Hybrid Inflation and Tachyonic Preheating

Hybrid Inﬂation and Tachyonic Preheating

Werkstattseminar Wintersemester 2010/11

Kai Schmitz and Gilles Vertongen (Dated: December 7, 2010)

Hybrid inflation [1,2,3] combines chaotic inflation with spontaneous symmetry breaking. At its late stages it is driven by the energy density of the vacuum and it ends in a rather unusual waterfall transition which is triggered by a tachyonic instability in the effective potential. Preheating in this scenario is not based on parametric resonance [4] but occurs due to the spinodal growth of long-wavelength Higgs modes [5,6]. It is so efficient that symmetry breaking completes within a single oscillation of the scalar fields. Particles coupled to the inflaton sector may be produced non- adiabatically as well [7].

I. OVERALL PICTURE 8 Φ 6 4 Φ The transition from the inflationary era to the stan- c 2 0 dard thermal FLRW Universe may proceed in various different ways depending on the details of the underlying 10 particle physics model. In our two talks in this seminar V HΦ,ΣL we wish to present the most important mechanisms on 5 V H0,0L the market that may be responsible for the reheating of the Universe after the end of inflation. 2 1 0 -1 0 Σ -2 Two weeks ago [4] we considered a simple chaotic sce- v nario with a convex potential (V,φφ > 0) for a coherently oscillating classical scalar inflaton field φ, FIG. 1: Effective scalar potential V (φ, σ) as in Eq. (3) with m = 100 GeV, M = 1011 GeV, λ = 10−1 and g = 1. 1 V (φ) = m2φ2 (1) 2 and encountered a stage of preheating during which tal quantum fluctuations δσ which will render the usual bosonic degrees of freedom coupled to the inflaton were perturbative picture of a classical field with small fluctu- excited due to parametric resonance. We, however, also ations on top of it inapplicable. Instead, we will find that noted that realistic theories of reheating require sponta- SSB occurs due to the spinodal growth of the fluctuations neous symmetry breaking (SSB) in the inflaton sector in such that any initial homogeneity soon is destroyed up order to allow for inflaton two-body decays. But as pre- to rather short scales. The purely classical description of heating in combination with SSB turns out to be much the fields in the inflaton sector thus has to be traded for more subtle we decided to postpone this issue. a semiclassical treatment of quantum fluctuations. Now we return to it and investigate preheating from One of the simplest cosmologically viable scenarios of a concave potential (V,σσ < 0) for a Higgs field σ that inflation featuring the concave potential in Eq. (2) is the breaks some global symmetry of the vacuum: standard hybrid inflation model which was proposed by Andrei Linde in the early 90s [1] and which represents a λ 2 V (σ) = σ2 − v2 ; M 2 = λv2 (2) mixture of the usual theory of SSB and chaotic inflation: 4

1 λ 2 1 In addition to changing the curvature of the effective V (φ, σ) = m2φ2 + σ2 − v2 + g2φ2σ2 (3) 2 4 2 scalar potential we choose completely different initial conditions. Last time we started with a homogeneous The first part of our talk outlines the characteristics of field at the Planck scale, φ & Mp. Now we take the this inflationary model. The second part is devoted to homogeneous component σ0 to be of the order of the ini- the dynamics of SSB and the related preheating process. 2

II. HYBRID INFLATION therefore assume the bare mass m to be small:

2 2 A. Motivation m H (8)

At the beginning of inflation, when the inflaton energy From the perspective of particle physics hybrid in- density still dominates over the vacuum energy density, flation is very attractive as it provides a natural set- 4 ting for SSB. Linde’s original motivation, however, was 1 2 M m φ (9) to construct a model in which inflation ends differ- 2 φ&Mp 4λ ently compared to the standard scenarios of a first-order we thus simply deal with ordinary chaotic inflation: phase transition and slow-roll motion becoming faster and faster. Indeed, as we will show, hybrid inflation may 1 2 2 V (φ & Mp, σ = 0) ' m φ (10) end in a waterfall regime which is characterized by a fast 2 rolling of the Higgs field σ triggered by the slow rolling of In his construction of the hybrid model Linde assumes in the inflaton field φ. Let us now formulate the conditions addition to Eq. (8) also that: for a successful waterfall transition and confront them with observational data. g2 m2 M 2 (11) λ

such that eventually inﬂation is driven by the vacuum B. Evolution of the scalar ﬁelds energy density:

