Tachyonic Preheating in Palatini R^2 Inflation
PPrepared for submission to JCAP
Tachyonic Preheating inPalatini R Inflation
Alexandros Karam, Eemeli Tomberg and Hardi Veerm¨ae
Laboratory of High Energy and Computational Physics, National Institute of ChemicalPhysics and Biophysics, R¨avala pst. 10, Tallinn, 10143, EstoniaE-mail: alexandros.karam@kbfi.ee, eemeli.tomberg@kbfi.ee, [email protected]
Abstract.
We study preheating in the Palatini formalism with a quadratic inflaton potentialand an added αR term. In such models, the oscillating inflaton field repeatedly returns tothe plateau of the Einstein frame potential, on which the tachyonic instability fragmentsthe inflaton condensate within less than an e-fold. We find that tachyonic preheating takesplace when α (cid:38) and that the energy density of the fragmented field grows with the rateΓ /H ≈ . × α . . The model extends the family of plateau models with similar preheatingbehaviour. Although it contains non-canonical quartic kinetic terms in the Einstein frame,we show that, in the first approximation, these can be neglected during both preheating andinflation. a r X i v : . [ a s t r o - ph . C O ] F e b ontents R models 23 Background evolution 4 The theory of cosmic inflation [1–6] is an attractive paradigm for the very early Universe sinceit provides a solution to some outstanding puzzles of the standard hot Big Bang cosmologysuch as the flatness and horizon problems. Most importantly, during this epoch of exponentialexpansion, quantum fluctuations seed primordial inhomogeneities which will grow into thelarge scale structure we observe today.The latest data from the Planck satellite [7] have severely constrained the allowed valuesfor the scalar spectral index n s and the tensor-to-scalar ratio r . As a consequence, simpleinflationary models with monomial potentials have been ruled out. However, adding an R term and studying these models in the Palatini formalism can significantly lower the value of r [8, 9] (see also [10–19]) and still lead to single-field inflation, unlike in the metric formalismin which adding an R term would introduce an additional scalar degree of freedom.In the Palatini approach [20, 21], the spacetime connection is not the usual Levi-Civitaone as in the metric formalism, but rather it is assumed to be an independent variable.Therefore, the action has to be varied with respect to both the metric and the connection.Although the two formulations are equivalent for general relativity, differences arise whenthe inflaton is non-minimally coupled to gravity [14, 22–57] or in f ( R ) theories of gravity [8–19, 58–65]. For example, while in the metric formalism non-minimally coupled Higgs inflation(omitting quantum corrections) predicts a value of the non-minimal coupling ξ ∼ O (10 ),the Palatini version of the model predicts ξ ∼ O (10 ). Similarly, as mentioned above, theeffect of an αR term in the Palatini formalism is to flatten the inflationary potential andreduce the scale of inflation (compared to the same model studied in the metric formalismwithout the αR term), thus modifying the prediction for the tensor-to-scalar ratio r .A typical feature of inflationary models in the Palatini approach is that the Einsteinframe potential becomes a plateau for large field values. As a result, it is possible to push– 1 –he Hubble scale below the frequency of inflaton oscillations so that, after inflation ends, theoscillating field can repeatedly return to the plateau. Because the field perturbations possessa strong tachyonic instability at the edge of the plateau, efficient production of inflatonparticles can take place and dominate preheating [41, 49]. Due to this flattening effect,Palatini R models provide a natural framework for tachyonic preheating.In this paper, we consider a minimal model in which the inflaton is governed by aquadratic potential and extend it with an R term in the Palatini formalism. While thequadratic model usually predicts r = 0 .
13, which is excluded by Planck [7], the effect ofthe αR term is to flatten the potential and lower the value of r as α increases, renderingthe model viable again [8, 10, 12, 19]. At the same time, a higher-order quartic kineticterm is generated in the Einstein frame. This term can be safely ignored during slow-rollinflation [8, 15], and we show that it is sub-dominant also during preheating. This is oneof the main results of our work. We study the preheating process numerically in the linearapproximation, complemented by analytical results, and compute the preheating timescaleas a function of α , as well as the spectrum of perturbations as a function of the comovingwavenumber k .The paper is organized as follows. In section 2, we review the general properties ofPalatini R models and introduce the specific model considered in this work. In section3, we review the inflaton field’s behaviour during inflation and discuss its evolution in thesubsequent preheating stage. In section 4, we study particle production during preheatingand present our main results. We also comment on the UV cut-off scale of the model andinteractions with Standard Model (SM) particles. Finally, we conclude in section 5. Sometechnical details are gathered in the appendices. We use natural units (cid:126) = c = k B = M = 1and the metric signature ( − , + , + , +). R models Let us start by considering a scalar field ϕ , the metric g µν , and a symmetric connection Γ ρµν ,treated as independent variables in the Jordan frame action S = (cid:90) d x √− g (cid:20) R + αR + 12 A ( ϕ ) R −
12 ( ∂ϕ ) − V ( ϕ ) (cid:21) . (2.1)Here g is the determinant of g µν and R = g µν R ρµρν (Γ , ∂ Γ) is the Ricci scalar (the Riemanntensor is built from the connection alone in the Palatini formulation). By introducing anauxiliary scalar χ , we can rewrite the R term as χ R − χ /
2. By performing a Weylrescaling of the metric g µν → Ω − g µν = [ αχ + A ( ϕ )] − g µν (2.2)and eliminating the auxiliary field χ , the action becomes [8, 9] S = (cid:90) d x √− g (cid:20) R −
12 1 − αU A ( ∂ϕ ) + α − αU (1 + A ) ( ∂ϕ ) − U (cid:21) , (2.3)where we defined U ≡ V (1 + A ) + 8 αV . (2.4) We will refer to the frame in Eq. (2.1) as the Jordan frame even when A = 0. – 2 – canonical (quadratic) kinetic term can then be obtained by redefining the field viad ϕ d φ = (cid:114) A − αU , (2.5)yielding the action S = (cid:90) d x √− g (cid:20) R −
12 ( ∂φ ) + 12 α − αU ( ∂φ ) − U (cid:21) . (2.6)The R term has been translated into a higher-order kinetic term and a modification of theJordan frame potential. Because a negative α would lead to negative kinetic energy, implyingan unstable system, we take α >
0. In this case, we see that, when the Jordan frame potentialis positive, i.e., V ≥
0, then the Einstein frame potential is bounded as 0 < U < (8 α ) − .This property avoids singularities in the quartic kinetic term and leads to the flattening ofany monotonously growing Jordan frame potential at large field values.We will keep the theoretical discussion as general as possible. However, for the sake ofconcreteness, in the numerical examples we will consider the case of a minimally coupled,free scalar field [5, 66, 67] V ( ϕ ) = 12 m ϕ , A ( ϕ ) = 0 . (2.7)With a minimal coupling to gravity, this is the only stable potential which gives the correctCMB predictions and is renormalizable in the Jordan frame. This is because adding an R term in the Palatini formulation does not change n s [8] and, therefore, a quartic potentialwould be in conflict with observations as it predicts a scalar spectral index n s ≈ .
