Alexei Starobinsky "New results on inflation and pre-inflation in modified gravity"
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1. The document summarizes new results on inflation and pre-inflation in modified gravity theories. 2. It discusses the simplest inflationary models in f(R) gravity, where f(R) is a function of the Ricci scalar R. Models with f(R) proportional to R2 can produce inflation. 3. Perturbation spectra during inflation are calculated for slow-roll f(R) models where f(R) is proportional to R2 times a slowly varying function. Scalar and tensor spectra depend on the Ricci scalar and its derivatives evaluated when modes cross the Hubble radius.
Alexei Starobinsky "New results on inflation and pre-inflation in modified gravity"
- 1. New results on inflation and pre-inflation in modified gravity Alexei A. Starobinsky Landau Institute for Theoretical Physics RAS, Moscow - Chernogolovka, Russia SEENET-MTP Workshop BW2018 Field Theory and the Early Universe Niˇs, Serbia, 11.06.2018
- 2. Inflation and two relevant cosmological parameters The simplest one-parametric inflationary models R2 inflation as a dynamical attractor for scalar-tensor models Isospectral f (R) inflationary models Constant-roll inflation in f (R) gravity Generality of inflation Formation of inflation from generic curvature singularity Conclusions
- 4. Inflation The (minimal variant of the) inflationary scenario is based on the two cornerstone independent ideas (hypothesis): 1. Existence of inflation (or, quasi-de Sitter stage) – a stage of accelerated, close to exponential expansion of our Universe in the past preceding the hot Big Bang with decelerated, power-law expansion. 2. The origin of all inhomogeneities in the present Universe is the effect of gravitational creation of pairs of particles - antiparticles and field fluctuations during inflation from the adiabatic vacuum (no-particle) state for Fourier modes covering all observable range of scales (and possibly somewhat beyond). NB. This effect is the same as particle creation by black holes, but no problems with the loss of information, ’firewalls’, trans-Planckian energy etc. in cosmology, as far as observational predictions are calculated.
- 5. Outcome of inflation In the super-Hubble regime (k aH) in the coordinate representation: ds2 = dt2 − a2 (t)(δlm + hlm)dxl dxm , l, m = 1, 2, 3 hlm = 2R(r)δlm + 2 a=1 g(a) (r) e (a) lm e l(a) l = 0, g (a) ,l el(a) m = 0, e (a) lm elm(a) = 1 R describes primordial scalar perturbations, g – primordial tensor perturbations (primordial gravitational waves (GW)). The most important quantities: ns(k) − 1 ≡ d ln PR(k) d ln k , r(k) ≡ Pg PR
- 6. In fact, metric perturbations hlm are quantum (operators in the Heisenberg representation) and remain quantum up to the present time. But, after omitting of a very small part, decaying with time, they become commuting and, thus, equivalent to classical (c-number) stochastic quantities with the Gaussian statistics (up to small terms quadratic in R, g). In particular: ˆRk = Rk i(ˆak−ˆa† k)+O (ˆak − ˆa† k)2 +...+O(10−100 )(ˆak+ˆa† k)+, , , The last term is time dependent, it is affected by physical decoherence and may become larger, but not as large as the second term. Remaining quantum coherence: deterministic correlation between k and −k modes - shows itself in the appearance of acoustic oscillations (primordial oscillations in case of GW).
- 7. New cosmological parameters relevant to inflation Now we have numbers: P. A. R. Ade et al., arXiv:1502.01589 The primordial spectrum of scalar perturbations has been measured and its deviation from the flat spectrum ns = 1 in the first order in |ns − 1| ∼ N−1 has been discovered (using the multipole range > 40): < R2 (r) >= PR(k) k dk, PR(k) = 2.21+0.07 −0.08 10−9 k k0 ns −1 k0 = 0.05Mpc−1 , ns − 1 = −0.035 ± 0.005 Two fundamental observational constants of cosmology in addition to the three known ones (baryon-to-photon ratio, baryon-to-matter density and the cosmological constant). The simplest existing inflationary models can predict (and predicted, in fact) one of them, namely ns − 1, relating it finally to NH = ln kB Tγ H0 ≈ 67.2.
