From the Big Bounce to the Near de Sitter Slow Roll

Transcription

1 p. From the Big Bounce to the Near de Sitter Slow Roll Abhay Ashtekar Institute for Gravitation and the Cosmos, Penn State Understanding emerged from the work of many researchers, especially: Agullo, Barrau, Bojowald, Cailleteau, Campiglia, Corichi, Grain, Henderson, Kaminski, Lewandowski, Mielczarek,Nelson, Pawlowski, Singh, Sloan, Taveras, Wilson-Ewing... CFT Workshop: Physics with a Positive Cosmological Constant, May 9-11, 2012

2 p. Physics With Λ > 0: Early History Einstein (1917): Static Universe Cosmological Considerations Concerning General Relativity" de Sitter (1917) : Solution B (Einstein s universe: Solution A) On Einstein s Theory of Gravitation and its Astronomical Consequences" Einstein e reaction: Mathematically fine but not physical because there was no matter. Mach s principle this curvature cannot be physically relevant. de Sitter: Since matter density is so low, can be neglected in the first approximation. Friedmann ( ): Expanding solutions with (and without) Λ. But primarily of mathematical interest (e.g. he also considered solutions with negative energy density). Einstein s reaction: Mathematical error. Meeting with Yuri Krutov meeting in Leiden (1923). Publicly retracts. But does not discuss expanding solutions in his 1929 Encyclopedia article. Lemaitre (1927): Independently rediscovered Friedmann solutions. Thought about them physically (remarkably close to the modern view). Found a solution which starts out being Einstein s static universe but evolves because of perturbative instability and evolves to de Sitter in the distant future. The paper contained Hubble s law. Einstein accepted mathematical correctness. But did not think expanding solutions were physical. ( C est abominable!")

3 p. Physics With Λ > 0: Early History 1930: Things start getting heated up: Cosmological models; Λ, Static versus Dynamics. But Friedman and Lemaitre s works were largely ignored. Leading models: Einstein s Static Universe and De Sitter s Dynamical one. 1930: In a January RAS Meeting, Eddington: Shall we put a little motion into Einstein s world of inert matter or a little matter into de Sitter Premium Mobile?" Lemaitre reminds Eddington of his 1927 paper and sends a copy again. Eddington embarrassed (McVitte). Readily acknowledged the importance of Lemaitre s paper. Sent to de Sitter who wrote a paper discussing "Lemaitre s solution" (1930). 1931: Einstein retracts his support for Λ. 1947: "I am unable to think that such an ugly thing should be realized in Nature". By contrast: Lemaitre thought Λ is a natural and indispensable part of cosmology. Already in 1933 clearly stated that it should be regarded as vacuum energy density with negative pressure. Modern resurgence: Late universe (supernovae) AND also very early universe (inflation). This talk will focus on the second.

4 p. Inflationary Paradigm Major success: Prediction of inhomogeneities in CMB which serve as seeds for structure formation. Observationally relevant wave numbers in the range (k, 200k ) (Radius of the observable CMB surface 10λ ). Rather minimal assumptions: 1. Some time in its early history, the universe underwent a phase of accelerated expansion during which the Hubble parameter H was nearly constant. 2. During this phase the universe is well-described by a FLRW background with linear perturbations. 3. A few e-foldings before the mode k exited the Hubble radius during inflation, Fourier modes of quantum fields describing perturbations were in the Bunch-Davis vacuum for co-moving wave numbers in the range (k, 200k ); and, 4. Soon after a mode exited the Hubble radius, its quantum fluctuation can be regarded as a classical perturbation and evolved via linearized Einstein s equations. Then QFT on FLRW space-times (and classical GR) implies the existence of tiny inhomogeneities in CMB seen by the 7 year WMAP data. All large scale structure emerged from vacuum fluctuations!

