Holography for inflation using conformal perturbation theory Paul - PowerPoint PPT Presentation

Holography for inflation using conformal perturbation theory Paul McFadden Perimeter Institute for Theoretical Physics 17.12.12 arXiv:1211.4550 with Adam Bzowski & Kostas Skenderis

Introduction Correlation functions of the primordial perturbations present some of our best clues to the physics of the early universe. Observationally, the power spectrum is consistent with a simple power-law � q � n s − 1 ∆ 2 S ( q ) = ∆ 2 S ( q 0 ) q 0 S ( q 0 ) ≈ 10 − 9 and spectral tilt n s ≈ 0 . 96 , i.e., nearly with amplitude ∆ 2 scale-invariant but with a slight red tilt. (Assuming no tensors or running.)

Introduction Where else in nature do we typically see power-law scaling of 2-point functions, with non-integer exponents? Critical phenomena: Systems undergoing continuous phase transition are described by a Euclidean QFT that flows to an IR fixed point. Universal scaling behaviour determined by operator dimensions in fixed point CFT.

Holographic cosmology Is there a connection? Holographic cosmology proposes that 4d cosmology admits a dual description in terms of a 3d non-gravitational QFT. Cosmic time evolution maps to inverse RG flow in the dual QFT: late times ↔ UV and early times ↔ IR. If the dual QFT is critical, then it will flow to a fixed point in the IR. Holographically, this is dual to a universe that is asymptotically de Sitter in the far past, i.e., that was inflating. The power-law scaling of 2-pt function in critical QFT translates to power-law scaling of cosmological power spectrum.

Refining the picture The RG flow nearest the IR fixed point is dominated by most nearly marginal irrelevant operator. Let’s assume this to be a single scalar operator O (= single-field inflation) of dimension ∆ IR = 3 + λ IR , where 0 < λ IR ≪ 1 . � d 3 x φ O . S = S CF T + We will see later that q 3 ∆ 2 φ 2 �OO� ∼ φ − 2 q − 2 λ IR , S ∼ n s − 1 = − 2 λ IR i.e., the tilt of the power spectrum on long wavelengths is controlled by the IR dimension of O .

Perturbative RG flows To compute behaviour on shorter wavelengths, need to understand flow further away from IR fixed point. One situation in which this is possible is when we have an RG flow between two closely separated fixed points. In this case, QFT correlators can be computed perturbatively in terms of the CFT correlators at either fixed point. (eg., Gaussian → Wilson-Fisher fixed point in d < 4 , calculable in either ǫ -expansion or 1 /N .)

Perturbative RG flows Crucially, the perturbation expansion is in the parameter controlling the separa- tion of the fixed points, which may be quite different from the QFT coupling. Here, this parameter will be n s − 1 ∼ λ IR ≪ 1 . Thus we can calculate even when the underlying QFT is strongly coupled, as will be the case for the QFT dual to conventional inflationary cosmology. In effect, we can simultaneously compute on both sides of the holographic correspondence!

Holography for inflation The dual QFT thus describes an RG flow between nearby UV and IR fixed points driven by a single nearly marginal scalar operator O . The corresponding cosmology is asymptotically de Sitter in the far past (inflationary epoch) and in the far future (dark energy?).

Holography for inflation Ultimately, a more complete holographic model would include radiation and matter dominated epochs, and presumably account for the large hierarchy between the inflationary and late-time dark energy scales. To do so would require changing the UV behaviour of the dual QFT, e.g., by allowing other operators besides O to influence RG flow. The simple single-field model we discuss today nonetheless captures the long-wavelength (IR) physics relevant for inflation.

Plan ➊ Perturbative RG flows, calculation of QFT correlators ➋ Holographic calculation of inflationary correlators ➌ Identifying the dual cosmology

References This talk is based on [1211.4550] with Adam Bzowski and Kostas Skenderis. Recent related work by Schalm, Shiu & van der Aalst [1211.2157] and also Mata, Raju & Trivedi [1211.5482]. Other relevant work: � γγγ � from CFT: Maldacena & Pimentel ’11, Bzowski, PM & Skenderis ’11. dS/CFT: e.g., Strominger ’01, Witten ’01, Maldacena ’02, Larsen, Leigh, van der Schaar ’02, van der Schaar ’04, Larsen & McNees ’04, etc. Conformal perturbation theory: Ludwig & Cardy ’87, A. Zamolodchikov ’87.

Perturbative RG flows Starting from the UV fixed point, we deform a 3d Euclidean CFT with a marginally relevant scalar operator O of dimension ∆ = 3 − λ where λ ≪ 1 � d 3 x ϕ Λ − λ O . S = S CF T + Here Λ is UV cutoff, ϕ is dimensionless coupling. The β -function may be found by demanding invariance of the partition function under changes of Λ β = − d ϕ d ln Λ = − λϕ + 2 πCϕ 2 + O ( ϕ 3 ) , where C is the OPE coefficient in the CFT α C O ( x 1 ) O ( x 2 ) = | x 12 | 2∆ + | x 12 | ∆ O ( x 2 ) + . . . as | x 12 | → 0 .

