Gravitational-Wave Mediated Preheating

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4-9-2015

Stephon Alexander Dartmouth College

Sam Cormack Dartmouth College

Antonino Marcianò Fudan University

Nicolás Yunes Montana State University - Bozeman

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Dartmouth Digital Commons Citation Alexander, Stephon; Cormack, Sam; Marcianò, Antonino; and Yunes, Nicolás, "Gravitational-Wave Mediated Preheating" (2015). Dartmouth Scholarship. 2024. https://digitalcommons.dartmouth.edu/facoa/2024

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Physics Letters B

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Gravitational-wave mediated preheating ∗ Stephon Alexander a, Sam Cormack a, , Antonino Marcianò b, Nicolás Yunes c,d a Center for Cosmic Origins and Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA b Center for Field Theory and Particle Physics & Department of Physics, Fudan University, 200433 Shanghai, China c Department of Physics, Montana State University, Bozeman, MT 59717, USA d Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA a r t i c l e i n f o a b s t r a c t

Article history: We propose a new preheating mechanism through the coupling of the gravitational field to both the Received 16 December 2014 inflaton and matter fields, without direct inflaton–matter couplings. The inflaton transfers power to the Received in revised form 4 February 2015 matter fields through interactions with gravitational waves, which are exponentially enhanced due to an Accepted 6 February 2015 inflation–graviton coupling. One such coupling is the product of the inflaton to the Pontryagin density, Available online 13 February 2015 as in dynamical Chern–Simons gravity. The energy scales involved are constrained by requiring that Editor: M. Cveticˇ preheating happens fast during matter domination. Keywords: © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license Preheating (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. Gravitational waves Chern–Simons

1. Introduction flation field and the graviton so that when the inflaton oscillates it causes resonances in the graviton’s modulation. We will show Inflation is the paradigm wherein the universe undergoes ex- that such couplings are possible via Chern–Simons modified gen- ponential expansion, resolving the horizon, entropy and structure eral relativity. As we shall see, no direct inflaton–matter coupling formation problems that plague the standard big bang scenario. will be necessary to obtain preheating in this gravitational-wave It is usually believed that inflation ends once the inflaton reaches mediated scenario. the bottom of its potential, at which point a new mechanism must act to transfer the inflaton’s kinetic energy into a process that 2. Action and evolution equations leads to particle creation. One such mechanism is preheating [1–4]: the inflaton enters a phase of parametric resonance, as it oscil- Many inflationary paradigms exist, but for concreteness con- lates around the minimum of its potential, and through a direct sider the following Chern–Simon extension to Chaotic Inflation matter–inflaton coupling, it leads to particle creation. There is a [5,6] large number of possible direct couplings between the inflaton and the standard model, and one must usually pick one somewhat ar- √ = 4 − L + L + L + L bitrarily. S d x g EH int φ χ , (1) But what if the coupling between the inflaton and the matter fields were indirect? Because of the equivalence principle, the R α LEH = , Lint = φR R, (2) graviton will interact with all matter fields and its coupling will be 16π G 4 non-arbitrary. Let us then consider the inflaton coupling to matter 1 μν 2 2 Lφ =− g ∂μφ∂νφ + m φ , (3) fields through a graviton intermediary. That is, consider the in- 2 φ flaton at the end of inflation depositing its kinetic energy in the where φ is the inflaton, χ is the matter field, R is the Ricci scalar graviton, which due to a direct graviton–inflaton coupling becomes parametrically excited, and then deposits its energy in the matter associated with the metric gμν , R R is the Pontryagin density, fields. This can happen if there are new couplings between the in- i.e. the contraction of the Riemann tensor with its dual, and α is a coupling constant with dimensions of inverse mass (we work here in natural units c = 1 = h). Except for the interaction term Lint, * Corresponding author. Eq. (1) is just a simple model for inflation with a quadratic infla- E-mail address: [email protected] (S. Cormack). ton potential, arising from a Taylor expansion about its minimum. http://dx.doi.org/10.1016/j.physletb.2015.02.018 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. S. Alexander et al. / Physics Letters B 743 (2015) 82–86 83