1 M 2 M 4 The evolution of the scalar fields is governed by the V (φ ' φ , σ = 0) ' m2 + (12) c 2 g 4λ classical EOMs in an expanding background: M 2 g2 M 4 = m2 + M 2 ' (13) 2g2 2λ 4λ φ¨ + 3Hφ˙ + m2 + g2σ2 φ = 0 (4) 1 σ¨ + 3Hσ˙ − ∇2σ + g2φ2 − M 2 + λσ2 σ = 0 (5) The Hubble parameter can then be estimated as: a2 8π M 4 2πM 4 At the early stages of inflation φ is very large, φ M , re- H2 ' H2 = · = (14) & p c 3M 2 4λ 3λM 2 eff. p p sulting in an effective Higgs mass mσ that is much larger eff. than the corresponding effective inflaton mass mφ : Omitting the negligible acceleration term the EOM for the inflaton field turns into: eff.2 2 2 2 m = V,φφ = m + g σ (6) φ m2 2 ˙ 2 ˙ eff. 2 2 2 2 3Hcφ + m φ ' 0 ⇒ φ ' −rHcφ ; r = (15) m = V,σσ = g φ − M + 3λσ (7) 2 σ 3Hc

Because of that σ initially rolls down to σ = 0 and re- Integration of Eq. (15) yields: mains there for the most part of inflation. As initial φ(t) = φ exp [−rH (t − t )] (16) conditions we can, hence, safely assume φ & Mp and c c c σ = 0. The end of inflation is initiated once φ has rolled down to the bifurcation point at the critical inflaton value Given the time dependence of the scale factor, a(t) ∝ eHct, we may rewrite φ(t) as a function of the number of φc = M/g. For φ < φc the Higgs potential exhibits a tachyonic instability at σ = 0, that is, the Higgs field e-folds N to the phase transition: possesses a negative effective mass squared, and inflation N = H (t − t); φ(N) = φ erN (17) quickly ends due to false vacuum decay. c c c

Before turning to the details of the waterfall regime let We find that for sufficiently small r the Universe also un- us first check how inflation in fact emerges in our model. dergoes a stage of accelerated expansion when φ is close Successful inflation requires a flat inflaton potential. We 3

of the Higgs ﬁeld in the absence of any self-interaction.

For momenta k < µσ the solutions of Eq. (19) are su- perpositions of exponentials that are either growing or

decaying at rate ωk:

p 2 2 σk(t) ∼ exp [±ωk (φ) t]; ωk (φ) = µσ (φ) − k (20)

The k = 0 mode has the largest growth rate, ω0 = µσ.

We may, thus, estimate tσ as the time scale on which the −1 increase of this mode sets in [2]: tσ = µσ (tσ). In other words: At time tσ the negative mass squared has become −1 that large that exactly the time µσ has passed. Let us 2 now estimate µσ (tσ) (for convenience we set tc = 0): FIG. 2: Probability distribution P (φ, |σ| , t) for ﬁnding the

field values φ and |σ| at a given point in space at time t, 2 2 2 2 2 d 2 µ (tσ) = − g φ (tσ) − φ ' −g φ · tσ (21) extracted from numerical 3D lattice simulations with M = σ c dt 1015 GeV and λ = g = 10−2. Figure taken from Ref. [5], an tc 2 animated GIF version can be found on the arXiv. 2 ˙ 2 m = − g 2φcφctσ = g 2φc φctσ (22) 3Hc 2 r 2 2 M 2 3λ Mp to φc. For instance, with r . 0.01 more than 70 e-folds = g m t (23) 3 g2 2π M 2 σ are generated during the period of vacuum domination. For comparision, recall that adiabatic density perturba- −1 Solving for tσ = µσ (tσ) leads us to: tions that we observe in the present Universe at galactic √ scales leave the Hubble horizon 50 to 60 e-folds before −3 2 tσ ∼ λm Mp (24) the end of inflation. We arrive at our first waterfall condition:

