270. Cosmological fluctuations from IR cascading during inflation

Posted by Neil Barnaby

February 18, 2009

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This is a guest blog post by Neil Barnaby who is working in the Lev Kofman’s group at CITA, the very same I worked 3 years ago. Dmitry.

First off, I want to thank Dmitry for inviting me to write about my recent paper, arXiv:0902.0615. This work was done in collaboration with Zhiqi Huang, Lev Kofman and Dmitry Pogosian. (In case your insomnia is incurable, my website is here.) The abstract is below.

Cosmological Fluctuations from Infra-Red Cascading During Inflation

Authors: Neil Barnaby, Zhiqi Huang, Lev Kofman, Dmitry Pogosian

Abstract: We propose a qualitatively new mechanism for generating cosmological fluctuations from inflation. The non-equilibrium excitation of interacting scalar fields often evolves into infra-red (IR) and ultra-violet (UV) cascading, resulting in an intermediate scaling regime. We observe elements of this phenomenon in a simple model with inflaton $\phi$ and iso-inflaton $\chi$ fields interacting during inflation via the coupling $g^2 (\phi-\phi_0)^2 \chi^2$ . Iso-inflaton particles are created during inflation when they become instantaneously massless at $\phi=\phi_0$ , with occupation numbers not exceeding unity. We point out that very quickly the produced $\chi$ particles become heavy and their multiple re-scatterings off the homogeneous condensate $\phi(t)$ generates bremschtrahlung radiation of light inflaton IR fluctuations with high occupation numbers. The subsequent evolution of these IR fluctuations is qualitatively similar to that of the usual inflationary fluctuations, but their initial amplitude is different. The IR cascading generates a bump-shaped contribution to the cosmological curvature fluctuations, which can even dominate over the usual fluctuations for . The IR cascading curvature fluctuations are significantly non-gaussian and the strength and location of the bump are model-dependent, through and $\phi_0$ . The effect from IR cascading fluctuations is significantly larger than that from the momentary slowing-down of $\phi(t)$ . With a sequence of such bursts of particle production, the superposition of the bumps can lead to a new broad band non-gaussian component of cosmological fluctuations added to the usual fluctuations. Such a sequence of particle creation events can, but need not, lead to trapped inflation.

The paper is about a new mechanism for generating cosmological fluctuations during inflation. This is interesting in its own right, but I think that there are some aspects of our analysis that may have applications also outside of the context of inflationary cosmology. The mechanism produces some observable signatures in the CMB (we’re currently working to understand the precise signal in more detail) and is expected to arise naturally in a number of well-motivation particle physics models of inflation. However, for me, the most interesting result of this work is the observation that interactions during inflation involving an insignificant amount of very massive particles can lead to dramatic observational consequences in the CMB. In this sense, our work provides an example the sensitivity of inflationary cosmology to extremely UV physics.

Before I get into the details of the mechanism, let me make a few general comments about inflation and cosmological perturbations to set the stage. I expect that most readers of NEQNET have already heard that the inflationary paradigm is very successful. This scenario provides an elegant explanation for the observed homogeneity of our large-scale universe, while simultaneously providing a mechanism to explain the origin and properties of the small inhomogeneities that are present. In the minimal construction (a single scalar field slowly rolling down a flat potential) these inhomogeneities originate from the quantum vacuum fluctuations of the scalar field that drives inflation. Such fluctuations are “born” on small (sub-horizon) scales but get stretched out by the expansion of the universe. Once a given fluctuation mode crosses the horizon ( $k \leq a H$ ), it becomes classical and “frozen”, inducing a large-scale cosmological curvature perturbation. These primordial cosmological perturbations are observable in the present epoch as temperature fluctuations in the CMB.

