JHEP02(2018)084 a Springer June 22, 2017 : January 25, 2018 February 14, 2018 December 11, 2017 : : : Received , Revised Accepted Published and Tomer Volansky a Published for SISSA by https://doi.org/10.1007/JHEP02(2018)084 Lorenzo Ubaldi a,b kohsakut@post.tau.ac.il , [email protected] , . 3 Kohsaku Tobioka, a 1706.03072 The Authors. Beyond Standard Model, Cosmology of Theories beyond the SM, Higgs c

The cosmological relaxation of the electroweak scale has been proposed as a , [email protected] Rehovot 76100, Israel E-mail: [email protected] Raymond and Beverly SacklerTel-Aviv School 69978, of Israel Physics and Astronomy,Department Tel-Aviv of University, Particle Physics and Astrophysics, Weizmann Institute of Science, b a Open Access Article funded by SCOAP ArXiv ePrint: its minimal form, theditional model ingredients, does such not as generate a sufficient curvaton field, curvature are perturbations needed. andKeywords: ad- Physics, Nonperturbative Effects reheating mechanism, which relies on themagnetic gauge-boson fields, production and leading proceeds to via strong thethe electro- vacuum Schwinger production effect. of electron-positron We pairscosmological refer through dynamics to of this the mechanism modelother as and Schwinger experiments. the reheating. phenomenological We constraints We find discuss from CMB that the and a cutoff close to the Planck scale may be achieved. In mechanism to address theion, hierarchy rolls problem down of its theHiggs, potential Standard relaxing it and, Model. to into A a doing an field, parametrically so, inflaton. small the scans value. relax- Wedissipation the couple mechanism In squared it which this slows to mass work, down Abelian we parameter the promote gauge of field the bosons, the in relaxion thereby the introducing last the stages. necessary We describe a novel Abstract: Walter Tangarife, Dynamics of relaxed inflation JHEP02(2018)084 11 30 12 3 11 35 – 1 – 19 13 15 17 m 7 14 34 26 33 24 32 34 34 26 28 2 9 23 C.2 Short wavelength D.1 Coherence D.2 Size and direction B.1 Regime 1 B.2 Regime 2 C.1 Long wavelength 4.2 Regime 1:4.3 slow-roll on a Regime gentle 2:4.4 slope slow-roll via photon Inflation production exit and relaxation 4.1 Conditions on the slope E Estimate of curvature perturbations D Electric field C Thermal effects on gauge-field production 9 Summary A A clockwork model B Slow roll conditions 6 A model with a dark7 photon Cosmological perturbations 8 Constraints and relevant scales 5 Schwinger reheating 3 A relaxed inflation model 4 Dynamics Contents 1 Introduction 2 Axion inflation and photon production JHEP02(2018)084 ]. 24 , 19 ]. 35 ]. The relevance of this type of 34 – 31 The photons that are produced in the 1 the end of inflation. This is described with – 2 – after ], the entire scanning takes place in the background of infla- 1 . The last two regimes share similarities with models of axion (20) e-folds of inflation, where slow roll is due to dissipation via 4 O and 3 ]. One pleasant consequence of this promotion is that we can indeed achieve 30 ] have elaborated on various aspects of this framework. 29 – 2 ], a new mechanism was proposed as an alternative solution to the hierarchy problem. 1 An important aspect of our work is the actual reheating mechanism, which, to the best The relaxion dynamics proceeds in three regimes. The first consists of a long period In the original proposal [ As we were finishing this work, another thermalization mechanism, involving scattering between the 1 of our knowledge, haslast not stages been of inflation explored have before. a large occupation number andinflaton very and low the momentum. gauge This bosons, is was a presented in the context of axion-inflation models [ and sufficient to allow forstage the of field relaxation of to the settledetail EW on scale in the occurs sections correct localinflation minimum. that Thus, have the been final dissipation, studied in previously the in context refs. of [ relaxion models, has been also discussed in refs. [ is a region of parameter space where this isof avoided. inflation, withcorresponds standard to slow the last rollgauge-boson due production. to At this thewhere stage, reheating the flatness takes relaxion of place. keeps The the rolling, third the potential. is friction after from reheating, The gauge-boson production second is still present process ends, preventing the overshootingfrom of the the tachyonic production EW of minimum.the gauge Here rolling bosons, we relaxion. due consider tointeresting friction Not the and time-dependent only novel background mechanism this of formechanism slows is reheating. down that A one the possible risks field overproducing issue efficiently, the related but cosmological to perturbations. also such We provides a show friction an there some of the restrictionsinflaton be itself? ameliorated In by this promotingpresented article, in the we [ relaxion answer to this playa question higher the positively, extending cutoff role the compared of discussion toof the the the previous relaxion proposal. require The a price friction to mechanism pay that is can that remain the efficient dynamics after the reheating works [ tion, which provides constantroll Hubble and friction eventually necessary stop at forthe the the local cutoff relaxion minimum. to scale This maintain setup significantly slow is below rather constraining, the rendering Planck scale. A natural question to ask is: can The proposal relies onan an effective axion-like Lagrangian. field, Themass, dubbed relaxion, relaxion, and during coupled finally its towhich cosmological settles the is evolution, on parametrically Higgs scans smaller field a the thancutoff, in Higgs local the the cutoff minimum more of where the successful effective the the Lagrangian. mechanism Higgs The in larger has the addressing the the observed hierarchy mass, problem. Several During the past fewput decades, forth. numerous ideas Theprotect to the majority Higgs solve of mass. theat Such these hierarchy symmetries the problem ideas lead electroweak to have require (EW) the been ref. the scale, prediction [ of but introduction new none of degrees of of new them freedom symmetries has yet to been found in experiments. In 1 Introduction JHEP02(2018)084 . ]. is g − 1 34 φ √ (2.1) (2.2) – 31 = 0123  , ) 2 , ) d~x φ ( . We find that a cutoff − V 2 3 − dτ )( µν τ ˜ ( F 2 a the gauge field, and µν − F µ f A φ = 4 2 γ c d~x ) which thermalize, reheating the universe. − ) t − , with ( e µν 2 σρ + , coupled to an Abelian gauge field, in a Fried- a F – 3 – e φ F ]. In this way, the energy of the electromagnetic + µν pairs via the Schwinger mechanism, and the subse- 2 F 37 , µνσρ 1 4 − dt  e 36 1 2 − − + e φ = = µ ν . Below, we summarize the main aspects of the gauge-field µν the conformal time. The Lagrangian reads φ∂ dx ˜ C F µ µ τ ∂ , µ 1 2 and the results are presented in figure dx A − 8 µν ν g ∂ = ≡ − L ) will be specified in the next section. The equation of motion for 2 ν φ ( A ds µ V ∂ and in appendix = 5 µν the cosmic time and F t We consider a pseudo-scalar inflaton, The available parameter space for the above scenario, a-priori rather large, is reduced where The potential mann Robertson Walker (FRW) metric, with of inflation. Once afollows large through number the of production coherent of photonsquent are thermalization of produced, the the system. reheating Afterthermal process that effects happens, in it is the importantin gauge-boson to section take production. into account We discussproduction Schwinger at zero and temperature. thermal The effects interested reader can find more details in refs. [ The inflaton will alsoonly play interested the in role the dissipation of mechanism thegauge due fields relaxion to particle to in production. the the inflaton, We nextof couple whose Abelian section, gauge time field but evolution leads quanta. for to nowinflaton This the we and production non-perturbative are slows has production it two important down, effects: (2) it (1) provides it a backreacts mechanism on to the reheat the universe at the end scale close to the Planck scale may be achieved2 in this relaxed inflation Axion scenario. inflation andIn photon this section production we review some aspects of axion inflation that are relevant to our framework. by several theoretical constraints thatby force phenomenological bounds. relations among These the includement the different for validity parameters, of a and the effective relaxation theory,and the at various require- the limits from correct colliders, scale, astrophysics,are cosmology the discussed and suppression in 5th-force of experiments. section cosmological They perturbations, by the Standard Model (SM), however,above it the does not electron seem mass, possible whichcircumvent to the is reheat issue to too is a low temperature to forturn couple, Big instead, has Bang the a Nucleosynthesis relaxion small to (BBN).paper, kinetic a A that mixing massless way the with dark to latter photon, the scenario which leads SM in to photon. successful reheating We and show, relaxation in of the the EW second scale. part of the field within the horizonto is very constant, large to values. a Eventually,through this the good allows Schwinger for approximation, mechanism vacuum and [ creationfield its is of strength transferred electron-positron to can pairs relativistic grow We particles refer ( to this mechanism as “Schwinger reheating.” If the photons are those described coherent collection that is best described classically as an electromagnetic field. The electric JHEP02(2018)084 )  0. A τ, ~x (2.6) (2.8) (2.9) (2.3) (2.4) (2.5) ( is the (2.10) τ < ~ i A ~ B · ~ E . Then, h  ~ | 0. Furthermore, k ~ | i . Note that 1 ∓ ξ < − , ) = and the mean field ap- i .  c t aH . ~ ( 0 and × 2 + h , , k ~ ' − ˙ ~x φ < i · k τ ~ ~ B i ) = 0 , (2.7) = 0 e · ]. To set our conventions, we will τ ~ , λ k ~ ( A ~ . E 31 a ) implies that low-momentum (long 0], so 0 k  = 0 and ~ h 1  ) γ . We promote the classical field 0 A − , . τ f = 0 and > > c ) 2.7 τ (  ) A ) 2 λ k ~ ~ Pl ∇×  φ ξ τ φ k = ( ~ 0 ˙ A aH ( φ M is assumed to dominate the energy density, 0 · k f ( f H ) φ ) V − 3 | 2 φ 2 k V k ~ ~ φ γ ξ ( ( ξ τ √ p γ 1 c | λ  ∂φ – 4 – c into annihilation and creation operators 2 k ~ 0. Eq. ( 2 − ' ∂V h ˆ 2 ~ ≡ k A 2 > 2 >  ξ + / H 1 ≡ 3 k ˙ γ − ) + φ 2 = 0, we have 3 c − ∇ k π d ) Ω H ~ A 2 τ 2 (2 ( 2 ∂  k ∂τ ~ + 3 ~ polarization, satisfying ), which during inflation is are such that ∇ · Z 0 ∂τ . The inflaton A ¨  φ t 2  − ~ (  B ∂ = ~ · A X λ /a 0 ~ E = dt ) and decompose t ˆ ~ A R τ, ~x ≡ ( is convenient because it stays almost constant when the term ) and any additional source terms, the temporal and the Coulomb gauges are equivalent, τ ˆ ~ ξ φ ), so the Hubble parameter is given by A i φ ∂ ( ]. is the reduced Planck mass. V 38 rolls from positive to negative [i.e. Pl  φ M 2 ˙ φ 0 by definition, and we take The equations of motion for the gauge field are more conveniently written using the Neglecting ( 2 develop a tachyonic instability and grow exponentially. This condition can be rewritten as see e.g. ref. [ τ < wavelength) modes of the where we have defined The parameter dominant dissipative force inassume the inflaton dynamics [ where the helicity vectors must satisfy the equation to an operator Choosing the Coulomb gauge where a prime denotes a derivative with respect to where conformal time where the dot denotesproximation a is derivative used with for respectwith to cosmic time given by JHEP02(2018)084 (2.14) (2.15) (2.16) (2.12) (2.13) (2.11) ), with the 2.3 . , ! 2 1

is dominated by the is the comoving hori- , k − ~ , . > | | γ 1 A ξ ξ |  | | ρ

− ξ 2 π π (10), the comoving wave- | )

2 2 2 are not enhanced and we k e e O k − ~ 4 3 4 4 aH + A | | + , .

ξ ξ A H H ]. | | ) 2 | , 0

− ξ 4 4 | 39 k − ~ − − 2 k τ k,τ

| A ξ 10 10 k + Ω( ~ | 0 2 , while ( ∂ A ∂τ × × −

dτ −

τ 4 4  √ , and therefore A . .

