SLIDE 1
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
SLIDE 2 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 ∆2
S(q) = ∆2 S(q0)
q q0 ns−1 with amplitude ∆2
S(q0) ≈ 10−9 and spectral tilt ns ≈ 0.96, i.e., nearly
scale-invariant but with a slight red tilt. (Assuming no tensors or running.)
SLIDE 3
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.
SLIDE 4
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.
SLIDE 5 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 + λIR, where 0 < λIR ≪ 1. S = SCF T +
We will see later that ∆2
S ∼
q3 φ2OO ∼ φ−2q−2λIR, ns − 1 = −2λIR i.e., the tilt of the power spectrum on long wavelengths is controlled by the IR dimension of O.
SLIDE 6
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.)
SLIDE 7
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 ns−1 ∼ λIR ≪ 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!
SLIDE 8
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?).
SLIDE 9
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.
SLIDE 10
Plan
➊ Perturbative RG flows, calculation of QFT correlators ➋ Holographic calculation of inflationary correlators ➌ Identifying the dual cosmology
SLIDE 11
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.
SLIDE 12 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 S = SCF 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 = − dϕ d ln Λ = −λϕ + 2πCϕ2 + O(ϕ3), where C is the OPE coefficient in the CFT O(x1)O(x2) = α |x12|2∆ + C |x12|∆ O(x2) + . . . as |x12| → 0.
SLIDE 13
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 ϕ = ϕ1 + O(λ2), ϕ1 = λ 2πC ≪ 1. About IR fixed point β = λ(ϕ − ϕ1) + 2πC(ϕ − ϕ1)2 + O(ϕ − ϕ1)3, thus ∆UV = 3 − λ (relevant) while ∆IR = 3 + λ + O(λ2) (irrelevant).
SLIDE 14 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(ϕ3) leaving β(ϕ) = −dϕ/d ln Λ = −λϕ + 2πCϕ2. Integrating, we find ϕ(Λ) = ϕ1 1 + (ϕ1/φΛλ) where the constant of integration φ parametrises the asymptotic behaviour ϕ → φΛλ as Λ → 0. Equivalently, φ is the dimensionful renormalised coupling in the UV QFT
SLIDE 15 Correlators
Correlators may be computed perturbatively in λ starting from those in the CFT (denoted with a subscript zero). For example, O(x1)O(x2) = O(x1)O(x2) exp
∞
(−ϕΛ−λ)n n! In where In =
- d3z1 . . . d3znO(x1)O(x2)O(z1) . . . O(zn)0.
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: ϕn ∼ λn, In ∼ 1/λn
SLIDE 16 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 dIn/dΛ in terms of In−1. e.g., for I1 =
- Λ d3z1O(x1)O(x2)O(z1)0 we find
dI1 dΛ = −2(4πΛ2) C Λ3−λ I0 + . . . 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
- f n-integrated insertions.
SLIDE 17 Correlators
Skipping over the technical details (see paper), we can write final result as a sum of CFT correlators with shifted dimensions O(x1)O(x2) = 1 6
∞
(n + 3)(n + 2)(n + 1)(− φ ϕ1 )nO∆′(x1)O∆′(x2)0, where ∆′ = 3 − λ(n + 2)/2. We may then resum the binomial series and Fourier transform:
= π2 12 αq3−2λ 1 + φ ϕ1 q−λ−4 . Similar methods may be used to compute 3-point functions, as well as correlators involving the stress tensor Tij.
SLIDE 18
Plan
➊ Perturbative RG flows, calculation of QFT correlators ➋ Holographic calculation of inflationary correlators ➌ Identifying the dual cosmology
SLIDE 19
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.
SLIDE 20
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.
SLIDE 21
Holography for cosmology
Work with Skenderis ’09-’11, Bzowski ’11
SLIDE 22 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 κ2V → −κ2V, q → −iq. where q =
- q 2 is the magnitude of 3-momentum on spatial slices and
κ2 = 8πGN. ✦ 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.
SLIDE 23
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.
SLIDE 24 Holographic formulae
Complete holographic formulae were calculated in [1011.0452, 1104.3894] with
- Skenderis. Here, these become:
- ζ(q)ζ(−q)
= 1 2λ2φ2 O(q)O(−q) ,
= δss′ A(q),
= φ O(q1)O(q2)O(q3) − 3
j=1
O(qj)O(−qj)
j=1
O(qj)O(−qj)
= −λφ O(q1)O(q2)T (s3)(q3) + 3 O(q)T (s)(−q)
O(q1)O(−q1)
A(q3) , where A(q) =
s=±
T (s)(q)T (s)(−q) .
