Stochastic growth of quantum fluctuations during slow-roll inflation
We compute the growth of the mean square of quantum fluctuations of test fields with small effective mass during a slowly changing, nearly de Sitter stage which takes place in different inflationary models. We consider a minimally coupled scalar with a small mass, a modulus with an effective mass (with the Hubble parameter) and a massless non-minimally coupled scalar in the test field approximation and compare the growth of their relative mean square with the one of gauge invariant inflaton fluctuations. We find that in most of the single fields inflationary models the mean square gauge invariant inflaton fluctuation grows faster than any test field with a non-negative effective mass. Hybrid inflationary models can be an exception: the mean square of a test field can dominate over the gauge invariant inflaton fluctuation one on suitably chosen parameters. We also compute the stochastic growth of quantum fluctuations of a second field, relaxing the assumption of its zero homogeneous value, in a generic inflationary model; as a main result, we obtain that the equation of motion of a gauge invariant variable associated, order by order, with a generic quantum scalar fluctuation during inflation can be obtained only if we use the number of e-folds as the time variable in the corresponding Langevin and Fokker-Planck equations for the stochastic approach. We employ this approach to derive some bounds for the case of a model with two massive fields.
The theory of quantum fields in an expanding universe has evolved from its pioneering years (1) into a necessary tool in order to describe the Universe on large scales. The de Sitter background - characterized by the Hubble parameter being constant in time (for the flat spatial slice), where is the scale factor of a Friedmann-Robertson-Walker (FRW) cosmological model - has been the main arena to compute quantum effects even before becoming a pillar of our understanding of the early inflationary stage and of the recent acceleration of the Universe.
However, while for any inflationary model, may not become zero in a viable model, apart from some isolated
moments of time. Indeed, the standard slow-roll expression for the
power spectrum of the adiabatic mode of primordial scalar
(density) perturbations becomes infinite, i.e., meaningless, if
becomes zero during inflation.
In this paper we wish to tackle in more detail the moduli problem issue. On one hand we wish to extend our results in (7) to different inflationary models and to different types of test fields, not only to massive minimally coupled scalar fields, but also to massless non-minimally coupled scalar fields and moduli with an effective mass . We investigate effects and we therefore need to consider both these extensions. On the other hand, it is known that the mean square of gauge invariant inflaton fluctuations grows (5) and the stochastic description for this effect for a general potential has been established (7); it is therefore interesting to compare the quantum amplification of test fields not only with the background inflaton dynamics, but also with the stochastic growth of gauge invariant inflaton fluctuations. This comparison aims for a self-consistent understanding of quantum foam during inflation.
We then discuss the diffusion equation for general scalar fluctuations in a generic model of inflation. On using the results obtained by field theory methods, we show that the diffusion equation for the gauge invariant variable associated with this generic scalar fluctuation should be formulated in terms of the number of e-folds .
The paper is organized as follows. In Section II we choose four representative cases of the inflationary “zoo” on which we focus in this paper. In Section III we compute the stochastic growth of a minimally coupled scalar with a small mass, a modulus with an effective mass , and a massless non-minimally coupled scalar in the test field approximation for the four inflationary models considered. In Section IV we review the result obtained in Ref. (7) for gauge invariant inflaton fluctuations and compare it with the growth of the test fields in the four different inflationary models considered. In Section V we examine the stochastic approach for a two field model for two generic self-interacting potentials, and we compare our solution, obtaining some constraints on the parameters, for a particular two quadratic field model. In Section VI we illustrate our conclusions.
Our paper does not include a derivation of the stochastic approach. Indeed an introduction of the stochastic method is given in Refs. [14,18] and a comparison between stochastic methods and quantum field theory results is done in Refs. [14,18] and in our previous paper .
Ii Inflationary Models
The detailed evolution of the expansion during the accelerated stage depends on the inflaton potential and so does the growth of quantum fluctuations. For this reason we consider in the following four different potentials which are representative of the “inflationary zoo”. Since we shall study the growth of quantum fluctuations as a function of the number of e-folds
we shall give the evolution of the Hubble parameter as a function of .
During the slow-roll trajectory we have:
where (these formulas were obtained already in (20) in the context of a closed bouncing FRW universe with two quasi-de Sitter stages during contraction and expansion). Let us note that in the numerical results presented in the figures all dimensional quantities have been rescaled w.r.t. . We then consider the case of a quadratic potential (of arbitrary sign) uplifted with an offset :
With the positive sign the potential in Eq. (5) is an approximation for the simplest model of hybrid inflation well above the scale of the end of inflation; in this case decreases during the inflationary expansion. With the negative sign the potential in Eq. (5) is a simple small field inflation model, again far from the end of inflation; in this case increases during the inflationary expansion. In the following we use the approximate solution for the square of the inflaton as:
valid when , i.e.