12 C. Waterfall conditions −6 4 2 6 M tσ ∼ λm Mp H ∼ 3 6 (25) λ Mp In order to figure out for which parameter choices infla- which is equivalent to: tion ends in an almost instantaneous waterfall transition we have to determine two time scales. tσ: The time it 3 2 M λmMp (26) takes until the negative mass squared term becomes efficient such that SSB sets it. And tφ: The time after which In order to estimate t suppose now that one Hub- the inflaton field has settled at the minimum of the ef- φ ble time H−1 t has passed since t . At this time fective potential. We will speak of a successful waterfall c σ c symmetry breaking has already occured and the Higgs transition once the following hierarchy is realized: field distribution has reached the minimum of the effec- t , t H−1 (18) tive potential. To a good approximation the peak of the σ φ c 1/2 distributionσ ¯ ∼ δσ2 minimizes V (φ, σ) at fixed φ:

Given the initial conditions σ0 = 0 andσ ˙ 0 = 0 it is 2 2 2 2 clear that SSB can only be accomplished by growing V,σ (φ, σ¯) = 0 ⇒ g φ − M + λσ¯ = 0 (27) 1 quantum fluctuations δσ. Neglecting the expansion of ⇒ σ¯2 = M 2 − g2φ2 (28) the Universe and the Higgs self-interaction for a moment λ the mode equations for these fluctuations read as follows: The distribution of the fields φ and σ evidently moves along an ellipse in the (φ, σ)-plane: 2 2 σ¨k + k − µσ (φ) σk = 0 (19) λ φ2 + σ¯2 = φ2 (29) 2 2 2 2 g2 c Here µσ = M − g φ denotes the negative mass squared 4

2 After one Hubble time the inflaton field value has de- with ρ0 = hρi. The power spectrum δρk follows from a creased by ∆φ. From the inflaton EOM we infer: decomposition of the mean square fluctations δρ2 on a logarithmic momentum scale: ∆φ −1 ' −rHcφc ⇒ φ H ' φc (1 − r) (30) H−1 c Z k c δρ2 (t, ~x) = d ln δρ2 (41) H k Within the first Hubble time after tc the effective mass eff. squared of the inflaton mφ grows significantly and it is At the onset of galaxy formation, when the energy in cold reasonable to estimate the time scale on which inflation matter has come to dominate, δρk can be estimated by eff. −1 will end tφ by 1/mφ at t = Hc [1]. According to standard methods [3]: Eq. (6) we then have: √ 3/2 δρk 16 6π V = 3 (42) −2 2 2 2 −1 ρ0 5M V,φ tφ = m + g σ¯ Hc (31) p k=aH g2 = m2 + M 2 − g2φ2 H−1 (32) In hybrid inflation for σ = 0: λ c 2 g 2 4 3/2 2 2 2 2 √ 1 2 2 M = m + M − g φc (1 − r) (33) m φ + λ δρk 16 6π 2 4λ = (43) g2 ρ 5M 3 m2φ ' m2 + · 2M 2r (34) 0 p λ k=aH 2 2 2 2 2g M m = m + · 2 (35) As mentioned at the end of Sec. II B the galactic scales λ 3Hc ! cross the Hubble horizon during vacuum domination: 2g2M 2 3λM 2 = m2 1 + · p (36) √ 4 6 3λ 2πM δρk 2 6π M φc ' √ (44) ! 3 2 2 2 2 2 2 ρ0 5Mp λ λm φc φ k=aH 2 g Mp g m Mp = m 1 + 2 ' 2 (37) πM πM rN r r With φ/φc = e = (ac/a) → (kc/k) we finally obtain: The second waterfall condition demands that t be much √ φ δρ 2 6π gM 5 k r −1 k ' √ (45) smaller than Hc : 3 2 ρ0 5Mp λ λm kc g2m2M 2 4 −2 p 2 M As a very usual feature of the hybrid inflation model tφ ∼ 2 Hc ∼ 2 (38) M λMp we note its blue spectrum. The scalar spectral index ns which is equivalent to: always is larger than 1: √ d ln δρ2 2m2 λm2M 2 3 2 n − 1 = k = 2r = = p (46) M λgmMp (39) s 2 4 d ln k 3Hc πM