R i 2 2 1 e 2 2 ∂ ) ∂τ |− ~ ' ' B ξ 3 | h | | dkk 2 π ξ ξ ], the authors considered the regime in which | | | k, τ e ξ π π 1 – 5 – dkk | Z 4 2 2 ), one can compute 32 / 8 7 e e 4 1 2Ω( 4 3 Z 4 4 a | | 2.7  4 p 2 ξ ξ 1 ], the authors considered the regime where the term 0. | H H | | a π is dictated by the equation of motion ( ξ i ' kτ 2 2 | 31 2 4 2 1 ' ' π π 2 φ ] − π ~ ) E 7! 6! + , meaning that the dissipation mechanism that ensures 4 = h 21 19 τ 0  31 ( 2 2 A i − V k 2 k − ~ ~ ' 2 1 1. The WKB solution for the tachyonic modes holds in the = A B , where it can be written as i ' γ i √ | + ρ ~ ξ ~ B  B | 2 · ' · 2

~ E ) 2 ~ ~ E h E τ < h Ω h 1 2 ( | / ), one finds [ 0 k − ~ = kτ Ω A |

γ 2.12 ρ < 1 ) can be solved analytically. However, it is more illuminating to use an ap- − balances the slope ) | is the comoving wavelength of the mode 2.7 ξ i | 1 ~ B − · k The evolution of the inflaton Using eq. ( With the explicit solutions to eq. ( Eq. ( ~ E h term typically negligible. In ref. [ γ f ¨ c φ slow roll is duequanta to on gauge-boson the production. inflaton Inobservations. produces such On perturbations a the that case, other are the hand, backreaction too in of large, ref. the and [ gauge excluded by CMB Incidentally, one can showelectric that field contribution. In the last expression, we took and the photon energy density and the exponential enhancementignore them is in explicit. what follows. The modes valid as long as range (8 in the CMB power spectrum have been discussed inproximate ref. solution, [ which can be derived from the WKB approximation, zon, which shrinks during inflation.and We shorter see wavelength become that, tachyonic. aslength inflation Since typically of proceeds, the modes exponentially with enhanced shorter horizon. modes has Note a that typical only sizement, comparable one a to polarization consequence the of of comoving the parity violation photon in experiences the exponential system. enhance- The signatures of parity violation Here JHEP02(2018)084 φ drops (2.19) (2.20) (2.21) (2.22) (2.17) (2.18) φ is roughly ξ decreases as . The produced H H 3 slow-rolls because of a φ . Although we use the WKB . 7 ,  .  ). This is when we enter the 4 0 4 γ H c 1, and H 2.3 , γ fV , γ c i | f c 0 4 f ) ξ ). When the potential of ~ 4 grows large enough for the backre- B 7 ) | − . φ |  − · 4 φ ' | and the equation of motion is given by ( ξ ) 0 ( | ξ 10 | 0 ~ ) | 10 φ E ˙ 2.19 φ V ( h H φ | V 0 × i ' ( 3 × γ f 4 c V H ∂φ ~ B 4 . ) is roughly constant and . · 2 ∂V 2 φ '  inside the logarithm. We see that ( ~  0 E |  ) and ( + ) ' − h 4 – 6 – ¨ ln φ V ˙ | φ ˙ ln ξ | φ φ ( ξ γ 7 | π ∂φ H 2.16 1 4 now decreases with decreasing 2 3 fH πc ∂V | ). The approximate solution is ˙ ' φ ), ( | γ ' − ' − ρ , the photon energy density becomes dominant, and we 7 for the later discussion in section 2.15 ξ / γ ˙ 2.15 φ ρ ∼ ) φ ( V increases to the point where | increases slowly since ˙ is negligible for most of the observable e-folds and slow-roll is solely due given by eq. ( φ | | ˙ i i φ | ~ ~ B B · · ~ E ~ E h h γ f c below the value exit inflation. photons have an energy density that remains roughly constant Here we have used eqs. ( constant in this regime, and we have Unlike the previous regime, with where we have neglected a factor of action of the photonssecond to regime become described important by in the equation eq. of ( motion Note that rolls down its potential. hence nearly flat potential. In this regime, Eventually, Initially, the photon production is negligible, In the scenario we investigate in this paper, inflation proceeds in the following steps: We keep track the coefficient of 4 2. 1. 3 approximation here, we have checkedcoefficient that with the only full 15% solution discrepancy. based on the Coulomb function reproduces this to Hubble friction.imprints They on showed the that CMB even that in can this be case measured. the photon production can leave the term JHEP02(2018)084 (3.1) (3.2) (3.3) (3.4) (3.5) , and (2.23) roll 1 due to V production  for further 3 − e D H + , γ e 0 , is /q , V a k γ ) ρ = , φ ) + MeV. At the same time γ φ H q ( (  V wig γ (see appendix V q − γ , ρ µν ) + ˜ √ F φ ··· ( , µν 2 + F roll playing the role of the inflaton. Our | ∼ . Λ 3 f V ~ φ E φ 4 φ 5 | − + 3 γ 2 c 2 at the end of this section. We omit terms H. Λ m that plays a crucial role in setting the VEV | ) − mφ ξ 1 6 | H h † wig is a dimensionless parameter of order one, and g wig µν + – 7 – < V H h 2 F ( γ g φ ). This system is best described classically as an q λ ) = µν 2 φ F + ( . m is the relaxion/inflaton field, and 2 1 4 2.16 1 2 φ f H µ φ † − + H φ cos ) φ µ 2 φ ( Λ in eq. ( wig 4 2 φ∂ | µ µ m ξ ∂ | π 1 2 2 e ) = ) = Λ ) = − φ φ ( ( , φ = H L roll wig ( V V V ) it follows that their typical physical momentum, 2.10 is the SM Higgs doublet, -dependent squared mass parameter of the Higgs potential. The Higgs bare mass Λ φ H Λ. We comment on the parameter Λ MeV close to the end of inflation, which in turn implies So far, we have described the generalities of The problem with this scenario is that the produced photons have extremely long wave-   is the is the cutoff of them effective Lagrangian, Here, with, coupled to SM photons.during inflation This and pseudoscalar acts dominatesAdditionally, both the there as is energy the a density periodic inflaton ofof potential and the the as Higgs universe the after field reheating. that The scans effective the Lagrangian Higgs for our mass. model is given by to add the relaxiondetail ingredients, the that whole cosmological come evolution next. of the In relaxion/inflaton the field. rest3 of the paper, A we relaxed explain inflation in The model first model we consider consists of an axion field on a very flat potential, discussion). This electric fieldvia grows the strong Schwinger enough mechanism. to Thisdescribed allow changes above. dramatically for We the vacuum discuss picture it in in the detail second in regime section main purpose is to use this inflaton to relax the electroweak scale and, to do so, we need H these photons have a highthe occupation large number exponential in theelectromagnetic Hubble field. volume One can showelectric that the field photons within add up the coherently horizon to form with a magnitude constant length and do not thermalizeFrom eq. via ( perturbative scattering processes to reheat the universe. As we describe in more detail in later section, the relaxation mechanism requires values of JHEP02(2018)084 ] ], µν 40 ˜ φ < Z 41 . (3.6) (3.7) (3.8) , µν A Z 10 ) , 9 , φ/f 0 , a nearly-flat settle to their such that the V φ . The potential 0 0 after integrating φ + V φ 2 and ( f δφ 2 Z and µν f + ˜ m F h 0 φ µν Z 0, the model possesses the ) is not well defined, as each ) , from the broken phase ). On the other hand, there once 0 cos φ ( 3.3 → 4 . φ/f , it is convenient to expand the 4 wig . We choose 0 wig 0 m φ V φ φ > φ meV ] and our model inherits some of the mδφ 1 ) + Λ h ∼ |  g ) and expand around δφ , δφ | obs + 3.3 cc 2 m V 0 ) = h Λ φ g ( δφ 2 , a choice that is justified only once we consider expansion in eq. ( ( – 8 – ≡ φ Λ 2 0 µ m δφ , m φ 4 + and + . A potential of this kind was first used by Abbott [ 0 4 and m φ λh πnf , where we also show how to map its parameters to the = H 1 4 φ A +2 , the small + 0 φ 2 φ h → is justified by the fact that as ) φ δφ m and generically order one corrections are expected. In what follows, ( terms do not affect the photon production. 2 4 ] that can be circumvented with a clockwork axion model [ linear term in the potential ( . Second, since the dynamics of inflation and relaxation end at almost µ is still a classical field, not a quantum fluctuation. Λ 5 4 2 1 for simplicity. In particular, there are ( φ W Λ ∼ δφ ), and the periodic (“wiggle”) potential Z ∼ φ ) = ( ) = 0. It separates the unbroken EW phase, ], we use it for the EW scale instead. As written, the model poses some 0 roll 1 and φ V is the radial mode of h, δφ ( (  2 h µ V . Similarly, W As most of the interesting dynamics happens near For the given coordinate, a special point in field space is The smallness of The relaxion potential here is the same as in ref. [ Z We stress that 4 . For field values of order 0 where cosmological constant has the observed value We keep only the then reads term is of order we will only keep the terma linear UV in completion of this model, such as thepotential clockwork around axion this discussed point. in We appendix define where φ following [ theoretical issues [ which we present inones appendix used in this section and in the rest of the paper. a form of dissipationthis distinct extra from source the of Hubble dissipation, friction. and The offers coupling a to noveldiscrete opportunity photons shift symmetry for provides reheating. in an attempt to explain dynamically the smallness of the cosmological constant. Here, potential are sharp differences that lead tothe stark relaxion contrast with is the the original inflaton proposal. itself,same which First, order allows in the as our energy case density ofthe the same universe time, to the be classical of rollingscale is the is automatically settled. a good Finally, the description relaxion when stops the after electroweak the end of inflation, and therefore we require terms that are gauge invarianttheir but effect the only photon appears production fromout is dimension not 8 affected operators by suppressed them by because properties of that scenario. These include a trans-Planckian field range for with JHEP02(2018)084 (4.1) (4.2) (3.9) is the (3.10) (3.11) (3.12) ˙ a a ] because . For the 1 µ ≡ A (the lightest = 2 case, the H u n and the energy , which provides a y 0. i R > ∼ , and q δφ 2 L h y ¯ q µ , and attains the observed , h t v δφ , i = 1, ~ B n · . depends on the Higgs VEV, ~ E λ W h obs (4.3) cc 4 wig , γ √ m V This case is excluded [ f c 5 + ' , = = 0 ) n 4 W ) = 0 , λ ) ) − . 4 τ 4 m ( EW k  ~ m 2 W λ M + δφ h h, δφ h, δφ A n ∂h ( m ( ( g ∂δφ 2 ) 2  Λ µ ξ τ h yv ∂V ∂V g ( – 9 – 2 W − k ' − + + 2 m ∼ . . This must happen when r ˙ h φ as a homogeneous classical field, but we must treat  EW + does not break the electroweak symmetry, we have a 2 = δφ 2 4 wig 2 4 a h ∇ δφ v Hδ δφ k Λ g Λ wig  M < v + , there is a singular term in the first derivative of the potential, = = V − 2 + 3 + is a lot smaller compared to the other two terms. In what ˙ / h i i 1 ¨ φ ) ] also in the unbroken electroweak phase, and the relaxation = h ) 0 H (the pion decay constant). δ h τ δφ V 0 5 ( h , 2 π V the scale factor. Since inflation is driven by k  2 ~ mδφ f + 3 to ( can be written generically as ∂τ a A 3 ¨ h 2 ∼ ∂ M obs cc wig to be effectively zero. M V 1 and thus is plagued by the strong CP problem. In the ∼ some fixed mass scale. The fact that Λ grows, the amplitude of the wiggles becomes larger and larger up to the obs ∼ cc 4 wig M v V = 0. The singularity is evaded thanks to the quark condensate, QCD δφ θ as quantum fields. The equations of motion are 0 and , at µ A n > /∂δφ The parameter Λ In this case, Λ 5 4 wig and , is crucial: as Λ Hubble parameter, with ∂ tadpole for the Higgs potential and results in a small, but non zero VEV even for Here, an overdot denotes a derivative with respect to cosmic time purpose of our study, weh can treat two-loop wiggle-potential [ mechanism works, provided that 4 Dynamics In this section, we discuss the cosmological evolution of the fields point where they stopvalue the of rolling 246 GeV. of Thequark case yukawa), and of the QCDit results axion in corresponds to sector responsible for generating with v We have then The contribution of follows we take VEVs: h (3.9) (3.15) (3.16) (3.17) (3.10) (3.11) (3.13) (3.14) (3.12) h = 2 case ] because 5 (3.9) n (3.15) (3.16) (3.17) (3.10) (3.11) (3.13) (3.14) (3.12) (the lightest , = 2 case u ] because , 0 5 y n ) V (the lightest ⇠ , u + , 0 ( y ) y V attains the observed ⇠ wig + ( v y V f attains the observed + . wig = 1, . 0 v 0 ⇠ V f + )+ n | is a = 1, . m 0 2 W 2 depends on the Higgs VEV, (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) . h < )+ n µ h v | m m and absorb the constant terms 2 W g 4 0 ( g depends on the Higgs VEV, . cos h ⇡ v 0 m and absorb the constant terms ⇤ g 4 0 ( 4 0 cos 4 h ⇡ ] roll 0 ) ⇤ 4 0 ⇤ 4 roll V ) ' ⇤ , 37 .

