Constraints on Axion Inflation from the Weak Gravity Conjecture

Transcription

1 Constraints on Axion Inflation from the Weak Gravity Conjecture T.R., /hep-th T.R., /hep-th Ben Heidenreich, Matt Reece, T.R., In Progress Department of Physics Harvard University Cornell Particle Theory Seminar April 17, 2015

2 Outline Inflation 1 Inflation 2 3

3 Section 1 Inflation

4 The CMB Inflation Figure 1 : The CMB, as measured by the Planck collaboration [1]. Image of last scattering surface, remnant of t = 380, 000 years. Note large-scale homogeneity, large-scale correlations.

5 The Horizon Problem Comoving Particle Horizon: / a 1 2 (1+3w) = ( a a 1/2 RD MD (1) Many causally disconnected patches at last scattering! Figure 2 : Standard Big-Bang model results in the horizon problem.

6 Solution: Inflation Inflation Inflation (violation of SEC w 1, exponential expansion a(t) e Ht, shrinking comoving horizon d(ah) 1 dt < 0) solves this problem: Extra conformal time pushes back standard big bang, allowing time before = 0. Figure 3 : Inflation offers a solution to the horizon problem.

7 Slow-Roll Inflation Inflation can be thought of as the theory of a ball rolling down a hill with friction. Figure 4 : The inflaton rolling down its potential. Slow roll parameters encode relevant features of potential: V = M2 p 2 V 0 ( ) V ( ) 2, V = Mp 2 V 00 ( ) V ( ). (2)

8 Inflationary Observables CMB correlations functions tell us information about the CMB on scales k = ah, where 1/(aH) is the comoving horizon size today. These modes exited the horizon e-folds before inflation ended, at log a end /a Measurable quantities include the ratio of the size of tensor perturbations to the size of scalar perturbations r, and the running of the scalar perturbations with k, n s. These quantities are determined by the slow-roll parameters, r 16 V, n s 1 2 V 6 V. (3)

9 Planck and BICEP2 Data Figure 5 : Planck and BICEP2 measurements give a best fit value of r = r < at 95% CI when lensing+ CDM+noise+dust are taken into account. r > 0 at 92% CI [1, 2, 3].

10 Future Experiments By the end of 2015, r = 0.10 will be a 3 detection, r = 0.05 will be a 2 detection (per Keck data). By 2017, r = 0.10 will be a 5 detection, r < 0.03 will be excluded at 95% CI (per Keck/BICEP3 data). Other upcoming experiments will improve these bounds further still [4].

11 Implications of a Large r A large r implies a large first derivative of the potential, and hence a fast-moving inflaton. r can therefore be related to the size of the traversal during inflation by the Lyth bound [5], & r 1/2 Mp. (4) 0.01 A detectable tensor-to-scalar ratio implies a trans-planckian traversal of the inflaton during the course of its slow-roll.

12 EFT of Inflation Inflation EFT for inflaton [6]: L eff = L l ( )+ X i i c i i (5) When is of order M p, higher dimension terms cannot be neglected. One expects large corrections to the potential on scales M p, destroying the flatness needed for slow-roll inflation. Solution: impose a shift symmetry! + a, potential vanishes.

13 Possible Solution: Axions Shift symmetry broken to discrete subgroup by instanton effects, V ( )= 4 (1 cos f )+( (2) ) 4 (1 cos 2 f )+... (6) (2) 2 M p, so these higher harmonics can be neglected provided M p.

14 Axions in String Theory Axions are ubiquitous in string compactifications, arising from integrating p-forms over p-cycles. Type IIB on Calabi-Yau Z gives N = 2 theory with axions b i, c i, # i, related to the NS-NS 2-form B 2, R-R 2-form C 2, and R-R 4-form C 4 by, B 2 =! i b i, C 2 =! i c i, C 4 = i # i, (7) where {! i } forms a basis of H 1,1 (Z ), { i } forms a basis of H 2,2 (Z ). But...axion decay constants in string theory are constrained to be O(M p ) or smaller [7], making them unsuitable for inflation.

15 Three Popular Solutions: V ø Figure 6 : N-flation [8] Figure 7 : Decay Constant Alignment [9] Figure 8 : Axion Monodromy [10] ø

16 Section 2

17 Global Symmetries in Quantum Gravity Conventional wisdom holds that there are no global symmetries allowed in a consistent theory of quantum gravity [11]: Consider a multi-particle state consisting of many particles of mass m, transforming under a representation r of the global symmetry.

