Inflationary Effects from High-Energy Physics Mark G. Jackson Institut D’Astrophysique Paris / Laboratoire AstroParticule et Cosmologie Paris Centre for Cosmological Physics Collaborators: K. Schalm, B. Wandelt, F. Bouchet, D. Langlois HKUST-IAS February 18, 2013
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Inflationary Effects from High-Energy Physics!Mark G. Jackson!Institut D’Astrophysique Paris / !Laboratoire AstroParticule et Cosmologie!Paris Centre for Cosmological Physics!Collaborators: K. Schalm, B. Wandelt,!! ! F. Bouchet, D. Langlois!
HKUST-IAS!February 18, 2013!
Inflation is phenomenologically successful, but...! Ideally it should be embedded in a quantum theory of
gravity, such as superstring theory. !
Outline!
The Opportunity of Cosmology! Modified Vacua! Correlation Function Evaluations! Position-Space Correlations! Future Directions!
Effective Actions (in Cosmology)! To test high-energy theories at low energy
we rely upon Wilsonian effective actions:!
Unfortunately, standard techniques rely upon energy conservation, which is absent during cosmological expansion!
To put inflation on a fundamental basis, we need to construct effective actions in a cosmological background.!
Removing high-energy modes!
The Opportunity of Cosmology: Sensitivity to High Energies!
Scale of new physics!
Scale of inflation!
either lack a fundamental theory, rely upon energy conservation, or assume a particular vacuum state.!
Observed CMB fluctuations today have low energy, but this is due to the cosmological redshifting.!
They should be sensitive to high-energy physics, possibly even the Planck scale, as!
Previous effective descriptions, !
1. New Physics Hypersurface,!2. Boundary Effective Field Theory,!3. Weinberg; Senatore & Zaldarriaga!
The power spectrum is simply the 2-pt correlation function of inflaton field fluctuations:!
Without specifying any particular model, what does dimensional analysis suggest the magnitude of corrections are?!
The Primordial Power Spectrum, and other Correlation Functions!
Inflaton Field Effective Action! Consider the effective action for φ:!
The freezeout scale is p=H, thus the 2-pt function is!
Only even powers of p are allowed in Seff, so we have an expansion in (H/M)2.!
! Which is disastrous, since H/M ~ 0.01!! (Brandenberger, Burgess, Cline, Danielsson, Easther, Greene,
Lemieux, Kaloper, Kinney, Kleban, Lawrence, Martin, Schalm, Shenker, Shiu, v.d. Schaar, Susskind)!
Fortunately, there appears to be a loophole!! (Easther, Greene, Kinney, v.d. Schaar, Schalm, Shiu).! Note that time-localized (‘boundary’) terms are one energy-
dimension lower, and thus would scale only as H/M:!
This changes the boundary condition of the inflaton, much like QM scattering from a δ-function potential:!
A Possible Solution:Vacuum State Modification!
Bunch-Davies boundary conditions
Re(φ)
Excited state boundary conditions
A Possible Solution:Vacuum State Modification!
In the Hamiltonian description, the Bunch-Davies vacuum is simply the familiar condition!
! which now becomes generalized to a squeezed coherent state :!
Such a modified state has a very characteristic signature!!
These ‘wiggles’ are a generic, model-independent feature of quantum gravity*, with all new physics encoded in β.!
Effect of Vacuum Choice onPower Spectrum!
Easther, Greene,!Kinney, Shiu ‘01!
Corrections of order |β|!
Ps(k)!
ln k!
Power-law result!
*And e.g. axion monodromy inflation by Silverstein and Westphal ‘08; !
Flauger, McAllister, Pajer, Westphal, Xu ‘09!
Parametrizing TP Physics ! Martin and Ringeval ‘06 searched for these
oscillations in the WMAP3 data using a template:!
Oscillations controlled by these parameters !
Parametrizing TP Physics!
They found suggestions of such oscillations ! In the fortunate event of detection, what is the theoretical
implication?! Martin and Ringeval ‘06!
Resonant frequencies!
Power Spectrum Corrections from a Toy UV Model!
We (MGJ, Schalm ‘10) recently developed the procedure to evaluate correlation functions from high-energy physics.!
Begin with inflating system,!
! and add (for example) Yukawa interactions to a heavy field χ : !
The power spectrum can then be computed using the in-in formalism:!
Note that this can be interpreted as an in-out correlation using!
This suggests we should transform into the new ‘Keldysh’ field basis given by!
In this basis the action is now!
The free field solutions are!
,!
Power Spectrum Corrections from a Toy UV Model!
Feynman Rules in Keldysh Basis! The correlations can now be evaluated using these:!
The interactions are given by:!
g! g! g!
Power Spectrum Corrections! 2-pt correlation can then be computed using normal
methods, producing four Feynman diagrams:!
Which are significant? We need some good approximations!!
Power Spectrum Corrections! Each vertex is an integral over the time of interaction, and has
the following form:!
This admits a stationary phase approximation near the moment of energy-conservation,!
The vertex (to leading order in H/M) is then simply!
τ
k1!
k2!
k1+k2!
k1!
k2!k1+k2!
momentum!energy!
Power Spectrum Corrections! Now things simplify immensely:!
These cancel!!
P(A) = - 2 P(B)!
Consider this diagram, with loop momentum q:!
k and q will determine the time of interaction τ via the stationary phase approximation. Let us also define an energy Eq for this light field,!
Inverting this allows us to transform the q-integral into (τ,Eq):!
How do you impose an energy cutoff with no energy conservation?!
q!k! k!
τ! τ!
Phase space of energy cutoff!
Power Spectrum Corrections!
