Quantum Gravity from the QFT Perspective

Ilya L. Shapiro

Universidade Federal de Juiz de Fora, MG, Brazil

Friedrich-Schiller-Universitat,¨ Jena, 08/01/2015

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Contents of the mini-course

GR and its limits of applicability. Why should we quantize gravity.• Semiclassical approach and higher derivatives.

Gauge-invariant renormalization. Power counting. Quantum GR• vs Higher derivative QG. How we can (if so) live with ghosts? More HD’s and more ghosts. Superrenormalizable QG.

Renormalization Group (RG). For which versions of QG can RG• be used? Decoupling: what remains in the IR?

Alternative approaches: Effective approach to QG and Induced• Gravity paradigm.

Bibliography I.L. Buchbinder, S.D. Odintsov, I.Sh., Effective Action in Quantum Gravity. (1992 - IOPP). I.Sh., Class. Q. Grav. 25 (2008) 103001 (Topical review); 0801.0216.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Lecture 1.

GR and singularities. Limits of applicability, Planck scale.

Dimensional approach and Planck scale.

Quantum & gravity or Quantum Gravity (QG)?

Quantum Field Theory in curved space: hints for a complete QG.

QG: how we can do it? Linearized parametrization vs covariance. Locality.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Classical Gravity – Newton’s Law,

G M M G M M ~ 1 2 ˆ 1 2 F12 = 2 r12 or U(r) = . − r12 − r

Newton’s law work well from laboratory up to the galaxy scale.

The “Pioneer anomaly” which put shadow to the Newton’s law for the last decades, has been resolved recently by elementary (albeit not simple!) approach. S. Turyshev et al., .... Support for the thermal origin of the Pioneer anomaly. PRL 108 (2012); arXiv:1204.2507 [gr-qc].

For galaxies and their clusters one needs, presumably, to introduce a Dark Matter, which consists from particles of unknown origin. Or modify gravity in some way.

Deﬁnitely, Newton’s laws do not “ask” for being quantized.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 General Relativity and Quantum Theory

General Relativity (GR) is a complete theory of classical gravitational phenomena. It proved valid in the wide range of energies and distances.

The basis of the theory are the principles of equivalence and general covariance. There are covariant equations for the matter (fields and particles, fluids etc) and Einstein equations for the gravitational field gµν 1 G = R Rg = 8πGT Λ g . µν µν − 2 µν µν − µν We have introduced Λ, cosmological constant (CC) for completeness.

The most important solutions of GR have speciﬁc symmetries.

1) Spherically-symmetric solution. Planets, Stars, Black holes. 2) Isotropic and homogeneous metric. Universe.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 I. Spherically-symmetric solution of Schwarzschild.

For a point-like mass in the origin of the spherical coordinate system, the solution is (Schwarzschild, 1916)

2 2 rg 2 dr 2 ds = 1 dt r dΩ , rg = 2GM − r − 1 rg /r − − r = 0 singularity is physical and indicates a serious problem.

An object with horizon may be formed as a consequence of the gravitational collapse, leading to the formation of physical singularity at r = 0.

Assuming GR is valid at all scales, we arrive at the situation when the r = 0 singularity becomes real.

Then, the matter has inﬁnitely high density of energy, and curvature invariants are also inﬁnite. Our physical intuition tells us that this is not a realistic situation.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 II. Standard cosmological model

Another important solution of GR is the one for the homogeneous and isotropic metric (FLRW solution).

dr 2 ds2 = dt2 a2(t) + r 2dΩ , − · 1 kr 2 − Radiation-dominated epoch is characterized by the EOS µ p = ρ/3 and Tµ = 0. The Friedmann equation is

8πG ρ a4 a˙ 2 = 0 0 = 3 a4 ⇒ The solutions becomes singular at t 0. → 1/4 4 4 1 a(t) = 8πGρ0a √t , H = a˙ /a = 3 · 0 × 2t The situation is qualitatively similar to the black hole singularity.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Applicability of GR

The singularities are signiﬁcant, because they emerge in the most important solutions, in the main areas of application of GR.

Extrapolating backward in time we ﬁnd that the use of GR leads to a problem, while at the late Universe GR provides a consistent basis for cosmology and astrophysics. The most natural resolution of the problem of singularities is to assume that

GR is not valid at all scales. • At the very short distances and/or when the curvature becomes very large, the gravitational phenomena must be described by some other theory, more general than the GR.

