Quite a lot activity in the last couple of years. Complementary to Emiliano. Marko & Gabriele
I talk about the CMB, but also the other experiments…
I talk about the CMB, but also the other experiments…
Scale invariance is easy to guess looking around. No antropic from the tilt.
Without reference to models. Symmetries of dS -> Symmetries of correlation functions (think about Minkowski). 6+1+3.
It could be broken to discrete subgroup,as in models with oscillations.
No anthropic for NG
Rare decay in SM. We are entering in the regime probing all these models
Dilations move a slide to the next, so with shift-symmetry…
Easily n-point function
No slow-roll approx: also in the presence of feautures. Holds also for non-inflationary models.
No tilt suppression!
Actually we did not consider the graviton exchange
But without CFT! Strong coupling?
Eventually we will be interested in quasi massless fields as they survive. 3pf is fixed by conformal invariance.
All the standard c_s shapes are not conformal invariant.
How to parametrize a general conf-invariant 4-point function?
Is there some non-linearly realized symmetry?
P. Creminelli: Symmetries of Cosmological Perturbations
Paolo Creminelli, ICTP TriesteSymmetriesofcosmological perturbationsPC, 1108.0874 (PRD)with J. Noreñ a and M. Simonovi , 1203.4595 (JCAP)ćwith A. Joyce, J. Khoury and M. Simonovi , 1212.3329 (JCAP)ćwith R. Emami, M. Simonovi and G. Trevisan, 1304.4238ćwith J. Noreñ a, M. Simonovi and F. Vernizzi, in progressć
Origin of scale invarianceInflation takes place in ~ dS:Hyperboloid in 4+1 dimIsometry group: SO(4,1)• Translations, rotations: ok• Dilations scale-invarianceConstant out of H-1
+ (generalized) slow-rollWe are expanding around a time-dependent backgroundWe need parameters to change slowly:1. Deviation of metric from dS. (One can consider the limit at fixed)2. Breaking due to scalar backgroundE.g.Dilation x Shift Symmetry Diagonal subgroupObserving an approximate dS, coming to anend|ns – 1| ~ 1/Ne ~ % is a general prediction of inflation
Slow-roll = weak coupling = GaussianityCompare with Higgs: λ ∼ 0.12We were born Gaussian
• Any modification enhances NG1. Modify inflaton Lagrangian. Higher derivative terms (ghost inflation, DBIinflation), features in potential2. Additional light fields during inflation. Curvaton, variable decay width…3. Alternatives to inflation• Potential wealth of informationAny signal would be a clear signal of something non-minimalWhat are symmetry properties of these functions?Smoking gun for "new physics"
Special conformalThe inflaton background breaks these symmetries
Non-linearly realized symmetriesThe inflaton background breaks the symmetry. Spontaneously.We expect the symmetry to be still there to regulate soft limit (q 0) ofcorrelation functions (Ward identities)For example. Soft emission of πsFor space time symmetries:number of Goldstones broken generators≠Manohar Low 01We expect Ward identities to say somethingabout higher powers of q
3pf consistency relationSqueezed limit of the 3-pointfunction in single-field modelsMaldacena 02PC, Zaldarriaga 04Cheung etal. 07The long mode is already classical when the other freeze andacts just as a rescaling of the coordinatesSimilar to the absence ofisocurvature
3pf consistency relationSingle-field 3pf is very suppressed in the squeezed limit
Phenomenologically relevant1. A detection of a local fNL would rule out any single-field model1. Some of the experimental probes are sensitive only to squeezed limits• Scale dependent bias• CMB µ distorsionDalal etal 07Pajer and Zaldarriaga 12
Extension to the full SO(4,1)A special conformal transformation induces a conformal factor linear in xPC, Noreñ a and Simonovi 12ć
Adiabatic mode including gradientsAdiabatic modes are… nothing (locally). They can be constructed from unfixed gaugetransformations (k=0)In ζ gauge:• Cannot touch t• Conformal transformation of the spatial coordinates:• Impose it is the k 0 limit of a physical solution• b and λ are time-independent + need a time-dep translation to induce the NiWeinberg 03Long wavelength approx of an adiabatic mode up to O(k2)
Conformal consistency relations(Assuming zero tilt for simplicity)2- and 3-pf only depends on moduli and qiDi reduces to:The variation of the 2-point function is zero: no linear term in the 3 pfPC, DAmico, Musso and Noreñ a, 11Hinterbichler, Hui and Khoury 12Kheagias, Riotto 12Goldbeger, Hui and Nicolis 13Goldberger, Hinterbichler, Hui, Khoury in progressConformal consistency relations as Ward identities and with OPEmethods
Small speed of sound4pf: scalar exchange diag.sdo not contribute to squeezed limit• At the level of observables, the non-linear relation among operators in the Lagrangian• Squeezed limit is 1/cs2while the full 4pf is 1/cs4• A large 4pf cannot have a squeezed limit
We are looking at the tail of CMB…The future is in LSS (hopefully):much larger volume
Consistency relations for Large ScaleStructuresThe construction of an adiabatic mode works outside the sound horizonDuring MD (or with Λ) we can study squeezed limit of correlation functionsinside H-1Kehagias Riotto 13Peloso Pietroni 13A constant gravitational field can be reabsorbed by achange of coordinatesNon-relativistic limit:Non-perturbative in the shortmodes !
Connecting to initial conditionsRelativistic formulation, encompasses the out of H evolutionMD:Non-relativisticConformaltransformationPhysically: long mode has been a coordinate transformation, since inflation,until nowPC, Noreñ a, Simonovi and Vernizzićin progress
Conclusionso Planck• Tilt of the power spectrum• Very Gaussian initial conditionso Symmetries constrain soft limits of cosmological correlation functions• Consistency relation for primordial correlators• Consistency relations for the late Universe
Scale Conformal invarianceIf perturbations are created by a sector with negligible interactions with the inflaton,correlation functions have the full SO(4,1) symmetryThey are conformal invariantIndependently of any details about this sector, even at strong couplingSame as AdS/CFTAntoniadis, Mazur and Mottola, 11Maldacena and Pimental, 11PC 12Kehagias, Riotto 12Mata, Raju, Trivedi 12Curvaton, modulated reheating…
Scale Conformal invarianceWe are interested in correlators at late timesThis is the transformation of a primary of conformal dim ∆Example:
Massless scalarsZaldarriaga 03Seery, Malik,Lyth 08Everything determined up to two constantsIndependently of the interactions!The conversion to ζ will add a local contribution:
4-point functionNot so obvious it is conformal invariant…I can check it in Fourier space Maldacena and Pimental, 11In general: 2 parameters instead of 5
ThereforeIf we see something beyond the spectrum• Something not conformal would be a probe of a "sliced" de Sitter• Something conformal would be a probe of pure de Sitter
Non-linear realization of dS isometriesNotice the two meanings of SO(4,1):• Isometry group of de Sitter• Conformal group of 3d EuclideanIn decoupling + dS limit: the inflaton breaks spontaneoulsy SO(4,1).It is still non-linearly realized
Conformal consistency relations with tilt• Dilation part evaluated on a non-closed polygon• Verified in modes with oscillations in the inflaton potential
Generalizations• Graviton correlation functions:• Soft internal lines• More than one q going to zero togetherInduce long graviton withNot more than one…