94, whichis excluded [7]. The Einstein frame field, defined through (2.5), is φ = φ sinh − ( ϕ/φ ) , φ ≡ m √ α , (2.8)where the scale φ roughly determines the onset of the plateau. The Einstein frame poten-tial (2.4) thus reads U = 18 α tanh ( φ/φ ) . (2.9)This potential is depicted in Fig. 1 for m = 7 × − and α = 10 . As discussed above,the effect of the R term is to produce an Einstein frame potential (solid line) by flatteningthe Jordan frame potential (grey dashed line) to (8 α ) − , while preserving the shape of thepotential at low field values, i.e., when U (cid:28) (8 α ) − . Due to this flattening, the effective mass U (cid:48)(cid:48) of perturbations can acquire large negative values triggering a tachyonic instability. Inour current example, U (cid:48)(cid:48) reaches its minimal value − m / α , the frequencyof oscillations, currently O ( m ), can significantly exceed the Hubble rate H ≈ (24 α ) − / . Asshown in Fig. 1, in this case, slow-roll inflation ends while the field is essentially still on theplateau. Thus, as the field starts to oscillate, it can repeatedly re-enter the tachyonic regionresulting in a very effective preheating.The potential (2.9) has been considered in several earlier works both in the context ofPalatini gravity [9, 12, 13, 19] and in the context of the metric formulation [68–70], whereit is usually referred to as the T-model, but the single-field tachyonic preheating stage weencounter has not been studied before in detail for this potential. As discussed above, com-pared to metric theories, the Einstein frame action in the Palatini R case has a manifestly– 3 – ��� ���� ���� ���� �� αϕ � ϕ �� ϕ � � � ��� =- � � � α = �� �� � � = � × �� - � ��� > � ��� < � Figure 1 . The potential as a function of the value of the inflaton field. The Einstein frame potential(2.9) with m = 7 × − and α = 10 is shown by the solid line, while the dashed line depicts theJordan frame potential m φ / φ is the field value corresponding to 50 e -folds of inflation. Theregion in which the tachyonic instability is active (grey shaded area) overlaps with the region in whichthe inflaton will oscillate (solid red line). φ max shows the maximal amplitude of oscillations. Theleftmost arrow shows the point at which the tachyonic instability is maximal, U (cid:48)(cid:48) = − m / non-canonical kinetic term implying modifications to the speed of sound. However, as we willfind, the quartic kinetic term does not significantly affect the phenomenological predictions. Before considering particle production, we first study the evolution of the homogeneous andisotropic background field during inflation and preheating. Initially, the field is far on theplateau, and the Universe undergoes a period of slow-roll inflation. After its end, the fieldbegins to oscillate around the minimum of the potential. When α is sufficiently large, inflationcan end while the field is still on the plateau and therefore the field may repeatedly returnto the plateau in the oscillating regime, as we will see shortly. This means that the field willspend a significant fraction of time in the tachyonic instability region boosting the growthof fluctuations, as seen previously in [41, 49]. We will outline the classical evolution of thebackground without accounting for the feedback from these fluctuations.In the spatially flat Friedmann–Robertson–Walker (FRW) Universe, the expansion iscontrolled by the Friedmann equation 3 H = ρ . The energy density and pressure of the field,derived from the action (2.6), are ρ = 12 (cid:18) B (cid:19) ˙ φ + U, P = 12 (cid:18) B (cid:19) ˙ φ − U , (3.1)and the field equation reads ¨ φ + 3 Hc s ˙ φ + 1 + 3 B B U (cid:48) = 0 , (3.2)where we defined B ≡ α ˙ φ − αU , c s = 1 + B B . (3.3)– 4 –he parameter B quantifies the relative contribution from the quartic kinetic term: when B (cid:28) B (cid:29) c s corre-sponds to the speed of sound of perturbations [71]. Slow-roll inflation is characterized by the slow-roll parameters (cid:15) U ≡ (cid:18) U (cid:48) U (cid:19) , η U ≡ U (cid:48)(cid:48) U , (3.4)which are small on the flat inflationary plateau, and the slow-roll approximation is valid.The slow-roll approximation gives the number of e-folds of expansion from field value φ tothe end of inflation, N ≈ (cid:90) φ d φ √ (cid:15) U ≈ φ φφ ⇒ φ ≈ φ Nφ , (3.5)where we took the limit φ (cid:28)
1. This limit is needed for successful preheating, as we willsee in section 4. Regarding the inflationary observables, as was shown for a general casein [8], the amplitude of the scalar power spectrum A s and the scalar spectral index n s remainunaffected by the αR term up to first order in the slow-roll parameters: n s = 1 − (cid:15) U + 2 η U ≈ − N ∗ , A s = U π (cid:15) U ≈ m N ∗ π . (3.6)Here N ∗ ≈
50 at the CMB pivot scale, so that the spectral index n s ≈ .