18 Global Symmetries in Quantum Gravity Conventional wisdom holds that there are no global symmetries allowed in a consistent theory of quantum gravity [11]: Take enough particles so as to form a black hole.

19 Global Symmetries in Quantum Gravity Conventional wisdom holds that there are no global symmetries allowed in a consistent theory of quantum gravity [11]: Let the black hole decay via Hawking radiation.

20 Global Symmetries in Quantum Gravity Conventional wisdom holds that there are no global symmetries allowed in a consistent theory of quantum gravity [11]: When the Schwarzschild radius is O(l p ), the Hawking process stops, leaving a stable black hole remnant still charged under the global symmetry.

21 Global Symmetries in Quantum Gravity Conventional wisdom holds that there are no global symmetries allowed in a consistent theory of quantum gravity [11]: Such remnants spell trouble, for they violate the covariant entropy bound/will drive the renormalized Newton s constant to 0 [12].

22 The weak gravity conjecture proceeds along the same lines for U(1) gauge symmetries that are almost" global i.e. have g 1. Extremal black hole of charge Q = M (in appropriate units) cannot decay if no state has charge to mass ratio q/m 1 in appropriate units (otherwise any emitted radiation would push it past extremality ) naked singularity). In other words, [13] Any consistent theory with a U(1) gauge field admitting a UV completion with gravity must contain a state with charge to mass ratio q/m 1 in appropriate units.

23 (cont.) The weak gravity conjecture also applies to magnetically charged black holes and magnetic monopoles: m mag. g mag M p 1 g el M p. The mass of a monopole is at least the order of the energy stored in its magnetic field, m mag & gel 2. Thus,. g el M p (8)

24 The Generalized Weak Gravity Conjecture We have so far dealt with 1-form gauge fields in 4d. It is natural to generalize this to arbitrary p-forms and d spacetime dimensions. The Generalized Weak Gravity Conjecture Consider a p-form Abelian gauge field in any number of dimensions d. Then, there exist electrically and magnetically charged p 1 and d p 1 dimensional objects with tensions, g 2 1/2 1 1/2 T el., T mag. G N g 2. G N

25 Axions and the Weak Gravity Conjecture Consider the case of a 0-form (i.e. an axion) in 4d. The generalized WGC then says that there must exist a 1-dimensional object (instanton) with tension, T. M p. (9) f But, this is just the instanton action S, which shows up in the potential as V e S cos /f +... (10) Thus, the generalized WGC applied to 0-form fields gives precisely the result of [7]: axion decay constants larger than M p are forbidden.

26 The N-Species Weak Gravity Conjecture Suppose we have not 1, but N 1-form gauge fields. Naïvely, would expect the WGC to postulate the existence of a particle of mass m i, charge q i with q i /m i < 1/M p for i = 1,...,N. Instead, find a more stringent condition: the charge-to-mass vectors ±~z i := ± ~ q i m i M p, ~q i =(q 1i,...,q Ni ) must span the unit ball [14].

27 The N-Species Weak Gravity Conjecture (cont.) Figure 9 : The N-species weak gravity conjecture holds that the convex hull of the charge-to-mass vectors ±~z i = ± ~q i m i M p must contain the N-dimensional unit ball.

28 Section 3

29 The N-Species Axion WGC So far, we have seen two extensions of the WGC: The generalized" WGC generalized the WGC to arbitrary p-form gauge fields (at the expense of losing direct contact with black hole remnants). The N-species" WGC extended the WGC to N-species of 1-form gauge fields (still using constraints from black hole remnants). It is natural to consider: what happens when we put these two together to form the N-species, 0-form WGC (N0WGC)?

30 Axion Inflation Models and the N0WGC The Main Point Vanilla" models of N-flation and decay constant alignment are both ruled out by the N0WGC.

31 N-flation and the N0WGC Consider a theory with N axions i and decay constants f i. Set the instanton action S i & 1 for each of the axions. The N0WGC is then the condition that the charge vectors ~z i := Mp f i S i ~e i contain the unit ball, NX fi S 2 i < 1. Or, r = i=1 NX i=1 M p f 2 i! 1/2. M p

32 Decay Constant Alignment and the N0WGC Consider a theory with just two axions and non-diagonal decay constant matrix, V e S 1 cos( 1 f f 12 )+e S 2 cos( 1 f f 22 ). Set the instanton action S i & 1 for each of the axions. The N0WGC is then the condition that the charge vectors ±~z i = ± P M p j f ij S i ~e j contain the unit ball, ( ~z 1 2 1)( ~z 2 2 1) (1 + ~z 1 ~z 2 ) 2. Which yields, after some calculation, r. M p.