5 p. Inflationary Paradigm: Incompleteness Particle Physics Issues: Where from the inflaton? A single inflaton or multi-inflatons? Interactions between inflatons? How are particles/fields of the standard model created during reheating at the end of inflation?... Quantum Gravity Issues: Fate of the initial singularity (Borde, Guth & Vilenkin): Is the infinite curvature really attained? What is the nature of the quantum space-time that replaces Einstein s continuum in the Planck regime? In the systematic evolution from the Planck regime in the more complete theory, does a slow roll phase compatible with the WMAP data arise generically or is an enormous fine tuning needed? In classical GR, if we evolve the modes of interest back in time, they become trans-planckian. Is there a QFT on quantum cosmological space-times needed to adequately handle physics at that stage? Can one arrive at the Bunch-Davis vacuum (at the onset of the WMAP slow roll) from more fundamental considerations?

6 p. Inflationary Paradigm: Incompleteness Quantum Gravity Issues: Fate of the initial singularity (Borde, Guth & Vilenkin): Is the infinite curvature really attained? What is the nature of the quantum space-time that replaces Einstein s continuum in the Planck regime? "One may not assume the validity of field equations at very high density of field and matter and one may not conclude that the beginning of the expansion should be a singularity in the mathematical sense." A. Einstein, 1945 In the systematic evolution from the Planck regime in the more complete theory, does a slow roll phase compatible with the WMAP data arise generically or is an enormous fine tuning needed? In classical GR, if we evolve the modes of interest back in time, they become trans-planckian. Is there a QFT on quantum cosmological space-times needed to adequately handle physics at that stage? Can one arrive at the Bunch-Davis vacuum (at the onset of the WMAP slow roll) from more fundamental considerations?

7 p. 1. Singularity Resolution: Loop Quantum Cosmology Difficulty: UV - IR Tension. Can one have singularity resolution with ordinary matter and agreement with GR at low curvatures?(background dependent perturbative approaches have difficulty with the first while background independent approaches, with second.) (Green and Unruh) These questions have been with us for years since the pioneering work of DeWitt, Misner and Wheeler. WDW quantum cosmology is fine in the IR but not in the UV. In LQC, this issue has been resolved for general homogeneous cosmologies as well as Gowdy Models. Physical observables which are classically singular (eg matter density) at the big bang have a dynamically induced upper bound on the physical Hilbert space. Mathematically rigorous and conceptually complete framework. (AA, Bojowald, Corichi, Pawlowski, Singh, Vandersloot, Wilson-Ewing,...) Emerging Scenario: In simplest models: Vast classical regions bridged deterministically by quantum geometry. No new principle needed to join the pre-big bang and post-big-bang branches.

8 p. The k=0 FLRW Model FLRW, k=0 Model coupled to a massless scalar field φ. Instructive because every classical solution is singular. Provides a foundation for more complicated models φ *10 4 2*10 4 3*10 4 4*10 4 5*10 4 Classical Solutions In the WDW theory: Good infrared behavior but persistence of singularity in the ultraviolet. Precise Statement provided by the consistent histories approach (Craig & Singh). v

9 p. The WDW Theory versus LQC In strikign contrast to the WDW theory, the singularity resolved in LQC. Why? In LQG we have a mathematically rigorous kinematical framework uniquely selected by the requirement of background independence (Lewandowski, Okolow, Sahlmann, Thiemann; Fleishchhack). This descends to LQC in a well defined manner. (AA, Henderson). This kinematics is distinct from the Schrödinger representation used in the WDW theory. In particular, in the WDW equation, 2 (Ψ(v, φ))/ v 2 = l 2 PĤφΨ(v, φ), the differential operator fails to be well-defined on the LQC Hilbert space and is naturally replaced by a difference operator: C + (v) Ψ(v + 4, φ) + C o (v) Ψ(v, φ) + C (v)ψ(v 4, φ) = l 2 P ĤφΨ(v, φ) (The Step size is determined by the area gap of Riemannian quantum geometry underlying LQG) Good agreement with the WDW equation at low curvatures but drastic departures in the Planck regime precisely because the WDW theory ignores quantum geometry (the area gap).