As another large field inflationary model we consider an exponential potential
This potential leads to a power-law expansion given by:
with (21) (we consider for the sake of simplicity). Such a solution is stable for and the slow-roll conditions are well satisfied for . In particular one obtains
Iii Growth of test fields with small effective mass in the stochastic approach
We shall consider a test scalar field with a small effective mass and a zero homogeneous
expectation value on an inflationary background driven by an inflaton with
potential in the slow-roll approximation.
The evolution equation for the renormalized mean square
(the pedix will denote
renormalized in the following)
in the next three subsections
(Eqs. 11, 16 and 22) follows in a straightforward manner from our previous paper .
Note that the right hand side of Eqs. (11,16,22) representing the contribution of created
fluctuations (”particles”) is obtained under the natural
assumption of the absence of particles in the in-vacuum state,
more exactly that each Fourier mode of the quantum field
was in the adiabatic vacuum state deep inside the Hubble
radius and long before the first Hubble radius crossing during
inflation, i.e., when its energy was much larger than
iii.1 Growth of scalar fields with in the stochastic approach
The stochastic equation is:
Its general solution is
which is just the integral form of Eq. (13) of Ref. (7) generalized to an arbitrary inflaton potential.
For the quadratic inflaton case we report here the solution given in
where we have assumed (we shall adopt the same choice afterwards if not otherwise stated). We then consider the potentials in Eq. (5), in the lowest order approximation; that is, for , we have
Let us note that the corrections induced by a non-zero term are typically small both for the case of hybrid inflation as well as for small field inflation (as long as the field does not grow too much due to instability). The corresponding analytic expressions, obtained using Eq. (6), can be written in terms of hypergeometric functions but we do not report them here. For the exponential potential we obtain:
where is the exponential integral function (see, for example, (23)).
iii.2 Growth of moduli fields with in the stochastic approach
If , the stochastic equation takes the form:
Its general solution is
For we obtain:
is equal to maximal number of possible
e-folds in this chaotic model. In the limiting case and
at the end of inflation (), we recover the
For the power-law inflation case we obtain:
Let us note that for this particular model, so the results above are also valid for the case with .
iii.3 Growth of non-minimally coupled scalar fields in the stochastic approach
The stochastic equation is now:
where is the non-minimal coupling to the Ricci scalar and we assume that (however, may be large). Indeed, the term in the action proportional to gives an effective time dependent mass for : where .
Its general solution is
If we again consider the chaotic scenario induced by a massive inflaton field as in the previous subsections, the integral can be easily computed in a closed form in terms of the exponential integral function . Assuming initially, one finds
One can verify that in the limit at the end of inflation and a fixed large value for , the result of Eq. (19) for a massless modulus is again reobtained.
For the potential in Eq. (5) we have, in the same approximation as in the two previous subsections,
As before, let us finish with the case of a power-law model of inflation. We obtain:
Iv Comparison with the growth of inflaton fluctuations
The results of the previous section should be compared with the growth of gauge-invariant inflaton fluctuations , the Mukhanov variable (24) which is used to canonically quantize the Einstein-Klein-Gordon Lagrangian. The evolution equation for found in (7) can be re-written as:
In Eq. (27) the positivity of is not determined by the convexity of the potential, i.e. , as we would expect in the absence of metric perturbations. The threshold corresponds to the following condition on the potential:
With the use of the slow-roll expressions for the scalar spectral index and the tensor-to-scalar ratio , one obtains:
and Eq. (27) can be rewritten as:
Both Eqs. (29) and (32) tell us that power-law inflation, for which holds, lies at the threshold between two opposite behaviors. Power-law inflation with is allowed at the 95 confidence level (25). We note that Eq. (32) is the same for a modulus with the mass and : below the power-law inflation line inflaton fluctuations behave as a modulus with negative .
The solution of Eq. (27) is:
For the quadratic chaotic potential the solution was found in (7):
For the potential in Eq. (5) we obtain the following in the lowest non-trivial approximation for small -dependent corrections in the potential:
where, as discussed previously, the case with a minus sign in the exponent refers to the hybrid model whereas the other case is associated with the small field inflationary model. Let us note that for a small value of the exponent (an almost constant potential) or for very small (), by expanding up to the linear order, one obtains almost the case of a de Sitter background (), with linearly growing in . In this approximation we see that the hybrid model is characterized by bounded as . This statement remains true even after dropping our approximations (see below).