Thus high-energy physics produces a nearly scale-invariant shift in the power spectrum.!
The integral over loop momentum smears out oscillations into an overall enhancement!
It is of magnitude ~ HΛ3/M4, which is ~ H/M!
+!
3-pt Correlation Function:The Bispectrum!
We may then consider the 3-pt correlation, !
A free field theory will produce a vanishing 3-pt correlation, and so the bispectrum measures interactions: (Creminelli ‘03) !
The shape of the momenta triangle indicates when it was produced (Babich, Creminelli, Zaldarriaga ‘04) and can serve to easily differentiate models (eg Khoury and Piazza ‘08)!
Adding!
Very small!
Potentially very large!
k2 k3
k1
Bispectrum Analysis, TNG! Bispectrum analysis has recently gotten more
sophisticated, without the need for templates...!
! Bispectrum of WMAP5 (left) and prediction for a sample inflationary model (right) in Fergusson, Liguori and Shellard. Density contours (light blue positive and magenta negative) are shown for l ≤ 500.!
High-Energy Corrections to the Bispectrum!
Nearly identical to power spectrum evaluation:!
These cancel!!
B(A) = - 2 B(B)!
Thus high-energy physics produces a nearly scale-invariant enfolded-triangle bispectrum!
This enfolded shape was precisely the shape predicted by !! R. Holman and A. Tolley ‘07; D. Meerburg, P. Corasaniti, MGJ, J.
P. van der Schaar ‘09! The loop momentum again smears any oscillations into an
overall enhancement.!
+!
where!
Heavy Field Corrections to the Bispectrum!
There are three tree-level trispectrum corrections: !
Naïve estimate:!
Actual answer: !
All states being on-shell (no loops) means wrong-way frequencies, and all interactions happen in imaginary time!!
Heavy Field Corrections to the Trispectrum!
τ
k1!
k2!
k1+k2!
k1!
k2!k1+k2!
wrong sign!!
Effective Action Construction! What is the quasi-de Sitter analogue of this?!
The effective φ4 interactions can be obtained by integrating out the χ-field, producing 3 terms:!
Similar to correlation function, except we don’t contract external φ-fields !
What is the low-energy approximation? !
Effective Action Construction!
Effective Action Construction!
Note that this is non-local! !
Oscillations (?)! So integrating out heavy modes does not produce
oscillations.! But there are other mechanisms which could produce
them, such as sharp turns in field-space!
! (Achucarro, Patil, Palma, Langlois, Gao, Mizuno, et al;! work in progress with Langlois) !
But What If…! If we assume oscillations are there, via some other
mechanism…what is the most efficient way to observe them?!
(Inverse)!Fourier!
Transform!
k! x!
The temperature fluctuations are approximately!
Consider a scale-invariant power spectrum with oscillations,!
The power spectrum is then!
Direction-Space Power Spectrum!
The majority of the contribution will come from the extremum,!
The fluctuations near this point are!
The power spectrum is then!
Direction-Space Power Spectrum!
Logarithmic oscillations smear signal out on all signals, so there is no peak in N-space!
Now reconsider the Trispectrum with excited initial state:!
What is the most efficient way to look for this signature in e.g. Planck data? !
Trispectrum of!Modified Initial State!
Phase oscillates!as ~ M/H !
Direction-space Trispectrum! The position-space trispectrum is:!
! where ! !
Using the Fourier representation of the δ-function!
allows us to put it in the form!
~!
A huge simplification occurs by noticing that the problem factorizes into two ‘halves’ coupled only via w:!
Solving the Constraints, 1
We can thus solve each half separately, then combine later via w-constraint, or conservation of momentum. The left-half consists of k1, k2 -constraints:!
left! right!
k1!
k2!
k3!
k4!
An exact solution exists:!
The identical result applies for the right half!
Solving the Constraints, 2
Color indicates!time of interaction!
Solving the Constraints, 3 Substituting these expressions back into F gives!
Taking the equation of motion dF/dw yields !
Which is equivalent to the conservation of momentum,!
Remember: this needs!! to satisfy both sides of!! the system!!
Solving the Constraints, 4! In general this is very hard to solve, but there is one
family of solutions which is easy to construct:!
Several Directions ofFuture Exploration!
Effective Field!Theory of Inflation!
Parameterization for Planck data (with F. Bouchet and B. Wandelt)!
Stability of Vacuum (with B. Greene, D. Kabat , K. Eckerle, A. Brainerd)!
Effective Description of Hybrid Inflation (with D. Langlois)!
Renormalization Group Flow (with K. Schalm)!
Conclusion! We are anticipating the deluge of upcoming
precision cosmological data in several ways: !1. Effective Field Theory in Inflation can now be
performed for fundamental theories!2. This produces calculable corrections to the
spectrum, bispectrum, and trispectrum, as well as effective action !
3. Currently working with the Planck team for search templates.!
4. We will soon be using this to explore theoretical issues such as vacuum structure and general classes of inflation models.!
谢谢!!
Effective Action Construction! Fourier transform into energy basis:!
One can then use the stationary-phase approximation like before:!
The first diagram is just:!
Re-write using!
! to obtain the approximation!
! !! where!
Effective Action Construction!
(MGJ ‘12)!
Effective Action Construction! One can define the energy at the moment of
interaction as!
! The energy transferred is then!
! !
Effective Action Construction!
Rapid phase!oscillations!
Most likely near!U/u << 1, !
Effective density matrix coupling is!
(MGJ ‘12)!
When is this Valid?! The energy transferred through the χ-field is!
! As long as this is small, the effective action is trustworthy.!
! The (ΔE/M)2 correction is easy to calculate, and is!
(MGJ ‘12)!
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