But, due to success of GR, we expect that this unknown theory coincides with GR at the large distance & weak ﬁeld limit.

The most probable origin of the deviation from the GR are quantum effects.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Dimensional arguments.

The expected scale of the quantum gravity effects is associated to the Planck units of length, time and mass. The idea of Planck units is based on the existence of the 3 fundamental constants: c = 3 1010 cm/s , ·

− ~ = 1.054 10 27 erg sec ; · ·

− G = 6.67 10 8 cm3/sec2 g . · One can use them uniquely to construct the dimensions of

1/2 1/2 −3/2 −33 length lP = G ~ c 1.4 10 cm; ≈ ·

1/2 1/2 −5/2 −43 time tP = G ~ c 0.7 10 sec; ≈ ·

−1/2 1/2 1/2 −5 19 mass MP = G ~ c 0.2 10 g 10 GeV . ≈ · ≈

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Three choice for Quantum Gravity (QG)

One may suppose that the fundamental units indicate to the presence of fundamental physics at the Planck scale.

We can classify the possible approaches to three distinct groups, namely we can:

Quantize both gravity and matter ﬁelds. This is, deﬁnitely, •the most fundamental possible approach.

Quantize only matter ﬁelds on classical curved background (semiclassical• approach).

Quantize something else. E.g., in case of (super)string theory• both matter and gravity are induced.

Which approach is “better”?

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 QFT and Curved space-time are well-established notions, which passed many experimental/observational tests.

Therefore, our ﬁrst step should be to consider QFT of matter ﬁelds on classical curved background.

Different from quantum theory of gravity, QFT of matter ﬁelds in curved space is renormalizable and free of conceptual problems.

Moreover, understanding renormalization of semiclassical theory can be very helpful in order to impose some constrains for a complete QG.

Metric enters the generating functional of the Green functions as an external parameter,

Z (J, g ) = dΦ exp iS(Φ, g )+ iΦJ . µν { µν } Z

We arrive at the Effective Action of the mean ﬁeld(s), Γ[Φ, gµν ].

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 The main speciﬁc feature of QFT in curved space is that the Effective Action depends on the background metric, Γ[Φ, gµν ]

In terms of Feynman diagrams, one has to consider graphs with internal lines of matter ﬁelds & external lines of both matter and metric. In practice, one can consider gµν = ηµν + hµν .

→ + +

+ + + + ...

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 An important observation is that all those “new” diagrams with hµν legs have superﬁcial degree of divergence equal or lower that the “old” ﬂat-space diagrams.

Consider the case of scalar ﬁeld which shows why the nonminimal term ξRϕ2 is necessary for a scalar ﬁeld:

→ + +

+ + + ... .

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 It is easy to see that the theory can be renormalizable only if we include higher derivative terms in the vacuum sector.

Renormalization involves ﬁelds, couplings, masses, non-minimal parameter ξ and vacuum action parameters.

Introduction: I.L. Buchbinder, S.D. Odintsov & I.Sh. (IOPP, 1992).

Relevant diagrams for the vacuum sector

+ + + + ... .

All possible covariant counterterms have the same structure as

Svac = SEH + SHD

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 The renormalizable QFT in curved space requires introducing a generalized form of the gravity (external ﬁeld), “vacuum action”.

Svac = SEH + SHD

1 4 where SEH = d x√ g R + 2Λ . − 16πG − { } Z is the Einstein-Hilbert action with the cosmological constant.

SHD includes higher derivative terms. The most useful form is

4 2 2 SHD = d x√ g a1C + a2E + a3R + a4R , − Z where C2(4)= R2 2R2 + 1/3 R2 µναβ − αβ is the square of the Weyl tensor in d = 4, E = R Rµναβ 4 R Rαβ + R2 µναβ − αβ is integrand of the Gauss-Bonnet term (topological term in d=4 ).

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Observation about higher derivatives.

The consistent theory can be achieved only if we include

4 2 2 SHD = d x√ g a1C + a2E + a3R + a4R , − Z C2(4)= R2 2R2 + 1/3 R2 is the square of the Weyl tensor. µναβ − αβ In quantum gravity such a HD term means massive ghost, the gravitational spin-two particle with negative kinetic energy. This leads to the problem with unitarity, at least at the tree level.