The minimal scenario (single field, slow roll) is still consistent with the data, however, there is a lot of interest in exploring non-minimal models. There are a couple of reasons why I think this is a worthwhile exercise. The first reason relates to our experience with trying to embed inflation into realistic particle physics models. One of the lessons we’ve learned from this kind of model-building is that naively simple models (like chaotic inflation) can be non-trivial to realize in realistic settings. On the other hand, models that naively seem more complicated (for example hybrid inflation) can arise very naturally. (There are a now number of controlled realizations of hybrid inflation from string theory and SUSY whereas it took much longer to construct large field models on a similar footing.) Thus, until we have a good metric for what “simple” means in this context, it’s worth keeping an open mind. The second reason why I think it’s worth considering non-minimal models is simply the fact that there will be lots of new data coming in from future missions and it would be nice to be armed with predictions.

“Non-minimal” as I was using the word above, could mean a lot of different things (multi-field, non-standard kinetic terms, etc). Let me discuss, in particular, some non-minimal mechanisms for generating cosmological perturbations during inflation. The literature contains several examples including the curvaton mechanism and inhomogeneous reheating. In both cases, one relies on some additional (non-inflaton) light field(s) being present during inflation. These acquire large-scale iso-curvature fluctuations during inflation (as in process described above) and at some point (usually after inflation) there is a transfer to the curvature fluctuation. Below, I’m going to discuss a qualitatively new mechanism that instead uses the non-equilibrium dynamics of interacting scalar fields during inflation.

The model we consider is very simple and generic. Suppose the rolling inflaton $\phi(t)$ is coupled to some iso-curvature mode $\chi$ (the iso-inflaton) as follows:

${\cal L}_{\rm int}=-\frac{g^2}{2} (\phi-\phi_0)^2\chi^2$

Here $\phi_0$ is some value that the rolling inflaton $\phi(t)$ crosses during the observable range of e-foldings. Of course, this is a very generic scenario that will admit lots of different microscopic realizations; I’ll discuss some of my favorites later on. (This kind of coupling is typical for studies of preheating, moduli trapping and is also relevant to the thermalization of QFTs.)

The interaction above induces a mass for the iso-inflaton of the following form:

$m_\chi^2=g^2 (\phi-\phi_0)^2$

The $\chi$ particles become instantaneously massless at some point during inflation when $\phi=\phi_0$ . For interesting values of the coupling g^2 the iso-inflaton remains light only for a very brief moment (much less than an e-folding). During this brief moment $\chi$ particles are produced quantum mechanically by draining energy from the condensate $\phi(t)$ . This burst of particle production is completely analogous to what occurs during preheating and during moduli trapping. (There’s also a very nice analogy with Schwinger pair production in a strong electric field.)

Before going any further, it’s useful to compare this situation to what occurs during preheating after inflation. A key difference is that in our scenario there is just one burst of particle production so the number of produced $\chi$ particles is small (the occupation number $n_\chi(k) \leq 1$ ). On the other hand, for broad resonance preheating one has many, many bursts of particle production so that a huge number of particles builds up over many oscillations of the background field.

What is the fate of these produced iso-inflaton particles? Very shortly after being created they acquire a huge mass become non-relativistic. At late times their number density thus dilutes as $a^{-3}$ with the expansion of the universe. However, producing this gas of massive particles drains energy from the condensate, so the background inflaton $\phi(t)$ must slow-down a little bit; see the figure. Previous studies have focused on this momentary slowing-down so I’ll quickly describe the effect, however, we’ll soon see that this is not the most important aspect of the dynamics.

270. Cosmological fluctuations from IR cascading during inflation

Shortly after the burst of particle production and associated dip in $\dot{\phi}$ the inflaton quicklysettles back onto the attractor and the produced $\chi$ particles are diluted. However, the momentary violation of slow-down induces a ringing in the spectrum of fluctuations for modes leaving the horizon at the time when $\phi=\phi_0$ . This story is very similar to what happens if you put a step (or a sharp feature) in the inflaton potential. In both cases, the ringing in the power spectrum is somewhat analogous to the pattern one observes for Fresnel diffraction at a sharp edge; see the picture below.

$Fresnel diffraction at a sharp edge$

The similarity to models with steps in the potential suggests another motivation for our analysis: you can think of this paper as a warm-up exercise for the case where the rolling inflaton induces a second order phase transition in some iso-inflaton field during inflation. Such inflationary phase transitions are rather popular in the literature and provided the motivation for putting steps in the inflaton potential in the first place.

