Minimal Inflation Luis Alvarez-Gaume BSI August 29th, 2011                                                         Work in ...
Inflation is 30 years old                                            Originally Inflation was related to the horizon, flatnes...
Summary of FRLW                                            Matter is well represented by a perfect fluid                   ...
Accelerating the Universe                                               We need to violate the dominanat energy           ...
Origins of Inflation                                            The number of models trying generating inflation is         ...
Basic properties                                            Enough slow-roll to generate the necessary number             ...
Supersymmetry is our choice                                            In the standard treatment of global supersymmetry t...
SSB Scenarios                                                                                            Observable Sector...
Flat directions                                            One reason to use SUSY in inflationary theories is the abundance...
Important properties of SSB                                            The Akulov-Volkov-type actions provide the correct ...
General Lagrangian                                               4             2          2                               ...
Some IR consequences                                                                                                      ...
Coupling goldstinos to other                                                                                              ...
Some details                                             An important part of our analysis is the fact that the graceful e...
The full lagrangian Luis Alvarez-Gaume BSI August 29th, 2011                                                              ...
Primordial density fluctuations Luis Alvarez-Gaume BSI August 29th, 2011                                                   ...
Choosing useful variables                                                                                              √  ...
Cosmological equations                                                  The full equations of motion, neglecting fermions ...
The scalar potential                                                                                                      ...
Attractor and inflationary trajectories                                                  Β                                 ...
Decoupling and the Fermi sphere                                                                          mINF 2           ...
Summary of our scenario                                            We take as the basic object the X field containing the G...
Luis Alvarez-Gaume BSI August 29th, 2011Monday, 29 August, 2011                                                           ...
Luis Alvarez-Gaume BSI August 29th, 2011Monday, 29 August, 2011                                                           ...
Graceful exit                                            The theory is constructed with a Kahler potential violating expli...
Summary                                            The chiral superfield X is essentially defined uniquely in the ultraviole...
Vanishing cosmological constant                                                                                           ...
...continued                                                                X     =     M (α + i β)                       ...
Slow roll conditions                                                                 2                                    ...
Some numbers                                                                                                       √ 1/2  ...
Comments                                            Supersymmetry breaking can be the driving force of inflation. We have u...
Slow roll, more details                                                                                            8πG    ...
Power spectrum                                            In flat De Sitter coordinates we can Fourier transform F.T. To co...
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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

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