96 is compatiblewith the Planck results [7]. The measured power spectrum strength A s = 2 . × − yields m = 7 × − (3.7)and we are left with a single free parameter α (or equivalently φ ). The αR term affects theexpression for the tensor-to-scalar ratio r , which becomes r = 16 (cid:15) U ≈ N ∗ + 4 N ∗ /φ ≈ φ N ∗ , (3.8)where the last approximation holds when 4 N ∗ (cid:29) φ or, equivalently, α (cid:29) − m − ≈ × as the mass is fixed. For smaller values of α , the effect becomes negligible and we recover r = 8 /N ∗ ≈ .
16, i.e. the prediction of standard quadratic inflation which is excluded. Thetensor-to-scalar ratio r is compatible with the Planck constraint r < .
056 when α (cid:38) × ,thus inflation must take place in the regime where the αR term is relevant.Above, we neglected the quartic kinetic term in (2.6) and set B = 0—it was suggestedin [8] that this is a good approximation during slow-roll. For the potential (2.7) we find that,in the slow-roll approximation and in the φ (cid:28) N (cid:29) B ≈ N (cid:28) . (3.9)The effect of the quartic kinetic term on the inflationary observables is thus indeed small.However, it may still affect preheating dynamics after inflation. We will return to this pointin section 4. – 5 – .2 Post-inflationary oscillation After inflation, the inflaton starts to oscillate around the minimum of the potential. We willsolve this phase numerically in section 4, but let us first gain some analytical understandingof the oscillations.As the numerics show, in the limit of small φ (large α ), the frequency of oscillations ω is much bigger than the Hubble rate, H (cid:28) ω . The expansion of space can be neglectedduring a single oscillation, and we can work in the Minkowski limit H →
0. The effect of theexpansion on the energy density can be studied by considering temporal averages of the fieldequations. These can be recast in terms of the continuity equation˙ ρ + 3 H ( ρ + ¯ P ) = 0 , (3.10)where barred quantities denote time averages.Before continuing, let us estimate the lower bound on α for the condition H (cid:28) ω to besatisfied after inflation ends. In the model determined by Eq. (2.7), ω = O ( m ) is a reasonableguess. Then, using the fact that on the plateau H ≈ (24 α ) − / , we obtain α (cid:29) . In thefollowing, we will show that α (cid:38) to satisfy the requirements for tachyonic preheating.The evolution of an homogeneous field on a flat background can be described as a me-chanical system with the action (cid:82) d tP ( φ, ˙ φ ) and a corresponding conserved energy ρ ( φ, ˙ φ ) =˙ φ∂P/∂ ˙ φ − P , which, for the model under consideration, are given by Eq. (3.1). Due to energyconservation, we can express the derivative of the field as ˙ φ ( φ, ρ ), where ρ is now a constantof integration. Integrating the field equation gives the half-period of oscillations, and thetime average of the pressure as T / ( ρ ) ≡ (cid:90) d t = (cid:90) φ φ d φ ˙ φ ( φ, ρ ) , ¯ P ( ρ ) ≡ T / ( ρ ) (cid:90) φ φ d tP ( φ, ˙ φ ( φ, ρ )) , (3.11)where the turning points φ i ( ρ ) are defined as the field values where the velocity vanishes.The second expression determines the effective equation of state. It is also useful to definethe abbreviated action between the turning points, W ( ρ ) ≡ (cid:90) φ φ d φ ∂P∂ ˙ φ , (3.12)since it can be used to compute all other relevant quantities: T / = ∂ ρ W, ¯ P = W/T / − ρ. (3.13)Moreover, using these relations in the time averaged continuity equation (3.10), we find that W is conserved in a comoving volume,˙ W + 3 HW = 0 ⇒ W ∝ a − . (3.14)It follows that the fractional change in the energy density within a half-period is simply∆ ρρ ≈ T / ˙ ρρ = − WH . (3.15) This follows by noting that the integrand vanishes at the boundaries and that ∂ ρ ( ∂ ˙ φ P ) = ˙ φ − . – 6 – � �� �� �� �� �� - ���� � � / � π ϕ / ϕ � � ��� �� �� α = �� �� � � = � × �� - � ��� > � ��� < � ��� < � Figure 2 . Time evolution of the background φ (solid) just after the end of inflation with m = 7 × − and α = 10 . The thick dashed line shows the envelope of oscillations form the analytic estimate(3.24) while the thin dashed line shows the envelope assuming ρ ∝ a − . The tachyonic instability isactive (inactive) in the shaded (unshaded) areas. Applying this formalism to Palatini R models, we first note from Eq. (3.1) that˙ φ = 43 ( ρ − U ) (cid:20)(cid:114) − ρ − ρρ − U − (cid:21) , (3.16)where we defined ρ ≡ (8 α ) − as the initial energy density during inflation, equal to themaximum value of the potential. In the limiting cases U → ρ or ρ (cid:28) ρ we find ˙ φ = 2( ρ − U ),which shows that the quartic kinetic B -term in (3.1) can be neglected when the motion ispotential dominated or when the field has fallen off the plateau. In general, when requiring U ≤ ρ ≤ ρ , Eq. (3.16) gives˙ φ ≤ − αU α ⇒ B ≤ , ≤ c s ≤ , (3.17)so the quartic term will also not dominate the oscillations. Numerical computations showthat, in the model considered here, B never saturates the upper bound (3.17).For our specific model with the potential (2.9), and neglecting the subdominant B , wefind that W ≈ π (1 − (cid:112) − ρ/ρ )4 mα , (3.18)and T / ≈ πm (1 − ρ/ρ ) − / . (3.19)We are interested in the oscillations that follow inflation, that is, we work in the regime ρ ≈ ρ . In order for the formalism to apply, the energy density cannot be strongly dampedby Hubble friction. Applying (3.15), we see that the fractional change in the energy densitywithin a half-period is ∆ ρρ ≈ − W √ ρ ≈ −√ πφ . (3.20)– 7 –hus, self-consistency requires that φ (cid:28) .