In the semiclassical theory gravity is external and unitarity of the gravitational S-matrix does not matter.

The consistency criterium includes: physically reasonable solutions and their stability under small perturbations.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 The standard “old” approach to perturbative QG is based on the assumptions:

Quantization can be performed using the variables • gµν = ηµν + hµν . (1) It is assumed that this choice of the quantum metric is the “right” one. This choice enables one to deal with the well-deﬁned object such as S - matrix.

In case of quantum GR (1) corresponds to so-called Gaussian• expansion of the action. The last means we can take

gµν = ηµν + κhµν . (2) and then, for

1 4 2 SEH = d x√ g R + 2Λ , κ = 16πG (3) − κ2 − { } Z and the second order (Gaussian approximation) of the action becomes κ- independent.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Finally, the expansion around the ﬂat background means we •have Newton constant as a coupling.

In the recent (20 years) literature one can meet a lot of works about “non-Gaussian ﬁxed point” in QG. Such a ﬁxed point was originally conjectured by S. Weinberg as a hypothesis of “asymptotic safety”, which is an alternative to renormalizability.

If being someday conﬁrmed, such a discovery would mean that• we do not need to construct the quantum theory around ﬂat space. Instead, this would mean that the theory should be constructed with some other background in mind.

But then, what is the sense of the S - matrix approach, •becomes unclear.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Further arguments:

One should deﬁnitely quantize both matter and gravity, for •otherwise the QG theory would not be complete.

The diagrams with matter internal lines in a complete QG are be• exactly the same as in a semiclassical theory.

This means one can not quantize metric without higher derivative• terms in a consistent manner, since these terms are produced already in the semiclassical theory.

Indeed, most of the achievements in curved-space QFT are •related to the renormalization of higher derivative vacuum terms, including Hawking radiation, Starobinsky inﬂation and others.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Once again: what is bad in the higher-derivative gravity?

For the linearized gravity

gµν = ηµν + hµν , (1) we meet 1 1 1 Gspin−2(k) , m MP . ∼ m2 k 2 − m2 + k 2 ∝ This means that the tree-level spectrum includes massless graviton and massive spin-2 “ghost” with negative kinetic energy and huge mass.

Interaction between ghost and gravitons may violate energy conservation• in the massless sector (M.J.G. Veltman, 1963).

In classical systems higher derivatives generate exploding instabilities• at the non-linear level (M.V. Ostrogradsky, 1850).

Without ghost one violates unitarity of the S -matrix.

Ilya Shapiro,• Lectures on QG from QFT perspective. Jena - January, 2015 Let us classify all assumptions we made to condemn higher derivative theory:

One can draw conclusions about the theory of gravity using •linearized gravity approximation. Furthermore, the S -matrix of gravitons should be the central object of our interest.

Massless and massive (ghost) modes can be separated to perform•• the analysis of the particle content and stability.

Ostrogradsky instabilities or Veltman scattering are relevant••• independent on the energy scale, in all cases they produce run-away solutions and the Universe “explodes”.

The standard conclusion is that one can treat higher derivative terms only as small corrections, in a perturbative framework.

The last assumption is especially risky and should be veriﬁed. The safe criterion••• is the stability of low-energy solutions when the higher derivatives are present.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 There is a simple way to check whether all these assumptions are correctly working or not.

Let us take some higher derivative theory and verify the stability with respect to the linear perturbations on some, physically interesting, dynamical background.

If all three assumptions are correct, we will observe rapidly growing modes even for the low-energy (slowly changing, or low-curvature) background.

If all but the last (energy-scale independence) assumptions are right, we are going to observe no growing modes until the 2 typical value of curvature becomes of the order of MP .

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 For the sake of generality, let us consider not only classical HD terms, but also take into account the semiclassical corrections, derived via conformal anomaly.

Consider the anomaly-induced effective action of gravity. • Derive gravitational waves in the quantum-corrected theory. • Verify the stability of classical solutions. •

Program realized in:

J. Fabris, A. Pelinson, F. Salles, I.Sh., JCAP 02 (2012); F. Salles and I.Sh., Phys. Rev. D89 (2014), arXiv:1401.4583.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Semiclassical theory of classically conformal free ﬁelds.