- 8. Direct approach: comparison with simple smooth models 0.94 0.96 0.98 1.00 Primordial tilt (ns) 0.000.050.100.150.200.25 Tensor-to-scalarratio(r0.002) Convex Concave Planck 2013 Planck TT+lowP Planck TT,TE,EE+lowP Natural Inflation Hilltop quartic model Power law inflation Low scale SSB SUSY R2 Inflation V ∝ φ3 V ∝ φ2 V ∝ φ4/3 V ∝ φ V ∝ φ2/3 N∗=50 N∗=60
- 9. Combined results from Planck/BISEP2/Keck Array P. A. R. Ade et al., Phys. Rev. Lett. 116, 031302 (2016) φ2 0.95 0.96 0.97 0.98 0.99 1.00 ns 0.00 0.05 0.10 0.15 0.20 0.25 r0.002 N=50 N=60 ConvexConcave φ Planck TT+lowP+lensing+ext +BK14
- 10. The simplest models producing the observed scalar slope L = f (R) 16πG , f (R) = R + R2 6M2 M = 2.6 × 10−6 55 N MPl ≈ 3.2 × 1013 GeV ns − 1 = − 2 N ≈ −0.036, r = 12 N2 ≈ 0.004 N = ln kf k = ln Tγ k − O(10), HdS (N = 55) = 1.4 × 1014 GeV The same prediction from a scalar field model with V (φ) = λφ4 4 at large φ and strong non-minimal coupling to gravity ξRφ2 with ξ < 0, |ξ| 1, including the Brout-Englert-Higgs inflationary model.
- 11. The simplest purely geometrical inflationary model L = R 16πG + N2 288π2PR(k) R2 + (small rad. corr.) = R 16πG + 5 × 108 R2 + (small rad. corr.) The quantum effect of creation of particles and field fluctuations works twice in this model: a) at super-Hubble scales during inflation, to generate space-time metric fluctuations; b) at small scales after inflation, to provide scalaron decay into pairs of matter particles and antiparticles (AS, 1980, 1981).
- 12. The most effective decay channel: into minimally coupled scalars with m M. Then the formula 1 √ −g d dt ( √ −gns) = R2 576π (Ya. B. Zeldovich and A. A. Starobinsky, JETP Lett. 26, 252 (1977)) can be used for simplicity, but the full integral-differential system of equations for the Bogoliubov αk, βk coefficients and the average EMT was in fact solved in AS (1981). Scalaron decay into graviton pairs is suppressed (A. A. Starobinsky, JETP Lett. 34, 438 (1981)).
- 13. Possible microscopic origins of this phenomenological model. 1. Follow the purely geometrical approach and consider it as the specific case of the fourth order gravity in 4D L = R 16πG + AR2 + BCαβγδCαβγδ + (small rad. corr.) for which A 1, A |B|. Approximate scale (dilaton) invariance and absence of ghosts in the curvature regime A−2 (RR)/M4 P B−2 . One-loop quantum-gravitational corrections are small (their imaginary parts are just the predicted spectra of scalar and tensor perturbations), non-local and qualitatively have the same structure modulo logarithmic dependence on curvature.
- 14. 2. Another, completely different way: consider the R + R2 model as an approximate description of GR + a non-minimally coupled scalar field with a large negative coupling ξ (ξconf = 1 6 ) in the gravity sector:: L = R 16πG − ξRφ2 2 + 1 2 φ,µφ,µ − V (φ), ξ < 0, |ξ| 1 . Geometrization of the scalar: for a generic family of solutions during inflation and even for some period of non-linear scalar field oscillations after it, the scalar kinetic term can be neglected, so ξRφ = −V (φ) + O(|ξ|−1 ) . No conformal transformation, we remain in the the physical (Jordan) frame!
- 15. These solutions are the same as for f (R) gravity with L = f (R) 16πG , f (R) = R − ξRφ2 (R) 2 − V (φ(R)). For V (φ) = λ(φ2−φ2 0)2 4 , this just produces f (R) = 1 16πG R + R2 6M2 with M2 = λ/24πξ2 G and φ2 = |ξ|R/λ. The same theorem is valid for a multi-component scalar field. More generally, R2 inflation (with an arbitrary ns, r) serves as an intermediate dynamical attractor for a large class of scalar-tensor gravity models.