Many possible graviton–inflaton couplings could be considered, To first-order in λ, the equations for the metric tensor pertur- but the one presented above, Lint, is well-motivated. Such a cou- bation become [13] pling arises naturally in a variety of frameworks: (i) in heterotic string theories upon 4-dimensional compactification and a low- 16π G lm ¨ ˙ ˙ ˙ h ij = [(φ − Hφ)h j)l + φ h j)l],m energy expansion [7,8]; (ii) in loop quantum gravity when the a (i Barbero Immirzi parameter is promoted to a field and coupled to 2 + 16π Ga pχ h ij, (8) fermions [9,10]; (iii) in effective field theories of inflation [11]; (iv) in dynamical Chern–Simons gravity [12]. Let us emphasize, while, neglecting scalar metric perturbations (we are looking at however, that the gravitational wave-mediated preheating mech- modes shorter than the Hubble scale), the equation for the inflaton anism proposed here does not depend on this particular coupling. perturbation is Regardless of the motivation, the theory described above should be considered effective, a truncated low-energy expansion of a 1 δφ¨ +3Hδφ˙ − ∇2δφ =−m2 δφ, (9) more fundamental theory that is thus valid only up to some a2 φ energy cut-off . The effective theory ceases to be a valid de- where ∇2 and are the Laplacian and wave operators in a homo- scription when the interaction term Lint becomes comparable to geneous and isotropic FLRW background, p is the pressure of the the Einstein–Hilbert term LEH. The former can be written as a χ ≡ −2 total derivative if φ is constant. Therefore, to estimate its size χ field and h ij a hij. Notice again that the Pontryagin density we should first integrate by parts, moving a derivative from RR does not enter the evolution equation of the inflaton perturbation, to φ. The interaction term then becomes comparable to LEH when since it also vanishes identically to linear order in λ. ˙ ∼ 2 −1 = −1/2 αφ M p(h0 f ) , where M p G is the Planck mass, f is the We can simplify the evolution equation for the metric pertur- gravitational wave frequency and h0 is the gravitational wave am- bation through order reduction. As discussed in [14–16], we de- −1 ˙ plitude. Saturating α at , φ at HMp , h0 at unity and f at H, compose the metric perturbation into a general relativistic piece 2 GR = GR + 2 where H is the Hubble parameter, one finds = (H/M p) M p , hμν and a deformation δhμν , namely hμν hμν α δhμν . Note 2 which of course satisfies M p . Another consequence of the that the deformation is proportional to α because Eq. (4) is pro- truncation of the effective theory at this order is that the terms portional to αCμν , Cμν is proportional to φ, and φ is proportional 4 4 neglected in the expansion, such as (∂φ) / , are indeed small to α due to Eq. (5). Using this decomposition, we can order re- and ignorable. A consequence of all of this is that the interaction duce Eq. (8): the left-hand side is proportional to δhμν , while L term int acts as a small perturbation to whichever inflationary the right-hand side is proportional to a differential operator act- mechanism one wishes to consider, and thus, it does not spoil (or GR ing on