JHEP02(2018)084V | ' + (3.9) ⇠ , and . as EW + ⇠ + , (3.6) (3.7) (3.8) (3.9) EW (3.15) (3.16) (3.17) (3.10) (3.11) (3.13) (3.14) (3.12) + | , ], that can be ⇤ | M< (3.10) (3.11) 4 W n M< , that is 4 W EW µ 6 ) n EW is a lot , (4.4) (4.5) (4.1) (4.2) (4.3) 4 2 4 W 4 ectively 4 W ( .The 2 ( m = 2 case 2 m A 0 , ⌧ ] because µ 4 0 4 4 t 4 2 2 ↵ m ⇤ ) V m ⇤ 5 . What V µ ( n µ + ], and the smallness + 0 M . m M m , + m 5 (the lightest , . For the n 2 + , 0 ⇤ n 2 2 ) to roll and , 4 0 )+ µ , + ⇤ ) settle to their H 4 ]. We present u , r ). h , 4.3 + h ⇤ such that the V V 0 ( 4 , does not break the electroweak symmetry, and yv –5– , r A g h | y 2 W (4.4) h ( h . The potential ) EW ( 4 h , does not break the electroweak symmetry, and yv V –5– 2 W = g 2 ] also in the unbroken electroweak phase. In this 38 2 W to be e V 0 0 obs 1 4 EW + . m ( cc h 8 . This must happen when (4.31) (4.23) (4.25) (4.26) (4.27) (4.28) (4.29) (4.30) (4.32) 4.4 ) is the 2 W = ), expand around ⇤ 2 ] also in the unbroken electroweak phase. In this v ⇠ , ⇠ V m such that the cosmological constant is zero once wig wig 1 4 1 4 h V + . , m and the photon 8 + . This must happen when ( V ), expand around g 4 0 ⇤ = V v 2 W 6 0 ⇠ m such that the cosmological constant is zero once and wig y = = obs 1 4 h 10 3.3 ⌫ 2 and , + cc attains the observed ⇤ V 0 ), with a velocity + m V g 4 0 µ = h 2 W f ) is not well defined, 6 0 i i ' , + m . 2 , from the broken phase V V wig h = = f 3.3 h 2 ˜ + h + 9 v 0 ⇤ ) black V )+ h F 0 0 0 V + h m , h f i i f ' ⌫ 0 + + 2 V ) 3.3 h , from the broken phase 2.17 µ ( = 1, h . ) h EW 0 , 0 0 ). We enter this regime + h (the pion decay constant). This case is excluded [ > F m : 2 2 )+ ⌧ , ( n ) cos ), the photons are h once 0, the model possesses the h roll f ⇡ 2 m ⇤ . 2 W | ⇤ g ( 4 0 depends on the Higgs VEV, EW a . V f 4 cos h µ 2 ↵ (the pion decay constant). This case is excluded [ ⌧ 2 1 > EW ⇤ m m v v , . 2 2.19 m and absorb the constant terms g 4 0 1 2 µ a ( ( ! cos cos W + h ⇠ 3) are the three goldstone modes that get obs | h i ⇡ . . , ⇡ blue 2 cc . ⇤ 0 . We choose , ⇤ g ( i , it is convenient to expand the 0 4 4 wig 4 0 2 2 0). When the parameter . We choose ]. We present the clockwork model in the ) and ( m p 4 (4.37) (4.38) (4.39) (4.40) (4.32) (4.34) (4.35) (4.36) (4.41) f ~ m V 0 µ 2 2 0 ⌫ B ) roll , the more successful is this mechanism in (20) e-folds. When the 1 and thus is plagued by the strong CP problem. In the ) 1 2 )= )= )= 7 h m meV M ⇤ ⇤ m , as representative of the electroweak scale. ⇤ ⇤ )= is already very close to , , and we stay only for the last µ m · some fixed mass scale. The fact that 1 2 µ g V ··· he ⇤ ' H < ⇠ can be written generically as O h + ◆ =0 . ' ⇠ W . F † 0 grows the amplitude of the wiggles becomes larger and larger up to the ⇠ 2 , a choice that is well justified only + slowroll g corresponds to tuning the cosmological constant. This is crucial for the ~ . We choose ( ( , m 2.16 0 E EW + ⌫ m 1 + grows larger than one and –4– ) and ( , ⌘ ) for our model, we assume M EW ⌧ ) )+ δφ =1 1 h m h h, v H p 0 µ V h . 4 W ⇤ are the Pauli matrices. 1 and thus is plagued by the strong CP problem. In the h, the energy density of the produced M< )= )= )= 2 4 W ( M g | ( | 0 n corresponds to tuning the cosmological roll EW 0 the model possesses the discrete shift symmetry )= expansion in ( wig ( V g 2 a , and some fixed mass scale. The fact that ) and expand around F 4 f ◆ obs 1 (4.24) ↵ ↵ 4 ) 4 W a cc V QCD V 4 0 V )= ( . .Once m ( can be written generically as 0 2 . The contribution of m = , V 2 EW 1 4 . ⌧ . ⇠ ✓ 0 4 ⌧ 4 grows the amplitude of the wiggles becomes larger and larger up to the 2.18 2 V = 0 and 0, in the unbroken electroweak phase. h + 9 corresponds to tuning the cosmological constant. This is crucial for the ! ⇤ V ( ( a 4 0 m | ), with the photons still providing ⇢ m m m fH ⇤ 2 W + 0 0 and )= = h, W ( + 3.3 Pl ⇤ m ⌧ µ @ + M f 0 , H 2 0 and h + ( h ( h 2 m 2 h, v . is the relaxion/inflaton field, ⇤ 2 W 4 3 0 ). M 00 ⇤ 0 V mass, h, 2 1 . > ( ) m f m m V m ( . g ⇠ g . )= h H 2 µ roll 1 + ⇤ M wig ⇠ 2 0 ( n V 2 Pl V 2 m ˙ ( red 2 † , ) i g m Pl 2 φ/ is ) 3 roll V f QCD 2 µ settle to their VEVs: V ( V f ⇤ 4 ⇠ + ), all the way into the broken EW + @ µ + = ⇤ 2.6 V V δ 2 h ✓ 2 W f n> µ , H r V h M . The potential then reads , , h ◆ ⇠ the scale factor. As inflation is driven by ( . In the second ( p ⌧ ↵ 1 2 2 γ . g M 4 –8– ) We keep only the linear term in ( For future convenience we rewrite the potential as 30 0 , Pl ⌘ , does not break the electroweak symmetry, and 2 W VV f 0 yv –5– g , )= @ 2 W ⇤ 2 ' c h, + 2 EW | ( switch 1 h V ⇤ | ˙ p 0 and ⇠ 8 7 4 0 cos 2 2 W a = µ V 0 2 2 ] also in the unbroken electroweak phase. In this as is the cuto ( @ µ ⇠ h, δφ ( m ✓ M 1 4 ⇢ 10 2

⇤ V . m m , is crucial: as Pl

| r 2 h @ . ◆ ⇡↵ 4 0 1 (4.33) m< –6– 8 regime 1 regime ⇠ ( . This must happen when 2 4.2 | ↵ ⇤ ), expand around 3 ⇤ m v ⇠ with v point where they stopvalue the of rolling 246 of GeV.quark The yukawa), case and of theit QCD results axion in corresponds to the sector responsible forwe generating have a two-loop wiggle-potentialcase [ the relaxation mechanism works provided that The parameter in success of the mechanism we describe in the rest of the paper. with and We have Choosing this with V m such that the cosmological constant is zero once µ wig ⌧ 3 ( ) . = 2 1 2 1 4 ⇤ h ' + ✓ (4.4) (4.5) (4.1) (4.2) (4.3) ), the Friedmann equation yields ⇤ µ m , 2 EW + 2 + V Pl + ⇢ ' ⇠ + 2 3 2 @ M settle to their VEVs: 0 ↵ = .The roughly constant, the energy density of the Pl = V √ g 4 0 = A . A potential of this kind was first used by Abbott [ V 2 W ⌧ | 6 0 , the small a , ) t r 2 0 0 4 3 = = 0 3.3 n> . 00 µ 4 2 4 V ξ 0 2 . The potential then reads V + h 2 and +2 = . What , where we show how to map its parameters to the ones ⇤ v V ξ Pl 0 p 2 1 2 0 ( gauge bosons. | M Pl as a classical field, but we must treat We keep only the linear term in ( For future convenience we rewrite the potential as m M 4 0 + 2 0 h h 0 ' Choosing this g 2 f Pl 2 i i ' = ⇤ V f Pl nf + 2 f N > 2 H + V )= )= )= f + . The inflaton field starts rolling from a point A . For the V 2 ⇤ V h, δφ f L ( 0 ⇠ ⇤ h 2 M , M ↵ ⇡ Z ⇠ ) = = Pl f h 0 ' ⌧ ↵ roll ˙ and 2 f 2 = M H µ , ( 2 VV 1 – 10 – f fV + ˙ h Pl h 1 4 M 1 , is crucial: as ( ( .Wedefine 2 , ⇡↵ | Pl that increases as the field rolls. We stay in this 2 V 2 ˙ ⇢ 7 8 4 0 ) 2 2 m i i H ) is still a classical field, not a quantum fluctuation. Excuse the notation Jack. 2 h A 0 | ✓ V ⇢ The parameter with v point where they stopvalue the of rolling 246 of GeV.quark The yukawa), case and of theit QCD results axion in corresponds to the sector responsible forwe generating have a two-loop wiggle-potentialcase [ the relaxation mechanism works provided that in with and with We have Choosing this success of the mechanism we describe in the rest of the paper. h 4 0 2 ⇠ M f Pl H ⇠ ↵ . ⇡↵ and generically order one corrections are expected. In what = 246 GeV, while ⇠ ↵ | ⇠ = +2 ↵ H H ⇤ h M ↵ | V . As written the model poses some theoretical issues [ H EW M ⌧ + 0 roll ˙ 2 wig fV ( + ✓ h + , with ⇤ (the pion decay constant). This case is excluded [ 2 φ and ' | 2 Pl m ' ⇠ 2 3 v 4 M 0 ↵ 2 Pl 2 –9– V V V 2 | γ h f ( δ ⇠ and the photon cos 2 0 ↵ V f 0 h V dynamics 2 | ) | V 2 +2 = ⇡ ⇤ Pl ⇠ 2 2 and ⇤ h M ↵ h, δφ ), the second is small for 2 +3 M kf g ( 2 +3 Pl 0 4

✏ ' | 2 =

V f /c H f ), with a velocity ( W ! 2 regime f ' ' photon-driven µ 2 is justified by the fact that as V ) ⇠  2 1 2 ⇠ m M ↵ ' Pl ↵ ¨ ⇤ m h h f 2 ⇠ 2 ˙ 1 fV Pl RH V ⇡ ¨ 1 2 µ M ⇡↵ | H 0 2 ⇢ 2 2 , where we took the ⇠ , 4 wig ⇠ ⇠ M ⌧ ↵ H . We enter this regime when ⇠ ↵ ˙ ↵ 2 ⇠ m ↵ M 4.25 = 0. It separates the unbroken EW phase, | V f . We choose ⌧ is the real degree of freedom that eventually gets a vacuum expectation value 2.17 0 2 ( fV V H Λ + ' 2 ✏ H H 2 is the SM Higgs doublet, | ⇤ ). We enter this regime –9– 2 +2 2 =2 ( f -dependent squared mass parameter of the Higgs potential. slow2 : 1 and thus is plagued by the strong CP problem. In the V f )= )= )= /f V M ) 3 radiation ξ = linear term in the potential ( h )= ), the second is small for 2 µ µ i 2 Pl ✏ V dominates some fixed mass scale. The fact that is still a classical field, not a quantum fluctuation. 2 ' ' =2 2 V H can be written generically as V is justified by the fact that as  ⇠ 2 1  ↵ ⌧ ⇠ ˙ RH B V V EW grows the amplitude of the wiggles becomes larger and larger up to the A special point in field space is The relaxion potential here is the same as the one used in Ref. [ The Higgs bare mass M . 0 2.19 . Check! Agree?] corresponds to tuning the cosmological constant. This is crucial for the ˙ ∼ | 2 ( ( m ⌧ We stress that is 0 ⇠ ⌘ ↵ · 2 0 ˙ 0 i ⇠ . 4.34 and the photon production is negligible. The motion is described f M 1 , and we remain there only for the last ↵ m ! h V H h, v above it is easy to very that this condition is again satisfied 2 W 0 ✏ H 0). When the parameter H ¨ ) and ( ⇤ h, V ✏ ~ =2 ( ( V g h B V roll 3 wig ( E )= V ✏ , and we stay only for the last where (VEV) equal to eaten by the of circumvented with a clockworkAppendix axion and model we show [ thereand how in to the map rest its of parameters the to paper. the ones we use in this section As most of the interestingpotential dynamics around happen near this point, it is convenient to expand the We write the Higgs doublet as Here is the dimensionless parameter. Them goal is to generateThe dynamically larger a the small hierarchyaddressing the between SM hierarchy problem. where 3 The model We consider the following lagrangian m · h is satisfied, that is =2 2 V V QCD EW V V ) = 0. It separates the unbroken EW phase, < .  =0 , with a speed V V γ ✓ =2 | ˙ 0 ~ 2.16 ⌘ E 2 0 c ), grows larger than one, we smoothly switch into the second regime, which is A ⇠ ) and ( ) ) ↵ EW δφ )= h above it is easy to very that this condition is again satisfied H 0. Then ¨ . ✏ the energy density of the produced is the radial mode of is the Hubble parameter, with ( V 0 and h, ¨ 3 ✏ f ↵ ] and use it instead for the EW scale. As written, the model poses some theoretical ] that can be circumvented with a clockwork axion model [ ) 2 ' ( 4 0 2.7 . ˙ / a a ), all the way into the broken EW phase ( . For field values of order . 1 ⇠ 5 ( h =2 µ ) 2.18 = 0 and 0, in the unbroken electroweak phase. h ( 0 0 ⇤ 4 0 V is already very close to the end of its run, ) we can compute = h, Pl 2 0. Then 30 settle to their VEVs: φ @ 00 f ( ( ¨ 0 ⌘ ⇤ V 2 W ). ( 0). Eventually, the parameter As most of the interesting dynamics happens near For the given coordinate, a special point in field space is The smallness of wig > n> ) µ f ⇠ 0 . The potential then reads M We stress that ⇠ V V 10 For future convenience we rewrite the potential as We keep only the linear term in ( V i 0 m 3 is 3 roll 4.29 < We stop, as long as the kinetic is small enough, when the condition | We must have at least one wiggle between ⇠ V @ H 00 2.6 V V the scale factor. As inflation is driven by In the next subsections we are going to give the quantitative details of this picture. A qualitative overview of the dynamics, shown in the cartoon of Fig. , goes as follows. p 2 0 ) we can compute ). V . Sketch of the different stages in our relaxation mechanism. The first ( We have then cosmological constant is to beVEVs: the observed value then reads where potential around this point. We define We keep only the constant. This ispaper. crucial for We discuss the this successsmaller tuning of further compared the in to mechanism Section the wezero. other describe two in terms. the In rest what of follows the we take as each term is offollows, order we will onlyonce keep we the consider term a linear UVAppendix completion in of this model, such as the clockwork axion discussed in [TV: Changed where in an attempt to explainfollow [ dynamically the smallness ofissues the [ cosmological constant. Herethe we clockwork model in Appendix used in this section and in the rest of the paper. discrete shift symmetry , is crucial: as h, + 2. 1. ⇤ p a ( case the relaxation mechanism works provided that value of 246 GeV.quark The yukawa), case and of theit QCD results axion in corresponds to the sector responsible forwe generating have a two-loop wiggle-potential [ with v point where they stop the rolling of The parameter success of the mechanism we describe in the rest of the paper. We have Choosing this with in with and @ V δφ < 0 h m< ' − –6– ). V N> 0, in the unbroken electroweak phase. In the first regime, the slow roll is due to the m V 0 2 ' At the end of theconditions rolling on the the relaxion/inflaton slope. must stop on the wiggles. This implies two relaxion keeps rolling. Asfinally the stops universe at cools the down, point the in wiggles which it reappear sets and the the correct relaxion 4.1 electroweak scale. Conditions on the slope dominated by photon production andwhen described by ( few e-folds. When thephotons inflaton becomes potential dominant reaches andfollows we is reheat a to period a of temperature radiation just domination slightly in above which the wiggles at first disappear and the In the first regimeproduction the is slow negligible. rollthat is The increases due motion as to the is( field the described rolls. smallness by of We ( defined stay the in in ( slope this regime for a very large number of efolds In the equations above anequation overdot of denotes motion a for derivative with the respect photon to is cosmic given time inThe ( inflaton field starts rolling from a point purpose of our studyquantum we fields. can The treat equation of motion for Here we have 4 Inflationary dynamics In this section we discuss the cosmological evolution of the fields The equation for 2 V From ( ˙ 4.20 4.25 @ φ V We now give some quantitative details of each stage in this simplified picture. In this The qualitative overview of the dynamics is similar to that described in section ⌧ a ) V 0 > r δ ⌧ 4.34 ⌧ i as a classical field, but we must treat V 1 + ˙ ⌧ 2 + 1 ( ⌧ ˙ 2 ˙ produced photons becomes dominantradiation and domination in we which exit theuntil inflation. relaxion it keeps stops What slowing on down follows the due is growing to wiggles a photon to dissipation, period setsection of the we observed neglect electroweak thermal scale. effects, with the aim of keeping the discussion clearer. As we regime for a veryphase ( large number ofwe smoothly efolds switch ( into thedescribed second regime, by which is dominatedthe by end photon of production its and inflaton run, potential reaches and is illustrated inδφ the cartoon of figure smallness of the slope by density of the universe is dominated by Figure 1 standard slow-roll regime, asresponsible for described the dissipation in in subsection theFinally, inflaton/relaxion dynamics, the which last is discussed stage indissipation subsection of and relaxation allowing occurs the after relaxion to reheating get ( trapped in the wiggle potential (see subsection ˙ ˙ ⌧ h ⌧ ˙ m H H ⌧ ¨ H 1 2 H ¨ H 3 Using the definition of given ( In our model From ( satisfied for that is until we exit inflation and reheat. The first term is small due to ( ✏ This is satisfied for 1 2 3 Using the definition of given ( In our model satisfied for that is until we exit inflation and reheat. The first term is small due to ( This is satisfied for ✏ Before reheating the slow-roll conditions are Before reheating the slow-roll conditions are +3 4. 3. 2. 1. 4. 3. 2. 1. +3 ¨ h

