χχχχχχχχχχχχ RESearch Center for the Early Universe The University of Tokyo

Scalar fields could be the origin of everything! Large Homogeneous, Isotropic, & Flat Universe Density fluctuations & CMB Anisotropy Radiation Baryon Asymmetry Dark Matter Inflation driven by a scalar field called the Inflaton Quantum fluctuations of the Inflaton field Reheating by Inflaton’s decay Affleck-Dine scalar fields in SUSY ？ Q-balls ？ But some other scalar fields could be harmful: Moduli

Today’s talk Behavior of an oscillating scalar field in a thermal bath The case decay products of the oscillating scalar field (in particular, the inflaton) has a larger thermal mass than the oscillation frequency (or the inflaton mass). Application to the cosmological moduli problem

Inflaton φ slow rollover Reheating V[φ] BEGINNING?? END?? Λ But little is known about the beginning and end of inflation. Slow-roll phase is now probed by astronomical observations. Klein Gordon Equation Einstein Equation Cosmic Scale Factor Hubble parameter

The Origin of the Hot Big Bang Universe Reheating Processes After Inflation ＝ Entropy Production through the decay of the Inflaton, a scalar field which drives inflation.

New/Topological inflation Hybrid inflation φ V Ψ “Energy history” of the inflationary universe Exponential expansion Potential energy Slow rollover Kinetic + potential energy Rapid field oscillation Preheating (parametric resonance) Reheating (perturbative decay) Radiation dominated stage Reheating temperature The maximum temperature after inflation is much higher than the reheating temperature in general. The inflaton decays in a thermal bath. Chaotic inflation φ V[φ] １

An interesting possibility If the would-be decay product of the inflaton acquires a thermal mass which is larger than the inflaton’s mass, its decay is temporarily suspended. (Linde 1985, Kolb, Notari, & Riotto 2003) other scalar particles fermions or thermal mass Phase space is closed and the scalar field cannot decay if would-be decay products have a thermal mass larger than !? The decay rate of the inflaton to two massive particles with mass. Decay rate to two massless particles. : the inflaton mass thermal mass Finite-Temperature Effective Potential

φdominant radiation dominant conventional reheating φdecays completely at If so, thermal history after inflation is drastically changed. field oscillation radiation Reheat temperature Thermal history after preheating in conventional theory with

If inflaton’s decay rate vanishes at high temperature, thermal history is drastically changed. conventional reheating ρ φ =ρ r This would affect the relic abundances of gravitinos, superheavy particles etc. new reheating scenario φ decays gradually, keeping. huge discrepancy discrepancy

Thermal mass is different from intrinsic mass. Coherent field oscillation is different from a collection of particles. Here, we consider ① Nonequilibrium field theory for the oscillating scalar field ② The case decay products do not have any thermal masses. ③ The case decay products have a large thermal mass. Assumptions & Conditions: ① Neglect cosmic expansion ② The would-be decay products of the oscillating field are in thermal equilibrium at a fixed temperature. ③ The oscillating field is in nonequilibrium and oscillating. ④ Parametric resonance ineffective (after the preheating stage, if any).

Fromalism to analyze behavior of oscillating scalar fields in a thermal bath

Time ordered & anti-ordered product Heisenberg picture time flow Coherent field oscillation behaves almost classically. But its decay is of course a quantum process. Derive an effective equation of motion for the expectation value of the scalar field φ by calculating its effective action Γ(φ). cf Quantity calculated in ordinary quantum field theory: Transition Amplitude What fraction of the initial state goes to the final state? time flow Time ordered product

oscillating scalar field (inflaton) φ→χχ φ→ψψ interacting field χ in a thermal state with temperature Model Lagrangian

Generating functional +branch -branch Effective Action in terms of the Legendre Transform Field variables also have suffices ±, and + fields interact with – fields, although they should be regarded as the same field in the end. time

Calculate the Effective action perturbatively. f f using finite-temperature propagators in the closed-time path formalism represent interactions between and.