10 p. 1 The k=0 FLRW Model FLRW, k=0 Model coupled to a massless scalar field φ. Instructive because every classical solution is singular. Provides a foundation for more complicated models φ *10 4 2*10 4 3*10 4 4*10 4 5*10 4 Classical Solutions In the WDW theory: Good infrared behavior but persistence of singularity in the ultraviolet. Precise Statement provided by the consistent histories approach (Craig & Singh). v

11 p. 1 k=0 LQC Ψ(v,φ) * * * *10 4 v 2.5* * * * φ Absolute value of the physical state Ψ(v, φ) (AA, Pawlowski, Singh)

12 p. 1 k=0 LQC 0 LQC classical φ *10 4 2*10 4 3*10 4 4*10 4 5*10 4 v Expectations values and dispersions of ˆV φ & classical trajectories. The wave packet is peaked at an effective trajectory that incorporates the key quantum geometry effects. (AA, Pawlowski, Singh)

13 p. 1 LQC: k=0 Results No unphysical matter. All energy conditions satisfied. But the left side of Einstein s equations modified because of quantum geometry effects (discreteness of eigenvalues of geometric operators.): Main difference from WDW theory. Effective Equations: To compare with the standard Friedmann equation, convenient to do an algebraic manipulation and move the quantum geometry effect to the right side. Then: (ȧ/a) 2 = (8πGρ/3)[1 ρ/ρ crit ] where ρ crit 0.41ρ Pl. Big Bang replaced by a quantum bounce. Observables: The matter density operator ˆρ has an absolute upper bound on the physical Hilbert space (AA, Corichi, Singh): ρ sup = 3/16π 2 γ 3 G ρ Pl! Provides a precise sense in which the singularity is resolved. Mechanism: Quantum geometry creates a brand new repulsive force in the Planck regime, neatly encoded in the difference equation. Replaces the big-bang by a quantum bounce.

14 p. 1 Generalizations More general singularities: At finite proper time, scale factor may blow up, along with similar behavior of density or pressure (Big rip) or curvature or their derivatives diverge at finite values of scale factor (sudden death). Quantum geometry resolves all strong singularities in homogeneous isotropic models with p = p(ρ) matter (Singh). Beyond Isotropy and Homogeneity: Inclusion of a cosmological constant and the standard m 2 φ 2 inflationary potential. Inclusion of anisotropies. k = 1 closed cosmologies. The Gowdy model with inhomogeneities and gravitational waves. Singularities are resolved and Planck scale physics explored in all these cases. (AA, Bentevigna, Pawlowski, Singh, Vandersloot, Wilson-Ewing,...) Summary: Cumulative evidence strongly suggests that all strong curvature, space-like singularities resolved by quantum geometry effects. For the k=0, FLRW background used in inflation, big bang replaced by a quantum bounce.

15 p. 1 Inflation φ e+10 1e+20 1e+30 1e+40 1e+50 1e+60 1e+70 1e+80 1e+90 1e+100 Expectations values and dispersions of ˆV φ for a massive inflaton φ with phenomenologically preferred parameters (AA, Pawlowski, Singh). v

16 p. 1 Inflationary Paradigm: Incompleteness Quantum Gravity Issues: Fate of the initial singularity (Borde, Guth & Vilenkin): Is the infinite curvature really attained? What is the nature of the quantum space-time that replaces Einstein s continuum in the Planck regime? In the systematic evolution from the Planck regime in the more complete theory, does a slow roll phase compatible with the WMAP data arise generically or is an enormous fine tuning needed? In classical GR, if we evolve the modes of interest back in time, they become trans-planckian. Is there a QFT on quantum cosmological space-times needed to adequately handle physics at that stage? In the more complete theory, is the Bunch-Davis vacuum at the onset of the slow roll compatible with WMAP generic or does it need enormous fine tuning?