¨

This is a regime of warm inflation, with constant photon temperature we reheat and the universe becomes radiation dominated. we reheat and the universe becomes radiation dominated. Once is h

|is satisfied, that is

), grows larger than one, we smoothly switch into the second regime, which is ) is the Hubble parameter, with 2.7 ˙

a a ), all the way into the broken EW phase ( ( is already very close to the end of its run, 30 0 ⌘ wig 10 V We stop, as long as the kinetic is small enough, when the condition | We must have at least one wiggle between H In the next subsections we are going to give the quantitative details of this picture. A qualitative overview of the dynamics, shown in the cartoon of Fig. , goes as follows. 2. 1. N> 4.1 Conditions onAt the the slope end of theconditions rolling on the the relaxion/inflaton slope. must stop on the wiggles. This implies two photons becomes dominant andfollows we is reheat a to period arelaxion of temperature radiation keeps just domination rolling. slightly in above As whichfinally the the stops universe wiggles at cools at the down, first point the disappear in wiggles and which it the reappear sets and the the correct relaxion electroweak scale. ( defined in ( dominated by photon production andwhen described by ( few e-folds. When the inflaton potential reaches The inflaton field startsIn rolling the from a first point regimeproduction the is slow negligible. rollthat is The increases due motion as to the is field the described rolls. smallness by of We ( stay the in slope this regime for a very large number of efolds Here we have In the equations above anequation overdot of denotes motion a for derivative with the respect photon to is cosmic given time in ( The equation for 4 Inflationary dynamics In this section wepurpose discuss of the our cosmological study evolutionquantum we fields. of can the The treat equation fields of motion for JHEP02(2018)084 (4.8) (4.9) (4.5) (4.6) (4.7) (4.11) (4.12) (4.10) 1, and 0 it never |  ξ | , thus W δφ < ). Then we have m 4.1 is small, ˙ φ δ in eq. ( i . . , ~ B ) · 2 ~ δφ E increases. We can introduce as = 0  ( h ) | i ) is zero, while for γ ˙ f 2 φ c 0 can never exceed i roll h δφ δ . δφ 2 | h ( V 2 ( 0 h must exist, . . . ). Pl h m V wig V ) 4 wig f h f 2 W 4 W M g .  2 2 4.5 Λ 0, f EW and therefore δφ 1 2 π m m Λ ), Λ Λ ( ˙ 2 π ˙ Pl 1 ∼ φ δφ δφ √ ¨ m + H δφ 4 2 16 M 0. In this first regime 3 H H 2 3.12 δφ > Λ 3 Λ m < m < ≡ > ' – 11 – − m decreases so i m )  2 ' − ˙ H ¨ δφ ˙ φ + H H δφ φ exit1 = 0 and ( ˙ δ V φ  δφ m & ≡ − ≡ Hδ 1, is satisfied so long as ) ) ) < δφ + 3 δφ δφ ( ) is satisfied. Consequently, the relation ( ( ¨ φ  V | η δ δφ ) ( .  δφ ( 5 , which is much smaller than Λ 0 2 W wig V , this bound is stronger than ( m W m | ' | ) & δφ ( 0 roll V For Λ | is implied. As we mention below eq. ( At least one wiggle between Assuming significant dissipation, the inflaton must halt when the condition During the slow-roll we also have 1. 2. One slow-roll condition, As the field rollsusual down the the slow-roll potential, parameters We can also safelygrows drop larger the than last term: for We assume that the rolling startswe from can ignore the photon production, dropping the term 4.2 Regime 1: slow-roll on a gentle slope we postpone to section 4.1 Conditions onAt the the slope end of the rolling,on the the relaxion parameters must stop of on the the model: wiggles. This implies two conditions will see, these effects have significant implications which require a careful treatment, that JHEP02(2018)084 ). 4.6 δφ < (4.15) (4.13) (4.14) , so it does not , 2 derived assuming Λ 4 wig | ˙ 2 W Λ φ becomes larger than δ 1, as we show in ap- m 2 | | Pl . ξ 2 | < ∼ M f  γ 4 has not moved much from ) c | ], that eventually results in H δφ ξ γ δφ 1 | ( c 45  2 4 – f . − ∼ 0 42 1 roll 10 V 0 < roll ) matches the × V 4 Pl . 4.9 2 Pl 2 f = 0 is that the exponential production of M  M γ c γ ) continues into the broken EW phase, δφ f ln c π | ) | ξ 4.9 γ √ | ξ 1 | – 12 – 2 γ 2 δφ from eq. ( ( ' πc ' H ) fH 3 ), γ ), we obtain / 1, is also satisfied when exit1 switch δφ 0 V ( roll V 2.21 ' − 2 4.12 < V ) | H γ ) 2 Pl | ' 0 δφ δφ ˙ roll ( φ ( M V δ ) and ( η 3 | | H ), is roughly constant. In the last equality we have used eq. ( 3 ≡ grows larger than one. At that point photon production becomes 4.14 | 2.20 ˙ φ δ , insufficient to reheat above the BBN temperature. Hf | switch γ 2 c 4 wig V crosses 0, an important phenomenon happens: the Higgs field experiences an = | δφ ξ . | , from eq. ( B ξ rolls to more negative values. The energy density associated with Higgs oscillations The slow-roll motion described by eq. ( Another consequence of the instability at Once = 0. From that point the dynamics of the Higgs are well captured by the evolution of its δφ where If we compare eqs. ( This happens when the potential is 4.3 Regime 2:We slow-roll switch via from photon the production one first and to the the increasing secondthe photon-driven regime friction, of eq. inflation ( when 0, until important. Neglecting thermaldissipation effects, is provided we by enter photonassociated a production dynamics rather second next. than regime Hubble of friction. slow-roll, We describe where the the short time because, as weits mentioned potential, above, at the which Higgs point isoff. the quickly driven tachyonic production, to The the and energycompared minimum therefore of to dissipated the the friction, by potential switches affect the energy the available relaxion at dynamics. via that point, this that mechanism is is absolutely negligible at the end ofsmaller the than run Λ the energy density of thetachyonic Higgs modes is of still the severaland Higgs orders provides field of another happens magnitude source at of the friction expense for of the the relaxion. relaxion kinetic This energy, friction is active for a very δφ zero mode, which oscillates aroundas the minimum. Meanwhile, thegrows minimum at grows deeper, the expensestore of enough the energy relaxion in energyThe the answer density. Higgs is One negative: to might allow the wonder relaxion for if dissipates reheating in most via of the its its end decays energy we via into Hubble SM friction, particles. and pendix instability, known as tachyonicthe or spontaneous spinodal breaking instability of [ the the field EW to symmetry. the minimum The of instability its develops mexican fast hat potential, and while drives The second condition, JHEP02(2018)084 . ) 2 ξ B ≡ ), are (4.19) (4.20) (4.16) (4.17) (4.21) (4.18) 4.16 (20) in the O with the correct 4 wig varies little from the ξ .  , leading to eq. ( 1). ) 0 4 Pl V δφ (  < M , and is typically 2 , keeps decreasing as the relaxion . 9 | V ˙ |  wig # φ 4 wig ¨ ) . φ δ ) as the result of the approximate | | δ Λ i 4 | 4 wig | ξ δφ f ~ | 2 B γ ( Λ γ ξ c · | c and Λ 0 . )], and therefore 6 2 ), so , roll ~ and , we exit inflation and enter a radiation E ξ ) below, and implies that we switch to the γ ∼ h | 3 γ φ 0 Pl δφ 5 ρ c fV ξ ( γ , f, | f V 4.16 2 c γ M 4 0 10 roll = c 4.17 − V " V ' < γ – 13 – 10 |  ) f ln 4 RH c ): γ ˙ ξ f φ | V c × π δφ 2 1 2 ( ξ 7 4 | . Hδ 4.18 ), we find 4 2 0 3 roll |  V ' − ' 4.4 ln 2 γ ξ ρ π 1 2 ) for the last expression. The photon energy density remains ), becomes ) in eq. ( ), we obtain that the energy density of the produced photons is ) and ( 4.6 4.1 ' − 4.14 ξ 2.16 is largely through ln[ 2.15 is still described by eq. ( ξ δφ ) and ( 4.16 ), using eqs. ( 4.16 The motion of At this stage, the energy density is still dominated by the inflaton potential. From In the second regime, the dissipation from photon production is important and the is given by the parameters of the model 2 dominated universe. However,malized, the hence photons we have cannot very talkmechanism low in about the momentum a next and reheat section. temperature are yet. not We ther- addressrolls. the When reheating the increasing amplitude of the wiggle potential reaches Λ balance between the exponentialradiation. production Once of the potential photons of and the the inflaton drops Hubble below dilution the of value this the energy density is no longer dominated by From eqs. ( where we have usedroughly eq. ( constant (up to a logarithmic variation of ξ parameter space of our interest. 4.4 Inflation exit and relaxation The dependence on beginning to the exit ofby the using second the regime. potential ( To be more accurate, we find this value ( eq. ( One can show thatsatisfied the for conditions Checking these conditions comes with some subtleties which are explained in appendix second regime while we are still slow-rolling from theequation first of ( motion, eq. ( The inequality is dictated by the condition ( JHEP02(2018)084 . ) ]. φ < H ~ 1 E 3 δφ |  ( π (5.1) ~ E 4 (4.22) (4.23) would | / roll e 2 2 V ˙ ) φ | δ ~ E 1 2 , can create | 2 e e ( m , , prompting the ∼ 2 e & ! | 2 e | ~ E πm | ~ E e | πm e − i ∼