- 16. Inflation in the mixed Higgs-R2 Model M. He, A. A. Starobinsky, J. Yokoyama, arXiv:1804.00409. L = 1 16πG R + R2 6M2 − ξRφ2 2 + 1 2 φ,µφ,µ − λφ4 4 , ξ < 0, |ξ| 1 In the attractor regime during inflation (and even for some period after it), we return to the f (R) = R + R2 6M2 model with the renormalized scalaron mass M → ˜M: 1 ˜M2 = 1 M2 + 24πξ2 G λ
- 17. Inflation in f (R) gravity Purely geometrical realization of inflation. The simplest model of modified gravity (= geometrical dark energy) considered as a phenomenological macroscopic theory in the fully non-linear regime and non-perturbative regime. S = 1 16πG f (R) √ −g d4 x + Sm f (R) = R + F(R), R ≡ Rµ µ Here f (R) is not identically zero. Usual matter described by the action Sm is minimally coupled to gravity. Vacuum one-loop corrections depending on R only (not on its derivatives) are assumed to be included into f (R). The normalization point: at laboratory values of R where the scalaron mass (see below) ms ≈ const. Metric variation is assumed everywhere.
- 18. Background FRW equations in f (R) gravity ds2 = dt2 − a2 (t) dx2 + dy2 + dz2 H ≡ ˙a a , R = 6( ˙H + 2H2 ) The trace equation (4th order) 3 a3 d dt a3 df (R) dt − Rf (R) + 2f (R) = 8πG(ρm − 3pm) The 0-0 equation (3d order) 3H df (R) dt − 3( ˙H + H2 )f (R) + f (R) 2 = 8πGρm
- 19. Reduction to the first order equation In the absence of spatial curvature and ρm = 0, it is always possible to reduce these equations to a first order one using either the transformation to the Einstein frame and the Hamilton-Jacobi-like equation for a minimally coupled scalar field in a spatially flat FLRW metric, or by directly transforming the 0-0 equation to the equation for R(H): dR dH = (R − 6H2 )f (R) − f (R) H(R − 12H2)f (R) See, e.g. H. Motohashi amd A. A. Starobinsky, Eur. Phys. J C 77, 538 (2017), but in the special case of the R + R2 gravity this was found and used already in the original AS (1980) paper.
- 20. Analogues of large-field (chaotic) inflation: F(R) ≈ R2 A(R) for R → ∞ with A(R) being a slowly varying function of R, namely |A (R)| A(R) R , |A (R)| A(R) R2 . Analogues of small-field (new) inflation, R ≈ R1: F (R1) = 2F(R1) R1 , F (R1) ≈ 2F(R1) R2 1 . Thus, all inflationary models in f (R) gravity are close to the simplest one over some range of R.
- 21. Perturbation spectra in slow-roll f (R) inflationary models Let f (R) = R2 A(R). In the slow-roll approximation | ¨R| H| ˙R|: PR(k) = κ2 Ak 64π2A 2 k R2 k , Pg (k) = κ2 12Akπ2 , κ2 = 8πG N(k) = − 3 2 Rk Rf dR A A R2 where the index k means that the quantity is taken at the moment t = tk of the Hubble radius crossing during inflation for each spatial Fourier mode k = a(tk)H(tk). NB The slow-roll approximation is not specific for inflation only. It was first used in A. A. Starobinsky, Sov. Astron. Lett. 4, 82 (1978) for a bouncing model (a scalar field with V = m2φ2 2 in the closed FLRW universe).
- 22. Slow-roll inflation reconstruction in f (R) gravity A = const − κ2 96π2 dN PR(N) ln R = const + dN − 2 d ln A 3 dN The two-parameter family of isospectral f (R) slow-roll inflationary models, but the second parameter affects a general scale only but not the functional form of f (R). The additional ”aesthetic” assumptions that PR ∝ Nβ and that the resulting f (R) can be analytically continued to the region of small R without introducing a new scale, and it has the linear (Einstein) behaviour there, leads to β = 2 and the R + R2 inflationary model with r = 12 N2 = 3(ns − 1)2 unambiguously.