Now, back to the topic at hand. As I alluded to above, the focus on the momentary velocity dip due to particle production misses some crucial physics. In particular, the discussion above doesn’t fully capture the inhomogeneous nature of particle production. One should account for fact that the produced iso-inflaton particles can generate inflaton fluctuations (particles). The most important interaction turns out to be the one in the figure below. (I hope that by focusing on this diagram I don’t give the impression that we have neglected all other interactions. Our lattice field theory simulations take all interactions/nonlinearity into account self-consistently.)

270. Cosmological fluctuations from IR cascading during inflation

This rescattering diagram describes the production of a $\delta\phi$ particle by an interaction between one of the produced iso-inflaton particles and the condensate. What’s nontrivial is how surprisingly efficient this rescattering process is and what a huge impact it has on the cosmological perturbations. Rescattering leads to a rapid build-up of power in long-wavelength inflaton fluctuations that we called IR cascading. IR cascading takes less than one e-folding to complete so it turns out that the $a^{-3}$ volume diltution of iso-inflaton particles is not very important. The efficiency of rescattering and IR cascading can be easily understood on physical grounds as follows:

The production of inflaton fluctuations deep in the IR is very energetically cheap (these are essentially massless compared to the iso-inflaton).
The same iso-inflaton particle can rescatter many times of the background condensate and generate lots of IR inflaton particles. This means that we can have very large occupation numbers for the IR cascading inflaton fluctuations, even though the we only had a small amount of iso-inflaton particles initially.

Long wavelength inflaton fluctuations play a special role in cosmology. These become frozen once their wavelength exceeds the Hubble radius and they lead to cosmological perturbations. The long-wavelength inflaton fluctuations from IR cascading are complimentary to the usual vacuum fluctuations from inflation. So, this process of IR cascading leads to observable features in the CMB. What’s surprising is how large these features are, even with modest values of the coupling constant.

The impact of IR cascading on the power spectrum is illustrated in the figure below. Just after the moment of particle production we see a bump-shaped contribution to the spectrum which is peaked inside the horizon. Very quickly multiple rescatterings lead to a rapid build-up of power in the IR causing the peak to grow and shift to longer and longer wavelengths. Within a single e-folding the peak has reached the horizon and frozen in. Depending on the value of the coupling, the fluctuations from IR cascading actually dominate over the usual vacuum fluctuations from inflation! Moreover, for all choice of g^2 this effect completely swamps the ringing induced by the momentary slow-down (discussed above).

270. Cosmological fluctuations from IR cascading during inflation

Additional bursts of particle production and IR cascading will lead to a additional bumps in the power spectrum. A sequence of densely spaced points $\phi_{0i}$ where new states become massless will lead to a sequence of bumps that can superpose to generate a broad-band spectrum.

We are currently working to understand in more detail the observational signal in the CMB. Of particular interest are the detailed properties of the induced nongaussianities. Preliminary analysis shows that the fluctuations from IR cascading are very nongaussian, however, the bispectrum peaks only over a narrow range of scales. (This type of nongaussianity is not very well constrained by observations.) Nongaussianity in the CMB is a topic close to my heart; I’ve been interested for a while in the idea of using higher order statistics like the bispectrum as a tool to discriminate between inflationary models. There are now a number of models in the literature that generate observable nongaussianity (of course everyone has their own personal favorite) and it will be very exciting to see what PLANCK discovers.

Now that I’ve fleshed out some of the details of IR cascading, let me say a few words about how typical this effect is in realistic particle physics models of inflation. The required interaction is fairly generic and there will certainly be many different microscopic realizations. Here I’ll just mention a few of my favorite models for the sake of explicitly demonstrating that there exist robust inflation models where $\phi_0$ is in the observable range. One might, for example, wish to consider models of inflation from string theory. In many such models the inflaton has a geometrical interpretation as the position of a brane moving around in the compact space (examples include brane-antibrane, D3/D7, DBI and brane monodromy models). In any such model there will be stretched strings between the inflaton brane and any other “spectator” branes living in the compactification. If the inflaton brane becomes coincident with a spectator brane during its inflationary rolling, some of those stretched string modes will become massless, emulating our prototype coupling. An explicit, controlled, stabilized implementation of this scenario was provided by Silverstein and Westphal using monodromy.