Since m2 H2, cf. Eq. (8), the deviation from 1 is D. Cosmological implications vanishingly small. But still, observations clearly indicate

ns < 1. The ΛCDM fit of the WMAP7 data yields: Confronting the power spectrum of adiabatic energy density perturbations predicted by hybrid inflation with ns = 0.963 ± 0.014 (47) observational data (CMB anisotropies, galaxy surveys, gravitional lensing, Ly-α forest) gives preference to cer- Moreover, notice that for fluctuations with very small k, tain regions in parameter space. We will now show that which left the Hubble horizon while the inflaton contribu- in these favored regions the end of inflation typically is tion to the potential energy density still dominated over described by the waterfall transition. the vacuum term, the spectrum begins growing again. First, let us write the cosmic density field ρ as:

ρ (t, ~x) = ρ0(t) + δρ (t, ~x) (40) Neglecting the k-dependence in Eq. (45) and using 5

Eq. (8) we may write: At the initial stages of Spontaneous Symmetry Break- √ ing (SSB), the Higgs modes follow the linear Eq. (19). In 5 1 2 6π gM 1 δρk · √ · (48) the symmetric phase σ = 0, i.e. when φ > φc, we take 3 2 Hc 5Mp λ λm m ρ0 the quantum fluctuations of the mode functions to be the √ r 5 3λ Mp 2 6π gM δρk same as the one for a massless field √ ⇔ 2 3 (49) 2π M 5Mp λ λm ρ0 1 −ikt ~ 3 δρk 5λ 2 σk(t) = √ e , k ≡ |k| . (52) ⇔ M · mMp (50) 2k ρ0 6g At t = 0, one “turns on” the term −M 2σ2/2, correspond- The typical scale of the amplitude of the power spectrum ing to a negative mass squared −M 2. Consequently, in has been measured by the COBE satellite, δρk/ρ0 ∼ 5 × this quench approximation, all modes k < M grow expo- 10−5. In order to generate density perturbations of the nentially right size the parameters of the hybrid inflation model thus have to satisfy: √ 1 t M 2−k2 σk(t) = √ e , (53) 2k λ M 3 5 × 10−5 · mM 2 (51) g p At latter stages, the mean value of the field stays equal to zero hσi = 0. The first non-vanishing quantity is the This means that for a wide range of values of the cou- second moment of σ, i.e. hσ2i. This is this quantity that pling constants λ and g the waterfall conditions (26) and we use to characterize the symmetry breaking process, (39) are automatically satisfied merely by virtue of the 2 2 2 2 stating that symmetry breaking occurs when hσ i = v . assumption m H and the requirement δρk/ρ0 ∼ 5 × 10−5. In cosmologically realistic scenarios hybrid in- Using the Heisenberg representation of the quatum flation hence tends to end in a waterfall regime. fluctuations of the σ field is