The vacuum action in the general case has the form

Svac = SEH + SHD ,

4 2 2 SHD = d x√ g αC + βE + γR + δR , − Z n o where E = R2 4R2 + R2 . µναβ − αβ

In the case of conformal theory, at the one-loop level, it is sufﬁcient to consider

4 2 S = d x√ g a1C + a2E + a3R . conf . vac − Z

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Conformal anomaly

Consider the theory which includes gµν and matter ﬁelds Φ. kΦ is the conformal weight of the ﬁeld.

The Noether identity for the local conformal symmetry is

δ δ 2 g + kΦ Φ S(g , Φ) = 0 . − µν δg δΦ µν µν µ It means Tµ = 0 on shell

2 δSvac (gµν ) µ gµν = Tµ = 0 . − √g δgµν

At quantum level Svac (gµν ) is replaced by effective action (EA)

Γvac(gµν ) = Svac (gµν ) + Γ¯vac(gµν ) .

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 The theory of matter includes N0 conformal real scalars , N1/2 Dirac spinors and N1 vectors. The expression for divergences 1 Γ = d 4x√g β C2 + β E + β R . div ε 1 2 3 Z 3N 18N 36N β1 1 0 + 1/2 + 1 where β2 = N0 + 11N + 62N1  −  360(4π)2  1/2  β3 2N0 + 12N 36N1 1/2 −     The anomalous trace is µ 2 ′ T = β1C + β2E + a R . h µ i − Useful conformal variables: 2 1 g = g¯ e2σ , ∆= 2 + 2Rµν R + ( µR) . µν µν · ∇µ∇ν − 3 3 ∇ ∇µ 2 2 √ g(E R)= g¯ E¯ ¯ R¯ + 4∆¯ 4σ . − − 3 − − 3 Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January,p 2015 Anomaly-induced action of vacuum

One can use T µ as an equation for the ﬁnite 1-loop EA h µ i 2 δ Γ¯ 1 g ind = T µ = ωC2 + bE + cR . ( ) − √ g µν δg h µ i − (4π)2 ∗ − µν Solution is straightforward Riegert, Fradkin & Tseytlin, PLB-1984.

The solution for the effective action is

1 4 ¯ 2 Γ¯ind = Sc[g¯ ] + d x g¯ ωσC (1) µν (4π)2 − { Z p 2 1 2 2 2 + bσ E¯ ¯ R¯ + 2bσ∆¯ 4σ (c + b)[R¯ 6( ¯ σ) (¯ σ)] ) , − 3 − 12 3 − ∇ − }

where Sc [g¯µν ]= Sc [gµν ] is the integration constant of (*).. The disadvantage is that this action is not covariant since it is not expressed in terms of original metric. gµν .

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 4 Introduce = d x√ g , ∆4 G(x, y)= δ(x, y). − Zx Z An equivalent covariant solution has the form

3c + 2b 4 2 Γind = Sc[g ] d x g(x) R (x) µν − 36(4π)2 − Z p ω b 2 2 + C2 + E R G(x, y) E R . 4 8 − 3 x − 3 y ZZ x y h i

One can rewrite this expression using auxiliary scalars,

3c + 2b 2 1 1 Γind = Sc[g ] R (x) + ϕ∆4ϕ ψ∆4ψ µν − 36(4π)2 2 − 2 Zx Zx a √ b 2 a + ψ C2 + ϕ − E R C2 . 8π√ b 8π − 3 − 8π√ b − − I.Sh. and A. Jacksenaev, Ph.L.B (1994).

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 An important application is the anomaly-induced inﬂation, or Modiﬁed Starobinsky Model Fabris, Pelinson, Solà, I.Sh. et al. (1998-2003, 2011-2012).

Consider a Cosmological Model based on the action

2 MP 4 Stotal = d x√ g (R + 2Λ) + Smatter + Svac + Γ¯ind . − 16π − Z

Equation of motion in physical time dt = a(η)dη

a¨ 3a˙ a¨˙ a¨2 4b a¨a˙ 2 2b a¨ + + 5 + 2k 1 + a a2 a2 − c a3 − c a3 M2 a¨ a˙ 2 k 2Λ P + + = 0 , −8πc a a2 a2 − 3 where k = 0, 1, Λ is the cosmological constant. ±

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Particular solutions Starobinsky, Ph.L.B., 1980

ao exp(Ht) , k = 0 a(t) = a cosh(Ht) , k = 1 ,  o ao sinh(Ht) , k = 1  −  1/2 MP 64πb Λ Hubble parameter H = 1 1 + 2 . √ 32πb ± s 3 MP ! −

Perturbations of the conformal factor: σ(t) σ(t)+ y(t). → The criterion for a stable inﬂation 1 1 c > 0 N1 < N + N0 , ⇐⇒ 3 1/2 18 in agreement to Starobinsky (1980).