Finally, let me point out that couplings of the type that we studied play a crucial role in a model called “trapped inflation”. In that case, one uses multiple bursts of particle production to slow down the fast rolling inflaton. This idea was first suggested in this paper and the details have been worked through in a very interesting recent paper by Green, Horn, Senatore and Silverstein. (Another model, called “warm inflation”, also uses dissipative dynamics to slow down the inflaton.) Our IR cascading mechanism can (but need not) be associated with trapped inflation.

That pretty much covers everything I wanted to say. The reader is invited to consult the actual paper for more details about how we did the calculation (both on the lattice and also using analytical QFT tools). For me, one of the most interesting aspects of this work is the fact that subtle QFT effects during inflation – involving an insignificant number of very massive particles out of equilibrium – can lead to such a dramatic observable signature in the CMB. (Indeed, you might argue that this effect probes Planck-scale physics since typically the iso-inflaton particles have a mass of order gM_p in the current epoch.) This sort of UV sensitivity in inflationary model building has driven a fruitful interaction between cosmology and particle physics in recent years. As more cosmological data comes in this interaction promises to become richer still.

Cosmological perturbations and large scale structure, Cosmology, Journal club, Master Mason, Quantum field theory

If you liked the post, please kindly consider to leave a comment, subscribe to the RSS feed or get new posts sent directly to your Inbox. If you want to chat with me in real time, you can find me on Twitter. The posts below are probably related to the subject of this one:

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21 Comments »

Comment by Dmitry

2009-02-18 16:49:12

Dear Neil,

thanks for the nice post! There is a part in your scenario which, I guess, I don’t understand: $\chi$ particles are produced in the same way as particles of any field which is lighter than in de quasi-Sitter space ( is the Hubble scale).

The characteristic wavelength of those produced modes is about inverse itself, i.e., particles are produced with wavelengths of the order of horizon. The amplitude of corresponding modes gets frozen very quickly after they cross the horizon (several efoldings), and they certainly don’t contribute into any rescattering whatsoever as long as $\lambda_{ph}\gg{}H^{-1}$ .

So, are you talking about $\phi$ particle production during rescattering with extremely long wavelength (horizon size) $\chi$ modes during that couple of efoldings that is needed for them to become classical? If yes, I have a impression that the mean free time (the time needed for rescattering) is extremely long for those modes ( $\tau\sim\frac{1}{n\sigma}\gg{}H$ since is small), and if so, rescattering cannot be efficient.

Could you provide an estimation for the mean free time?

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-18 17:12:09

Dear Dmitry,

Thanks again for the opportunity to post!

No: the $\chi$ particles are not produced in the standard way (dS vacuum fluctuations). Initially, $\chi$ is extremely massive and (obviously) has no large scale correlations. At the moment t=t_0 the iso-inflaton particles become instanteneously massless and get produced with characteristic wavelength $k_{star}^{-1}$ which is inside the horizon. (This moment of production is just like broad band preheating.) By the time rescattering starts to become important the $\chi$ particles have acquired a huge mass via the interaction with the inflaton. Rescattering generates inflaton modes with a whole spectrum of wavenumbers. Initially their power is peaked near $k_{star}$ but quickly the IR cascade moves the peak to where they freeze in and become classical. The characteristic time scale for the cascade is
$\Delta t \sim k_{star}^{-1} < H^{-1}$ so there should be lots of rescatterings per e-folding.

Best,

Neil

Neil, fixed your LaTeX and removed the second duplicate comment.

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Comment by Dmitry

2009-02-18 18:49:07

Dear Neil,

I went through the paper and a kind of see now what you mean.