This leads, by invoking the consistency of the perturbative expansion in field fluctuations through the condition , to the following hierarchy which the inflationary model has to satisfy:
where we have set . From this expression, when analyzing the hybrid inflation case, one can notice that the fluctuations have a maximum for a certain amount of e-folds and then decay to the asymptotic value for a large number of e-folds. Nevertheless, such a maximum is typically a few percent above the asymptotic value which has been already obtained above using a more crude approximation.
For the exponential potential we find
which at late times (see also (26)) becomes:
We show in Figures 1, 2, 3 and 4, one for each inflaton potential investigated, the mean square of quantum fluctuations of the three types of test fields together with the gauge invariant inflaton ones. For the hybrid model, quantum fluctuations of test fields with a small effective mass can dominate the gauge-invariant inflaton one, because of the presence of the leading constant term in the potential.
V Growth of quantum fluctuation in two field inflationary models
We now wish to consider a two field model in which an inflaton and a minimally coupled scalar field are present (see (27) for a different approach to the moduli problem). We shall neglect the energy density and pressure in the background FRW equations. We expand to second order in the uniform curvature gauge (UCG), in which the inflaton fluctuation coincides with the gauge-invariant Mukhanov variable, the Einstein and Klein-Gordon equations.
In the test field expansion , the homogeneous term satisfies
while fluctuations satisfy, order by order, for the leading order in the slow-roll approximation and in the long-wavelength limit (neglecting vector and tensor contributions), the following equations:
Following the consideration in section VI of Ref. (7), we wish to investigate which time variable in the stochastic equation should be chosen to re-obtain, order by order, the equation of motion for the test field starting from
and expanding order by order. For a general time variable , the equation becomes
As before, expanding to leading order in the slow-roll approximation, we obtain the following equation to the first and second order:
As is easy to verify, we recover the former result only for . So for the case of a test scalar field, evolving in a FRW inflaton driven space-time, in the UCG the right time to consider is the number of e-folds , this recovers the result obtained in (7) for the inflaton fluctuations. As for the case of the standard Mukhanov variable , which is defined, order by order, as the value of the inflaton perturbation in the UCG, we can define a generic gauge-invariant Mukhanov variable , associated with the perturbation of , as the value that this perturbation has in the UCG. In this way in the UCG and, as for the variable in (7), the equations above can be regarded as the gauge-invariant equations of motion, to first and second order, of this new Mukhanov variable, where one replaces with and with .
As for the case of the Mukhanov variable, this result can be considered as a starting point to study the fluctuations of in the stochastic approach for an arbitrary potential in the described background. The correct stochastic differential equation is obtained with respect to the number of e-folds () which appears to be the right evolution parameter. One starts from the slow-roll approximation to the Heisenberg equation, which can be interpreted in a general non-perturbative sense, for the large-scale quantum field
where is the stochastic noise term given, to the leading order in the slow-roll approximation, by
Thus, on expanding to the second order, one obtains the following stochastic equations for and
Let us consider the first order stochastic equation. Its general solution with the zero initial condition is given by
Taking into account that , it is easy to derive the expression for the mean square of the first order fluctuation :
where we have used the stochastic solution
with being the stochastic noise for the inflaton defined analogously to . Similarly, one can obtain the following solution for a vacuum expectation value of the second order fluctuation:
v.1 A working example: two field quadratic model
Let us now consider the particular case and . Classical slow-roll inflation in this model and the evolution of small perturbations in it were calculated in (28), but here we take into account the backreaction of the generated quantum fluctuations of these scalar fields on the evolution of their background values. By solving the background equations, one obtains the following zero order solution for the test field
It remains a test field for the whole duration of the inflation era if
for any value of (where ). For the case , we obtain the following limiting condition at the end of inflation ():
and for this particular background we can solve Eq.(51) obtaining
Thus, one obtains the term already considered in (7) plus a new term which depends on the background value of . At the end of inflation, the leading value of this second term is negligible with respect to the leading value of the first one, for , if
Analogously we can evaluate Eq.(53)
which to leading order gives
Motivated by previously found differences for gravitational particle production in the de Sitter background and in realistic inflationary models with , we have studied in detail the growth of quantum fluctuations for the latter case. We have selected four different potentials as representative examples of the inflationary zoo and different types of nearly massless fluctuations, including inflaton ones. We have rewritten in Eq. (32) the diffusion equation for the gauge invariant inflaton fluctuations found in (7), emphasizing the role of the slope of the spectrum of curvature perturbations and of the tensor-to-scalar ratio , i.e. the relevant observable quantities.