Since and are identified in the end, it is more convenient to define and set in the end. f f Equation of motion

induced by the interaction. From now on, I concentrate on the diagram related with the decay process Since φ is a real scalar field, we cannot obtain a sensible equation of motion by the variation of such a complex-valued effective action Γ. Its contribution is complex-valued which is a manifestation of the dissipative nature of this interaction. imaginary part Its real part and imaginary part are mutually related. Effective Action

As we often encounter a complex-valued effective action or effective potential even for a real scalar field, there is a known prescription to obtain a real-valued equation of motion. instability,dissipation ① Introduce a real-valued random Gaussian auxiliary field and rewrite the effective action as including a path integral of. ② is a probability distribution function defined by Gaussian with a dispersion ： the imaginary part of the effective action

（ N.B. ） If we performed path integral over using Gaussian integral, we would recover the original complex-valued effective action.

The expectation value of the scalar field evolves according to the above Langevin equation. ③ Here we take variation of the effective action as it is. Manifestly real-valued equation of motion! Auxiliary field is treated as a random Gaussian noise with a dispersion. ④ Equation of motion: a Langevin equation auxiliary stochastic field quantum correction Memory term depending on the past Real part of the effective action: Deterministic terms in EOM Imaginary part of the effective action governs a Stochastic Noise term.,

multiplicative noise f f (N.B.) If we incorporate other diagrams, the Langevin equation has both additive and multiplicative noises and dissipation terms.

The Langevin eq. can easily be solved via Fourier transform. spatial Fourier transform General solution The memory of the initial condition is erased after temporal Fourier transform pure imaginary real

and that the mean square amplitude of each Fourier mode averages to zero noise correlation relaxation time, inverse dissipation rate relaxes to a constant. Next we calculate From the solution we find only the following term survives at late time

Inflaton’s dissispation rate is given by evaluated at. So we calculate using

annihilation termscreation terms First line Second line ー The ratio takes a constant. Detailed balance relation

annihilation terms creation terms ＋ Similarly, Fourier transform of the noise dispersion reads Equipartition Law For We therefore find that is, the Fluctuation-Dissipation Relation holds for low-momentum modes.

For general case without the limit, we find So leads to The scalar field φ relaxes into the thermal equilibrium state. Energy density of k mode thermal particles vacuum contribution =+ (N.B.) We can also show that the final state is the thermal equilibrium state for the case of other interactions, too, including the case with multiplicative noises.

The dissipation rate of the homogeneous mode ( ) has a simple form One particle decay rate in the vacuum Induced emission This dissipation rate vanishes if. at high temperature This δfunction vanishes if. Decay rate through Yukawa coupling Pauli blocking Cf The decay rate to fermions is suppressed by Pauli blocking. f f

Incorporation of the thermal mass of the decay products

In our scheme, thermal mass is included in, if we incorporate the finite-temperature self energy of χ, Σ(T ), in its propagator, because is determined by the pole of the propagator …. Σ ＝ ＋ ＋ ＋ …… ΣΣΣΣΣ Full or ‘dressed’ propagator original propagator Resummation Does this apply to the large thermal mass as well? Does the dissipation rate vanish if ?

Full propagator in the Matsubara representation Σ ＝ ＋ ＋ ＋ …… ΣΣΣΣΣ self energy due to χ’s interaction includes a thermal mass term such as which depends on the nature of χ’s interaction. Apparently, high-temperature effect closes the phase space of φ’s decay. 0 ?

Then the δ function is replaced by the Breit-Wigner form It is nonvanishing even when. However, contains an imaginary part as well, and the full propagator has a complex phase.

Dissipation rate of the zero mode coherent field oscillation of the inflaton φ for When, the dissipation rate reads It is nonvanishing and proportional to. Imaginary part of self energy dissipation rate of the decay product χ, not the inflaton φ. It depends on interaction of χ which thermalizes it. dominant

For example, if χ thermalizes through interaction, we find for the imaginary part of χ’s self energy. As a result, the dissipation rate of the inflaton φ is given by for dissipation rate to massless particles at high temperature suppression factor depending on the form of coupling constants in general. The dissipation rate of the inflaton is finite even when its decay product, χ, acquires a larger mass than the inflaton in a high temperature plasma. reheat temperature when inflaton decays to massless particles suppression factor

conventional in case thermal mass prohibits decay suppressed by coupling constants of the decay product actual thermal history

Oscillating scalar fields can dissipate their energy even if thermal masses of the decay products are larger than the oscillation frequency. Not only the thermal mass, namely real part of the self energy, but also its imaginary part of the would-be decay product,,plays an important role. When, the reheat temperature is suppressed by a power of coupling constants which thermalizes the decay product χ. conventional in case thermal mass prohibits decay actual thermal history gravitino abundance depends on the physics of decay products. supermassive particles could be created.