- 23. For PR = P0N2 (”scale-free reconstruction”): A = 1 6M2 1 + N0 N , M2 ≡ 16π2 N0PR κ2 Two cases: 1. N N0 always. A = 1 6M2 1 + R0 R √ 3/(2N0) For N0 = 3/2, R0 = 6M2 we return to the simplest R + R2 inflationary model. 2. N0 1. A = 1 6M2 1 + R0 R √ 3/(2N0) 1 − R0 R √ 3/(2N0) 2
- 24. Constant-roll inflation in f (R) gravity Search for viable inflationary models outside the slow-roll approximation. Can be done in many ways. A simple and elegant generalization in GR: ¨φ = βH ˙φ , β = const The required exact form of V (φ) for this was found in H. Motohashi, A. A. Starobinsky and J. Yokoyama, JCAP 1509, 018 (2015). Natural generalization to f (R) gravity (H. Motohashi and A. A. Starobinsky, Eur. Phys. J. C 77, 538 (2017)): d2 f (R) dt2 = βH df (R) dt , β = const
- 25. Then it follows from the field equations: f (R) ∝ H2/(1−β) The exact solution for the required f (R) in the parametric form (κ = 1): f (R) = 2 3 (3−β)e2(2−β)φ/ √ 6 3γ(β + 1)e(β−3)φ/ √ 6 + (β + 3)(1 − β) R = 2 3 (3−β)e2(1−β)φ/ √ 6 3γ(β − 1)e(β−3)φ/ √ 6 + (β + 3)(2 − β) Viable inflationary models exist for −0.1 β < 0.
- 26. Generality of inflation Some myths (or critics) regarding inflation and its onset: 1. In the Einstein frame, inflation begins with V (φ) ∼ ˙φ2 ∼ M2 Pl . 2. As a consequence, its formation is strongly suppressed in models with a plateau-type potentials in the Einstein frame (including R + R2 inflation) favored by observations. 3. Beginning of inflation in some patch requires causal connection throughout the patch. 4. ”De Sitter (both the exact and inflationary ones) has no hair”. 5. One of weaknesses of inflation is that it does not solve the singularity problem, i.e. that its models admit generic anisotropic and inhomogeneous solutions with much higher curvature preceding inflation.
- 27. Theorem. In inflationary models in GR and f (R) gravity, there exists an open set of classical solutions with a non-zero measure in the space of initial conditions at curvatures much exceeding those during inflation which have a metastable inflationary stage with a given number of e-folds. For the GR inflationary model this follows from the generic late-time asymptotic solution for GR with a cosmological constant found in A. A. Starobinsky, JETP Lett. 37, 55 (1983). For the R + R2 model, this was proved in A. A. Starobinsky and H.-J. Schmidt, Class. Quant. Grav. 4, 695 (1987). For the power-law and f (R) = Rp , p < 2, 2 − p 1 inflation – in V. M¨uller, H.-J. Schmidt and A. A. Starobinsky, Class. Quant. Grav. 7, 1163 (1990).
- 28. Generic late-time asymptote of classical solutions of GR with a cosmological constant Λ both without and with hydrodynamic matter (also called the Fefferman-Graham expansion): ds2 = dt2 − γikdxi dxk γik = e2H0t aik + bik + e−H0t cik + ... where H2 0 = Λ/3 and the matrices aik, bik, cik are functions of spatial coordinates. aik contains two independent physical functions (after 3 spatial rotations and 1 shift in time + spatial dilatation) and can be made unimodular, in particular. bik is unambiguously defined through the 3-D Ricci tensor constructed from aik. cik contains a number of arbitrary physical functions (two - in the vacuum case, or with radiation) – tensor hair. A similar but more complicated construction with an additional dependence of H0 on spatial coordinates in the case of f (R) = Rp inflation – scalar hair.
- 29. Consequences: 1. (Quasi-) de Sitter hair exist globally and are partially observable after the end of inflation. 2. The appearance of an inflating patch does not require that all parts of this patch should be causally connected at the beginning of inflation. Similar property in the case of a generic curvature singularity formed at a spacelike hypersurface in GR and modified gravity. However, ’generic’ does not mean ’omnipresent’.
- 30. What was before inflation? Duration of inflation was finite inside our past light cone. In terms of e-folds, difference in its total duration in different points of space can be seen by the naked eye from a smoothed CMB temperature anisotropy map. ∆N formalism: ∆R(r) = ∆Ntot(r) where Ntot = ln a(tfin) a(tin) = Ntot(r) (AS, 1982,1985). For 50, neglecting the Silk and Doppler effects, as well as the ISW effect due the presence of dark energy, ∆T(θ, φ) Tγ = − 1 5 ∆R(rLSS , θ, φ) = − 1 5 ∆Ntot(rLSS , θ, φ) For ∆T T ∼ 10−5 , ∆N ∼ 5 × 10−5 , and for H ∼ 1014 GeV, ∆t ∼ 5tPl !