13. In this case, also during the subsequentoscillations only a small fraction of the energy density is lost via Hubble damping. Thetachyonic instability, U (cid:48)(cid:48) < U > / (24 α ) or ρ > ρ /
3. As during each n oscillations the amount of energy lost is roughly 2∆ ρ , we find that the number of oscillationsin the tachyonic regime is roughly n tach ≈ ρ | ∆ ρ | ≈ . φ , (3.21)implying that, since the mass is known (3.7), α (cid:38) is needed to make at least two oscil-lations before exiting the tachyonic regime. This value is 3 orders of magnitude larger thanwhat was obtained by naive considerations in the beginning of this section. Numerical solu-tions (see Fig. 2) show that (3.21) slightly underestimates the actual number of oscillationsin the tachyonic regime.The half-period of the first oscillation can be estimated by approximating the subsequentenergy density as ρ − ∆ ρ . We find that T / H − ≈ πρ m √ ρ ≈ − / (cid:112) πφ . (3.22)Thus a small φ will also guarantee that a large number of oscillations will take place duringone e-fold of expansion, as was assumed above. To estimate the initial amplitude of oscil-lations φ max , we will again use that, at the first turning point, i.e. when ˙ φ = 0, the energydensity is roughly ρ − ∆ ρ = U ( φ max ), which yields φ max ≈ − φ (cid:32)(cid:114) πφ (cid:33) . (3.23)For φ (cid:28)
1, we have φ max (cid:29) φ : the field returns to the plateau after the first oscillationand also repeatedly thereafter. These estimates are in good agreement with our numericalcalculations.In the long run, neglecting fragmentation, equations (3.14) and (3.18) imply that theenergy density evolves as ρ ≈ ρ (cid:18) aa (cid:19) − (cid:32) − (cid:18) aa (cid:19) − (cid:33) , (3.24)where a is the scale factor at the end of inflation. The last bracketed term rapidly ap-proaches a constant value and the Universe transitions to an effectively matter-dominatedphase within less than a single e -fold. There ρ (cid:28) ρ , and the half-period (3.19) matches thatof a harmonic oscillator. However, many oscillations can take place before this. The fieldrepeatedly returns to the plateau where it experiences a tachyonic instability. This fragmentsthe field condensate before the matter-dominated phase begins, as we will see in the nextsection.The envelope of the field implied by (3.24) is shown in Fig. 2 when α = 10 . Observethat even for such a large α , the field transitions to the behaviour ρ ∝ a − , characteristicof a massive non-interacting scalar, within the first few oscillations after inflation. For lowervalues of α , this transition is even faster, reducing the effectiveness of tachyonic preheating.Therefore, considering the background evolution, it is expected that tachyonic preheatingturns on around α ≈ . We will demonstrate this numerically in section 4.1 (see Fig. 5).– 8 – Preheating
An oscillating inflaton field generates time-dependent masses to all coupled fields and therebyinduces non-perturbative particle production. This process is called preheating [72–75], andwe will study it numerically by solving the background evolution and linear perturbationequations for the inflaton.We expand the inflaton perturbations in terms of Fourier modes δφ k . These follow,approximately, the mode equations¨ φ k + 3 H ˙ φ k + ω k φ k = 0 , ω k ≈ k a + U (cid:48)(cid:48) . (4.1)During preheating, these perturbations grow. To quantify this growth, we compute theenergy density in the perturbations, which can be estimated as δρ ≈ (cid:90) k max d k (2 π ) (cid:16) | δ ˙ φ k | + ω k | δφ k | (cid:17) . (4.2)Expressions (4.1) and (4.2) are only approximations, because they neglect the effect of metricperturbations, which are coupled to the inflaton perturbations already in linear order, andthe contribution from the quartic kinetic term of the inflaton. We have included these effectsin our numerical computations; the more accurate (but lengthy) equations are derived inappendix B. Nevertheless, equations (4.1), (4.2) provide a schematic understanding of thecomputation. In fact, they include the leading terms of the full expressions.Traditionally, the growth of perturbations is studied in terms of the number density ofparticles created with respect to the adiabatic vacuum. In our case, however, the concept ofparticle number is ill-defined, since the effective mass U (cid:48)(cid:48) < k max . In practice, our results are insensitive to questions about the correctvacuum or the regularization scheme, since δρ is dominated by growing modes below the cut-off scale k max . These modes grow exponentially, reaching values that far exceed the initialconditions.After preheating ends, the particle interpretation is again valid. Assuming fast preheat-ing and thermalization, the temperature of the particle bath is T ∼ ρ / ∼ (8 α ) − / . This islarger than the inflaton mass m = 7 × − (3.7) if α (cid:46) . We will show in section 4.2 thatthis bound is automatically satisfied when we demand tree-level unitarity. This means thatthe inflaton particles are relativistic and the universe transitions into radiation dominationwhen the inflaton fragments. Reheating is completed after the inflaton decays into standardmodel particles, as we discuss in section 4.3.As the inflaton’s total energy density must follow the continuity equation, the fractionof the energy density in the coherent scalar will decrease due to fragmentation. This feedbackcan be captured at the second order of perturbation theory, but a complete description offield fragmentation requires a dedicated lattice simulation. In the following, we will limitthe discussion to linear perturbations only. That is, we will establish the exponential rate ofgrowth of the energy density in perturbations and thus the timescale of fragmentation.