hμν . This differential operator will contain one term of the really affect) inflation, until inflation ends and the inflaton reaches form , which automatically vanishes when acting on hGR because the bottom of its potential. μν [ GR ] = Variation of the action in Eq. (1) with respect to all degrees of Rμν hαβ 0. Using this and going to the transverse-traceless (TT) freedom leads to the field equations [12] gauge [17] and in the left/right-circular polarization basis for the gravitational wave perturbation, Eq. (8) becomes Gμν + 16π G α Cμν = 8π GTμν, (4) 16π G ¨ ˙ ∂ ˙ 2 2 α 2hR = i α(φ − Hφ) hR + 16π Ga pχ hR . (10) φ − m φ + R R = 0, (5) a ∂z φ 4 − 2 = The equation for h can be obtained by taking i →−i and χ mχ χ 0, (6) L h → h . The amplitudes h and h are defined by L/R R/L L R where is the curved wave-operator, Gμν is the Einstein tensor, ⎛ ⎞ Tμν is the sum of the stress-energy tensors of the φ and χ fields, −(hL + hR ) i(hL − hR ) 0 and 1 ⎝ ⎠ h ij = √ i(hL − hR )(hL + hR ) 0 . (11) μν ≡∇ αβγ (μ∇ ν) +∇ ∇ β(μν)α 2 000 C αφ γ Rβ (α β)φR , (7) Notice that Eqs. (8) and (9) are not coupled and can thus be solved with Rβ(μν)α the dual Riemann tensor with indices symmetrized. independently, once the evolution of the background fields is ob- 3. Order reduction and perturbation theory tained. Let us now discuss the evolution of the matter fields. We Let us expand the equation for the metric tensor and the in- anticipate that the matter occupation number will be generated flaton about a fixed background: gμν = gμν + λhμν and φ(t, x) = through parametric resonance, so even a small perturbation of O | | φ(t) +λδφ(t, x), where λ is an order-counting parameter. The back- size ( hμν ), may have a large effect. We therefore treat the mat- ground gμν and φ(t) will be taken to be the flat Friedmann– ter field, χ , exactly, without a perturbative decomposition, while Lemaître–Robertson–Walker (FLRW) metric and a homogeneous the gravitational field is treated to first order in its perturba- and isotropic background field respectively, while hμν and δφ(t, x) tions, hμν . We obtain the equation to first order in hμν , are first-order perturbations. 1 The FLRW metric satisfies the background field equations ex- ¨ + ˙ − ∇2 + ij =− 2 χ 3Hχ χ h ∂i∂ jχ mχ χ. (12) actly for any homogeneous and isotropic background inflaton field. a2 The Hubble parameter is sourced by the energy density and pres- In the TT gauge and in a circular GW polarization basis, this equa- sure of this background field and the matter fields (for reasons that tion becomes will become clear later, we do not decompose χ ). The background 1 1 inflaton field satisfies the homogeneous and isotropic wave equa- ¨ + ˙ − ∇2 − √ [ 2 + 2] =− 2 χ 3Hχ χ Re hL ∂L hR ∂R χ mχ χ, (13) tion on an FLRW background with a mass potential. The Pontryagin a2 2a2 density does not contribute to the background evolution of the in- [ ] ≡ flaton, because this quantity vanishes exactly when evaluated for where Re x is √the real part of x and we have defined ∂L,R any spherically symmetric metric. (∂/∂x ∓ i∂/∂ y)/ 2. 84 S. Alexander et al. / Physics Letters B 743 (2015) 82–86