17 p Background Geometry in LQC How generic is the necessary slow roll inflationary phase? Even if a theory allows for inflation, a sufficiently long slow roll may need extreme fine tuning. To test this, we need a measure on the space S of solutions to the equations. Elegant solution: Use the Liouville measure to calculate a priori probabilities (Gibbons, Hawking, Page,...). They are useful, if extremely low or extremely high. Controversy in the literature. For the m 2 φ 2 potential, answers from probability close to 1 (Kauffman, Linde, Mukhanov) to e 165 (Gibbons, Turok)! Main Reason: The question is ill posed in general relativity. Problem: The Liouville volume of S is infinite! But the infinity is a gauge artifact (associated with the a λa rescaling freedom). But because of the Big Bang singularity, no natural way to factor out the gauge freedom. Vastly different answers stem from different gauge fixing. (AA, Sloan; Corichi, Karami; Shiffrin, Wald)

18 p Background Geometry in LQC In LQC, the Big Bang is replaced by the Big Bounce where the effective geometry and matter fields are all smooth. Provides a natural strategy to precisely formulate and answer: How generic is the desired slow roll phase? Start with generic data at the bounce. Evolve. Will it enter slow roll at the GUT energy scale determined by the 7 year WMAP data (ρ m 4 Pl )? Note: 11 orders of magnitude from the bounce to the onset of the desired slow roll! Answer: LQC φ B (0, ) m Pl. If φ B 1.06m Pl, the data evolves to a solution that encounters the slow roll compatible with the 7 year WMAP data sometime in the future. Thus, Almost every initial data at the bounce evolves to a solution with the desired slow roll sometime in the future. Fractional Liouville volume of data that does not achieve the slow roll compatible with WMAP data, in particular with 63 e-foldings < Result (AA & Sloan) much stronger than the attractor idea.

19 p. 1 Inflationary Paradigm: Incompleteness Quantum Gravity Issues: Fate of initial singularity (Borde, Guth & Vilenkin): Is the infinite curvature really attained? What is the nature of the quantum space-time that replaces Einstein s continuum in the Planck regime? In the systematic evolution from the Planck regime in the more complete theory, does a slow roll phase compatible with the WMAP data arise generically or is an enormous fine tuning needed? In classical GR, if we evolve the modes of interest back in time, they become trans-planckian. Is there a QFT on quantum cosmological space-times needed to adequately handle physics at that stage? Can one arrive at the Bunch-Davis vacuum (at the onset of the WMAP slow roll) from more fundamental considerations?

20 3. An LQG Extension of the Inflationary Paradigm LQG Strategy: Focus on the appropriate truncation/sector of classical GR and pass to quantum gravity using LQG techniques. (Has been successful in singularity resolution, BH entropy and the graviton propagator calculations.) Sector of interest for inflation: Linear Perturbations off FLRW background with an inflaton φ in a suitable potential as the matter source. Includes inhomogeneities, but as perturbations. Truncated Phase Space {(a, φ; δh ab (x), δφ(x)) and their conjugate momenta} Quantum Theory: Start with Ψ(a, φ; δh ab (x), δφ(x)) and impose appropriately truncated constraints (i.e. to quadratic order in first order perturbations). Then, for purely inhomogeneous lapses and shifts, one obtains: Ψ = Ψ(a, φ; S(x), T (I) (x)) Ψ(a, φ; S k, T (I) ); with I = 1,2. k i.e., states depend only on 3 gauge invariant DOF of perturbations. For purely homogeneous lapse and shift: i φ Ψ = ( Ĥ 2 o 2Ĥ2) 1 2 Ψ Dynamical equation for Ψ. Again, the constraint can be interpreted as an evolution equation in the relational time variable φ. Perturbations and background geometry both quantum, described by Ψ(a, φ; S k, T (I) k ). p. 2