~ E h e exp 2 ) | 3 is the component orthogonal to . ~ π E . | | ⊥ 4 e 2 γ 2 k ~ . The virtual pairs, produced in the ( ξ c | Λ 1 4 wig − e = 2 W ∼ Λ | m m 2 ⊥ ~ E | ] e πk EW ∼ − 46 has changed approximately e δφ – 14 – ⊥ EW f f k − δφ 2 δφ d RH − Z and the relaxion would overshoot the EW minimum, δφ | 2 e . In order to discuss thermalization in this case, we have ~ E | 2 e πm 4 wig − e | 3 , this implies that about one Hubble time has elapsed and the ~ E π γ | 4 e ]. In the presence of a constant electric field, the number of pairs = 37 , Hf/c − 2 e has only changed by an order-one amount. ξ 36 + V γ e ρ ' Γ ˙ φ δ is the electron (or positron) momentum, and k ~ 10, the electric field grows exponentially and reaches In axion inflation scenarios, like ours, one typically has very strong electric fields Quantum electrodynamics predicts that a strong electric field, Note this is an important difference with respect to the initial proposal of ref. [ . . So, in a Hubble time, a large number of pairs per unit volume 2 e | ξ where m is produced. In| the model we consider, close to the end of the first regime, with 1 Schwinger effect [ produced per unit volume per unit time is [ electron-positron pairs, provided that thethan characteristic wavelength the of the Compton photons wavelength isvacuum larger of polarization of the the electron states photon, if can they be can accelerated borrow apart enough and energy become from real the asymptotic electric field itself. This is known as the large occupation number of thefield, photons as implies we that explained they into form section a take classical into electromagnetic accountWe discuss an it important in this non-perturbativereheat section phenomenon: via and proceed SM the to photons. point Schwinger In out effect. the a next problem section that arises we when propose trying a to resolution with a dark photon. 5 Schwinger reheating The picture described in theof previous the section EW is scale, good butenergy for fails and a to successful the reheat dynamical system the relaxation cannot universe. be Each produced thermalized photon via carries perturbative very scattering little processes. The In that work,settles the down relaxation after of theboson the end production EW is of crucial scale inflation.inevitably in occurs grow this For during larger last this stage. inflation,causing than reason the while Without Λ the whole it, in friction mechanism the to provided ours kinetic fail. by energy gauge- Given that energy density and the relaxion stops at From the end of inflation to this point, value of the EW VEV, the slope of the wiggles counterbalances the linear slope of JHEP02(2018)084 , . . i 3 = / 2 H | 2 ~ D ) E ξ (6.1) (5.2) (5.3) h m , so the /H ∼ | 2 / D µν 1 a k D ) ˜ m | F ~ & E | pairs proceeds e D D,µν − F m e f + φ 4 e D γ is correlated with c i ~ − B · ~ . µν E , h 6 6 F ξ ξ in the exponent is present when 4 4 4 D 4 D D 4 H H m m D,µν 2 2 4 4 F π π /m ), (2) the reheat temperature would e e 4 2 κ 9 7 | | reaches the very large value ( H 4 wig ξ ξ − | | | Λ ξ 4 4 | µν D H H > F 4 2 4 D 2 D 2 ˙ H H φ m m – 15 – δ D,µν 4 3 1 2 F 1 1 π π 1 4 2 2 pairs per unit time via the Schwinger effect is roughly , − − ) e because, once this threshold is crossed, the temperature i ' i ' + µν 2 , φ ~ e B /e for details), ~ F E H · 2 e , and inverse Compton scatterings on the long-wavelength h . The thermalization of the produced ( 51 MeV for the electron mass. C µν ~ 1 2 . m − E V h γγ e F 1 4 + − = 0 e | ∼ e → e − ~ . E ψ m | e − µ φ e µ γ m e + ¯ e ψ ∼ These electrons and positrons inherit an energy of order ( φ∂ through the Schwinger effect, and thermal effect suppress the photon . The rate of such processes is faster than the Hubble expansion. µ µ 6 e T ∂ , which is below BBN temperature. One way to fix both problems is to eγ , and thus a big suppression of order e eA 1 2 m H m → − + ∼ =  eγ . This is a very efficient process: an order one fraction of the electric field energy T 2 L D / 5 6, is larger than the characteristic momentum of the instability, m ) | Now we have two issues: (1) because of the suppressed backreaction, the relaxion does Consequently, the electrons and positrons thermalize veryThe fast finite and temperature changes the the dispersion temperature relation of the photon, due to in-medium √ When pair production starts, the Higgs VEV is almost at its final value. For this reason, it is a good ~ 6 E | e approximation here to use We have seen that thedue scenario to where thermal the effects. relaxiondark In couples photon, this to we section the can we SM avoid showand photon those that reheating. issues is by and not coupling, We successfully viable consider instead, achieve the the relaxation following relaxion of Lagrangian to the a EW scale larger than the heightbe of of the order barriers, introduce a dark photon, as we describe in the next6 section. A model with a dark photon reaches production. On top of that,the since photon the friction size does of not the grow backreaction enough, unless not slow down enough and does not stop on the wiggles (its kinetic energy at the end is Here compared to the zero temperaturecannot case. go much This above tells us that the intensity of the electric field Accounting for these thermal effects,magnetic we fields arrive (see at appendix different expressions for the electric and quickly reaches effects, and the tachyonic instabilityeT/ is suppressed, especially when the Debye mass, energy density transferred to the ( density is transferred to via annihilations, photons, pair production. JHEP02(2018)084 , D 2 wig γ c Λ (6.4) (6.5) (6.6) (6.2) (6.3) 2 / → 1  , must be γ . | c D p 2 . γ ξ f | c . m  ` , the Schwinger κ | ∼ max D ~ E | . The equations derived D . We considered the cross is the visible electron, and γ 3 , ρ . e T 1 ψ √ H ! . | 2 e 8 > D / ~ | ∼ background, our choice of shifting 1 E . , since the latter is suppressed by | T D πm φ 2  2 . Note that the coupling of the dark ~ 2 | 2 E − α / κe e | D 1 and used the reheating temperature 2 4 T 1 α . scales as γ ξ ψ κ 2 2 e

| c  µ e 2 Pl | κ γ D  n 2 T ∼ e M γ ξ 2 | ¯ exp ∼ c / ψ D 1 2 > πm  D µ ∼ ) eγ eγ  | | , compared to the SM photon case, when | 2 e → → 3 D D – 16 – Pl 2 wig D ~ H D D π ~ eκA m E E wig 1 ~ Λ | | E 4 eγ eγ | Λ αM σ σ κe κe e & (  n removes the kinetic mixing and introduces a coupling of = κe . ≡ D µ . − )) and consequently, e p . κe f + V . κA e m Γ 4.16 − ` µ , rather than A . Also, we took 2 , since it changes only by an order-one amount between reheating 2 κ α 2 T → ) and ( κ wig denotes the massless dark photon. Here, can be used for this model simply with the replacements: µ Λ 4.6 , so that their number density 4 A ∼ D 4 e / 1 ), ( eγ ) distinguishes it from the visible photon. Since during the cosmic evolution and D → γ φ D 2 ), we took the electrons to be relativistic and in thermal equilibrium at a tem- 2.16 /c T > m eγ | σ 2 6.5 . In particular, because the coupling to electrons is suppressed by ξ | ( κe To avoid the complication we encountered with the suppressed tachyonic production of The relevance of the photons being dark clarifies when describing the end of inflation ∼ → In eq. ( perature section two extra powers of T and needs to hold until the relaxion settles down. This is satisfied as long as photon. To do so,out we of require thermal the equilibrium. dark Equivalently, the photonlarger dark than to photon’s the be mean Hubble free sufficiently radius, path, weakly and coupled therefore as it cannot to be stay refracted. Such a condition reads visible photons, we wish to ensure that there is no thermal mass associated with the dark It becomes effective at larger values of The maximum value the dark(see electric field eqs. can ( achieve is given by production rate is now only the visible photon in order to remove theand mixing reheating. proves They convenient. arepaper, produced and in the give rise same fashionin to as a sections described constant in dark the electrice first field part of the we assume there is nofield light matter redefinition content in the darkthe sector dark besides photon the to dark thephoton photon. visible to The electrons, only dark photons are produced in the time-dependent where the index JHEP02(2018)084 ), 6.6 (6.9) (6.7) (6.8) (6.10) (6.11) , 1 , ) and ( − 1 d 6.4 H  11 4 / 1 10  ∼ Pl 1 wig − M d ) in the inequality. Λ of energy to each electron, | H d 2 D ) 6.6 2 ξ | γ | / c 1 D α ~ = 100 GeV, and we again used  E  | directly by the Schwinger effect is . Pl 1 α wig κe − . Note that with these choices the wig M wig Λ , e . . 3 Λ +  / e 4 4.4 2 2 4 wig Pl / / 2 1  Λ 3 / M | 1  . Thus, the energy density transferred can − 2 D | 2 Pl | 1 / γ wig ξ α , with Λ D 2 5 D | c − γ can be quickly transferred to the SM radiation, ξ M ) Λ 2 Pl ~ | c E . , and we used eq. ( | H D – 17 –  2 ∼ κe 2  γ 1 / ( ) /M ρ 1 | − d D ) ∼ | D γ ∼ H 4 wig ρ ~ D E κe > 1 | 2 Λ ~ . Shortly after Schwinger creation, the number density E RH − |  d κe T ∼ ( H κe Pl 2 wig 2 / 2 | Pl ∼ 5 M Λ ) D t |  ~ E /M D ∆ | 3 ) ~ D E − γ | e ρ κe V + ( e κe , where the main friction force arises from dark photon production. energy is ( ( Γ ∼ 4 ∼  ∼ 2 e = d ) e H | n D ~ E D D | γ γ ) is conservative. ρ ρ κe Schwinger ( ρ 6.6 · ). This is very efficient, provided that ∆ e n 6.6 One can show that for values of the kinetic mixing bounded by eqs. ( However, the electric field can transfer an amount ( The absence of a thermal mass for the dark photons implies that we keep producing relaxation mechanism could be spoiled. 7 Cosmological perturbations Our model is similar togauge those bosons. of natural The inflation, where associated the cosmological axion perturbations field couples have to been Abelian largely investigated the dark photons never reach thermalthe equilibrium latter, with and the remain visibleits sector, cold. after production So reheating via far of we thehave have to relaxion. assumed be a small However, enough massless one in dark can order photon not give to to it maximize suppress significantly a its small production, mass. otherwise the Its mass would and implies that an order oneso fraction of the reheating temperature can reach where we took eq. ( of electrons is be estimated as where the typical by accelerating it over a distance The energy transfer from theinefficient, dark unlike electric in field the to SM photon case. In a Hubble time this can be estimated as bound ( efficiently the dark electric fielddescribed as in we enter section the secondWe saw regime that of in slow roll this for regime the the relaxion, amount of energy available in the dark electric field is and the end of relaxation, as explained in section JHEP02(2018)084 ) ∼ ) 7.1 k (7.2) (7.1) ( ] for a ζ P 34 ) holds for 7.3 GeV, and the , 13 4 4 / the energy density / 1 1 RH ] end increases, the power ρ ] . V (regime 1) (7.3) end | 9 33 remains quasi-constant 32 , ξ V ln − | GeV, but then inflation | ξ 1 3 ξ 31 | 10 13 π 4 − × 10 e 4 4 ) (regime 2) (7.4) 5 . Taking the highest value for / / . ξ 1. As ∼ 1 1 ( ∗ end leads to several features, which , combined with a very shallow 2 4 wig V V ' ] f H H = 2 ˜ Λ F 2 such that, thanks to the exponen- s 50 ∼ n | COBE φF + ln ξ | 'P COBE left the horizon, RH i | . The second equality in ( ρ available close to the end of inflation is of  P 32, while the other logarithms in eq. ( ξ 4 k 6 | / GeV ∗ 5 π ∼ 1 ' ∗  P 4 V /ξ − . The first one gives a power spectrum which, 4 e V 16 , the other is from fluctuations induced by the ) 10 − W end ξ 4 – 18 – f δϕ 10 H ( GeV / V × 10 m 1 2 ∗ f ln V 5 → 16 ∼ 48 ' P 10 − ∗ − ∼ ) V δA 0 ]. The coupling ξ 10 2 ( ) 1 + + H k 2 2 49 h < 0 f – 1 1 a 2 πξ δA − ˙ φ 4 s e-folds and most of the potential energy initially stored in the ln the energy density at reheating, and the subscript 0 refers to (2 47 2 ], n , and H π −  , is largely insufficient to explain the observed perturbations: 30 32 ' 4 1 RH 0 , we have ln 38 ) k E , ρ k − )], and the power spectrum saturates to [ k = W (  34 ζ here is a consequence of low-scale m P – P P ) = 62 4.19 k ∼ 31 P . For this reason, our model should be regarded as a low-scale inflation ( 4 W ) = N 002 Mpc wig k . [see ( ) increases exponentially. When we enter regime 2, ( ζ 2 < m ξ P 7.3 ), we match the observed power spectrum for curvature perturbations, of the inflaton, proportional to = 0 is the energy density when the mode , and for the sake of the estimate we took 30. 0 ∗ 4 wig 7.3 | k δϕ V ξ | ' , that is Λ This allows, in principle, to have a period around 30 e-folds from the end of inflation We have two sources for curvature perturbations: one is from vacuum quantum fluctu- In most models of natural inflation the Hubble scale is of order 10 ) k ( wig where we are stilltial in in ( regime 1, but with a large where large spectrum ( with value The smallness of potential. Including the second contribution in regime 1, we have [ ations inverse decay of photons [ as we show in appendix today’s value. In our caseΛ we have are roughly zero.N Therefore the observable number of observable e-folds in our model is where at the end of inflation, is much shallower. Atproceeds the for more beginning than we 10 scalar can field also is have dissipated.order The Λ energy density model. The number of observable e-folds is given by [ include the generation ofgravitational curvature waves, perturbations and and the nongaussianities, formation thereview of of production primordial these of black topics. holes (PBH). See ref. [ number of e-folds is roughly 60. The important difference in our model is that the potential in the literature [ JHEP02(2018)084 (8.1) changes = 30, we )] is 5, 15, e ζ P N 8.6 , that -� around �� 2 =�� COBE =�� ���� ζ P � � �� � = ζ P ). This indicates that it is difficult )], and fall exponentially in regime �� 7.3 7.4 , κ. D γ is so small, but a conclusive statement GeV. The three curves correspond to values =�� P �� � � � 16 � ]. Such a bound is typically very stringent for , f, c 49 wig , = 10 Λ – 19 – �� f 33 , . The independent parameters in our construction h 3 , g Λ �� ��� =� m, � , so that the number of e-folds in regime 2 [see eq. ( � D γ � c = 10 GeV, and wig -� -� -�� -�� -�� �� �� �� �� �� ζ � ]. Naively they are negligible, because )]. On the horizontal axis, time increases from right to left. The gray region shows the . We show the power spectrum as a function of the number of e-folds from the end of 51 . In practice, it still does not mean that this model is agreement with CMB observa- 7.3 Our current estimate does not take into account the modulation effects due to the COBE 8 Constraints and relevantWe scales are now in the positionton. of summarizing A the summary constraints plot on isare the given model in with figure the dark pho- wiggles [ requires a dedicated study, beyond theof scope the of parameter this paper. spacemodel, We in to leave a future relation more study. to detailed CMB study constraints, and a possible extension of this measurements of higher multipoles. Therefore,in we regime need to 2 roughly to havethe comply less observed than with amount 25 of CMB e-folds curvature bounds, perturbationsaddition in the the of consequence model another being as field, it that like stands. we a We do curvaton, note can that not the help produce in matching the CMB power spectrum. for the model as it stands to predict the observedP curvature perturbations. tions, unfortunately. The sameby exponential many implies, orders of as magnitude we within see a couple in of figure e-folds, which is in contradiction with CMB and 25, respectively. The curves1 are [eq. flat ( in regimebound 2 from [see eq. PBH, ( computednatural following inflation refs. models, [ butNote here that it while it is is easilysee numerically that evaded possible the due to lines to explain fall the very the steeply, observed significantly due lower to inflation the exponent scale. in eq. ( Figure 2 inflation. We have fixedof Λ the axion-photon coupling, JHEP02(2018)084 as D (8.5) (8.6) (8.2) (8.3) (8.4) γ c are related varies very f ξ ). We have seen and 10. Note that once 8.4 ), that , and the visible sector dV , wig is the Goldstone of a < 0 φ D φ 25 . This implies an upper D γ 4.18 γ c < c ξfV 2 2 7 > f | from eq. ( − RH f one would like a value of EW . 2 switch V Pl V , f (1). Since Pl Z δφ M f | , we must impose that the scale Λ, O 2 , . M f = f ). Requiring this to last for 25 e-folds | 2 = 3 2 D is a free parameter, this rather narrow 2 W | 1 ξ 2 γ f 2 | h c dV m ξ D g ). Incidentally, in this window we have 4.21 0 and on the cutoff Λ, independently of the ). We also require the presence of many | γ 4 < c f f . ˙ 8.4 H 4.6 φV D ' δ f γ . m < c – 20 – Λ . dV by eq. ( . RH switch V 3 V Pl 4 wig RH RH f Z can possibly be achieved in the clockwork framework, switch V V Λ . M from eq. ( V , that is we impose | = D 1 to the hidden photon coupling to 2 Z 2 . γ ξ 0 m 0 c | κ EW ) and D Hdt fV δφ γ is constrained to the window (10), which confirms, following eq. ( 2 c and ξ D 2 4.14 RH γ D t switch γ t as a constant, gives ∼ O c ' Z ξ f/c 2 2 | 2 Pl = 2 = 0 and N ξ M | 2 (20). To increase the allowed value of 2 D N δφ , as discussed at the end of section 2 γ f c , but for now we restrict our attention to the case W A ∼ given by eq. ( . The two conditions together give the window | ∼ O 2 we get directly an upper bound on m < m ξ | RH D varies only logarithmically in the short photon-dominated regime, and its value is γ wig | c switch /V , the cutoff is only limited by the upper bound on ξ V f | The goal of the whole mechanism is to achieve a cutoff Λ as large as possible. As The number of e-folds in regime 2 is The combination . switch typically big as possible. Largesee values appendix of we fix other parameters of the model. V little during this regime.window We leaves stress a that significant viable since parameter space, nonetheless. Λ that from which we obtain the lower bound of ( with at most, and treating The upper bound comesroll from regime, the while requirement the lower that boundthan we comes the enter from last asking the 25 that photon-dominated such e-folds, slow- a see regime figure does not last more with Λ This implies a lowerwiggles bound between on bound on to the wiggle potential, respectively. For the sakeglobal of simplicity, symmetry we spontaneously take brokenwhich at explicitly the breaks scale the symmetry, be smaller than The first 3 parameters are related to the shallow rolling potential, Λ JHEP02(2018)084 ] . < wig 52 , D (8.7) (8.8) (8.9) γ , (8.11) (8.10) 22 m vs Λ f GeV 8 10 / 1 ). ), and take for 3  3 | , are tiny. In this D 2 /f 8 ). Together, these γ ξ | c − Λ = 10 4 wig  10 6.10 2 / ). We also impose that . 1 , 100Λ ,  θ GeV, the relaxion mass is 6.6 9 Pl ∼ − , eq. ( 10 . wig − are allowed, which in turn can GeV 4 10 m e Λ / αM 1 + Pl 11 , . . To cover the whole region with e  . ] (orange in figure  f M | κ 3 W D 2 . 4 wig wig γ 22 ξ = 10 . , | c Λ Λ fm κe 4  21 10 / )) 5 ∼ 1 − . / 1  , say above 10 6.7 # 10
| 3 f D 4 e 2 (which means / γ ξ , θ 2 | c m f – 21 –  very close to 2  1 Pl , . 4 wig f Pl f Λ wig ∼ , we need to ensure that the dark photons create GeV. the allowed (white) region on the plane αM M GeV. Thus, we have Λ . ∼ , f 6 16 3 GeV  11 are negligible: they are small because they break the RH 2 φ > , T , Λ = 0 31 | m 2 − 2 roll / ξ wig = 10 | 1 V 10 26 GeV Λ / f  , | Pl = 1 GeV ∼ ∼ D 2 to the window, γ wig ξ M | c 2 wig RH . κe  Λ 10 GeV, values of 2 e ), while not acquiring thermal mass, eq. ( = 0 2 wig ∼ m Λ D 6.4 , γ , m ), provide interesting bounds for high in this range will be equally good for reheating. At the same time, they " wig 5 = 1 GeV and 3 ,T κ − (1) f/c max O 10 24 wig GeV their sensitivity would have to improve by a few orders of magnitude. For = ' GeV, the mass of the relaxion is above 10 keV. In this region of parameter space, the | ' h 14 D 9 2 pairs, eq. ( g γ ξ | c 10 10 Concerning the dark photon, there are almost no experimental constraints in our sce- The relaxion mass and its mixing angle with the Higgs are given by We provide a benchmark point to give an idea of the scales and numbers involved. Finally, as discussed in section − e . + f relaxion can be probed viaand cosmological there and astrophysical are processes, various or constraints in studied the in laboratories, refs. [ nario. This is because the dark photon has to be massless or extremely light, discrete shift symmetry.smaller For than high 1 values keV, of andrange its it couplings is to hard matter, toHiggs, suppressed detect it can by it be experimentally the as mediator(blue of a in a particle. long-range figure force. However, via Experimentalf its tests for > mixing fifth with force the [ Here the contributions from More generically, we showNote in that figure for Λ accommodate a cutoff as high as 10 First, we fix instance, Λ The reheating temperature we get is (see eq. ( Any value of yield a lower bound on Λ e the dark electric fieldrequirements transfers contrain sufficient energy to the JHEP02(2018)084 | 2 > ξ | wig