4. Behavior of solutions

Before attempting to solve the equations of motion, let us make some approximations. On the one hand, as the φ field oscillates, it will amplify and cause the gravitational waves to modulate, which in turn will drive the production of χ particles. The latter will therefore occur on the timescale τφ = 1/mφ . On the other hand, the expansion of space is governed by the Hubble parameter and the scale factor changes on the timescale τH = 1/H. When τφ τH , or equivalently H mφ , preheating occurs much faster than the expansion of space and one is justified in setting a = 1 and H = 0. Using that the Hubble parameter for a simple quadratic 2 2 = 4π 2 potential satisfies M p H 3 (mφ φ) , the requirement H mφ implies φ0 M p , where φ0 is the value of φ at the end of inflation. Even though it works well, this approximation cannot be claimed to be excellent, as we know that φ0 < M p . Neverthe- less, the approximation is correct, since the timescale involved in the growth of the particle number is smaller than the oscillation time. With this approximation, the equation of motion for the background inflaton field greatly simplifies to a that of a simple harmonic oscillator with frequency mφ , so

φ = φ0 sin(mφt + δ), (14) −1 Fig. 1. Top: Fourier transform of hR as a function of time in m units, for k values where and are constants of integration, i.e. the amplitude of φ φ0 δ of 50mφ (dashed dotted), 100mφ (dotted), 150mφ (dashed) and 200mφ (solid). We the background inflaton and a phase shift. here choose γ = 0.062, as this is the smallest value of γ for which particle produc- = The background inflaton sources the evolution of the metric tion occurs when k 200mφ is the largest wavenumber used. We assume an initial ˜ scale-invariant power-spectrum for hR . Bottom: Total particle number for χ , calcu- perturbation through Eq. (10). Using the approximation described ˜ above, and transforming to the Fourier domain, this equation be- lated from the hR in the upper-panel and summing over all wavenumbers. Most of the contribution to the sum comes from the largest wavenumbers, i.e. χ particles comes with large momenta are preferentially produced. Calculations have been done for qz from m to 100m , q = 0, q = 100m . ˜¨ ˜˙ 2 ˜ φ φ x y φ hR = γ sin(mφt + δ)khR − (k + 16π Gpχ )hR , (15) where the overhead tilde stands for the Fourier transform and we Let us contrast our matter production equation with the usual have defined way particle creation occurs via the Mathieu equation. In that case, the matter fields obey the following evolution equation φ m2 ≡ 2 = 0 φ γ 16π Gαmφφ0 16π , (16) M∗M2 p ¨ + 2 + 2 =− 2 2 2 χ (k mχ )χ g φ0 sin (mφt) χ, (18) which controls the strength of the oscillatory anti-damping term. The above equation of motion admits an exact solution through where g is the coupling constant between the χ and φ fields. a linear combination of Heun functions times an oscillatory term, Parametric resonance occurs because of the purely temporal oscil- which we confirm by numerically solving Eq. (15). We could ob- lations of the inflaton. In our case, however, the graviton couples tain a similar expression for the left-polarized mode, but we do not through spatial gradients of the inflaton, and thus, parametric res- need to. As found in [7,17,18], Eq. (10) leads to exponential ampli- onance occurs due to non-linear mode couplings and temporal fication/damping of right-/left-handed gravitational waves during oscillations in the gravitational wave’s source term. the inflationary epoch, so by the end of inflation, the left-handed gravitational waves are negligible and can be neglected during preheating. The left/right asymmetry is controlled by the sign of the 5. Parametric resonance and particle creation coupling constant α. Here we take α to be positive, which causes the left-handed gravitational waves to be damped. The evolution equations for the gravitational wave and the With the solution to the background inflaton and the metric mode functions, Eqs. (15) and (17), are those of a parametric oscil- perturbation, we can now solve for the evolution of the matter lator, i.e. a harmonic oscillator with parameters, like the damping fields. Working in the Heisenberg picture, we promote the mat- coefficient or the natural frequency, that oscillate in time at some ˆ ter field to a quantum operator χ(x, t) and expand the latter other frequency. For example, the damping coefficient in Eq. (15) is in Fourier modes with raising and lowering operators, as done a function of time with frequency mφ , while the natural frequency routinely in quantum field theory [19]. Using the flat-space ap- in Eq. (17) is also an oscillatory function of time. proximation described above, the evolution of the Fourier mode For the oscillatory damping to have an effect in the evolution functions obeys of the metric perturbation, the quantity γ k/mφ must be close to ¨ + 2 + 2 unity. The quantity γ k is compared to mφ since the damping must χ (k mχ )χ occur on timescales comparable to the oscillation of φ. This implies 1 =−√ + 2 ˜ − + that the energy scale M∗ = 1/α should satisfy dkz (kx iky) hR (kz kz) h.c. χ , (17) 4π i 2 where k is the magnitude of the 3-momentum k -vector and h.c. M p ∼ (19) stands for Hermitian conjugate. α φ0 . mφk S. Alexander et al. / Physics Letters B 743 (2015) 82–86 85