- 31. Different possibilities were considered historically: 1. Creation of inflation ”from nothing” (Grishchuk and Zeldovich, 1981). One possibility among infinite number of others. 2. De Sitter ”Genesis”: beginning from the exact contracting full de Sitter space-time at t → −∞ (AS, 1980). Requires adding an additional term Rl i Rk l − 2 3 RRk i − 1 2 δk i RlmRlm + 1 4 δk i R2 to the rhs of the gravitational field equations. Not generic. May not be the ”ultimate” solution: a quantum system may not spend an infinite time in an unstable state. 3. Bounce due to a positive spatial curvature (AS, 1978). Generic, but probability of a bounce is small for a large initial size of a universe W ∼ 1/Ma0.
- 32. Formation of inflation from generic curvature singularity In classical gravity (GR or modified f (R)): space-like curvature singularity is generic. Generic initial conditions near a curvature singularity in modified gravity models (the R + R2 and Higgs ones): anisotropic and inhomogeneous (though quasi-homogeneous locally). Two types singularities with the same structure at t → 0: ds2 = dt2 − 3 i=1 |t|2pi a (i) l a(i) m dxl dxm , 0 < s ≤ 3/2, u = s(2−s) where pi < 1, s = i pi , u = i p2 i and a (i) l , pi are functions of r. Here R2 RαβRαβ . Type A. 1 ≤ s ≤ 3/2, R ∝ |t|1−s → +∞ Type B. 0 < s < 1, R → R0 < 0, f (R0) = 0 Spatial gradients may become important for some period before the beginning of inflation.
- 33. What is sufficient for beginning of inflation in classical (modified) gravity, is: 1) the existence of a sufficiently large compact expanding region of space with the Riemann curvature much exceeding that during the end of inflation (∼ M2 ) – realized near a curvature singularity; 2) the average value < R > over this region positive and much exceeding ∼ M2 , too, – type A singularity; 3) the average spatial curvature over the region is either negative, or not too positive. Recent numerical studies confirming this for similar models in GR: W. H. East, M. Kleban, A. Linde and L. Senatore, JCAP 1609, 010 (2016); M. Kleban and L. Senatore, JCAP 1610, 022 (2016). On the other hand, causal connection is certainly needed to have a ”graceful exit” from inflation, i.e. to have practically the same amount of the total number of e-folds during inflation Ntot in some sub-domain of this inflating patch.
- 34. Bianchi I type models with inflation in R + R2 gravity Recent analytical and numerical investigation in D. Muller, A. Ricciardone, A. A. Starobinsky and A. V. Toporensky, Eur. Phys. J. C 78, 311 (2018). For f (R) = R2 even an exact solution can be found. ds2 = tanh2α 3H0t 2 dt2 − 3 i=1 a2 i (t)dx2 i ai (t) = sinh1/3 (3H0t) tanhβi 3H0t 2 , i βi = 0, i β2 i < 2 3 α2 = 2 3 − i β2 i 6 , α > 0 Nest step: relate arbitrary functions of spatial coordinates in the generic solution near a curvature singularity to those in the quasi-de Sitter solution.
- 35. Conclusions First quantitative observational evidence for small quantities of the first order in the slow-roll parameters: ns(k) − 1 and r(k). The typical inflationary predictions that |ns − 1| is small and of the order of N−1 H , and that r does not exceed ∼ 8(1 − ns) are confirmed. Typical consequences following without assuming additional small parameters: H55 ∼ 1014 GeV, minfl ∼ 1013 GeV. In f (R) gravity, the simplest R + R2 model is one-parametric and has the preferred values ns − 1 = − 2 N and r = 12 N2 = 3(ns − 1)2 . The first value produces the best fit to present observational CMB data.
- 36. Inflation in f (R) gravity represents a dynamical attractor for slow-rolling scalar fields strongly coupled to gravity. Inflation is generic in the R + R2 inflationary model and close ones. Thus, its beginning does not require causal connection of all parts of an inflating patch of space-time (similar to spacelike singularities). However, graceful exit from inflation requires approximately the same number of e-folds during it for a sufficiently large compact set of geodesics. To achieve this, causal connection inside this set is necessary (though still may appear insufficient). The fact that inflation does not ”solve” the singularity problem, i.e. it does not remove a curvature singularity preceding it, can be an advantage, not its weakness. Inflation can form generically and with not a small probability from generic space-like curvature singularity.