Z d3k h i σ(t, x) = a σ (t) e−ikx + a† σ∗(t)eikx , III. TACHYONIC PREHEATING (2π)3/2 k k k k (54) The basic form of the effective potential in hybrid inflation scenario is given by Eq. (3). As stated above, in- where x and k denote the position and momentum vec- flation in this model occurs while the φ field rolls slowly tors, one can deduce the initial (t = 0) dispersion of all in the σ = 0 valley towards the global minimum at φ = 0 growing fluctuations with k < M and |σ| = v. In realistic versions of this model, the mass 2 m and the velocity φ˙ are small after inflation. The fields hσ i ≡ h0|σ(x)σ(x)|0i , (55) 2 then fall along a trajectory φ(t), σ(t) in such a way that Z m dk2 m2 = 2 = 2 . (56) the initial trajectory is flat, then it rapidly falls down, and 0 8π 8π becomes flat again near the minimum. In between, the † curvature of the effective potential is initially negative, where the canonical commutation relation [ak, ak0 ] = and therefore, the field should experience a tachyonic in- δ(3)(k − k0) has been used. Using the 2-point correla- stability along the way. tion function Z dk sin(k|x − y|) hσ(x)σ(y)i = δσ2 , (57) k k|x − y| k A. Dynamics of the Symmetry breaking

one can deduce the average initial amplitude δσk of all The main goal of this section is to show that the tachy- fluctuations with k < M onic instability leads to SSB so efficient that it completes k typically within an oscillation. We will see that the quan- δσ = . (58) k 2π tum fluctuations of the Higgs field grow such that it pro- duces Higgs classical waves. At t > 0, the modes with k < M start to growth expo- 6 nentially, and their dispersion relation becomes Using Eq. (53), the occupation number

2 Z M 2 √ √ 2 2 dk 2t M 2−k2 1 1 t M 2−k2 1 2Mt −k2/(2k2) hσ i = e , (59a) n + = e ≈ e e ? , (62) 2 k 0 8π 2 2 2 e2Mt(2Mt − 1) + 1 = , (59b) 16π2t2 becomes exponentially large very quickly for the long wavelength modes, and drops abruptly for k > k? ≡ M(2Mt)−1/2. M 2 hσ2i ∼ e2Mt . (59c) 16π2 −3 −4 Example : For λ = 10 and v = 10 MP , the sym-

This means that the average amplitude δσk of quantum metry is broken within Mt? ∼ 6 and the typical cut-

ﬂuctuation with momenta k initially was δσk = k/2π and oﬀ frequency is k? ∼ M/3. The occupation numbers of √ t M 2−k2 then started to grow as e . modes with k < k? is exponentially large

1 16π2 To get a qualitative understanding, consider a single n (t ) ≈ e2Mt? ' ∼ 2 × 105 . (63) k ? 2 λ sinusoidal wave σ = ∆(t) cos(kx) with k ∼ M, and with an initial amplitude ∆(t) ∼ M/2π in 1D. The ampli- Importantly, one remarks that for small λ, the fluctua- √ tude of this wave grows until it becomes O(v) ∼ M/ λ. tions with k < M have very large occupation numbers, This leads to the division of the Universe in domains and therefore they can be interpreted as classical waves of size O(M −1), in which the field changes from O(v) of the field σ. to O(−v). The gradient energy density of domain walls separating areas will be ∼ k2σ2 = O(M 4/λ). This is of When the fields rolls down to the minimum of its ef- the same order as the total initial potential energy of the fective potential, its fluctuations scatter off each other as field V (0) = M 4/4λ. Because the initial state contains classical waves due to the λσ4 interaction. It is difficult many fluctuations with different phases growing at dif- to study this process analitycally, but fortunately, it can ferent rates, the resuling field is a Gaussian random field. be studied using lattice simulations. In Fig. 3 is plotted It cannot coherently give all its gradient energy back and the result of the evolution of the Higgs vev, together with return to its initial state σ = 0. This is one of the reason the approximate expression why SSB occurs within a single oscillation of the field σ. v M(t − t ) √ σ(t) ≡ hσ2(t)i1/2 = 1 + tanh ? . (64) At σ ∼ v/ 3, the curvature of the effective potential 2 2 vanishes, i.e. the Higgs mass becomes positive. In consequence, the tachyonic growth of all fluctuations with As can be seen, the vacuum expectation value never k < M continues until phδσ2i ∼ v/2. Subsequently, all comes close to the initial point σ = 0 again, which im- the modes oscillates. plies that symmetry becomes broken within a single oscil- The symmetry is broken when hσ2i ' v2. Using lation of the distribution of the field σ. After SSB, space Eq. (59c), one can estimate the time it takes for sym- becomes divided into domains with σ = ±v. The initial metry breaking to happen size of each domain is O(M −1), and inside each domain, the deviation from σ = |v| is much smaller than v. Grad- 1 32π2 ually, the size of each domain grows and the domain wall t? ' ln . (60) 2M λ structure becomes more and more stable.