]. The

1

W wig > Λ m ), Λ, is fixed by . The range of 8.1 2 | that were chosen to 2 ξ 10 D |

γ

/ TeV c

Pl

10 ≈ M 2

. RH T ] for bounds that extend to

5th force = 0 53

TeV

D

1 GeV γ

) is satisfied, which is always the case. ≈

GeV

100 RH

astrophysics f/c

1 cosmology 8.3 T

≈ 10

≈

RH 10

T

RH ). In such a region there is no viable value of (see e.g. [ T GeV] 8 [ beam dump and collider 8.4 − -2 -5 -8 -11 wig 10 ≈9 Λ ≈ 10 ≈ 10 ≈ 10 D ≈ 10 γ – 22 – D D D c D

γ γ γ

γ ]. The dark gray area corresponds to values of Λ κ < effect c c c c 22 1 ). The dashed blue contours depict the reheating temperature

8.4 Schwinger

), combined with eq. ( No 8.8 -1 ). The blue region is excluded by 5th force constraints, while the orange region does not need to be specified as long as eq. ( 10 6 4 8 16 14 12 10

8.9 m to allow at the same time for reheating via the Schwinger effect and for the dark photon 10 10 10

10 10 10 10

[ f ] κ GeV . Summary of constraints on our model. We fix ), and GeV, and the mixing very small, tum fluctuations, that the vacuum energy be dominated by thethat inflaton, the evolution of the relaxion be dominated by classical rolling rather than quan- and is excluded as it implies an unacceptable electroweak breaking scale. The light gray region In this relaxed inflation scenario, we can achieve a higher cutoff than in ref. [ CMB observables represent perhaps the most interesting arena for testing this frame- 4.6 14 1. 2. − W limiting factors in the original model were the conditions: work. The darkmordial photon black production holes can andcan lead gravitational produce to waves, measurable the modulations. whilebeyond generation the These the of wiggles features scope nongaussianities, deserve of of a the pri- the current dedicated relaxion work. study, potential which is eq. ( 10 this region of parameter space). colliders; these are explainedm in detail in [ is defined by the bound eq.the ( mixing to avoid a thermal mass. Note that of the dimensionful parameters listed in ( Figure 3 in this plot issaturate from the 19 lower to bound 26.given in The eq. in dashed ( eq. red ( contourscorresponds show to the a values of set of constraints from astrophysics, cosmology, beam dump experiments, and JHEP02(2018)084 ]. 1 . wig ], is also 1 for the wiggles to settles down when wig δφ pairs quickly thermalize, − e + e – 23 – GeV. It is obvious why condition 1 does not apply, as in our case 16 10 ∼ appear. that the Hubble parameter during inflation be smaller than Λ We have studied the phenomenologically viable parameter space, and showed that while The reheating process is an important novelty of this work. It first starts with the 3. cutoff close to the Planck scale, significantly above the one found in theAcknowledgments original proposal [ We would like tocollaboration thank at the Tim embryonic stages Cohen, of Erik this work. Kuflik, We benefited Josh from Ruderman, a multitude and of discus- Yotam Soreq for our scenario can evadequite CMB stringent, constraints it from is primordial difficultAn to black extra generate hole ingredient, the formation, like observedspectrum. a typically amount curvaton We of find field, curvature perturbations. that is the likely promotion needed of to the match relaxion the to measured an power inflaton can accommodate a dark photons which are only weakly coupledreheat to safely the above visible BBN sector. temperature, Weprovides find while enough that the this dissipation unsuppressed allows to for productionEW the of dark minimum. relaxion, photons which A slows detailedpresented down in study and future of settles work. this on reheating the correct mechanism is under study and will be production of very strongpositron electric pair production and via magnetic the Schwingerreheating fields, mechanism. the which The universe. allow forgauge To vacuum bosons achieve electron- cannot a be sufficientlyshut coupled high off strongly reheat the to temperature, non-perturbative the the photon thermal produced production. bath, as Here thermal we effects considered quickly the production of it is necessary tothis can introduce be an accomplished additionalinflation, by the dissipation coupling gauge-boson mechanism. the production relaxionallowing becomes We to for significant, have a gauge slowing new shown bosons. down reheating that the mechanism. In relaxion the and last stages of the inflaton. Twothe key potential ingredients and of theproportionally the presence to original of the a proposal Higgsthe periodic VEV. were The potential a motion wiggles (wiggles), very of provide with shallowthat the the amplitude backreaction slope the relaxion growing necessary of relaxion to and stop conditions. itself set The could the EW be scale observed must the be EW inflaton, set scale. after as the it end A of automatically shallow inflation satisfies and slope to the suggests avoid slow-roll overshooting either, as our wiggles reappear after reheating once the9 temperature drops below Summary Λ We have investigated a model in which the relaxion, originally proposed in ref. [ In the framework presented in thisachieve paper, a these cutoff three Λ conditions arethe not relaxion relevant, is so we the can inflatonthe universe itself. is not Condition de 2 Sitter is anymore but not radiation necessary dominated. since Condition 3 is not necessary JHEP02(2018)084 . , , ] j 1 +1 ˜ − +1 G 10 N 3 1 N ) (A.1) ˜ G π D G ( 2 f φ α +1 1 π 8 α N + 1)th site ∼ j ), except for G ) have charges , D φ F ( j  γ . A.1 π N c c 8 α . , responsible for the + h φ F +1 3 j 1 in eq. ( , which we identify with cos to fermions charged under
Φ † j 4 N N  1 3 +1 Φ is provided by the clockwork Φ N Φ  3  ...  1 9 0, all the U(1)’s are spontaneously =1 N j 1 3 X . Because of the suppressed overlap > 1 + 2 Φ wig µ  Λ 2 N | j  = Φ † j N φ Φ | ]). The construction relies on the potential [ – 24 – Φ 4 62 λ – + , due to the terms with 54 j , . The corresponding Nambu-Goldstone bosons (NGB) f couple to. Φ Φ Φ 41 † j  /λ Φ +1 √ ) 2 Φ ]. For example, by charging the fermions at the ( N 2 Φ µ + 1 field, the operator leads to the coupling µ 63 − (2 N  q +1 is a normalization factor. The relaxion has exponentially suppressed =1 j X N = under the unbroken U(1). As . Below the confining scale, the potential Λ N f to fermions charged under a non-Abelian gauge group that confines at the f ](see also refs. [ N 1 1 3  10 (Φ) = , f 9 V N ,..., . Via the one-loop triangle diagram the relaxion obtains the coupling 1 9 ’s are complex scalar fields. The terms in the first sum respect a global U(1) , j over a large range [ = 3 1 3 wig ) , F D ( We couple Φ γ c = 1 with rolling, emerges. By controllingcontrol which the of strength the of scalarsof the couple photon to coupling the to dark theunder the photon, relaxion, Abelian namely one gauge symmetry, one may the can relaxion-photon coupling set would the be value scale Λ which gives rise to the periodicanother wiggle potential. gauge We group couple Φ withof confining the scale relaxion Λ with the obtain a mass proportional to that associated with theat U(1) this which stage) is isthe not given relaxion. explicitly by Here broken. the combination overlap The with latter the NGB operators (massless Φ where Φ symmetry, while the second sumQ explicitly breaks it to abroken U(1). at The a fields scale Φ mechanism [ A A clockwork model A possible UV completion for the model presented in section Horizon 2020 Programme (ERC- CoG-2015German-Israeli - Foundation (grant Proposal No. n. I-1283- 682676 303.7/2014).in LDMThExp) part The and at work the by of Aspen the LU Center wasgrant for performed PHY-1066293, Physics, which and is was supported partially by supported National Science by Foundation a grant from the Simons Foundation. P. Fox, R. Harnik, A.L. Hook, McAllister, K. M. Howe, McCullough, S.Redigolo, S. Ipek, A. Nussinov, J. S. Romano, Kearney, Paban, R. H.part E. by Kim, Sato, the Pajer, G. L. I-CORE M. Marques-Tavares, Program Sorbo, Peskin,Foundation of (grant and G. the No. Planning Perez, M. Budgeting 1937/12), D. Takimoto. Committee by and the the This European Israel work Research Science Council is (ERC) supported under in the EU sions with P. Agrawal, B. Batell, C. Csaki, P. Draper, S. Enomoto, W. Fischler, R. Flauger, JHEP02(2018)084 and term 7 (A.7) (A.8) (A.9) (A.2) (A.3) (A.4) (A.5) (A.6) 2 π (A.10) (A.11) h ) φ ( wig V ) + between 0 and φ ( , roll φ/F V .  φ +  F . = 0: 4 W λ h EW . cos 4 F . m F. m 0 δφ − F F δφ φ +  − 0 4 1 . F φ EW f sin h F λh δφ h 1 g δφ 1 4 g   F sin 2 0 broken phase + 2 2 W + N = F φ 0 Λ Λ 2 Λ m h 0 φ h , we have g F φ – 25 – ∼ sin ,  , h δφ 1 ' cos Λ 4 N ) ' − g φ  F  cos D Λ ) + ( φ F > 1 determines the point at which we switch from the  as the point where 4 N 0 ˜ is expected to be of order Λ (up to a loop factor), as the F ' δφ EW cos ) φ 0 F ( φ N reads ) F > 2 D φ h cos wig ( δφ = . 0 g µ h Λ δφ F ) = Λ φ − g ( φ φ f cos − f φ φ cc ( 4 1 roll ) we find α anyway. cos V  D roll (  2 φ F 2 W V γ c Λ 4 N 4 wig m 1 2 + cos 4 2 Λ π ' − h 16 g ) = Λ ) = Λ ) = ) φ φ ( ( h, φ EW ( roll wig δφ V V ( is a dimensionless constant that we use to tune the cosmological constant to zero. −L 2 µ cc α The full clockwork-inspired Lagrangian for the relaxion that we consider is then In the absence of tuning, the scale Λ 7 which after expanding around is going to generate We want to tune the cosmological constant at this point: Setting Expanding around this point, With these conventions, we imaginerolls that down the to rolling the starts left. from We define to be smaller than the spontaneous breaking scale, so we have the following hierarchy The dimensionless parameter unbroken to the broken EW phase: Here, To make sense of the notion of pNGB, all the scales corresponding to explicit breaking have JHEP02(2018)084 (B.6) (B.1) (B.2) (B.3) (B.4) (B.5) (A.12) (A.13) (A.14) (A.15) , ) δφ ( wig as V ˙ φ . δ ) + 3 δφ 0 is satisfied. We saw that ( > roll . V , ˙ . as roughly constant, we have , 0 φ wig ¨ ) φ + 0 V 2 δ H F φ . Hδ = 0 ϑ + ( , 4 W λ ) = 0. With the boundary conditions  4 O 0 0 m sin ( roll wig ' − roll V V = 0 O . + 2 N + ˙ F