I still have a question: deriving the formula (4) in the paper for the occupation number of $\chi$ particles how do you take into account that you are in the quasi-de Sitter universe? Actually, the formula (4) looks like the Schwinger rate for me in the flat spacetime, and from the Appendix A I see that this is indeed what you do – take estimation the number of produced particles from the flat spacetime case. Why the large friction term is not important?

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-18 19:00:22

Dear Dmitry,

The initial burst of $\chi$ particle production happens quickly compared to the Hubble time and generates sub-Hubble modes, so there’s no problem with neglecting the expansion. So equation (4) (which is indeed the flat-space Schwinger result) is perfectly valid.

Next, you could ask about the effect of Hubble friction on the subsequent process of rescattering. The DEFROST calculation incorporates the expansion self-consistently, so we’re not neglecting anything. For the analytical work in appendix A we dropped the friction because the whole IR cascade only takes a single e-folding. The analytical and numerical results line up pretty nicely, justifying that approximation.

I should mention that there’s no obstruction to including the Hubble friction in the analytical calculation, I’ve managed to do this in an unpublished note.

Best,

Neil

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Comment by Dmitry

2009-02-18 20:28:55

Dear Neil,

I should mention that there’s no obstruction to including the Hubble friction in the analytical calculation, I’ve managed to do this in an unpublished note

If you already know the answer, could you write how finite constant affects the formula for the Schwinger rate?

Cheers,
Dmitry.

P.S. The reason I ask all these questions is that I always had a kind of different picture in mind. Namely, if there are light fields $\phi_n$ during inflation ( m<H ), their expectation values $\langle\phi_n^2\rangle$ grow stochastically, but the main contribution into this growth comes from superhorizon modes (occupation numbers there also become large compared to 1). I really want to understand where this intuition goes wrong in this particular example (or maybe just incomplete) and why the effect you are talking about is dominant.

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Comment by Neil Barnaby

2009-02-18 21:07:09

Dear Dmitry,

First consider effects of expansion on eqn (4). If you re-scale the modes by factors involving and work in conformal time it amounts to just adding a contribution like a'{}'/a to the frequency. This term is tiny over the time interval that $\omega_k$ varies non-adiabatically. The field re-scaling will just lead to the usual $a^{-3}$ dilution for non-relativistic particles.

For the inflaton modes the correction is less trivial (because you need to use a more complicated Green function) but the calculation is still tractable.

Best,

Neil

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Comment by Dmitry

2009-02-19 15:15:31

Hi Neil,

Thanks for the explanations! Not sure about smallness of a'{}'/a , this is basically the curvature scale, in quasi de Sitter case it is constant and large ( $H\gg{}m_{\rm eff}$ at least for the regime you are considering). I feel that ultimately I will believe in what you say after I explicitly solve F.-P. equation for the $\chi$ field with $g^2{}(\phi{}-\phi_0)^2\chi^2$ potential, that’s what I am going to do now…

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-19 15:56:52

Hi Dmitry,

No, the inflaton is light ( $H\gg m_\phi$ ) but for the $\chi$ particle the situation is opposite. The iso-inflaton is not a light field.

The $\chi$ particle has mass

$m_{\chi}^2=g^2(\phi-\phi_0)^2 \cong k_{\star}^4 t^2$

Because $k_{\star} > H$ this is less than H^2 only over a time interval $\Delta t \sim k_{\star}^{-1} < H^{-1}$ (less than an e-folding). So the scale factor $a(t)=e^{Ht}$ is almost constant during that brief massless interval.

In you prefer to work in conformal time, the effective frequency is:

$\omega_k^2 = k^2 + a^2 g^2 (\phi-\phi_0)^2 – a'{}'/a$

Up to slow roll parameters this gives:

$\omega_k^2 = k^2 + \frac{1}{\tau^2}\left[\frac{k_{\star}^4}{H^4}\ln^2(-1/H\tau) - 2\right]$

(I’ve set the origin of time so that $\tau=-1/H$ is the moment when $\phi=\phi_0$ .) Because $k_{\star} > H$ the term coming from a'{}' /a is subdominant.