Perturbative RG flows If C is positive and of order unity, we obtain an RG flow from the UV CFT at ϕ = 0 to a nearby IR fixed point at λ ϕ = ϕ 1 + O ( λ 2 ) , ϕ 1 = 2 πC ≪ 1 . About IR fixed point β = λ ( ϕ − ϕ 1 ) + 2 πC ( ϕ − ϕ 1 ) 2 + O ( ϕ − ϕ 1 ) 3 , thus ∆ UV = 3 − λ (relevant) while ∆ IR = 3 + λ + O ( λ 2 ) (irrelevant).

Perturbative RG flows Since ϕ is small throughout the flow, we may remove higher order terms in the β -function by a field redefinition ϕ → ϕ + O ( ϕ 3 ) leaving β ( ϕ ) = − d ϕ/ d ln Λ = − λϕ + 2 πCϕ 2 . Integrating, we find ϕ 1 ϕ (Λ) = 1 + ( ϕ 1 /φ Λ λ ) where the constant of integration φ parametrises the asymptotic behaviour ϕ → φ Λ λ as Λ → 0 . Equivalently, φ is the dimensionful renormalised coupling in the UV QFT � � d 3 x Λ − λ ϕ O → d 3 x φ O .

Correlators Correlators may be computed perturbatively in λ starting from those in the CFT (denoted with a subscript zero). For example, � d 3 x ϕ Λ − λ O �O ( x 1 ) O ( x 2 ) � = �O ( x 1 ) O ( x 2 ) exp � − � � 0 ∞ ( − ϕ Λ − λ ) n � = I n n ! n =0 where � d 3 z 1 . . . d 3 z n �O ( x 1 ) O ( x 2 ) O ( z 1 ) . . . O ( z n ) � 0 . I n = To evaluate correlators in perturbed theory, we thus need to sum up the entire series of CFT correlators with integrated scalar insertions! ⇒ All terms in sum contribute at leading order: ϕ n ∼ λ n , I n ∼ 1 /λ n

Correlators Amazingly, this can be achieved! The argument is subtle, but the basic idea is to use the OPE to write a differential equation for d I n / dΛ in terms of I n − 1 . Λ d 3 z 1 �O ( x 1 ) O ( x 2 ) O ( z 1 ) � 0 we find � e.g., for I 1 = d I 1 C dΛ = − 2(4 π Λ 2 ) Λ 3 − λ I 0 + . . . One can then integrate w.r.t. Λ and fix arbitrary function by requiring a smooth limit λ → 0 . We then remove the cutoff Λ → 0 . Can extend argument to case of n -integrated insertions.

Correlators Skipping over the technical details (see paper), we can write final result as a sum of CFT correlators with shifted dimensions ∞ �O ( x 1 ) O ( x 2 ) � = 1 ( n + 3)( n + 2)( n + 1)( − φ � ϕ 1 ) n �O ∆ ′ ( x 1 ) O ∆ ′ ( x 2 ) � 0 , 6 n =0 where ∆ ′ = 3 − λ ( n + 2) / 2 . We may then resum the binomial series and Fourier transform: � = π 2 1 + φ ϕ 1 q − λ � − 4 12 αq 3 − 2 λ � � �O ( q ) O ( − q ) � . Similar methods may be used to compute 3-point functions, as well as correlators involving the stress tensor T ij .

Plan ➊ Perturbative RG flows, calculation of QFT correlators ➋ Holographic calculation of inflationary correlators ➌ Identifying the dual cosmology

Holography for cosmology Having computed all QFT 2- and 3-point functions of interest, let’s now discuss holographic cosmology. Our tasks are twofold: ➊ To compute cosmological 2- and 3-point functions from QFT correlators ➋ To identify the corresponding slow-roll inflationary model Both are simply accomplished, but first let’s review briefly some relevant background.

Holography for cosmology From an observational perspective we are primarily interested in computing tree level cosmological correlators. This requires perturbatively quantising small fluctuations about an FRW background geometry. On the dual QFT side, this corresponds to working in large- N perturbation theory. A hologaphic map may then be constructed via a simple analytic continuation between perturbed FRW cosmologies and holographic RG flows.

Holography for cosmology Work with Skenderis ’09-’11, Bzowski ’11

Holography for cosmology Given a solution for a perturbed holographic RG flow in Euclidean signature, a perturbed FRW cosmological solution in Lorentzian signature is given by κ 2 V → − κ 2 V, q → − iq. q 2 is the magnitude of 3-momentum on spatial slices and � where q = � κ 2 = 8 πG N . ✦ For explicit proof at quadratic order in gauge-invariant perturbation theory, in case of gravity with minimal scalar and potential, see [1104.3894]. ✦ Bunch-Davies vacuum ↔ smooth in the interior.

Holography for cosmology On the dual QFT side, this is equivalent to performing the following analytic continuation on large- N correlators N 2 → − N 2 , q → − iq. Thus we first compute correlators in the regular QFT dual to the holographic RG flow, then continue. As we will be able to check explicitly, this prescription indeed yields the correct inflationary 2- and 3-pt correlators.