φ Λ 4 + /V + ¨ 0 φ (1) roll t ˙ δ 0 0 φ 00 roll V 0 ∼ wig λh 0 Hδ roll roll ˙ , V φ V 1 4 V 0 V ) ϑδ +

− F 2 φ Pl – 26 – + ) + + ≡ (0) ) = EW 2 ˙ M ˙ t φ φ δφ sin ϑ ( (0) 1. . ( = 0, and treating δφ ' ˙ 2  φ t Hδ (0) F Hδ δ Λ δφ h − δφ 3 ≡ (0)  < δφ at 0 = ) η ≡ f + 3 + δφ F φ ˙ remains smaller than 1 for values of the potential down to ( φ ¨ 0 φ H δφ m 0 δ 2 δ φ ( 3 F φ sin η / 2 0 cos F roll ˙ sin Λ H/H because V h N 4 g 4 wig − F Λ 1 2 (0) ¨ ≡ − , the equation of motion reads φ ) = δ ϑ 0 ˙ φ ) = ) = Λ ) = δφ ( = typically does not grow larger than 0.1. We expand  δφ δφ ( ( , we discussed the slow-roll conditions in regime 1, where the barriers from ˙ φ . In this appendix, we discuss in detail the other slow-roll parameter: ϑ h, δφ δ ( roll wig 0 roll 4.2 V V V V Pl M We start from the equation of motion = 0 and ∼ where we dropped δφ In regime 1, At zeroth order in and define the small parameter The last equality holds as long as In section the wiggles are not yet large,the namely parameter the condition V we can match this potential to the one given atB the end of Slow section roll conditions B.1 Regime 1 We see that, by identifying Putting all the pieces together we have JHEP02(2018)084 . } (B.7) (B.8) (B.9) 0 (B.15) (B.11) (B.12) (B.13) (B.14) (B.10) f φ , , 0 ˙ sin φ io 2 0 f ) φ πf/ Hf sin (3 = 2 Ht . We have 3 . f 2 − t − t 0 Λ f e 0 φ ˙ , W 2 φ − ϑ . f t m + cos , 0 ˙ φ 0 0 = 0 t ∼ ∼ ˙ f 0 + φ , φ . t ˙ 0 φ stays roughly constant because . 0 t 1 V φ 0 0 ˙ roll 0 wig ˙ φ f φ ˙ Hf + cos V ), we get φ V δ f 3  2 0 + sin = 0 f { ) 0 + h φ 0 ˙ B.7 φ 0 Hf φ Ht 0 ' − 0 wig 3 φ ˙ Hf ϑ 2 φ Hf cos V − πHf ) 3 e sin 2 (0) ∼ (1) ˙ + sin 0 φ ¨ − φ +(3 − ˙ Hf Hf φ i t 2 0 4 wig f (0) 3 (1) 0 ˙ (1) < ϑ 0 ϑδ 4 wig φ ˙ f ˙ ˙ Hδ f Λ φ φ ¨ φ φ φ

f Λ + +(3 ) into eq. ( f 4 wig f 0 − – 27 – t 2 0 ϑδ Hδ φ ˙ ' Λ cos ' − φ 0 Hϑδ B.6 (1) (1) ˙ ˙ φ ¨ ˙ Ht ' φ φ (1) φ cos ¨ 3 φ ' − ¨ φ 0 + 3 δ − ˙ ϑδ (1) ˙ φ Hδ e

φ ¨ ϑδ ¨ φ φ (1) Hϑδ − δ ¨ φ t Hf Hδ ϑδ 0 ' − ˙ + 3 φ can be estimated by taking one period ϑδ f η +3 + 0 is constant to a very good approximation. 1 also in this limit. Note that 0 t ˙ ' − (1) φ φ 0 ˙ ¨ ˙ φ φ φ < η δ f = | + 0 ϑδ cos η ˙ | φ φ h δ 0 ˙ φ sin , the equation of motion is n 1, and . . 2 0 | | ϑ ˙ φ 0 0 ˙ ˙ 4 wig φ φ |  Λ 4 wig f η | − Λ  |  | = = Hf Hf (1) (1) the relaxion does not gaindeviation net from kinetic energy from the wiggles. Indeed, the maximum and This proves that This is the more interesting limit, which corresponds to the end of regime 1. We have and Thus This corresponds to the beginning of regime 1, when ˙ ¨ φ φ 3 3 There are two limits to study: 2. 1. ϑδ ϑδ which can be solved analytically: Substituting the zeroth order solution ( At first order in JHEP02(2018)084 0 > 0 (B.19) (B.20) (B.21) (B.22) (B.16) (B.17) (B.18) V , that is RH V . = 2  γ 0 4 roll ρ . 2 V H , γ 0 ! f , γ πξ C 4 , V ρ 1 ξ − ˙ e V > γ φ  + 4 < 4 πδ Hf ln ξ 2 γ H 2 2 − Pl Pl 2 f , γ 0 e γ 2 ξ 1 4 M π M f ξ C ) Hf 2 2 γ <

. In what follows we justify these c H f = 2 | γ γ Pl 4 2 3 . ξ ˙ γ 0 φ f | roll ' − M . Recall that in our conventions ' 2 H | πδ V Pl 4 Hf = V f ˙ ξ | φ  ˙ | − 2 − Pl φ/ ξ < γ M ). | e 2 δ 1. Using the Friedmann equations we can ρ 10 2 ) ( M 4 γ 4 3 γ ) f 0 , δ  γ × < 4.14 ' f C H 4 γ + . 2 γ | f ˙ V 4 – 28 – 2 2 ˙ H ξHf f H φ ˙ = φ 2 H 0 0 (2 δ ' − const. roll ˙  V Hδ 1 2 φ/ 0 V 3 ∼ δ 2 = | C ( + '   , see eq. ( H ˙ ) is negligible. We have 0 γ φ 2 2 1 f Pl C ) we are keeping only the rolling potential and neglecting 0 0 ˙ 4 γ roll roll φ f V | V δ Hδ V M , and H ξ ' B.16 | 1 2 2 γ 2 Pl 0 γ f ] 2 ) c f B.17 M C 4 + 3 = we claimed that in regime 2 the EOM is well approximated by 31 ξ δφ ≡ ¨ φ  ( =  δ γ 0 f 4.3 in eq. ( ln roll V ˙ 1 π φ switch Hδ ' − ξ to the form [ V < V  0. In section ) is obtained by is slowly varying, that is ˙ The term 3 due to roughly until reheating. Then for the first term we have to impose bring The second term in parentheses is smaller than one for H The kinetic energy is smaller than the potential φ < B.17 δ First, note that in eq. ( • • • With this we check the following conditions: with the slow-roll conditionsstatements. satisfied when the wiggles. We checkeq. later ( what happens when we include the wiggles. The solution to and Let us rewrite the full EOM as where we have defined B.2 Regime 2 JHEP02(2018)084 ˙ φ 1, δ < ) with (B.28) (B.25) (B.26) (B.27) (B.23) (B.24) (1) γ ˙ φ ! B.7 ). At first Hf γ πϑδ 4 ) . We proceed /Hf B.17 (0) (0) 00 . ˙ wig ˙ φ φ . V 1 1 and and expanding δ ) we find 1 πδ (

− < < . e 0

0 0 H. roll wig | B.18 ϑ < V  (1) V V − (0) ˙ ξ φ

2 ˙ | φ , the second vanishes as = γ δ π  . ϑδ − is another small parameter. 4 = 2 t ) ). Again we can check what 0 0 /Hf roll wig ϑ (1) ) ' 00 (1) (0) ˙ 2 switch roll (0) (1) ˙ V V 0 φ ˙ φ ˙ ˙ roll φ φ φ (1) ! ˙ V δ δ ϑδ φ fH B.24 f ϑδ ' VV γ ϑδ γ ϑδ (0) t +  ˙ πc ! + φ + 0 ) is satisfied. (0) 2 V < V Hf Pl φ γ ˙ (0) (0) 2 πδ φ γ ˙ ˙ φ (0) φ δ f δ ˙ f φ ( Hf πM B.21 cos because of the nonzero δ π + πδ ( 4 + 0 − 0 f − e

− V φ (0) eff ˙ 4

φ 0 roll H (0) 3 δ γ – 29 – − 0 sin ˙ V 3 V roll φ f γ ,

8 δ V 2 Pl f γ 3 4 wig f 4 wig f ) is enough to guarantee slow-roll in this approxima- H 0 M roll 4 ). It is easy to verify that Λ Λ +  V 0 in two limits: ξHf 1 − V H

B.21 = 2 ' B.9 | ' | (1) = 2 ≡ ¨ ξ ' φ 3 | 0 (1) wig | 2 ¨ φ eff V ϑδ (0) ˙ (1) 0 φ ' H ϑδ + ¨ δ φ | V | ¨ → φ 0 (1) ϑδ roll → | δ ˙ φ . . | V | | could grow larger than 0 H ˙ φ ¨ is negligible. Taking the time derivative of eq. ( φ (0) (0) ˙ ˙ δ φ φ Hϑδ ¨ φ δ δ δ + 3 ). We have already solved the zeroth order EOM, that is eq. (  |  | (1) f f the EOM is = 0, the third is small as long as eq. ( B.4 ¨ φ eff eff ϑ 00 ϑδ roll H H Here we have Here we have Here, the firstV term is smaller than one for The term 3 3 • 2. 1. The solution then is thatwhich of confirms eq. the ( consistencyhappens of to our the expansion acceleration in eq. ( is just a modification ofthe the replacements Hubble friction term, and the EOM reduces to eq. ( We linearized the equation in theOne second can line check assuming this assumption is correct after finding the solution. Now the photon friction as in eq. ( order in We see that the conditiontion. of eq. Next ( we examine whatworry happens being when that we take into accountas also the in wiggles, the the previous main section, by defining the small parameter JHEP02(2018)084 ) ¨ φ δ (C.1) 0. At B.32 (B.31) (B.32) (B.33) (B.29) (B.30) < ). ) implies . This is is large at H | H | 0 ). Keeping ¨ (0) φ γ ξ ¨ 1 ˙ φ | δ V φ f | δ π δ 2 B.33 6 B.18 ≡ 1, based on the # & 0  t ξ < , | 0 4 0 ¨ roll φ leading to a different δ V V | H !# γ t 0 , | (0) f ˙ C ˙ 4 φ H φ 0 roll 0 ξ γ δ f V Hδ πV ) correction to 3 − | . 0 − | e ϑ 0 ( ξ | . 0 = 0 π roll O (20) e-foldings, eq. ( ˙ 2 γ (1). The contribution of eq. ( φ V − − O πδ O Hf e A (0) − ¨  φ = e ! t δ H. | | ˙ 4 (0) + 0 4 φ f ¨ φ ξ H ¨ 0 ξ φ δ V | roll γ H | γ δ f π (0) 0 0 πV roll γ ¨ 6 φ V f δ C − − akc e quickly becomes small because the exponential ∼ 1 + − | – 30 – = 0 ¨ +

