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# L. Alvarez-Gaume - Minimal Inflation

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The SEENET-MTP Workshop JW2011
Scientific and Human Legacy of Julius Wess
27-28 August 2011, Donji Milanovac, Serbia

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### L. Alvarez-Gaume - Minimal Inflation

1. 1. Minimal Inﬂation Luis Alvarez-Gaume BSI August 29th, 2011 Work in collaboration with C. Gómez and R. Jiménez 1Monday, 29 August, 2011
2. 2. Inﬂation is 30 years old Originally Inﬂation was related to the horizon, ﬂatness Luis Alvarez-Gaume BSI August 29th, 2011 and relic problems Nowadays, its major claim to fame is seeds of structure. There is more and more evidence that the general philosophy has some elements of truth, and it is remarkably robust... 2Monday, 29 August, 2011
3. 3. Summary of FRLW Matter is well represented by a perfect ﬂuid Tab = (p + ρ)ua ub + pgab 2 ˙ a k 8πG + 2 = ρ a a 3 The Einstein equations are ¨ a 4πG Gab = 8πG Tab = − (ρ + 3p) a 3 These are the possible ρ + 3H(ρ + p) = 0 ˙ k = ±1, 0 curvatures distinguishing the space sections p = wρ ds2 = −dt2 + a(t)2 ds2 ˙ a(t) 3 ρ+3 ˙ (1 + w)ρ = 0 a(t) a 3(1+w) Luis Alvarez-Gaume BSI August 29th, 2011 0 ˙ a(t) ρ k ρ = ρ0 H≡ Ω≡ 1−Ω = 2 2 a a(t) ρc a H 3H 2 ρc = 8πG 1 √ √ 1 ab S = −g(R − 2Λ) + −g − g ∂a φ∂b φ − V (φ) 16πG 2 1 ˙2 1 ˙2 ρ = φ + V (φ) p = φ − V (φ) 2 2 3Monday, 29 August, 2011
4. 4. Accelerating the Universe We need to violate the dominanat energy condition for a sufﬁciently long time. ¨ a 4πG = − (ρ + 3p) If we take during inﬂation H approx. constant, a 3 the number of e-foldings: 1 p − ρ 3 a(t) 0 ¨ |1 − Ω| ∝ a−2 ˙ Number of e-foldings could be a(tf ) N = log = H(tf − ti ) 50-100, making a huge Universe. a(ti ) We can use QFT to construct |1 − Ω(tf )| = e −2N |1 − Ω(ti )| some models 1 ˙2 1 ˙2 1 √ √ 1 ab ρ = φ + V (φ) p = φ − V (φ) S = 16πG −g(R − 2Λ) + −g − g ∂a φ∂b φ − V (φ) 2 2 2 Luis Alvarez-Gaume BSI August 29th, 2011 Slow roll paradigm ˙2 V V Hybrid inﬂation φ V (φ) 1, 1 V V (Courtesy of Licia Verde) 4Monday, 29 August, 2011
5. 5. Origins of Inﬂation The number of models trying generating inﬂation is enormous. Frequently they are not very compelling and with large ﬁne tunings. Different UV completions of the SM provide alternative scenarios for cosmology, and it makes sense to explore their cosmic consequences. Luis Alvarez-Gaume BSI August 29th, 2011 5Monday, 29 August, 2011
6. 6. Basic properties Enough slow-roll to generate the necessary number of e-foldings and the necessary seeds for structure. A (not so-) graceful exit from inﬂation, otherwise we are left with nothing. A way of converting “CC” into useful energy: reheating. Everyone tries to ﬁnd “natural” mechanisms within Luis Alvarez-Gaume BSI August 29th, 2011 its favourite theory. 6Monday, 29 August, 2011
7. 7. Supersymmetry is our choice In the standard treatment of global supersymmetry the order parameter of supersymmetry breaking is associated with the vacuum energy density. More precisely, in local Susy, the gravitino mass is the true order parameter. Having a vacuum energy density will also break scale and conformal invariance. When supergravity is included the breaking mechanism is more subtle, and the scalar potential far more complicated. Needless to say, all this assumes that supersymmetry exists in Nature Luis Alvarez-Gaume BSI August 29th, 2011 Claim: It provides naturally an inﬂaton and a graceful exit 7Monday, 29 August, 2011