Application to the cosmological moduli * problem *Here we mean scalar fields typically with weak or TeV mass scales which have Planck-suppressed interactions with other fields

Modulus field does not move due to a large Hubble friction until the Hubble parameter decreases to its mass scale. Evolution of moduli fields in the early Universe φ It has a very long lifetime because it interacts with other fields only with the gravitational strength, It starts oscillation around a potential minimum when. φ This is a coherent oscillation of zero-mode scalar field condensate. much severer than the gravitino problem demolishes primordial nucleosynthesis must satisfy

What all the previous studies have neglected…. φ The cosmic temperature at the onset of moduli oscillation was very high.

is the decay rate in the vacuum, modified at finite temperature. e.g. Decay into two bosons  through the interaction If  was in thermal equilibrium, the decay rate is enhanced by the induced emission. But it does not help, because.

In the thermal background,  acquires a large thermal mass. If we could replace by, the coupling of moduli with  could be significantly enhanced, leading a much larger decay rate. Moduli are coupled with kinetic terms as well such as It has been concluded, however, that such couplings lead to the decay rate similar to using equation of motion.

Taking we find. Moduli decay right after they start oscillation if ? In the presence of thermal background, moduli fields decay as soon as they start oscillation ?? due to the fact that the decay product has a large thermal mass. Decay rate through would be given by Calculate the modulus dissipation rate with the same method

Oscillating scalar field Moduli Standard field  interacting with  Simple Model One loop effective action relevant to dissipation φ χ thermalizing interaction

One-loop Effective Action in terms of Real part: Dissipation Imaginary part: Fluctuation Langevin type Equation of motion auxiliary stochastic field quantum correction Memory term depending on the past

Moduli’s dissipation rate is given by the Fourier transform of the memory kernel which is related to the imaginary part of  ’s self energy. In order to take thermal effects on  into account correctly, we should use the dressed propagator of  in the loop calculation. The dressed propagator is obtained by resummation.   Σ ＝ ＋ ＋ ＋ …… ΣΣΣΣΣ Full or ‘dressed’ propagator original propagator Resummation (Matsubara representation) self energy,which are determined by  ’s interaction

Temperature-corrected moduli dissipation rate (to the first order in ) : themal mass of  The last term could be larger than the other terms. number of decay modes much larger than the vacuum value but yet insufficient in the current version of the paper, I have incorrectly neglected in the denominator and got a very large dissipation rate in proportion to. This point was indicated by Boedeker recently, who inappropriately put and obtained a dissipation rate.

number of decay modes In reality,… This factor is bounded by unity and is maximal when. This gives the largest contribution if, when we find enhanced by rather than. we require when. Putting, we find Modulus may be dissipated due to thermal effects. the modulus oscillation can be dissipated through the proposed mechanism. If this dissipation rate is larger than the cosmic expansion rate,

Several comments: The proposed mechanism works well only at relatively high temperature. Gravitino problem may still be a problem. Entropy production of several order of magnitude is required. So the reheat temperature after inflation must be sufficiently high. In this mechanism, the moduli fields decay into only particles in a high-temperature thermal state. This means that gravitinos are not produced practically in this mechanism. This may be helpful to the “new gravitino problem” in the heavy moduli scenarios.

Several comments: Thermal effects may also shift the potential minimum.   large shift at high temperature The proposed mechanism anchors the modulus at the finite-temperature minimum. After that, traces adiabatically as the universe cools down, because the Hubble parameter is much smaller than by that time. So this does not cause any serious problem. Thermal effects may also induce a mass term proportional to. Then the modulus field oscillates with the frequency and our mechanism may work when.

Thermal effects can dissipate the dominant part of the cosmological moduli problem, provided and.

Thermal effects in the early Universe is nontrivial and could be very important !