SLIDE 25 Results
Plugging in our QFT correlators computed via conformal perturbation theory, the results precisely match those of an slow-roll inflationary cosmology with ǫ∗ = λ4 8π2C2 q q0 2λ 1 + q q0 λ−4 η∗ = −λ + 2λ
q q0 λ−1 Note ǫ∗ ≪ η∗. Find H∗ is one to leading order in λ. We have repackaged the arbitrary dimensionful QFT coupling φ into the arbitrary momentum scale qλ
0 = 2πCφ/λ.
SLIDE 26 Results
Explicitly:
= κ2H2
∗
4q3ǫ∗ ,
= 2κ2H2
∗
q3 δss′
= η∗
- i<j
- ζ(qi)ζ(−qi)
- ζ(qj)ζ(−qj)
- ζ(q1)ζ(q2)γ(±)(q3)
= κ4H4
∗
16 √ 2ǫ∗q2
3
1 ac3 (−a3 + ab + c)(a3 − 4ab + 8c), where κ2 = 12/π2α and a =
i qi, b = i<j qiqj, c = q1q2q3,
- Equilateral piece of ζζζ ∼ H4
∗/ǫ∗ ∼ λ−4 subleading to local piece above
∼ H4
∗η∗/ǫ2 ∗ ∼ λ−7, so need higher order calculations to see.
- ζγγ also higher order due to vanishing of certain OPE coefficient for
Einstein gravity.
SLIDE 27
Plan
➊ Perturbative RG flows, calculation of QFT correlators ➋ Holographic calculation of inflationary correlators ➌ Identifying the dual cosmology
SLIDE 28 Identifying the dual cosmology
What is the potential V (Φ) appearing in the bulk action? S = 1 2κ2
- d4x√−g[R − (∂Φ)2 − 2κ2V (Φ)].
Can compute systematically: ➊ Start from 1st order Friedmann equations: (since RG flow monotonic) H = −1 2W(ϕ), ˙ ϕ = W ′(ϕ), −2κ2V = W ′2 − 3 2W 2 dS asymptopia where V ′ = 0 correspond to either W ′ = 0 or W ′′ = 3W/2. Only the former correspond to stable RG flows ⇒ Choose W ′(0) = 0.
SLIDE 29
Identifying the dual cosmology
➋ We now Taylor expand: W(ϕ) = −2 + a2ϕ2 + a3ϕ3 + O(ϕ4) Fix coefficients via AdS/CFT: a2 maps to the operator dimension ∆ = 3 − λ and a3 maps to OPE coefficient C. ➌ Our cosmology then derives from the cubic superpotential W(ϕ) = −2 − 1 2λϕ2 + 2 3πCϕ3 + O(ϕ4) V (ϕ) is a sextic polynomial describing hilltop inflation:
SLIDE 30 Identifying the dual cosmology
Solving for background: ϕ = ϕ1 1 + (ϕ1/φ)eλt , a = φ ϕ1 + eλt−ϕ2
1/12
exp
1
12
12φϕ1eλt φ ϕ1 + eλt−2 . The asymptotic behaviour is t → ∞, ϕ → φe−λt, a → et t → −∞, ϕ → ϕ1 − ϕ2
1
φ eλt, a → ϕ1 φ ϕ2
1/12
exp
1
12
i.e., de Sitter in far past and far future but with ∆H = λϕ2
1/12 ∼ λ3.
SLIDE 31 Identifying the dual cosmology
Calculating the slow-roll parameters at horizon crossing, one recovers ǫ∗ = λ4 8π2C2 q q0 2λ 1 + q q0 λ−4 + O(λ7), η∗ = −λ + 2λ
q q0 λ−1 + O(λ4), H∗ = 1 + O(λ3). The slow-roll cosmological results coincide exactly with our leading order in λ holographic results obtained from the dual QFT.
SLIDE 32 Summary
➊ We constructed a holographic description of an inflationary cosmology in terms of a 3d dual QFT. The QFT is a deformation of a CFT by a nearly marginal operator generating an RG flow to a nearby IR fixed point. ➋ Even though the QFT is strongly interacting, the form of correlators is dictated by the perturbative breaking of conformal symmetry: O(x1)O(x2) = O(x1)O(x2)e
➌ We can calculate and compare results on both sides of holographic correspondence ⇒ sucessful test! The slow-roll parameters ǫ∗ and η∗ are fixed by the CFT data, i.e., the operator dimensions and OPE coefficients λ and C.
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Further directions
✦ Different OPE coefficients ↔ different bulk actions. ✦ Can we describe slow-roll with ǫ∗ ∼ η∗? ✦ Study constraints from broken conformal invariance from bulk perspective. ✦ Multi-scalar models: entropy perturbations ↔ operator mixing under RG flow.