Remark. The value of Λ and the choice of k = 0, 1 do not inﬂuence the stability condition. ±

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Unstable case: 1 1 N > N + N . 1 3 1/2 18 0 One can chose initial conditions such that the Universe performs sufﬁcient inﬂation and then gracefully exits to the FRW stage.

The Starobinsky model is based on the unstable case M.V. Fischetti, J.B. Hartle & B.L. Hu, PRD20 (1979); A.A. Starobinski, Ph.L.B 91 (1980) 99; •V.F. Mukhanov & G.V. Chibisov, JETP Lett. 33 (1981); JETP (1982); A.A.Starobinski, Let.Astr.Journ. 9 (1983); A. Vilenkin, PRD 32 (1985); P. Anderson, PRD 28 (1983).

Stable case: The Universe starts inﬂation from an arbitrary position at the phase plane, anomaly-induced inﬂation “kills” any matter content in a dozen of Planck times.

Pelinson, Takakura, I.Sh. NPB (PS) - 2003.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Simple test of the Modiﬁed Starobinsky Model. Pelinson, I.Sh. & Takakura IRGA-NPB(PS)- 2003.,

Consider late Universe, k = 0, H0 = Λ/3.

Only photon is active, N0 = 0 , N1/2p= 0 , N1 = 1 . −42 Graviton typical energy is H0 10 GeV , = all massive ≈ −12 ⇒ particles (even neutrino) mν 10 GeV decouple from gravity. c < 0 = today≥ inﬂation is unstable. ⇒ λt Stability for the small H = H0 case: H H0 + const e → · 2 2 3 2 3 2 (3c b)4H0 MP 32πbH0 + MP H0 λ + 7H0λ + − λ = 0 . " c − 8πc # − 2πc

3 MP The solutions are λ1 = 4H0 , λ2/3 = H0 i . − −2 ± 8π c | | Λ > 0 efﬁciently protects low-energy solution bfromp higher-derivative instabilities in this case.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Consider unstable case, matter (or radiation) dominated Universe, close to the classical FRW solution. Then

a¨ 3a˙ a¨˙ a¨2 4b a¨a˙ 2 2b a¨ + + 5 + 2k 1 + a a2 a2 − c a3 − c a3

2 2 MP a¨ a˙ k 2Λ 1 + + = ρmatter , −8πc a a2 a2 − 3 −3c The terms of the ﬁrst line, of quantum origin, behave like 1/t4.

The second line terms, of classical origin, behave like 1/t2.

After certain time the “quantum” terms become negligible.

Conclusion: Classical solutions are very good approximate solutions of the theory with quantum corrections and/or higher derivatives, as far as we deal only with the conformal factor dynamics.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Stability & Gravitational Waves

As far as classical action and quantum, anomaly-induced term, both have higher derivatives, an important question is whether the stability of classical solutions in cosmology holds or not.

Consider small perturbation

0 gµν = gµν + hµν , hµν = δ gµν ,

0 2 where g = 1, δij a (t) , µ = 0, 1, 2, 3 and i = 1, 2, 3. µν { − }

3 d k ·~ h (t,~r) = ei~r k h (t, ~k) . µν (2π)3 µν Z ij Using the conditions ∂i h = 0 and hii = 0 , together with the synchronous coordinate condition hµ0 = 0, we arrive at the equation for the tensor mode

Fabris, Pelinson and I.Sh., NPB (2001); Fabris, Pelinson, Salles and I.Sh., JCAP (2011).