33 Why should the 0-form WGC be true? The problems with black hole remnants also apply to general p-branes in general d dimensions except p = 0 (no Hawking process without time). Simple studies of axion moduli spaces in string theory agree with this bound [16, 17, 18]. But, can relate 0-form version to 1-form version in string theory using the duality web [15].

34 Example: Extranatural Inflation [19, 20] 5d theory with U(1) gauge field compactified on circle of radius R gives axion in 4d. Charged particle of mass m 5 in 5d gives instanton of action m 5 R, decay constant f, V e S inst cos /f Ordinary 1-form WGC in 5d is g 5 1 m 5 (M (5) p ). 3/2 In terms of 4d quantities this becomes f apple M p 2 S inst.

35 Loopholes in the N0WGC Figure 10 : A model with three charge vectors and two axions. Although the generalized weak gravity conjecture still constrains the size of moduli space, one could achieve a large inflaton traversal as long as the potential contributions from ~z 3 dominate those from ~z 2.

36 Loopholes in the N0WGC (cont.) A strong" form of the N0WGC holds that the particles of minimal action should satisfy the convex hull condition, which would close the loophole (but is it true?). In extranatural inflation, the requirement S > 1 is not necessary, leaving f unbounded. But the magnetic WGC can close this loophole in some cases.

37 Conclusions Inflation The N0WGC rules out vanilla models of N-flation and axion decay constant alignment. It is possible to relate this conjecture to the ordinary WGC, which derives from arguments regarding black hole remnants. There are loopholes which would allow natural inflation consistent with the N0WGC, and there are (unverified) conjectures that would close these loopholes. More work is needed to understand these mysterious conjectures and their applications to axions (axion monodromy inflation?). Forthcoming experimental data should shed light on axion inflation stay tuned!

38 For Further Reading I P.A.R. Ade et al. (Planck collaboration) (2013), arxiv: [astro-ph]. P.A.R. Ade et al. (Planck, BICEP2 collaboration) (2015) arxiv: [astro-ph]. P.A.R. Ade et al. (Planck collobration) (2013), arxiv: [astro-ph]. P. Creminelli, D.L. Nacir, M. Simonović, G. Trevisan, and M. Zaldarriaga (2015), arxiv: [astro-ph]. D.H. Lyth, Phys.Rev.Lett. 78, 1861 (1997), arxiv: [hep-ph].

39 For Further Reading II C. Cheung et al., JHEP 0803:014, (2008) arxiv: [hep-th]. T. Banks, M. Dine, P.J. Fox, and E. Gorbatov, JCAP 0306, 001 (2003), arxiv: [hep-th]. S. Dimopoulos, S. Kachru, J. McGreevy, and J.G. Wacker, JCAP 0808, 003 (2008), arxiv: [hep-th]. J.E. Kim, H.P. Nilles, M. Peloso, JCAP 0501, 005 (2005), arxiv: [hep-ph]. L. McAllister, E. Silverstein, and A. Westphal, Phys. Rev. D (2010), arxiv:0808:0706 [hep-th].

40 For Further Reading III T. Banks and N. Seiberg, Phys.Rev.D 83, (2011), arxiv: [hep-th]. L. Susskind, (1995), arxiv: [hep-th]. N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, JHEP 0706, 060 (2007), arxiv: [hep-th]. C. Cheung and G.N. Remmen, Phys. Rev. Lett. 113, (2014), arxiv: [hep-ph]. J. Brown, W. Cottrell, G. Shiu and P. Soler, (2015), arxiv: [hep-th]. T. Rudelius, (2014), arxiv: [hep-th]. T. Rudelius, (2015), arxiv: [hep-th].

41 For Further Reading IV T. Bachlechner, C. Long, L. McAllister, (2014), arxiv: [hep-th]. N. Arkani-Hamed, H.C. Cheng, P. Creminelli, and L. Randall, Phys. Rev. Lett. 90, (2003), arxiv: /hep-th. A. de la Fuente, P. Saraswat, and R. Sundrum, (2014), arxiv: [hep-th].