8. 8. SSB Scenarios Observable Sector Gauge mediation MEDIATOR Gravity mediation Anomaly mediation Hidden Sector It is normally assumed that SSB takes places at scales well below the Planck scale. The universal prediction is then the existence of a massless goldstino that is eaten by the gravitino. However in the scenario considered, the low-energy gravitino couplings are dominated by its goldstino component and can be analyzed also in the global limit. Luis Alvarez-Gaume BSI August 29th, 2011 This often goes under the name of the Akulov-Volkov lagrangian, or the non-linear realization of SUSY f µ2 µ → ∞ m3/2 = = Mp Mp M → ∞ m3/2 ﬁxed 8Monday, 29 August, 2011
9. 9. Flat directions One reason to use SUSY in inﬂationary theories is the abundance of ﬂat directions. Once SUSY breaks most ﬂat directions are lifted, sometime by non- perturbative effects. However, the slopes in the potential can be maintained reasonably gentle without excessive ﬁne-tuning. For ﬂat Kahler potentials, and F-term breaking, there is always a complex ﬂat direction in the potential. A general way of getting PSGB, the key to most susy models. The property below holds for any W breaking SUSY. Most models of supersymmetric inﬂation are hybrid models (multi-ﬁeld models, chaotic, waterfall...) Luis Alvarez-Gaume BSI August 29th, 2011 F = −∂ W (φ) . V (φ + z F ) = V (φ) V = ∂ W (φ) ∂ W (φ) 9Monday, 29 August, 2011
10. 10. Important properties of SSB The Akulov-Volkov-type actions provide the correct framework to analyse the general properties of SSB. We can use the recent Komargodski-Seiberg presentation. The starting point of their analysis is the Ferrara-Zumino (FZ) multiplet of currents that contains the energy-momentum tensor, the supercurrent and the R-symmetry current α ˙ Jµ = jµ + θα Sµα + θα S µ + (θσ ν θ) 2Tνµ + . . . ˙ √ X = x(y) + 2θψ(y) + θ2 F (y) Luis Alvarez-Gaume BSI August 29th, 2011 √ 2 µ α˙ 2 ψα = σ ˙S , F = T + i∂µ j µ 3 αα µ 3 α ˙ D Jαα = Dα X ˙ 10Monday, 29 August, 2011
11. 11. General Lagrangian 4 2 2 i ¯¯ ¯ ¯ ¯¯ S= d θK(Φ , Φ ) + d θW (Φ ) + d θW (Φi ) i i ¯˙¯ ¯˙ ¯ ¯ Jαα = 2gi (Dα Φi )(Dα Φ) − 2 [Dα , Dα ]K + i∂α (Y (Φ) − Y (Φ)) ˙ 3 1 2 1 2 X = 4 W − D K − D Y (Φ) 3 2 X is a chiral superﬁeld, microscopically it contains the conformal anomaly (the anomaly multiplet), hence it contains the order parameter for SUSY breaking as well as the goldstino ﬁeld. It may be elementary in the UV, but composite in the IR. Generically its scalar component is a PSGB in the UV. This is our inﬂaton. The difﬁculty with this approach is that WE WANT TO BREAK SUSY ONLY ONCE! unlike other scenarios in the literature Luis Alvarez-Gaume BSI August 29th, 2011 The key observation is: X is essentially unique, and: 2 = 0 X → XN L U V → IR XN L SP oincare/P oincare 11Monday, 29 August, 2011
12. 12. Some IR consequences L = d θ XN L X N L + 4 d θ f XN L + c.c. 2 G2 √ XN L = + 2θG + θ2 F 2F Luis Alvarez-Gaume BSI August 29th, 2011 This is precisely the Akulov-Volkov Lagrangian 12Monday, 29 August, 2011
13. 13. Coupling goldstinos to other ﬁelds: reheating We can have two regimes of interest. Recall that a useful way to express SUSY breaking effects in Lagrangians is the use of spurion ﬁelds. The gluino mass can also be included... msof t E Λ The goldstino superﬁeld is the spurion XN L 2 V 4 d θ m Qe Q + d2 θ XN L (B Q Q + AQ Q Q ) + c.c. f f Integrate out the massive superpartners Luis Alvarez-Gaume BSI August 29th, 2011 E msof t adding extra non-linear constraints For light fermions, and similar conditions XN L = 0, 2 XN L QN L = 0 for scalars, gauge ﬁelds,... Reheating depends very much on the details of the model, as does CP violation, baryogenesis... 13Monday, 29 August, 2011