– 9 – .00 0.01 0.02 0.03 0.04 - N l og δ ρ ρ bg α = , m = × - Figure 3 . Growth of the perturbation energy density δρ for α = 10 , measured in units of backgroundenergy density, as a function of e-folds N after the start of preheating. The strong features occurwhen the inflaton crosses zero and ω k behaves wildly; this includes regions where δρ <
0. The dashedline gives the average exponentially growing behaviour, denoted by δρ Γ in the caption of Fig. 4, withslope here equal to Γ / ( H ln 10), see Eq. (4.7). Preheating completes in three zero-crossings, in lessthan one e-fold. To calculate the energy density in scalar perturbations we will numerically solve the timeevolution over a range of k -modes, for different values of the free parameter α . Details canbe found in appendix B. We find the spectrum of perturbations as a function of k , and thegrowth rate of the perturbation energy density for different values of α . As already mentioned, the inflaton field probes the plateau repeatedly during its os-cillations during preheating. There the inflaton mass U (cid:48)(cid:48) is negative, leading to tachyonic,exponential growth of the perturbations, as previously studied in [41, 49]. All modes forwhich ω k from (4.1) is negative when the background field is at its oscillation amplitude φ max undergo tachyonic growth: they spend most of the time in the instability regime. Forlarge enough k , this is no longer true. The fastest-growing peak modes occur at the largest k -values which are still tachyonic, that is, k peak a ≈ (cid:112) − U (cid:48)(cid:48) ( φ max ) . (4.3)Using the analytical approximation (3.23) for φ max gives the estimate k peak aH ≈ × / √ mπα / ≈ . × α / . (4.4)By performing a numerical fit we find that, for large α , the peak frequency scales as k peak aH ≈ . × α . , (4.5) In section 3, we worked mostly in terms of φ , but from this point on, we find it more convenient to use α and m . The definitions and the value m = 7 × − give φ = 7 × α − / , so that the condition φ (cid:28) α (cid:29) × . – 10 – .5 2.0 2.5 3.0 3.5 - - -
505 log k / aH l og ( d | δ ρ | / d l og k ) α = , m = × - N Figure 4 . Rescaled energy spectrum of field perturbations at different times, with α = 10 . Theplotted rescaled perturbation energy density is δ ˜ ρ ≡ δρ/δρ Γ , that is, the late-time exponential growthshown in Fig. 3 is factored out. Colours correspond to different times, measured in e-folds N after thestart of preheating, followed until the breakdown of perturbativity and somewhat beyond for illustra-tive purposes. The peak of the spectrum lies deep inside the Hubble radius, around k ≈ aH , atthe scale where tachyonic growth on the plateau is shut off. Modes with smaller k grow tachyonically;modes with larger k undergo non-tachyonic parametric resonance with multiple narrow resonancebands. The dominant peak modes conform to the leading exponential growth quickly; other modesgrow slower, leading to a sharper and sharper peak as time goes on. The oscillations around theaverage behaviour seen in Fig. 3 produce the wide, overlapping ‘bands’ of uniform colour. which is in good agreement with the analytic estimate. In Fig. 4 we see that the spectrum ofenergy density falls off sharply on both sides of the peak. The momentum cutoff k max fromEq. (4.1) is chosen so that k max (cid:29) k peak and the full peak with all out-of-vacuum modes isaccounted for in the integrated perturbation energy density.Radiation formed at the peak scale will be redshifted and, at present, will carry thefrequency f = 2 πk peak aH ≈ . T ≈
10 GHz , (4.6)where T = 2 . ρ ≈ T after preheating, and a ∝ /T . All α -dependence cancels out. The violent preheating process may produce stronggravitational waves at this scale. The frequency is beyond the reach of current or near-future gravitational laser-interferometric GW detectors which are sensitive to frequencies upto 10kHz [76–78]. Currently, the most sensitive probe of such primordial GW signals is thenumber of effective relativistic degrees of freedom N eff during Big Bang nucleosynthesis [79].However, a GW background at 3-30GHz frequency range can also leave an imprint on theCMB Rayleigh-Jeans tail and may thus be probed via future 21cm physics experiments [80,81]. Integrating over the spectrum gives the total perturbation energy density, which alsogrows exponentially, see Fig. 3, and is dominated by the peak mode. For large α , the processis efficient: the inflaton condensate fragments within less than a single e-fold of expansionand takes only a few background oscillations. We estimate the timescale of fragmentation as– 11 – - α l og Γ / H Figure 5 . Growth rate of perturbations as a function of α , defined in Eq. (4.7) and demonstrated inFig. 3. Its inverse Γ − gives the timescale of preheating and background oscillations. There is a cut-offat α ∼ ( φ ∼ .