Cheers,

Neil

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Comment by Dmitry

2009-02-19 17:00:08

Dear Neil,

although k_*^4/H^4 is large, logarithm $\log^2(-1/H\tau{})$ is small in the regime your are interested in (and exactly equal to zero at $\phi=\phi_0$ ).

On the other hand, the term a'{}'/a is of the order H^2 near $\phi=\phi_0$ and cannot be neglected during the conformal time interval

$\delta\tau\sim{}H^{-1}\exp{}(-H^2/k_*^2)$ .

as follows from your formula for frequency.

Since k_* is not terribly larger than , this seems to be more or less the interval where you produce particles due to non-adiabaticity breakdown.

Cheers,
Dmitry.

P.S. Sorry for being buggy.

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Comment by Neil Barnaby

2009-02-19 17:47:23

The logarithm is small during an interval much less than an e-folding. You can use conformal time or whatever coordinate you like, the result is still that during the nonadiabatic regime the change in the scale factor is $\delta a / a < 1$ .

Explicitly: setting $\tau=-1/H+\delta\tau$ you find that the logarithm is subdominant over an interval
$H\delta\tau \sim 2 H^2/k_{\star}^2$ . During that interval the scale factor $a=-1/(H\tau) = 1/(1-H\delta\tau)$ changes by an amount $\delta a \sim H\delta \tau \sim 2 H^2/k_{\star}^2 \sim 0.002 < 1$ for $g^2\sim 0.1$ .

Obviously there will be some small corrections to the Schwinger formula near k=0 but those are irrelevant for our calculation because the power in produced chi peaks near $k_{\star}$ which is inside the horizon.

One could re-compute the Bogiliubov coefficients including the Hubble friction numerically and compare to eqn (4). We already did that and, as expected, the effect is small.

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Comment by Dmitry

2009-02-19 18:13:31

Neil,

you need to estimate a derivative, not $\delta{}a$ – that is, you need to divide small $\delta{}a$ by another small quantity – $\delta\tau$ . The result is not so small quantity which has the order of magnitude of , i.e., square root of de Sitter curvature ( a'{}'/a is also not small, especially compared to effective mass, it is of the order of dS curvature).

obviously there will be some small corrections to the Schwinger formula near

That was exactly my question from the beginning – why they are small and what is the small parameter that suppresses them. It seems to me that you neglect the curvature term exactly in the regime where it is important. Numerics is a good argument, I just don’t see physically why the curvature term can be neglected.

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-19 18:53:52

All I’m saying here is that if you consider very massive fields with sub-hubble wavelength over a time interval less than an e-folding then it will be a good approximation to ignore the expansion. I don’t think this is a controversial claim.

Think about comparing the $\ddot{\chi}$ and $3 H \dot{\chi}$ terms in the equation of motion. If the field varies appreciably over a time interval $\Delta t \sim k_{\star}^{-1} < H^{-1}$ then, to first approx, you can ignore the friction term.

The $\partial_\tau^2 a / a$ term gives the in my formula. For all but a tiny interval, this term is subdominant to the logarithm (from the mass). Even at the massless point, for the interesting modes, $H < k < k_{\star}$ , this term is subdominant. At k < H nothing interesting is going on. Remember that $\chi$ was heavy for some number of e-foldings before $\phi=\phi_0$ so it has no superhorizon fluctuations anyway.

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Comment by Dmitry

2009-02-19 19:24:19

Neil,

Let us work in conformal coordinates.

$\partial_\tau^2 a / a$ is equal to $- 2/\tau^2$ , that is, it is of the order 2H^2 near $\phi=\phi_0$ .

In principle, I could drop it out as you say since

$\phi_k'{}'\sim \frac{k_*^4}{H^2}\phi_k$ ,

while

$a'{}'/a\phi_k\sim{}H^2\phi_k$ ,

so the curvature term is suppressed by the power $\frac{H^4}{k_*^4}$ for modes peaked near $k\sim{}k_*$ .

It is however not suppressed for modes with $k\lesssim{}H$ , moreover, it introduces tachyonic mass for them, which leads to their growth. So, I disagree that nothing interesting happens for modes with k<H .