The exponential growth of ﬂuctuations can be interpreted as the growth of the occupation number, which is B. Particle production deﬁned by [8]

1 1 We now turn to the study of the production of scalars n + ≡ |σ∗(t)σ ˙ (t)| . (61) k 2 2 k k χ and fermions ψ, coupled to the Higgs via the usual 7

10 q(t) (Tanh) Bosons: Lattice 1/2/v (Lattice) Tanh nB(x100) (Tanh) Fermions: Lattice nB(x100) (Lattice) 1 Tanh 1 0.1 k (t) B 0.01 0.1

/v, n 0.001 1/2

(t)> 0.0001 2 0.01 q < Occupation number: n 1e-05

1e-06 0.001

1e-07 5 10 15 20 25 30 0.1 1 time: mt k/m

FIG. 3: Time evolution of the vacuum expectation value FIG. 4: Spectrum of occupation number for bosons and hσ2(t)i1/2/v (here φ) compared with the approximate solu- fermions in the true vacua, in both lattice and analytical ap- tion Eq. (64) with Mt? = 8. Figure taken from Ref. [7]. proximation. Here the parameters are ﬁxed to λ = 0.01, g = 0.5 and h = 0.5 Figure taken from Ref. [7].

and fermions through the Higgs mechanism following interactions 2 2 2 2 2 mB(t) = g hσ i , mF (t) = hhσ i . (68) 1 2 2 2 ¯ Lint = g σ χ + hσψψ . (65) 2 It will be responsible for the non-adiabatic production Since the backreactions of the produced χ and ψ modes of particles which can be studied using the formalism of on the evolution of the Higgs expectation value, is negli- quantum fields in strong background. gible, we can solve first the process of symmetry breaking (see below), and then take the resulting evolution of the Higgs as a background fields that induces particle pro- 1. Number densities duction.

The mode equations in terms of rescaled ﬁelds At ﬁrst, one can solve the mode equations Eqs. (67) within the approximation Eq. (64). For bosons, the mode 3/2 3/2 Xk(t) = a χk , Ψ(t) = a ψ , (66) functions Xk(t) are solutions of the oscillator Eq. (67a) 2 2 2 with time-dependent frequency ωk(t) = k + mB(t). It are can be rewritten

2 2 2 2 2 2 00 2 α mB 2 ∂t Xk + k + mB(t)a (t) Xk = 0 , (67a) X + ω (k) + (1 + tanh x) X = 0 (69) k − 4 k µ (iγ ∂µ − mF (t)a(t)) Ψ = 0 . (67b) √ where x = M(t − t )/2 and α ≡ g/ λ, while ω2 (k) = k √ ? − As we have seen previously, the quench approximation, 2 2 2 2 and ω+(k) = k + α M are the in/out asymptotic fre- i.e. the sudden appearance of the mass term, is valid quencies. Solving this equation in terms of hypergeomet- in hybrid inﬂation models fulﬁlling the waterfall condi- ric functions [7], one can deduce the boson occupation tions. In that case, the rate of expansion is typically number much smaller than the masses involved, so that we take √ 2 it constant here (a = 1) during SSB. The change of vac- B cosh[ 4α − 1] − cosh[2π(ω+ − ω−)/M] nk (k) = (70) uum induces a sudden change in the masses of bosons sinh[2πω−/M] sinh[2πω+/M]] 8