14. 14. Some details An important part of our analysis is the fact that the graceful exit is provided by the Fermi pressure in the Landau liquid in which the state of the X-ﬁeld converts once we reach the NL-regime. This is a little crazy, but very minimal however... ¯ a(X + X) bX X¯ c X +X 2 ¯2 ¯ X +X ¯ ¯ K(X, X) = X X 1+ − − + ... − 2M 2 log +1 2M 6M 2 9M 2 M W (X) = f0 + f X Luis Alvarez-Gaume BSI August 29th, 2011 K ¯ 3 1 V =e M2 −1 (KX,X DW DW ¯ − 2 |W |2 ) D W = ∂X W + ∂ KW 2 X M M 14Monday, 29 August, 2011
15. 15. The full lagrangian Luis Alvarez-Gaume BSI August 29th, 2011 15Monday, 29 August, 2011
16. 16. Primordial density ﬂuctuations Luis Alvarez-Gaume BSI August 29th, 2011 2 2 Mpl V nS = 1 − 6 + 2η, f ∼ 1011−13 GeV = 2 V , r = 16 nt = −2, 2 V V Mpl4 η= Mpl , ∆2 = . V R 24π 2 16Monday, 29 August, 2011
17. 17. Choosing useful variables √ z = M (α + iβ)/ 2 ds2 = 2gzz dsd¯ = ∂z ∂z K(α, β)M 2 (dα2 + dβ 2 ) ¯ z ¯ 3 3 1 ˙ S=L dta g(α, β)M 2 (α2 + β 2 ) − f 2 V (α, β) ˙ 2 Luis Alvarez-Gaume BSI August 29th, 2011 1 t = τ M/f S=L f 3 2 m−1 3/2 dτ a 3 g(α, β)(α2 + β 2 ) − V (α, β) 2 17Monday, 29 August, 2011
18. 18. Cosmological equations The full equations of motion, neglecting fermions for the moment are: a 1 α + 3 α + ∂α log g(α2 − β 2 ) + ∂β log gα β + g −1 Vα = 0 a 2 a 1 Plus fermion β + 3 β + ∂β log g(β 2 − α2 ) + ∂α log gα β + g −1 Vβ = 0 terms on the RHS a 2 a H 1 1 = =√ g(α2 + β 2 ) + V (α, β) a M3/2 3 2 Looking for the attractor and slow roll implies that the geodesic equation on the target manifold is satisﬁed for a particular set of initial conditions. This determines the attractor trajectories in general Luis Alvarez-Gaume BSI August 29th, 2011 for any model of hybrid inﬂation. Numerical integration shows how it works. We have not tried to prove “theorems’ but there should be general ways of showing how the attractor is obtained this way 1 ˙ DΦi /dt ∼ 0 H= 3V + 6V + 9V 2 18 18Monday, 29 August, 2011
19. 19. The scalar potential 1.0 Β 0.5 0.0 0.10 0.05 0.00 0.0 0.5 Α 1.0 a=0, b=1, c=-1.7 a=0, b=1, c=0 Ε 1.0 Β 0.5 Luis Alvarez-Gaume BSI August 29th, 2011 0.0 1.0 0.5 0.0 0.0 0.5 Α 1.0 19Monday, 29 August, 2011
20. 20. Attractor and inﬂationary trajectories Β 1.0 Β 0.5 0.8 0.4 0.3 0.2 0.1 0.6 Α 0.2 0.4 0.6 0.8 V 0.10 0.08 0.4 Luis Alvarez-Gaume BSI August 29th, 2011 0.06 0.04 0.02 0.2 t 0.05 0.10 0.50 1.00 5.00 Nearly a textbook example of inﬂationary potential Α 0.05 0.10 0.15 0.20 0.25 0.30 20Monday, 29 August, 2011
21. 21. Decoupling and the Fermi sphere mINF 2 η = ( ) m3/2 The energy density in the universe (f^2) contained in the coherent X-ﬁeld quickly transforms into a Fermi sea whose level is not difﬁcult to compute, we match the high energy theory dominated by the X-ﬁeld and the Goldstino Fock vacuum into a theory where effectively the scalar has disappeared and we get a Fermi sea, whose Fermi momentum is Luis Alvarez-Gaume BSI August 29th, 2011 f qF = η To produce the observed number of particles in the universe leads to gravitino masses in the 10 TeV region. 21Monday, 29 August, 2011
22. 22. Summary of our scenario We take as the basic object the X ﬁeld containing the Goldstino. Its scalar component above SSB behaves like a PSGB and drives inﬂation Its non-linear conversion into a Landau liquid in the NL regime provides an original graceful exit Reheating can be obtained through the usual Goldstino coupling to low energy matter In the simplest of all possible such scenarios, the Susy breaking scale is ﬁtted to be of Luis Alvarez-Gaume BSI August 29th, 2011 the order of 10^{13-14} GeV 22Monday, 29 August, 2011
23. 23. Luis Alvarez-Gaume BSI August 29th, 2011Monday, 29 August, 2011 Thank you 23
24. 24. Luis Alvarez-Gaume BSI August 29th, 2011Monday, 29 August, 2011 Back up slides 24