1) below which the inflaton field oscillates only near the potential’s quadraticminimum with no inflaton particle production. Above this, Γ /H ≈ . × α . , indicated by thedashed line. Our numerical calculations break down above α ∼ due to insufficient numericalaccuracy, but these values also run into problems with unitarity, see Eq. (4.9). the rate of the growth of perturbation energy density:Γ ≡ d ln δρdt . (4.7)This is essentially constant around the time when the perturbation energy density crossesthe background energy density. Note that Γ − gives the timescale of preheating, and thusthe timescale for the transition from inflation to radiation domination.In Fig. 5 we plot Γ /H as a function of α . It follows a broken power law. Above α ∼ ,Γ H ≈ . × α . . (4.8)Part of the α -scaling arises from H ∝ α − / at the end of inflation, while the rest is dueto the effect of α on the evolution of the oscillations. When α (cid:46) × , the backgroundfield no longer returns to the region at which U (cid:48)(cid:48) <
0, and the tachyonic production shutsoff. There is no fragmentation of the inflaton condensate, since the inflaton mass is constantnear the quadratic potential minimum, preventing parametric resonance. Reheating mustthen proceed through other channels, such as non-perturbative production of other fields orthe perturbative decay of the inflaton. These details are highly model-dependent. Effectivepreheating through inflaton fragmentation then requires α (cid:38) , in agreement with ourprevious estimates from section 3.2. Note that this gives r (cid:46) × − , which is too small tobe probed in the near future.We repeated the computation with the simplified expressions (4.1) and (4.2), droppingthe quartic kinetic term also from the background equations, and compared the obtainedΓ-values to the accurate results. For α (cid:38) , the difference is less than 10%, showingthat both the metric perturbations and the quartic kinetic term play subleading roles duringpreheating. – 12 –n summary, we have shown that the inflaton field φ quickly fragments into inflatonparticles and the universe transitions into radiation domination. Reheating can then becompleted through the decay of the inflaton into Standard Model particles. We will discussthis in section 4.3, after commenting on the UV cut-off scale of the model. The Lagrangian (2.6) for the scalar field is non-renormalizable due to the quartic kinetic termand the non-polynomial potential. The non-renormalizable contributions may cause problemssuch as violation of tree-level unitarity in high-energy scattering. The quartic kinetic term α ( ∂φ ) implies that, for the perturbations, the problems arise at energy scale α − / , whilethe relevant energy scale related to the potential U is φ . To describe preheating consistently,the energy scale of the produced particles should remain below both of those cut-off scales.The energy scale of the produced inflaton particles is, by Eq. (4.5), k peak /a ≈ . α − / .After thermalization, the typical energy scale of a particle is ρ / ≈ . α − / , assuming ther-malization is fast and the energy density does not dilute considerably during it. We see thatboth of these scales are below the cut-off scale arising from the quartic kinetic term (if onlybarely for ρ / ). To push them below the U -cut-off as well, we demand α − / (cid:46) φ ⇐⇒ α (cid:46) m − ∼ . (4.9)As long as this limit is obeyed, our preheating treatment is consistent. We remark that theupper bound on α implies that r > × − , and is thus in conflict with proposals relying onPalatini R inflation [12, 15] for accommodating the “Trans-Planckian Censorship” conjecturewhich suggests r (cid:46) − [82]. In Palatini R gravity, the conformal transformation (2.2) will couple the inflaton to everynon-conformally coupled field in the Einstein frame. In the SM, the only non-conformal fieldis the Higgs field. It is thus necessary to study its couplings.Since the scale of inflation is much higher than the electroweak scale, we neglect theHiggs mass term and take a Higgs potential of the form V ( h ) = λ h h + λ hϕ h ϕ , (4.10)where λ h is the quartic coupling of the Higgs and λ hϕ is a portal coupling between the Higgsand the inflaton. After transforming into the Einstein frame, the action for the inflaton-Higgssector is S = (cid:90) d x √− g (cid:26) R −
12 (1 − αU ) (cid:104) ( ∂ϕ ) + ( ∂h ) (cid:105) + α − αU ) (cid:104) ( ∂ϕ ) + ( ∂h ) (cid:105) − U (cid:27) , (4.11)with the potential U = m ϕ + λ h h + λ hϕ h ϕ α (cid:16) m ϕ + λ h h + λ hϕ h ϕ (cid:17) . (4.12)This is equivalent to the Jordan frame action (2.3) but with the Higgs field and a potentialadded on an equal standing with the inflaton. The inflaton’s kinetic term can be made– 13 –anonical with the same field redefinition as earlier, using h = 0 in the defining equation(2.5). For the Higgs, the full quadratic kinetic term reads −
12 ( ∂h ) × (cid:16) − αU + 2 α ˙ φ (cid:17) , (4.13)where U ≡ α tanh ( φ/φ ), the background potential without h or inflaton perturbations;similarly, ˙ φ is the background field velocity here and in the following paragraphs. We canmake the Higgs field (almost) canonical by the field redefinition s ≡ h (cid:113) − αU + 2 α ˙ φ . (4.14)The resulting kinetic term is not fully canonical, since φ depends on time, which results inextra terms of the form ˙ φ ˙ ss when taking time derivatives. However, this kinetic term issufficient for order-of-magnitude estimates. In particular, we can derive mode equations for s which take the standard form:¨ s k + 3 H ˙ s k + (cid:18) k a + m , s (cid:19) s k = 0 . (4.15)The complete expression for the effective mass squared m , s is messy, but it contains twotypes of terms: those arising from the potential, m U,s = λ hϕ U (1 − αU ) m (cid:16) − αU + 2 α ˙ φ (cid:17) > , (4.16)and terms arising from the kinetic sector, m , s ∼ ˙ φ . All terms containing ˙ φ are slow-rollsuppressed during inflation. To ensure the stability of the Higgs field there, we demand m U,s ≈ λ hϕ αm > H ≈ α ⇒ λ hϕ > m ∼ − . (4.17)After inflation, the kinetic mass terms are no longer suppressed, but they all have the sameform, m , s ∼ (cid:32) ˙ ff (cid:33) , ˙ f ∼ ˙ φφ f , f ∼ ⇒ m , s ∼ ˙ φ φ ∼ m αα ∼ m . (4.18)Here f refers to a class of hyperbolic and exponential functions, such as 8 αU = tanh ( φ/φ )and its derivatives, which follow the given properties during the oscillations. We also used˙ φ ∼ ρ ∼ /α : the kinetic energy of the ocillations is the same order as the potential energy.The m , s terms can be negative and cause a strong tachyonic instability to the Higgs (see[49] for a detailed study of a similar case). However, this tachyonicity can be shut off bymaking the positive mass term m U,s large enough: m U,s ∼ λ hϕ αm > | m , s | ∼ m ⇒ λ hϕ > m α ∼ α × − . (4.19)This condition ensures that no tachyonic Higgs production takes place. Higgses can stillbe produced through parametric resonance, but this is likely to be a subleading channel– 14 –ompared to the fast tachyonic production of the inflaton. Note that (4.19) can be satisfiedfor a perturbative value λ hϕ < α satisfies the condition (4.9).For the interactions, all new non-renormalizable terms involving the Higgs boson comefrom expanding 8 αU , see (4.11) and (4.12). These consist of renormalizable operators multi-plied by powers of αm ϕ , implying the cut-off scale ( αm ) − / ∼ φ , and powers of αλ h h and αλ hϕ h ϕ , with the implied cut-offs ( αλ ) − / > α − / for perturbative λ . These order-of-magnitude estimates do not change when switching to the Einstein frame fields φ and s ,since both the Jordan and the Einstein frame fields are of the same order during preheat-ing. These cut-offs are larger than those discussed in section 4.2, so the treatment is stillconsistent as long as the α -bound (4.9) is obeyed.The emerging picture is then the following: with the bounds (4.17), (4.19), Higgses arenot produced during inflation, and the violent tachyonic inflaton production still dominatespreheating. After preheating, the inflaton can decay into the SM through the Higgs portalcoupling and complete reheating. We have no upper bound for λ hϕ (beyond demandingperturbativity), so this decay can be efficient. We studied preheating in the Palatini formulation of general relativity, focusing on the modelwith a quadratic inflaton potential and an αR term in the action. In the Einstein frame,this is a plateau model compatible with the CMB observations. In this model, however, theinflaton’s kinetic term has a quartic component, making the analysis more involved than instandard plateau inflation.We showed that for sufficiently large values α (cid:38) the inflaton returns to the plateaurepeatedly during preheating, giving rise to a tachyonic instability similar to that of previousPalatini studies [41, 49]. This leads to a fast fragmentation of the inflaton field and the onsetof radiation domination in less than a single e-fold of spatial expansion. We obtained thespectrum of the perturbations numerically for different values of α . When α (cid:38) , thespectrum peaks at the comoving wavenumber k peak /aH ≈ . × α . , deep within the Hubble radius, and the energy density of the fragmented field grows expo-nentially with the rate Γ /H ≈ . × α . , which is considerably larger than the rate of expansion. Below α ≈ the growth rateof the perturbations begins to drop fast. In the range 10 (cid:46) α (cid:46) preheating is stillpossible, but with a greatly reduced rate. When possible interactions with the SM Higgs areincluded, the tachyonic inflaton production still tends to dominate preheating.Our numerical studies show that neglecting the Einstein frame quartic kinetic termdoes not significantly affect the evolution of the smooth background (both during inflationor the later coherent oscillations) or the dynamics of preheating. In particular, the effecton Γ is less than 10%. This complements the earlier results that the quartic kinetic termis subleading during slow-roll inflation [8, 15]. While the study of preheating dynamics wasmostly numerical, we provided a more complete analytic picture of the evolution of theoscillating background field. In particular, we prove analytically that the quartic kineticterm must always be smaller than the quadratic one.– 15 – general consequence of a large α in Palatini R models is a strong suppression ofthe tensor-to-scalar r . We show that unitarity considerations together with an effectivepreheating suggest the range 10 (cid:46) α (cid:46) , implying a tensor-to-scalar ratio 4 × − (cid:46) r (cid:46) × − , too low to be detectable in near-future experiments.Our study was limited to linear perturbations, omitting the interactions between theperturbations and the background. The inclusion of such effects requires a lattice simulation.Previous lattice studies of similar models [83–86], albeit without the quartic kinetic terms,support our results of a fast, efficient preheating process. Acknowledgments
This work was supported by the Estonian Research Council grants PRG803, PRG1055,MOBJD381 and MOBTT5 and by the EU through the European Regional DevelopmentFund CoE program TK133 “The Dark Side of the Universe.”
A Quartic kinetic term during inflation
In order to consistently neglect the contribution of the quartic kinetic term in (2.6) duringinflation, the function B (3.3) must be small. To estimate the size of B during slow-roll,consider the field equations in terms of e -folds N ≡ ln a . Since ˙ φ = Hφ (cid:48) and B ∝ H ,the Friedmann equation gives a quadratic equation for H that can be used to recast thedynamics in terms of φ , φ (cid:48) only. After some algebra we see that the (exact) evolution of thehomogeneous background is determined by φ (cid:48)(cid:48) + φ (cid:48) (cid:0) c s − (cid:15) (cid:1) + 1 + 3 B B U (cid:48) H = 0 , (A.1)with B = 8 αU − αU φ (cid:48) − φ (cid:48) (cid:32) (cid:115) − αU − αU φ (cid:48) (6 − φ (cid:48) ) (cid:33) − . (A.2)The Hubble parameter H and the first slow-roll parameter (cid:15) read H = U − φ (cid:48) / − Bφ (cid:48) / , (cid:15) ≡ − H (cid:48) H = 1 + B φ (cid:48) . (A.3)We remark that in case the field redefinition (2.5) does not admit an analytic form, then, inpractical computations, the dynamics is more conveniently described by first order equationsusing the phase space variables ϕ and y ≡ φ (cid:48) and the additional equation y = (cid:113) αU A ϕ (cid:48) .Returning to the model with a quadratic potential (2.7), the slow-roll approximationgives N = (cid:90) φφ e d φ √ (cid:15) U = φ φφ , (A.4)where φ e is the field value at the end of inflation. As a rough approximation we use φ e = 0 asthe corresponding error in N is less than an e-fold. Plugging φ ( N ) into (A.2) and evaluatingthe Pad´e approximant of the order [1/1] for large N gives B ≈ N + 3 φ − / . (A.5) In this appendix, prime denotes a derivative with respect to N , except in U (cid:48) , which still denotes d U d φ . – 16 –hus, the quartic kinetic term can be consistently ignored during inflation when N (cid:29) B Solving for cosmological perturbations