0.25 by

0.20 Z ρX 2gsλ 2 X = 2 dκ κ nk (α) ω+(κ, α) (72) ρ0 π 0.15 L Λ , Α H

f where κ ≡ k/M. Fitting functions to the ﬁnal energy Λ 0.10 densities are provided by

0.05 ρB −3 ' 2 × 10 gs λ f(α, 1.3) , (73) ρ0 0.00 0.0 0.2 0.4 0.6 0.8 1.0 ρF −3 ' 1.5 × 10 gs λ f(α, 0.8) , (74) g ρ0 √ √ FIG. 5: Evolution of the λf(α, γ) function with the coupling p 2 2 g (of equivalently h) for λ = 0.1, 0.01, 0.001 (from top to bot- where f(α, γ) ≡ α + γ − γ and α = g/ λ(h/ λ) tom). The case of bosons (fermions) is represented by the for bosons (fermions). The production of bosons and plain (dashed) lines. fermions from symmetry breaking is thus more efficient when the Higgs mass and thus λ is large, as is also clear A similar treatment can be done for the fermions, which from Fig. 5. Unless the couplings are unnaturally large, leads to the fractional energy density in bosons and fermions is always small. As a consequence, no backreaction on the F cosh[2πα] − cosh[2π(ω+ − ω−)/M] nk (k) = (71) evolution of the Higgs is thus expected, justifying a pos- 2 sinh[2πω−/M] sinh[2πω+/M]] teriori our initial assumption of dividing the problem √ with α ≡ h/ λ in this case. into two stages. Importantly, this also implies that at More precisely, the modes equations Eqs. (67) are the end of tachyonic preheating the Universe energy den- solved numerically, using the lattice Higgs vev evolution. sity is dominated be the non-relativistic classical waves The two options are illustrated in Fig. 4 where one can of the Higgs bosons. One would thus have to wait for see that they agree very well, except at large momentum, the subsequent decays for the Universe to be reheated. where parametric resonance effects are responsible for the The reheating temperature is then given by the usual excitation of the low wavelength modes (remember : the expression first band in the instability chart in the parametric reso- p T ' 0.2(200/g )1/4 Γ M . (75) nance case is sitting around the value k ∼ M). R ? σ P

2. Energy densities

The ratio of energy densities of particles created to the 4 initial false vacuum energy density ρ0 ≡ M /4λ is given

[1] A. D. Linde, “Hybrid inflation,” Phys. Rev. D 49 (1994) [5] G. N. Felder, J. Garcia-Bellido, P. B. Greene, L. Kof- 748 [arXiv:astro-ph/9307002]. man, A. D. Linde and I. Tkachev, “Dynamics of symmetry [2] J. Garcia-Bellido and A. D. Linde, “Preheating in hy- breaking and tachyonic preheating,” Phys. Rev. Lett. 87 brid inflation,” Phys. Rev. D 57 (1998) 6075 [arXiv:hep- (2001) 011601 [arXiv:hep-ph/0012142]. ph/9711360]. [6] G. N. Felder, L. Kofman and A. D. Linde, “Tachy- [3] A. D. Linde, “Particle Physics and Inflationary Cosmol- onic instability and dynamics of spontaneous symmetry ogy,” arXiv:hep-th/0503203. breaking,” Phys. Rev. D 64 (2001) 123517 [arXiv:hep- [4] K. Schmitz and G. Vertongen, “Reheating and preheat- th/0106179]. ing,” Werkstattseminar WS 2010/11. [7] J. Garcia-Bellido and E. Ruiz Morales, “Particle produc- 9

tion from symmetry breaking after inﬂation,” Phys. Lett. cannot be used since ωk then becomes imaginary. B 536 (2002) 193 [arXiv:hep-ph/0109230].

[8] Here, the standard deﬁnition of the occupation number nk