25. 25. Graceful exit The theory is constructed with a Kahler potential violating explicitly the R-symmetry which phenomenologically is a disaster. We also input our conservative prejudice that the values of the inﬂaton ﬁeld should be well below the Planck scale. The general supergravity lagrangian up to two derivatives and four fermions is given by a very simple set of function: The Jordan frame function The Kahler potential The superpotential (and the gauge kinetic function and moment maps) Luis Alvarez-Gaume BSI August 29th, 2011 K ¯ 3 1 V =e M2 −1 (KX,X DW DW ¯ − 2 |W |2 ) D W = ∂X W + ∂ KW 2 X M M 2 V ¯ ¯ K(X, X) = X X + . . . η = 1 + ... η = Mpl , V 25Monday, 29 August, 2011
26. 26. Summary The chiral superﬁeld X is essentially deﬁned uniquely in the ultraviolet. It has the following properties: In the UV description of the theory, it appears in the right hand side the supercurrent equation (FZ) where it represents a measure of the violation of conformal invariance. The expectation value of its F component is the order parameter of supersymmetry breaking. When supersymmetry is spontaneously broken, we can follow the ﬂow of X to the infrared (IR). In the IR this ﬁeld satisﬁes a non-linear constraint and becomes the “goldstino superﬁeld. The correct normalization is 3X / 8 f. At low energies this ﬁeld replaces the standard spurion coupling. Luis Alvarez-Gaume BSI August 29th, 2011 XN L = 0, 2 G2 √ XN L = + 2 θ G + θ2 F. 2F 26Monday, 29 August, 2011
27. 27. Vanishing cosmological constant √ √ 2 2 (2h− 2α)(−72+α( 2+2α)(18+5α )) 1− √ 3 √ 2 72(1+ 2α) − −2h + 2α + α2 2 =0 4 1+ 3 + √ 2 (1+ 2α) 0.9980 0.9975 Luis Alvarez-Gaume BSI August 29th, 2011 0.9970 h 0.9965 0.9960 0.10 0.15 0.20 0.25 0.30 Α 27Monday, 29 August, 2011
28. 28. ...continued X = M (α + i β) V = f 2 (1 + A1 (α2 + β 2 ) + B1 (α2 − β 2 ) + . . .) 2 f2 2 f2 m2 α = (A1 + B1 ), m2 β = (A1 − B1 ). M2 M2 One could be more explicit, and choose some supersymmetry breaking superpotential. leading to an effective action description of X for scales well below M. It is however better to explicit examples of UV-completions of the theory. The potential is taken to be stable A - B 0. We consider the beginning of inﬂation well below M, hence the initial conditions are such that a,b are much smaller than 1. In fact, since b is the lighter ﬁeld, we take this one to be the inﬂaton. The inﬂationary period goes from this scale until the value of the ﬁeld is close to the typical soft α, β ∼ f /M breaking scale of the problem, where we certainly enter the non-linear, or strong coupling regime, the ﬁeld X become X_{NL} and behaves like a spurion. Its couplings to low-E ﬁelds is (not including gauge couplings) Luis Alvarez-Gaume BSI August 29th, 2011 X N L 2 2 V XN L 1 L = − d4 θ ¯ f m Qe Q + d2 θ f 2 Bij Qi Qj + . . . + h.c. Once we reach the end of inﬂation, the ﬁeld X becomes nonlinear, its scalar component is a goldstino bilinear and the period of reheating begins. The details of reheating depend very much on the microscopic model. At this stage one should provide details of the ``waterfall that turns the huge amount of energy f^2 into low energy particles. Part of this energywill be depleted and converted into low energy particles through the soft couplings, we need to compute bounds on T-reheating 28Monday, 29 August, 2011
29. 29. Slow roll conditions 2 2 Mpl V nS = 1 − 6 + 2η, = , 2 V r = 16 nt = −2, 2 V V Mpl4 η= Mpl , ∆2 = . V R 24π 2 2 = 2 ( (A1 − B1 )β ) + . . . V = f 2 (1 + A1 (α2 + β 2 ) + B1 (α2 − β 2 ) + . . .), η = 2 (A1 − B1 ) + . . . , Delta^2 is the amplitude of the initial perturbations. Since a,b are very small, the value of epsilon is also very small. For eta we need to make a slight ﬁne-tuning. In fact it is related to the ratio between the inﬂaton and the gravitino masses. Now we can compute some cosmological consequences. For instance, the number of e-folding to start 1 dx βf dβ 1 N = √ = √ N = √ log βf . m3/2 = f M 2 βi 2 2|A1 − B1 | βi Luis Alvarez-Gaume BSI August 29th, 2011 M 2 1 m2 1 m2 √ m3/2 |A1 − B1 | = β 2 , |A1 + B1 | = α 2 , N = 2 log βf 2 m3/2 2 m3/2 mβ βi 1/4 V f 1/2 = 1/2 = .027M, WMAP data 21/4 (|A1 − B1 |β) 29Monday, 29 August, 2011