21 p. 2 Implications of Truncation i) Can take Ĥ o as the main term and Ĥ 2 as a small correction, i.e., ignore terms (Ho 1/2 Ĥ 2 Ho 1/2 ) n, for n 2; ii) Can assume Ψ factorizes as: Ψ(a, φ; S(x), T (I) (x)) = Ψ o (a, φ) ψ(s k, T (I) k, φ) Idea: Ψ o (a, φ) is sharply peaked at an effective LQC solution gab o of part 1, which will eventually encounter the slow roll compatible with WMAP data as in part 2. ψ propagates on the quantum geometry determined by Ψ o : QFT on QST a la (AA, Lewandowski, Kaminski) Then Ψ o (a, φ) evolves via i Ψ o = Ĥ o Ψ o ; and the evolution of ψ turns out to be sensitive to only certain features of the background quantum geometry of Ψ o (a, φ). Surprising consequence: Complete mathematical equivalence with quantum fields Ŝ, ˆT I on a smooth geometry ḡ ab : ḡ ab dx a dx b = ā 2 ( d η 2 + d x 2 ) with dη = Ĥ 1 o [ Ĥ 1/2 o â 4 Ĥo 1/2 ] 1/2 dφ; ā 4 = ( Ĥ 1/2 o â 4 Ĥo 1/2 )/ Ĥ 1 o Because of this, the mathematical machinery of adiabatic states and renormalization of the Hamiltonian can be lifted to the QFT on cosmological QSTs under consideration. Result: Full mathematical control to compute the CMB power spectrum, and spectral indices starting from the bounce.

22 p. 2 Key Questions 1. Take Ψ o peaked at a generic effective LQC solution g o ab which undergoes slow roll compatible with WMAP as in part 2. Are there a large class of initial states ψ of perturbations at the bounce for which ρ Pert ρ BG from the bounce to the onset of the slow roll compatible with WMAP (so that our truncation strategy is justified by self-consistency)? 2. At the end of the WMAP compatible slow roll, do we recover the inflationary power spectrum: 2 R (k, t k) H2 (t k ) πm 2 Pl ǫ(t k)? (t k is the time the mode k exits the Hubble horizon during slow roll) If so, we will have a quantum gravity completion of the inflationary paradigm. 3. Does ψ(s k, T (I) ( k) ; φ B) evolve to a state which is indistinguishable from the Bunch Davis vacuum at the onset of slow roll or are there deviations with observable consequences for more refined future observations (e.g. non-gaussianitities in the bispectrum (Agullo, Shandera) )?

23 p. 2 An LQG Completion of the Inflationary Paradigm Analysis involves several conceptual and technical subtleties. Result: If ψ(s k, T (I), φ) is close to (the infinite dimensional space of) translationally k invariant states at the bounce the answers to first two questions is in the affirmative. (Agullo, AA, Nelson) Trans-Planckian Frequencies: In LQG, largeness of frequencies by itself is not directly relevant. For example: In the treatment of the background quantum geometry, p φ is highly trans-planckian (typical values: > Planck units!!). But quantum theory completely well-defined and ˆρ is bounded above by 0.41ρ Pl. Thanks to the background quantum geometry, trans-planckian modes can be readily incorporated provided the test field approximation holds: ρ Pert ρ BG. Careful analysis was needed to show that this is the case. The evolved state at the onset of slow roll is indistinguishable from the Bunch-Davis vacuum if Φ B 1.2m Pl. If φ B 1.1, conflict with observations. In the narrow window in between, state at the onset of inflation is populated with particles. No problem with WMAP data. But potential for interesting predictions of non-gaussianties. Concrete implications of the initial state at the bounce on observations and vice versa.

24 p. 2 Predictions for the Power spectrum Mathematical power spectra for the scalar mode. Red: standard inflationary scenario using the Bunch-Davis vacuum at the onset of slow roll. Blue: LQG power spectrum. For modes whose wave length at the CMB surface is less than or equal to the size of the observable universe at that time, the two are indistinguishable.

25 p. 2 An LQG Completion of the Inflationary Paradigm Non-triviality of the result: Trans-Planckian problem in GR; Superinflation and the subtle behavior of the Hubble radius in LQG.

26 p Summary Can one provide a quantum gravity extension of the inflationary paradigm all the way back to the Big Bounce? Background geometry: Big Bang singularity: In LQG, quantum geometry creates a brand new repulsive force in the Planck regime, replacing the big-bang by a quantum bounce. Repulsive force rises and dies very quickly but makes dramatic changes to classical dynamics. (AA, Pawlowski, Singh,...) New paradigm: Quantum space-times may be vastly larger than Einstein s. The horizon problem dissolves. UV and IR Challenge: Singularity resolution and the detailed recovery of classical GR at low curvatures/densities? Met in cosmological models. Singularities analyzed are of direct cosmological interest. Do need extreme fining tuning to get agreement with the 7 year WMAP data? Question has a well posed formulation: because of singularity resolution, can focus on initial data at the bounce. LQC φ B (0, 10 6 ). If φ B 1.06m Pl, the (effective) solution admits a slow roll phase compatible with the the 7 year WMAP data. (AA, Sloan)