φ t 2 δ V 0 | H roll k 0 H roll γ 0 , 0 0 roll 4 4 roll roll γ

f V f πV V V H πV H H (0) term, and neglect again the time-dependence of the γ γ + 0 0 e + ¨ φ f f ˙ δ φ C C ¨  4 4 φ − 2 is large when we transition from regime 1 to 2, and perhaps ξ ξ δ A ¨ " " φ 2 Hδ ∂τ + 1 δ ∂ ln ln ). However, (0) 0 γ γ roll ¨ φ V δ π π Hf Hf  ). Despite regime 2 only lasts B.23 ). At last, we examine a possible = . To see this effect, we consider the following differential equation ) = ) = 4.14 | t t 0 ¨ φ ( ( δ V ˙ B.18 ¨ | φ φ δ δ ) remains negligible. becomes negligible sooner than one e-fold after the transition ( = 0 corresponds to the time of transition from regime 1 to 2, and | term in the EOM introduces extra time-dependence of B.16 0 ¨ φ t δ V ¨ | φ So far we have checked the self-consistency conditions, δ At zero temperature, thenentially equation enhanced of reads motion (EOM) for the polarization that gets expo- Here we used eq. ( that C Thermal effects on gauge-field production is not included in eq.damping ( factor is much larger than where the transition the acceleration could be sizable, For simplicity we dropprefactor on the the 3 r.h.s. ,solution keeping is only the stronger time-dependence in the exponent. The the beginning of regime 2,the the solution quickly converges tocontribution the to one of eq. ( in eq. ( solution of eq. ( because one could worry that it is not a good approximation to neglect it in the EOM. We show that even if We conclude that even when taking into account the wiggle potential, the acceleration JHEP02(2018)084 ) is , the (C.6) (C.7) (C.8) (C.9) (C.2) (C.3) (C.4) (C.5) t (C.10) (C.11) C.3 0, of the , D iω > m  a k , , .  , 0. The easiest way to find , 2 . k k D 2 , 0. Written in terms of

| 2 ) ) < | ) the cosmic time. An overdot 0 m + − ) = 0 2 t ω ω ˙ k,τ φ < ω k, τ  − k, τ ( k, τ ( Ω( ln A 0 ( a k − − −  . dτ A ! A 2 ˙ 2 | ˙ τ A φ f , with φ f . | ∂ k R , ω ∂τ 2 γ γ ∂ 1 e dt

∂τ c , 2 c k ) − 3 2 2 a k a k T 2 D 6  = 1 2

g m + k, τ +  2 dkk dkk dkk 1 2 1 2 2 2 1 2 = Ω = adτ a k a k – 31 – Z Z Z 2Ω( ˙ + φ f Ω 2 D

4 4 4 ∂τ = ∂ γ p a a a m k ω c

2 2 2 + 1 1 1 2 2  . In our conventions a k ' π π π a t ω − 4 4 4 ˙ ) k ω − A 2 D = = = 2 2 H k, τ a k i i i m ( 2 2 + ~ B − ~ ~ − = E B · − h h A 2 ˙ 2 ¨ φ f ~ A 2 1 1 2 E a ω h γ c ) is via the WKB approximation: a k C.1 − 2 2 a k experiences tachyonic enhancement when ] . This approximation holds as long as we satisfy the adiabatic condition − 64 − 2 2 iω a A ω ≡ is the conformal time, defined as the U(1) gauge coupling. We want to find tachyonic solutions, Ω = τ g At finite temperature, in the long wavelength limit, the dispersion relation ( that solves eq. ( − In these expressions with equations above. while in the short wavelength limit it is modified to [ Then one can compute where, Ω The mode A denotes a derivative with respectEOM to is from which we can read off explicitly the dispersion relation Here JHEP02(2018)084 (C.20) (C.15) (C.16) (C.17) (C.18) (C.19) (C.12) (C.13) (C.14) . 0 this equation is ! 2 2 D : , . kτ > . | H . m | ξ 1 6 , − k . Ω | . ξ π ξ | . < 2 D 2 = 2 4 . In this limit the solution 2 D 4 D

= 0 | − m = 0 x H H H 2 m . m ξ = 0 ) 2 2 | π ) 4 2 k Ω Ω x < 3 x x π 4 x ξ 2 D π x  x = 2 D close to 2 2 D m + | − ' m x 0 and | − H D 2 3 | k m ξ | − iω ξ 2 | ξ π | + 32 Ω ξ | m 2 π < | 2 D + (2 4 k (2 + ˙ 2 dx x m φ f ˙ φ ˙ x Hf + φ f 2 γ , and the dispersion relation to π kx 2 16 c 2 γ 2 D max ξ 2 2 D γ 2 D c x a k x min < x < 2 c − H m x a m k H m 2 ) reduces to using again the WKB approximation. We have 2 2 3 Z – 32 – 4 + 2 D π = k 4 D 1) = 2 π

+ 1 k − 2 H 2 ξ m H − m ' + C.9 2 2 a k = A 4 2 2 a k π ξ

π ln( k + Ω = 2 Ω = 2 1 + 1 r 2 2 Ω + | − 2 k 2 a ω Ω

dτ 2 ξ Ω a | Ω D 2 2 Ω ∂τ ∂ Z x k

m 4 ) reduces to Ω = C.9 ) first. , we see there is no real and positive solution (no tachyonic modes) D C.9 m  a k k k   Now that we have Ω we can get The adiabatic condition is This is satisfied only in a very narrow range of In the casessimplifies we to are interested in, we have Ω is positive (we have tachyonic modes) as long as with solution Using the dimensionless variables The right hand side term in eq. ( so the dispersion relation becomes Given for Ω in this limit. Therefore weΩ turn to the opposite limit: Ω The r.h.s. of eq. ( • • Let’s consider eq. ( C.1 Long wavelength JHEP02(2018)084 (C.25) (C.26) (C.27) (C.28) (C.21) (C.22) (C.23) (C.24) , the width ). Since the 2 ). Then the 2 D | H ξ m | C.8 4 π ξ = 2 ), ( , . x ' 6 1 / C.7 1 aH dx ), ( , the equation of motion = 0  , , . 7 ), we estimate the integrals 0 V 6 6 | , as a small perturbation in 6 1 ξ ξ C.6 a ξ ξ | H 4 4  4 D 4 D 4 . 4 D D 4 7 0 H H m m H 6 | m C.19 = 1 now defined without the minus ξ m ξ 2 2 H 4 4 2 | 4 π π 4 γ 4 D π fV , we see that at finite temperature a x dt 4 c e e 0 H e m 7 9 | | | 2 2 . The results are H 2 D 4 ξ 1 | fH ξ ξ Z π . γ | . We estimate fV | | ξ H c m e | i 4 4 x 5 = D 3 | ~ 2 B 2 D | ∝ H H H π = ξ to ˙ m · ), with | φ 2 H 2 4 m 2 D 4 D | 4 . We derived this result assuming inflation, x k  k ~ 3 3 E π H H 4 m m , τ / h H π ln 2 3 4 2 t γ 2 C.17
f 1 r c 2 2  1 1 1 π π π , we can treat – 33 – 4 D π 2 2 8 a k H ' ' m =  ) 0 3 = ln ) we can approximate Ω with Ω(  / V i ' i ' i ' 2 6 2 2 | k, x ~  B ( D ξ ~ ~ ,H E B C.4 | · − D h h m 4 2 D 4 H ~ 1 2 2 1 E A / m h 1 H m   2 4 0 t π t γ ), and we substitute we have c Hf  is positive in R.D. The rest of the derivation then follows. 2 T τ C.19 = ) = | t ˙ ( φ | a grows large enough to become comparable to i ~ B , because · ~ E kτ ). We then find ourselves in a situation similar to the zero temperature case. h γ ≡ f c Note the E field is suppressed compared to the B field at finite temperature. as follows. We first change variableof from the interval ( WKB solution is With this, we can compute theWKB E approximation and is B valid fields in using a eqs. very ( narrow range ( In the denominator of eq. ( x C.10 If sign, C.2 Short wavelength In the shorteq. wavelength ( limit, but it holds also in the radiation dominated (R.D.) era. Indeed during R.D. we have and one can check that we get again eq. ( Compared to the zero temperaturethe case, velocity where is enhanced by a factor of from which we find of the inflaton becomes In this regime at finite JHEP02(2018)084 . # 1 = ∼ . → ) c − ∗ . k ) ∗ k ( k k (D.5) (D.2) (D.3) (D.4) (D.1) coh R ( V f + h R 2 ) f

1 ~x k − ~ ) is a func- + k ~x ∂τ ( ( · ∂A R k

~ i f 3 e ) k k λ ~ 3 π a d (2 k λ ~ ∂τ Z . Since ∂A , roughly, we expect zero 6 4 , 1 ) 1 k ~ a ) − ∗ λ, k k . ~ kR . We study the dispersion of ( = ( 1 " ρ ~x 2 / ) ~x 2 | · 2 /  k k ~ ~ 3 )] i k ) | 3 − ξ 3 | from π 1 − | d | kR π ~x ξ · ∗ 2 (2 | R k ~ k e i e cos( Z 2 ∼

1 as 2 )Θ(2 ). The occupation number is given by the x R k − ~

kR 3 2 ∗ V

H IR k − ∂τ ~ d | − ∂A k R ξ A )

| V – 34 – /∂τ 3 Z ∂ ) | − ∂τ k = kR k − ~

3 k π ~ λ | ∗ d ∗ X γ (2 k ∂A q

1 Θ( 2 ω , for simplicity we treat it as a step function, a 3 Z 1 − ' − − 2 ) 2 = 9 [sin( x R i R

coh 3 0 V 2 ) = V | d k − ~ 1 aH )   ( 1 ~x ~x x ∂τ · x R ( ∂A k k ~ + 3 i

V 2 R e d ~ E x R | R t, ~x 3 V V 0 ( d h Z ~ E R x 1 V λ 3 V R x R X d 3  V 4 , instead. Thus, at a comoving scale larger than d 1 a Z ∗ R ≡ k V = = ) 8 Z ). Also, we approximate , i is volume inside a sphere with the radius k , k ) ( 3 R ≡ R R − (the physical momentum is − ) f V ) t,~x 1 ( − We can make these statements more explicit by using an averaged electric field within 2 /τ R t, ~x | We thank Masahiro Takimoto for suggesting this quantity. aH ( ~ R | 8 E ξ R h ξ | ~ Θ( where tion damping quickly for the averaged electric field, a radius E where result in a zerohigh net momentum, but electric we have field. seen thatmomentum, Randomized those produced photons exponentially in in ourelectric a model field, have microscopic low but scale we must will have have a non-zero field when we zoom into scales smaller than This number is significantly larger thana 1, classical implying field. that the photons are coherent and form D.2 Size andEven direction if numerous photons are produced, one might wonder if their random directions −| number of photons in( the coherent volume (within the de Broglie wavelentgh), We discuss here somenumber properties of of photons. the classical electricD.1 field formed by Coherence the exponential The comoving momentum of photons with the largest tachyonic enhancement is D Electric field JHEP02(2018)084 . ) 1 (E.5) (E.1) (E.2) (E.3) (E.4) (D.6) − , ) ) ). The ∗ 1 = 0) and k − IR φ (2 2.16 , , is negligible. ) ∗ R > k R < k ( . 2 2 2 switch / i 3 2 V k > # E 1 − 0h ( 2 N 1 ( V ]. They studied a potential ( ' + 8 can match the observed one, 2 0 ) 43 2 , , as given in eq. ( k V P at that time. We are in regime .  ( γ . 2 1 P 42 ρ ˙ R 2 φ 1 P Pl f M ) N V f M M 2 k 2 0 γ 3 ( i ∼ and c ρ . The appearance of a direction of the = V 2 ) . = v  | 2 P ~ E , H k 2 2 ~ h − 2 2 ˙ 1 R M φ 4 ξ 2 = dV V − | " H π + 2 φ 0 3 4 P UV 1 V – 35 – M k = switch V ,H 3 V 0 2 − P Z switch H )Θ( ), there is no net electric field, while at small scales V V 3 2 0 1 10 IR − IR − = V k ' 1 2 P 2 1 = : 2 0 1 V ˙ M φ | − 3 V N C k is the potential when we switch to regime 2. Hence we have ~ R > k | 6 = P V 0 ˙ M φ Θ( V in this regime, before we switch to the one dominated by photon 0 2 2 3 P H 1 ) V k − 3 ) we can estimate π N M d 10 γ to zero, going from small to large scales, is expected to be smooth. f (2 dV c . First ,we need to estimate and other patches have i | 9 E.2 2 2 − v ξ 1 Z | with a homogenous initial condition in the symmetric phase ( ~ , but cannot probe directions. In order to observe the direction, one needs E 1 P' 2 i 4 h 10 switch V = 2 1 2 a V ) ), there is a strong electric field = ~ 2 φ × Z E 1 v h − 5 . = ) i ' − ∗ ) 1 k 2 switch = 2 N φ (2 V t, ~x ( ( 4 λ 2 R The direction of the electric field can appear as a consequence of quantum fluctuations ~ E = h COBE R < With this and eq. ( where the potential as a function of 1, with The number of e-folds backreaction, is contribution from gauge fields, is We want to check ifP at 30 e-folds from the end of inflation patches have electric field is analogous to this inhomogeneity. E Estimate of curvatureThe perturbations power spectrum from the usual vacuum fluctuations of the inflaton, neglecting the ties, such as a lattice simulation, whichtachyonic instability, is simulations beyond were performed the inV scope refs. [ of this paper.initial quantum fluctuations. For a Then a similar tachyonic situation instability drives of the inhomogeneity: some Averaging over large scales( ( transition from which grow exponentially. Our analytic approach is limited to estimating quadratic quanti- The IR cutoff iscases, needed with because microscopic there and are macroscopic no scales zero momentum photons. We examine two because the production of non-tachyonic photons, with momentum JHEP02(2018)084 ] 02 (E.6) ]. (2015) (2016) Relaxion 12 (2015) ]. JHEP 01 (10), so the SPIRE , Cosmological IN . Thus ]. 115 . P ][ JHEP ∼ O SPIRE , 1 JHEP IN arXiv:1511.02858 ]. , N SPIRE COBE ][ [ (2016) 1650215 ]. IN ][ (2015) 077 < f < M  P SPIRE 11 A 31 SPIRE IN W W ]. f IN ][ m Phys. Rev. Lett. ][ , < m 48 JHEP − arXiv:1511.01827 (2016) 103527 SPIRE , Relaxion monodromy and the weak [ wig 10 IN Is the relaxion an axion? ][ ' arXiv:1508.03321 D 93 [ W arXiv:1511.00132 f 10. 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