30. 30. Some numbers √ 1/2 2 √ m3/2 2 βf m3/2 f mβ N = 2 log , 21/4 = 0.027, η = mβ βi mβ M m3/2 µ √ ≈ 5.2 10−4 η µ= f mβ = m3/2 η, M η ∼ .1 µ ∼ 1013 GeV βi ∼ 1013 /M, βf ∼ 103 /M N ∼ 110 With moderate values of eta, we can get susy breaking scales in the range 10^{11-13} without major ﬁne Luis Alvarez-Gaume BSI August 29th, 2011 tunings. We get enough e-foldings and the inﬂaton is lighter than the gravitino by sqrt{eta}. Reheating depends very much on the details of the theory, but some estimates can be made 3/2 TRH = 10−10 f /GeV GeV 107 TRH 109 nχ ∼ 10 70−90 Sf /Si = 10 ( f /GeV )−1/2 7 30Monday, 29 August, 2011
31. 31. Comments Supersymmetry breaking can be the driving force of inﬂation. We have used the unique chiral superﬁeld X which represents the breaking of conformal invariance in the UV, and whose fermionic component becomes the goldstino at low energies. Its auxiliary ﬁeld is the F-term which gets the vacuum expectation value breaking supersymmetry. To avoid the eta problem it is crucial to have R-symmetry explicitly broken. The imaginary part of X plays the role of the inﬂaton. It represents a pseudo-goldstone boson, or rather a pseudo-moduli. There are many model in the market with such constructions (Ross, Sarkar, German...). We have been avoiding UV completions on purpose. Similar to the inﬂation paradigm, the susy breaking paradigm has some universal features, and this is all we wanted to explore. Better and more detail numbers require microscopic details. Luis Alvarez-Gaume BSI August 29th, 2011 An interested consequence (work in progress) is the detail theory of density perturbations in terms of X_{NL} correlators. This is related to current algebra arguments and hence it is quite general. If these comments are taken seriously, we may have plenty to learn about Susy from the sky! 31Monday, 29 August, 2011
32. 32. Slow roll, more details 8πG ˙ a H2 = ρ H= ρ ≈ V (φ) 3 a 2 ¨ ˙ ˙ 2 MP l V 2 V φ + 3H(t) φ + V (φ) = 0 → 3H(t) φ ≈ −V (φ) = 2 V η = MP l V During inﬂation, it is a good approximation to treat the inﬂaton ﬁeld as moving in backgroun De-Sitter space, with H constant. The quantum ﬂuctuations of the inﬂaton are the source of primordial inhomogeneities. The existence of this quantum ﬂuctuations are similar to the Hawking radiation in dS-space. Schematically the picture shows what happens to the ﬂuctuations. Remember that in dS there is a particle horizon 1/H H2 φ(x) = 2 Luis Alvarez-Gaume BSI August 29th, 2011 2π Lphy = a(t) Lcom ds2 = −dt2 + e2Ht (dx2 + dy 2 + dz 2 ) 32Monday, 29 August, 2011
33. 33. Power spectrum In ﬂat De Sitter coordinates we can Fourier transform F.T. To compute the power spectrum, we quantize the ﬁeld in dS space. Then, expanding the ﬁeld in oscillators: 3 ∞ k dk Pφ (k) = V 2 φ(k) φ(k)† φ(x)2 = Pφ (k) 2π 0 k 2 H Precisely: Pφ (k) = 2π aH=a0 k Finally, the curvature perturbations can be computed using the perturbed Einstein equations, and a number of laborious and subtle steps, we get the power spectrum for the induced density perturbations (at horizon exit) 2 2 H H 1 H2 PR (k) = = ˙ ˙ Luis Alvarez-Gaume BSI August 29th, 2011 φ 2π aH=a0 k 4π 2 φ t=tk Using the slow-roll equations and deﬁning the spectral index: d log PR (k) ≡ n(k) − 1 d log k 1 1 V3 1 1 V k n−1 PR (k) = = PR (k) ∼ ( ) 12π 2 MP l V 2 6 24π 2 MP l 4 kp n=1 is Harrison-Zeldovich 33Monday, 29 August, 2011