Scalar perturbations of a spatially flat FRW metric in the longitudinal gauge are capturedby [87] d s = (1 + 2Φ) d t − a (1 − δ ij d x i d x j , (B.1)where Φ and Ψ are the Bardeen potentials and the perturbations of the scalar field are δφ ≡ φ − ¯ φ , where ¯ φ denotes the background field. In terms of the Sasaki-Mukhanov variable v ≡ (cid:16) Φ +
Hδφ/ ˙ φ (cid:17) z and the conformal time d τ = d t/a , the dynamics of scalar perturbationsis described by the action [71] S = 12 (cid:90) (cid:16) v ,τ − c s ( ∇ v ) + z ,ττ z v (cid:17) d τ d x , (B.2)where z ≡ a ˙ φc s H √ B = aφ (cid:48) √ B . (B.3)The momentum eigenmodes obey the Sasaki-Mukhanov equation v ,ττ + ω k v = 0 , ω k ≡ c s k − z ,ττ z , (B.4)where the expression for z ,ττ /z can be expressed as z ,ττ /z = a H (cid:18) − (cid:15) + 32 (cid:15) − s − (cid:15) (cid:15) + (cid:15) s − (cid:15) s + 14 (cid:15) + s + 12 (cid:15) (cid:15) − ˙ sH (cid:19) , (B.5)where (cid:15) ≡ − ˙ HH = − d ln H d N , (cid:15) n +1 ≡ d ln (cid:15) n d N = ˙ (cid:15) n H(cid:15) n , s ≡ ˙ c s Hc s . (B.6)Sufficiently far past in the inflationary epoch any mode is within the horizon, k (cid:29) aH . Inthis limit, ω k ≈ c s k and, assuming a slowly varying speed of sound c s , the Mukhanov-Sasaki equation (B.4) is solved by v ( τ ) = e − ic s kτ / √ c s k , defining the so-called Bunch-Davies vacuum. These equations provide both the primordial power spectrum of curvatureperturbation generated during the inflationary epoch, P R ( k ) ≡ k π (cid:12)(cid:12)(cid:12) vz (cid:12)(cid:12)(cid:12) , (B.7)as well as the preheating dynamics when the perturbations are in the linear regime.We compute the energy density of field perturbations in the FRW background, definedas δρ = T | pert = − √− g δSδg (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) pert , (B.8)expanded to second order in field perturbations and with the homogeneous background con-tribution removed. We neglect metric perturbations when calculating δρ . Using the expec-tation value of the corresponding quantum operator in the Bunch-Davies vacuum (see [88]),– 17 –he first-order terms vanish and the second-order terms can be written in terms of the Fouriermode functions of the field, φ k . We have that (cid:104) δρ (cid:105) = (cid:90) k max k min d ln k k π δρ k , (B.9)where δρ k ≡
12 (1 + 9 B ) | ˙ φ k | + 12 B ˜ B Re φ k ˙ φ ∗ k + 12 (cid:20) k a (1 − B ) + U (cid:48)(cid:48) (1 + 3 B ) + 24 B ˜ B (cid:21) | φ k | (B.10)and we defined ˜ B ≡ α ˙ φU (cid:48) / (1 − αU ). The integral is regulated by cut-offs which removethe adiabatic high- k modes ( k > k max ) and the frozen super-Hubble modes ( k < k min ) whichare not excited during preheating. To study how the inflaton condensate fragments intothe perturbations, the resulting quantity can then be compared to the background energydensity.Expression (B.9) does not include metric perturbations and is therefore not a fullyrigorous description of the energy density of the perturbation. However, it should be agood measure of the fragmentation of the inflaton condensate when the field perturbation iscomputed in the spatially flat gauge, where metric perturbations are essentially minimized.Our numerical results show that (B.9) is almost identical to the ‘naive’ perturbation energydensity introduced in Eq. (4.2). This equivalence holds up to fluctuating order-one correctionswhich are not important when determining the exponential growth rate Γ in (4.7).To compute (B.9), we numerically solve the mode equation for φ k in the spatially flatgauge in which the curvature perturbation is zero, so that the Sasaki-Mukhanov variable is v = a δφ √ B . We derive the mode equation¨ φ k + f ˙ φ k + ω k φ k = 0 (B.11)from the Sasaki-Mukhanov equation (B.4). The coefficients read f = 3 H B (1 + 3 B ) − B )(1 − B ) ˜ B (1 + 3 B ) ,ω k = k a c s + 1 + 3 B B U (cid:48)(cid:48) − ˙ φ H (1 + B ) + 2 ˙ φU (cid:48) H (1 + B )(1 + 5 B )(1 + 3 B ) + 3 ˙ φ (1 + 6 B )(1 + B ) (1 + 3 B ) − H B ˜ B (1 + 3 B ) − B )(1 − B ) ˜ B (1 + 3 B ) . (B.12)Numerical tests confirm that the leading contribution to ω k coincides with the ‘naive’ ex-pression k /a + U (cid:48)(cid:48) in Eq. (4.1). The other terms, arising from the non-canonical kineticterms and the coupled metric perturbations, are sub-leading and can be neglected to a goodaccuracy when calculating the growth rate Γ and the shape of the spectrum. Similarly, thefriction term with f is negligible. Thus, the non-canonical kinetic terms are unimportant dur-ing preheating, just as they were shown to be unimportant during slow-roll in [8, 15]. Theirrelevance of the metric perturbations was also noted in the context of Higgs inflation in[41], suggesting that neglecting the metric perturbations when computing the energy density(B.9) is justified. – 18 –o solve for the time evolution of the energy density of the perturbations, we solve themode equation (B.12) with the Bunch-Davies initial conditions during inflation for a range ofmodes between k min and k max and use these to compute the integral (B.9). Its time evolutionis followed until it becomes comparable to the background energy density. From this timeevolution the growth rate Γ (4.7) is extracted via a numerical fit, see Fig. 3. This process isrepeated for different values of the α -parameter to produce Fig. 5. References [1] A. A. Starobinsky,
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