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-19 19:46:40

Hi Dmitry,

It’s very misleading to refer to that term as “tachyonic”. Such a term dominates the effective frequency of the large scale inflaton modes but it doesn’t lead to uncontrolled super-horizon growth.

Best,

Neil

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Comment by Dmitry

2009-02-19 21:28:52

Such a term dominates the effective frequency of the large scale inflaton modes but it doesn?t lead to uncontrolled super-horizon growth.

Ok, sure. What I want to say is that therу was some interesting near-horizon effect (as for any light field in dS background), but for a very short time – as long as the field was effectively massless. The effect was extremely weak, but on the other hand you also talk about exponentially small effect (Schwinger rate is exponentially suppressed). So, one has first to compare two small particle production effects and see which one is more important.

Suppose that your effect is more important. You say that it is anyway irrelevant, since what affects dynamics of the inflaton is rescattering. Let me now understand this statement, if you did not get bored too much by the discussion.

Particle production rate of $\delta\phi$ from the diagram you draw is proportional to the particle number density of $\chi$ generated from the Schwinger process. This particle number density is a) small from the very beginning (according to (4) $n_k\lesssim{}1$ for interesting modes), b) number density is actually proportional to $a^{-3}\sim\exp{}(-3Ht)$ . Why is there some nontrivial kinetics before the particles of $\chi$ get exponentially diluted?

Cheers,
Dmitry.

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Comment by Dmitry

2009-02-19 23:09:18

Ok, really stupid question. I went through the paper again – the answer is that dynamics happens at time scales smaller than 1 efolding anyway, that’s why you don’t have any redshifting in your formulae. The overall effect is of course small – suppressed by Schwinger exponent, one power of g^2 and small phase volume of the IR fluctuations – but that what it is.

Thanks a lot, Neil! Everything is much clearer to me now.

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-19 23:20:58

Great, I’m glad we agree on this.

For the record, the expansion is consistently included in the numerics, we just left it out of the analytical calculation in appendix A for simplicity.

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Comment by Dmitry

2009-02-19 23:27:14

Ok, I’ll have to check out how Frolov treats superhorizon modes in DEFROST

(Comments won't nest below this level)

Reply here

Comment by Neil Barnaby

2009-02-19 22:58:52

First off, the Schwinger result is not exponentially suppressed in the interesting regime $H<k<k_{\star}$ . It is exponentially suppressed deep in the UV $k\gg k_{\star}$ but that’s fine, nothing interesting happens there.

Next, on scales $k \ll H$ there were no super-horizon fluctuations of $\chi$ for $\phi > \phi_0$ because it was massive. Recall that the mode function for a massive field in dS damps exponentially on large scales. Of course the mass is decreasing and the field is effectively light very near to $\phi=\phi_0$ . But 1/10 of an e-folding is not enough to generate any significant effect outside the horizon. We’ve already verified this by computing the Bogoilubov coefficients numerically.

Next, onto the issue of rescattering. What’s important is the power in produced IR inflaton modes. That this is larger than one might naively expect is precisely the central claim of the paper. Intuitively you can understand this by noting (1) that production of very IR inflaton modes is very cheap, and, (2) that the same iso-inflaton can produce many inflaton modes. The time scale for this cascading is $k_{\star}^{-1}$ which is fast compared to the Hubble time. So there is lots of interesting dynamics before the dilution becomes important.

Obviously this intuitive argument is not completely convincing. That’s why in the paper we backed that argument up in two ways: (1) analytical computation of the correlators using QFT tools, and, (2) fully nonlinear lattice field theory simulations. Both agree with each other (see the power spectrum above, the dots are numerics and the solid lines are analytics). Moreover, both agree with the intuitive argument I made above.

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Comment by Dmitry

2009-02-19 23:24:00

Just wrote a reply to myself above But let me argue with you a bit more for the sake of arguing…

First off, the Schwinger result is not exponentially suppressed in the interesting regime $H<k<k_{\star}$ .

No, but n_k is of the order 1 for the modes you are interested in. There is no kinetic amplification of rescattering familar to us from the physics of preheating, just rigid gas of $\chi$ particles. But as I said above – the effect is what it is and it is definitely there.