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 ...... f2 2 f2 2f1 + h + 3H 4f1 + f2 + 4f˙1 + f˙2 h + 3H 6f1 + 4f3 2 2 − h i h ˙ 9 ˙ ¨ 1 ¨ 3 2 2 +H 16f1 + f2 +6H˙ f1 f3 +2f1 + f2 +f0 +f4ϕ¨ + f4Hϕ˙ f5ϕ˙ h¨ 2 − 2 2 − 3 i 2h¨ 1 3 4f1 + f2 ∇ + H˙ 4f˙1 6f˙3 21HH˙ f2 + 2f3 H¨ f2 + 6f3 − a2 − − 2 − 2 h 2 1 3 3 3 2 +3H 4f˙1 + f˙2 4f˙3 9H f2 +4f3 +H 4f¨1 + f¨2 + f4ϕ˙ 3H +H˙ 2 − − 2 2 2 ˙ 3 2 1 ... 4 h +H 3f4ϕ¨+ f0 2f5ϕ˙ + f4 ϕ f5ϕ˙ϕ¨ h˙ H 4f1+f2 +4f˙1+f˙2 ∇ 2 − 2 −3 − a2 i ... ...... 2 2 + 5f4H ϕ +f4 ϕ 36HH˙ + 18H˙ + 24HH¨ + 4 H f1 + f2 + 3f3 − h 2 HH˙ 32f˙1 + 36f˙2 + 120f˙3 8H¨ f˙1 + f˙2 + 3f˙3 H 4f¨1 + 6f¨2 + 24f¨3 − − − ¨ ¨ ¨ 3 2 3 ˙ ˙ ˙ 4H˙ f1+f2+3f3 9f4ϕ˙ H +HH˙ +f4ϕ¨ 3H +5H˙ H 8f1+12f2+48f3 − − − Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 2 1 2 1 2 1 2 1 ... 2 +f5ϕ˙ H + H˙ + f5Hϕ˙ϕ¨ f5ϕ¨ + f5ϕ˙ ϕ h + f0 2H˙ + 3H h 2 3 3 − 6 3 i h i 1 + H 2f˙ + f˙ + 2H˙ f + f + 3f 1 2 2 1 2 3 h 2 4 1 1 2 h 1 h ¨f2 + f4ϕ¨ + f0 + 3f4Hϕ˙ f5ϕ˙ ∇ + 2f1 + f2 ∇ = 0 , −2 − 3 a2 2 a4 where the f - terms are deﬁned asi h i 2 MP b + ω ω f0 = ; f1 = a1 + a2 ϕ + ψ ; − 16π − 2√ b 2√ b − − ω + 2b ω f2 = 2a1 4a2 + ϕ ψ; − − √ b − √ b − − a1 3c + 2b 3b + ω ω f3 = + a2 ϕ + ψ; 3 − 36 − 6√ b 6√ b − − 4π√ b 1 f4 = − ; f5 = . − 3 2

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Qualitative results were achieved by using both

I. Analytical methods. We can approximately treat all coefﬁcients as constants. There is a mathematically consistent way to check when (and whether) it works. And, with the Wolfram’s Mathematica software, manipulating our equation is not too difﬁcult, indeed.

II. Numerical methods. Modern methods to study the CMB spectrum (CMBEasy software) can be pretty well applied to gravitational waves and give the result which is perfectly consistent with the output of method II.

Net Result: The stability does not actually depend on quantum corrections. It is completely defined by the sign of the classical coefficient a1 of the Weyl-squared term. The sign of this term defines whether graviton or ghost has positive kinetic energy!

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 We can distinguish the following two cases:

The coefﬁcient of the Weyl-squared term is a < 0 Then • 1 1 1 1 Gspin−2(k) , m MP , ∼ m2 k 2 − m2 + k 2 ∝ there are no growing modes up to the Planck scale, ~k 2 M2 . ≈ P For the dS background this is in a perfect agreement with Starobinsky, Let.Astr.Journ. (in Russian) (1983); Hawking, Hertog and Real, PRD (2001).

The classical coefﬁcient a > 0 , also G < 0 . • 1 1 1 1 Gspin−2(k) , m MP . ∼ m2 m2 + k 2 − k 2 ∝ and there are rapidly growing modes at any scale.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015 Last discussion concerning ghosts

In fact, the result is natural. The anomaly-induced quantum 3 4 correction is (R....) and (R....), Until the energy is not of the Planck order ofO magnitude,O these corrections can not compete with classical (R2 ) - terms. O ....

For a1 < 0 there are no growing tensor modes in the higher derivative gravity on cosmological backgrounds.

Massive ghosts are present only in the vacuum state. We just do not observe them “alive” until the typical energy scale is below the Planck mass.

All in all, massive ghosts do not pose real danger below the •Planck scale. Above MP we need new ideas to ﬁght ghosts.

Ilya Shapiro, Lectures on QG from QFT perspective. Jena - January, 2015