27 p. 2 Summary (contd) Perturbations: Since they propagate on quantum geometry, using QFT on cosmological quantum geometries (AA, Lewandoski, Kamniski), trans-planckian issues can be handled systematically. (Agullo, AA, Nelson) If Φ B > 1.2m Pl and ψ(s k, T (I), φ) is close to (the infinite dimensional k space) of translationally invariant states, then modes of observational interest are all in the Bunch Davis vacuum at the onset of the WMAP slow roll Predictions of the standard inflationary scenario for the power spectra, spectral indices & ratio of tensor to scalar modes recovered starting from the deep Planck era. (Agullo, AA, Nelson) Non-Gaussianity: If φ B 1.05m Pl, then the state at the onset of inflation has excitations over the Bunch-Davis vacuum. These give rise to specific non-gaussianities which are furthermore important for the bias relating the dark and visible matter power spectra. So could be observed in principle: Link between observations and the initial state! A window to probe the Planck era around the LQC bounce. (Agullo, AA, Nelson, Shandera)

28 p. 2 Final Remarks Deeper, physical understanding of the boundary conditions at the bounce from physics of the contracting branch? Semi-heuristic considerations suggest an avenue that may naturally lead to these conditions. But much more work is needed to see if these ideas can be made precise. LQG does not imply that inflation must have occurred because it does not address particle physics issues. The analysis simply assumed that there is an inflaton with a suitable potential. But it does show concretely that many of the standard criticisms (e.g. due to Brandenberger) of inflation can be addressed by providing a self-consistent theory starting from the Big Bounce to the onset of slow roll, facing the Planck regime squarely. Background References: Short Review of LQC: Proceedings of the Cosmology Conference, Paris, arxiv: Detailed Review of LQC: AA & Singh: CQG 28, (2011) Short Review of LQG: arxiv:

29 Supplementary Material p. 2

30 p. 3 k=0 Model with Positive Λ 0.8 classical LQC φ *10 3 4*10 3 6*10 3 8* * * *10 4 v Expectations values and dispersions of ˆV φ & classical trajectories. (AA, Pawlowski)

31 p. 3 The k=1 Closed Model: Bouncing/Phoenix Universes. Another Example: k = 1 FLRW model with a massless scalar field φ. Instructive because again every classical solution is singular; scale factor not a good global clock; More stringent tests because of the classical re-collapse. ( Lemaître, Tolman, Sakharov, Dicke,...) classical φ *10 4 2*10 4 3*10 4 4*10 4 5*10 4 6*10 4 7*10 4 8*10 4 9* *10 5 Classical Solutions v

32 p. 3 k=1 Model: WDW Theory WDW classical φ *10 4 2*10 4 3*10 4 4*10 4 5*10 4 6*10 4 7*10 4 8*10 4 9* *10 5 v Expectations values and dispersions of ˆV φ.

33 p. 3 k=1 Model: LQC 0 LQC Effective Classical -1-2 φ *10 4 4*10 4 6*10 4 8* *10 5 v Expectations values and dispersions of ˆV φ & classical trajectories. (AA, Pawlowski, Singh, Vandersloot)

34 p. 3 Merits and Limitations of QC One s first reaction: Symmetry reduction gives only toy models! Full theory much richer and much more complicated. But examples can be powerful. Full QED versus Dirac s hydrogen atom. Singularity Theorems versus first discoveries in simple models. BKL behavior: homogeneous Bianchi models. Do not imply that behavior found in examples is necessarily generic. Rather, they can reveal important aspects of the full theory and should not be dismissed a priori. One can work one s way up by considering more and more complicated cases. (e.g. recent work of the Madrid group on Gowdy models which have infinite degrees of freedom). At each step, models provide important physical checks well beyond formal mathematics. Can have strong lessons for the full theory.