We have found that for most of the inflationary models, the mean square of the gauge invariant inflaton fluctuations dominates over those of moduli with a non-negative effective mass. Hybrid inflationary models can be an exception: the mean square of a test field can dominate over that of the gauge invariant inflaton fluctuations on choosing parameters appropriately. Our findings show that the understanding of inflaton dynamics including its quantum fluctuations is more important than the moduli problem in most of the inflationary models.
We have then discussed the stochastic approach for general scalar fluctuations, which may have a non zero homogeneous mode, in a generic model of inflation. We show, by using the field theory results as a guideline, that the stochastic equations for the gauge invariant variable associated with such scalar fluctuations are naturally formulated as a flow in terms of the number of e-folds . Finally we have studied the particular case of a massive inflaton and a second massive scalar field , for which we show how to extract some bounds for the homogeneous mode of and for its mass.
G.M. was supported by the GIS ”Physique des Deux Infinis”. A.S. was partially supported by the Russian Foundation for Basic Research, Grant No. 09-02-12417-ofi-m. This work was started during his visit to Bologna in 2009 financed by INFN: we thank INFN for support.
- This is why the statement sometimes found in literature that, for , a flat (Harrison-Zeldovich) spectrum is generated is also meaningless. Actually, it is a inflaton potential that leads to (and ) in the slow-roll approximation, see (2) for the exact solution for without using this approximation.
- For a double well SSB potential , which may be approximated by Eq. (5) for small field values, the condition becomes together with , which again means that is always considered far from the minima.
- This assumption also means the absence of the so called trans-Planckian particle creation, see (22) and references therein for discussion why trans-Planckian particle creation should be absent in the standard QFT.
- For the particular value we obtain
- This corresponds to the massless limit of moduli production computed in Eq. (15) of (7) for .
- N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982).
- A. A. Starobinsky, JETP Lett. 82, 169 (2005).
- O. Seto, J. Yokoyama and H. Kodama, Phys. Rev. D 61, 103504 (2000).
- F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi, Phys. Rev. D 65, 103521 (2002).
- F. Finelli, G. Marozzi, G. P. Vacca, and G. Venturi, Phys. Rev. D 69, 123508 (2004).
- F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi, Phys. Rev. D 74, 083522 (2006).
- F. Finelli, G. Marozzi, A. A. Starobinsky, G. P. Vacca and G. Venturi, Phys. Rev. D 79, 044007 (2009).
- Y. Urakawa and K.-I. Maeda, Phys. Rev. D 77, 024013 (2008); Phys. Rev. D 78, 064004 (2008).
- T. M. Janssen, S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quant. Grav. 25, 245013 (2008).
- T. M. Janssen and T. Prokopec, arXiv:0906.0666.
- A. D. Linde, Phys. Lett. B 116, 335 (1982).
- A. A. Starobinsky, Phys. Lett. B 117, 175 (1982).
- A. Vilenkin and L. Ford, Phys. Rev. D 26, 1231 (1982).
- A. A. Starobinsky, in Field Theory, Quantum Gravity and Strings, eds. H. J. De Vega and N. Sanchez, Lect. Notes in Physics 246, 107 (Springer, New York, 1986).
- A. D. Linde, Phys. Lett. B 175, 395 (1986).
- A. D. Linde, D. A. Linde and A. Mezhlumian, Phys. Rev. D 49, 1783 (1994).
- N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 724 (2005) 295.
- A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50, 6357 (1994).
- E. O. Kahya and V. K. Onemli, Phys. Rev. D 76, 043512 (2007).
- A. A. Starobinsky, Sov. Astron. Lett. 4, 82 (1978).
- F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 (1985).
- E. W. Kolb, A. A. Starobinsky and I. I. Tkachev, JCAP 0707, 005 (2007).
- I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic Press, New York, 1980).
- V. F. Mukhanov, JETP Lett. 41, 493 (1985); Sov. Phys. JETP 68 1297 (1988).
- F. Finelli, J. Hamann, S. M. Leach and J. Lesgourgues, arXiv:0912.0522 [astro-ph.CO].
- G. Marozzi, Phys. Rev. D 76, 043504 (2007).
- M. Lemoine, J. Martin and J. Yokoyama, Phys. Rev. D 80, 123514 (2009); Europhys. Lett. 89, 29001 (2010).
- D. Polarski and A. A. Starobinsky, Nucl. Phys. B 385, 623 (1992).