV. Is there another consistency relation for

the trispectrum?VI. Conclusion and further work

Overview

2Verifying a second consistency relation for the trispectrum. Jonathan Ganc 10/12/2009

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.

I. Cosmic 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.

What produced inflation?


For a large class of single field inflationary models, we can write the field Lagrangian as ℒ=P(X,φ), where X≡-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.

10/12/2009Verifying a second consistency relation for the trispectrum. Jonathan Ganc 5

“Slow-variation” inflationChen et al 2007

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).

Inflation seeds large-scale fluctuations through quantum fluctuations


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.

Curvature perturbation ζ


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

Inflation produces large-scale fluctuations cont’d

A straightforward calculation yields the power spectrum Pζ(k) of ζ:

where(originally calculated, for cs≠1, by Garriga and Mukhanov

1999)


II. Characterizing inflation: the power spectrum

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.

Characterizing inflation: non-Gaussianity


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).

Bispectrum calculations


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.

III. Bispectrum consistency relation

1210/12/2009

power spectrum spectral tilt

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!

Bispectrum consistency relation: the significance

13

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).1410/12/2009

Maldacena 2003

Verifying a second consistency relation for the trispectrum. Jonathan Ganc

Bispectrum consistency relation: the argument

Any measurable quantity f can ultimately only be a function of physical (not comoving) distance, so:

Figure adapted from a talk by Komatsu 2009.

Bispectrum consistency relation: the argument cont’d

Remember: aeff(t)=eζk3(x)

a(t)

Creminelli and Zaldarriaga 2004, Cheung et al 2008

Expanding in terms of the background field ζk3


Creminelli and Zaldarriaga 2004, Cheung et al 2008Bispectrum consistency

relation: the argument cont’d

Fourier Transform

|Δx| ≈ 1/k1,1/k2

Finally, we correlate the result with ζk3:

as desired.



relation: the argument cont’d

The important thing to note is that we made no assumptions except that we could expand <ζk1ζk2> in terms of a single background field ζk3.

Thus, the relation holds for any single field inflation model.



relation: summary

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).

IV. The trispectrum


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:

Trispectrum shapes

2010/12/2009

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.

Maldacena-like 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.


V. Is there another trispectrum consistency relation?

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)

Seery et al 2009 argument: scalar exchange


ns-1=O(ε); ε≈10-2

Diagram:

An essentially identical argument for graviton exchange yields:

This term goes as O(Pζ3ε), so it’s

dominant over the scalar exchange term (O(Pζ

3ε2)). (χ12,34≡ φ1 - φ3 is the angle between the projections

of k1 & k3 on the plane orthogonal to k12)

Seery et al 2009 argument: graviton exchange


Diagram:

r=scalar-tensor ratio =O(ε)


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 must give:

Thus <ζ4>SE+GE ∝ Pζ(k12) ∝ k12-3.

For local form:ζ= ζg+3/5fNLζg

2+9/25gNLζg3

we find τNL=36/25fNL2. If the contact

interaction is sufficiently small, then fNL

2=25/64r cos2χ12,34.

=O(ε2) =O(ε)

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 ∝ k12

0. However, they don’t claim that CI term

will be negligible in more general models.

What about the contact interaction (CI)


...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...

I noticed that...


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.


Reviewing in-in formalism

scalar exchange

graviton exchange

contact interaction

the 3 connected tree diagrams correspond to terms from the in-in formalism:


Applying the in-in formalism

Look at SE term:

The bracketed term equals the sum of all fully contracted terms, where (Chen et al 2009):

How does a k12-3 factor arise in

exchange terms?


Note that the time variable t is uniquely given by the momentum variable (e.g. p’⇒(p’,t’) or k⇒(k,t))

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.


exchange terms? (cont’d) Scalar exchange

1 2 34

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.


exchange terms? (cont‘d)


GE

SE

For connected terms, every ζ’pi contracts 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.

But, contact interaction has no propagator


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.

But, can the time integral blow up?


h(η) = some scalar function of η

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).)

For slow-variation inflation, the time integral can’t blow up.


From earlier:

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).

For slow-variation inflation, the time integral can’t blow up.


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.

Non Bunch-Davies vacuums


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.

Non Bunch-Davies vacuum (cont’d)


folded limit

Squeezed limit:

True consistency relation: will always hold.

Folded limit:

Will hold for slow-variation inflation and a Bunch-Davies vacuum.

VI. Summary of results for trispectrum


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?

Further work



Questions?

Documents

Investigating a second consistency relation for the trispectrum