Next, onto the issue of rescattering. What?s important is the power in produced IR inflaton modes. That this is larger than one might naively expect is precisely the central claim of the paper.

What is naive estimation? The particle production rate is proportional to g^2 , to initial occupation number of $\chi$ particles, and to the phase volume k^3 (just write a collision integral)/tex. That is exactly your formula (8).

But 1/10 of an e-folding is not enough to generate any significant effect outside the horizon. We?ve already verified this by computing the Bogoilubov coefficients numerically.

As I said above, effect is certainly small but your effect is also small, so one needs to honestly compare them. Regarding the Bogolyubov coefficients – did you calculate them with fixed ? If yes, I thought I already convinced you above that this is not very good idea since a'{}'/a is of the order H^2 .

Well, thanks for the discussion anyway, it was really great. Without exaggeration, I enjoyed it a lot.

Cheers,
Dmitry.

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Comment by Neil Barnaby

2009-02-20 17:37:06

Hi Dmitry,

A few more comments:

(1) Right, $n_\chi(k) \leq 1$ for all and, for the modes of interest, $n_k \sim 1$ . The large effect on the power spectrum isn’t because we have a lot of $\chi$ particles, it’s because of the IR cascade. We’re not at all claiming that the production of iso-inflaton modes is particularly violent. The effect is all coming from what happens after the iso-inflatons are produced. The same $\chi$ can produce lots of IR $\delta\phi$ by multiple rescatterings in less than an e-folding. Because of multiple rescatterings, you can have $n_\phi(k) > 1$ even though $n_\chi(k)\leq 1$ . (Recall that the large contribution to the power spectrum is from the cascading inflaton modes, not from the iso-inflaton particles.)

(2) If you agree with formula (8) in the text then the naive estimation is trivial. Set m=0 to make life simpler. Now set $k \sim H$ and $\Delta t \sim H^{-1}$ (because we’re talking about modes near the horizon after about an e-folding). Then you trivially find $P_k / H^2 \propto g^2 k_\star^3 / H^3$ which is very large. Putting the factors of $2\pi$ (which make a difference) you find $P_{\mathrm{resc}} / P_{\mathrm{vac}} \sim 10$ which is consistent with fig 3 up to factors order unity, as it should be.

(3) For the issue of generating super-horizon $\chi$ at the moment $\phi=\phi_0$ we may have to agree to disagree. I have already given the intuitive argument for why nothing interesting happens. Also, I have explained that we already checked by computing the Bogoliubov coefficients numerically.

NOTE: Of course we did that check in de Sitter space. You don’t need a computer to solve for the Bogoliubov coefficients with a=1 , it’s a text-book case. We didn’t put this in the paper because it’s not at all interesting.

However, I want to impress upon you that even if you did somehow generate super-horizon fluctuations in the tiny fraction of an e-folding that $\chi$ was light, it still wouldn’t change our final answer. There are several reasons:

FIRST, note that all the contribution to the IR inflation power comes from $\chi$ modes near $k=k_\star$ (see equation 6). This is what I mean when I say that all the interesting modes are near $k=k_{\star}$ . The fact that the $\chi$ power peaks near $k=k_{\star}$ is precisely the reason that you get the factor of $k_{\star}^3$ in equation (8).

SECOND, remember than $\chi$ was MASSIVE for some number of e-foldings before $\phi=\phi_0$ . I really want to stress this: heavy fields in dS have exponentially damped fluctuations on large scales! So any non-trivial effect would have to be exponentially large to overcome this damping. Heavy fields in dS behave very differently from light fields on large scales. Heavy fields do not acquire super-horizon correlations.

THIRD, even if you somehow did generate a lot of superhorizon $\chi$ fluctuations (you don’t, but let’s pretend) it would make our effect even stronger; you would get get a much larger effect on the large scale inflaton power spectrum.

(4) Thanks again for the opportunity to post and for all the discussion. Jolly good fun!

270. Cosmological fluctuations from IR cascading during inflation

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