We can compare this with the condition that Lint remains small Before presenting a solution to the evolution equations, let us ˙ compared to LEH. By taking φ ∼ mφφ0 and identifying f and k we define a good measure of particle production: the occupation num- can write this condition as ber. Following [4], this quantity in a given mode function is given 2 by M p α φ (20) 0 |˙ |2 mφkh0 ωk χ 2 1 n = +|χ| − , (24) k 2 We can satisfy this equation if the gravitational wave amplitude 2 ω 2 k satisfies h0 1, which is simply equivalent to the condition that where as usual ω2 = k2 + m2 . hμν is a small metric perturbation. k χ We should also address the possible appearance of ghost insta- We now have all the ingredients we need to explore parti- bilities in the theory. The analysis of ghost instabilities in Chern– cle production due to mediated parametric resonance. We solve Simons gravity has been performed by Dyda et al. [20]. They find Eqs. (15) and (17) numerically through the Dormand–Prince ˜ that at the linearized level, a ghost instability coming from the Runge–Kutta method [21]. We ignore the back-reaction of χ on h, third derivative term in Eq. (8) is only present if the momentum and thus first solve Eq. (15) and then use this solution to solve cutoff for the theory is greater than a mass scale they call the Eq. (17). The integral term in Eq. (17) is treated through the con- = 2 ˙ ˙ ≤ Chern–Simons mass, defined as mCS M p /8παφ. In our work, φ volution theorem, using a fast Fourier transform. The numerical mφφ0, so, in terms of our parameter γ , we have mCS ≥ (2/γ )mφ . code is gridded in momentum from a minimum value of mφ to In our numerical solution we use a cutoff of = 200mφ while 200mφ . Any mode with momentum k < mφ will not have time our Chern–Simons mass is mCS ≥ 32mφ . If this situation persists to oscillate much during the period of preheating, and thus can to the present day then Dyda et al. show that too many photons be ignored. The gravitational waves are assumed to start with a will be produced by vacuum decay. However, the Chern–Simons scale-invariant power-spectrum, mass is not constant and, in our case, the situation > mCS oc- 10π AT curs only during preheating. During slow roll inflation, the value |h˜ |= (25) R 3/2 of φ˙ will generically be smaller than during preheating, and the k Chern–Simons mass will hence be greater. By the end of preheat- but with large amplitude, AT = 1/20π , since the right-handed ˙ ing, φ → 0so mCS →∞ and we return to the regime of standard waves are exponentially amplified during inflation. The parameter general relativity. The ghost instability is therefore present only AT is the standard gravitational wave spectrum amplitude. The χ during the very brief period of preheating and possibly the very field starts with Gaussian fluctuations with variance proportional end of inflation. to 1/k. The presence of the ghost during preheating will mean that the Fig. 1 shows our numerical results. The left panel shows the vacuum is unstable to decay to light particles. Instability of the amplitude of the Fourier transform of hR as a function of time −1 vacuum in the presence of a large classical field is not necessar- in units of mφ for different values of k. The right panel shows ily a fundamental problem. For instance, the Schwinger effect can the corresponding growth in particle number, calculated by sum- be thought of as an instability of the vacuum in the presence of ming over wavenumber in Eq. (24). Observe that there are large a large electric field. In our case the classical field is φ and the jumps in particle number when the amplitude of the gravitational strength of the instability is controlled by φ˙. waves becomes large. Energy must be conserved in the production We have checked that vacuum decay will not be the dominant of χ particles, and this energy must come from that contained in particle production effect in our scenario, and this provides an ad- gravitational waves. The stored gravitational wave energy is pro- ditional constraint on the energy scales of the model. The vacuum portional to the square of their amplitude, so the energy carried decay rate to photons is given by [20] by them peaks just as the χ particles are being created. If we in- 5 clude the effect of χ back on h, the peaks in amplitude will be mCS ∼ . (21) much smaller. 2 M p The number density, and hence the energy density, of χ par- −1 ticles increases by approximately ten orders of magnitude during The duration of preheating is of the order 10mφ , and let us as- each jump. We therefore conclude that this process is much more sume that all of the produced photons have energy . Then, using efficient than the vacuum decay mentioned earlier and that it will = ∼ −1 200mφ , our expression for mCS, and γ 10 , we get an esti- drain the energy from the inflaton field in only a few jumps. This mate for the energy density of photons from vacuum decay during supports our assumption that the expansion of the universe can be preheating of neglected during the preheating phase. We note that the amount 6 of particle production from resonance increases with increasing mφ ρ ∼ 1016 . (22) momentum cutoff, similar to the vacuum decay rate (see Eq. (21)). decay 2 M p This suggests that the particle production is related to the ghost This should be much less than the energy initially stored in the in- instability, though it is a nonlinear rather than perturbative effect. flaton field otherwise it will overwhelm any other particle production mechanism. The energy density in the inflaton field initially is 6. Conclusion and discussion ∼ 2 2 ρφ mφφ0 and imposing ρdecay ρφ requires that We have presented a gravitational wave-mediated preheating m2 scenario where during its oscillatory phase, the inflaton deposits φ −8 10 (23) its energy to the graviton and the latter then excites matter pro- M φ p 0 duction through parametric resonance. This mediated mechanism which does not contradict any of the other assumptions in our eliminates the need for any ad hoc direct interaction between the work. We will find that matter particles are produced exponen- inflation and the matter fields, obviating issues of fine tuning the tially via parametric resonance so that particle production from inflaton’s interactions with those of the standard model. In our vacuum decay will be insignificant as long as the above condition case, the minimal coupling between the standard model and grav- holds. ity is non-arbitrary. We have provided a concrete realization of 86 S. Alexander et al. / Physics Letters B 743 (2015) 82–86 the mechanism in the context of Chaotic Inflation with a Chern– [7] S.H.S. Alexander, M.E. Peskin, M.M. Sheikh-Jabbari, Leptogenesis from grav- Simons coupling to the inflaton field. These models are promising ity waves in models of inflation, Phys. Rev. Lett. 96 (8) (2006) 081301, because they have already been implemented to generate lepton http://dx.doi.org/10.1103/PhysRevLett.96.081301, arXiv:hep-th/0403069. [8] S.H. Alexander, J. Gates, S. 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