Higher order terms in the inflaton potential and the lower bound on the tensor to scalar ratio r
The MCMC analysis of the CMB+LSS data in the context of the Ginsburg-Landau approach to inflation indicated that the fourth degree double–well inflaton potential in new inflation gives an excellent fit of the present CMB and LSS data. This provided a lower bound for the ratio of the tensor to scalar fluctuations and as most probable value , within reach of the forthcoming CMB observations. In this paper we systematically analyze the effects of arbitrarily higher order terms in the inflaton potential on the CMB observables: spectral index and ratio . Furthermore, we compute in close form the inflaton potential dynamically generated when the inflaton field is a fermion condensate in the inflationary universe. This inflaton potential turns out to belong to the Ginsburg-Landau class too. The theoretical values in the plane for all double well inflaton potentials in the Ginsburg-Landau approach (including the potential generated by fermions) fall inside a universal banana-shaped region . The upper border of the banana-shaped region is given by the fourth order double–well potential and provides an upper bound for the ratio . The lower border of is defined by the quadratic plus an infinite barrier inflaton potential and provides a lower bound for the ratio . For example, the current best value of the spectral index , implies is in the interval: . Interestingly enough, this range is within reach of forthcoming CMB observations.
- I Introduction
- II Physical parametrization for inflaton potentials
- III Higher–order even polynomial double-well inflaton potentials
- IV The quadratic plus the th order double-well inflaton potential
- V The quadratic plus the exponential potential.
- VI Dynamically generated inflaton potential from a fermion condensate in the inflationary stage.
- VII The Universal banana region
The current WMAP data are validating the single field slow-roll scenario (1). Single field slow-roll models provide an appealing, simple and fairly generic description of inflation (2); (3). This inflationary scenario can be implemented using a scalar field, the inflaton with a Lagrangian density
where is the inflaton potential. Since the universe expands exponentially fast during inflation, gradient terms are exponentially suppressed and can be neglected. At the same time, the exponential stretching of spatial lengths classicalize the physics and permits a classical treatment. One can therefore consider an homogeneous and classical inflaton field which obeys the evolution equation
in the isotropic and homogeneous Friedmann-Robertson-Walker (FRW) metric
which is sourced by the inflaton. Here stands for the Hubble parameter. The energy density and the pressure for a spatially homogeneous inflaton are given by
The scale factor obeys the Friedmann equation,
In order to have a finite number of inflation efolds, the inflaton potential must vanish at its absolute minimum
These two conditions guarantee that inflation is not eternal. Since the inflaton field is space-independent inflation is followed by a matter dominated era (see for example ref. (7)).
Inflation as known today should be considered as an effective theory, that is, it is not a fundamental theory but a theory of a condensate (the inflaton field) which follows from a more fundamental one. In order to describe the cosmological evolution it is enough to consider the effective dynamics of such condensates. The inflaton field may not correspond to any real particle (even unstable) but is just an effective description while the microscopic description should come from a Grand Unification theory (GUT) model.
At present, there is no derivation of the inflaton model from a microscopic GUT theory. However, the relation between the effective field theory of inflation and the microscopic fundamental theory is akin to the relation between the effective Ginsburg-Landau theory of superconductivity (4) and the microscopic BCS theory, or like the relation of the sigma model, an effective low energy theory of pions, photons and nucleons (as skyrmions), with the corresponding microscopic theory: quantum chromodynamics (QCD).
In the absence of a microscopic theory of inflation, we find that the Ginsburg-Landau approach is a powerful effective theory description. Such effective approach has been fully successful in several branches of physics when the microscopic theory is not available or when it is very complicated to solve in the regime considered. This is the case in statistical physics, particle physics and condensed matter physics. Such GL effective theory approach permits to analyse the physics in a quantitative way without committing to a specific model (4).
The Ginsburg-Landau framework is not just a class of physically well motivated inflaton potentials, among them the double and single well potentials. The Ginsburg-Landau approach provides the effective theory for inflation, with powerful gain in the physical insight and analysis of the data. As explained in this paper and shown in the refs. (6)-(7), the analysis of the present set of CMB+LSS data with the effective theory of inflation, favor the double well potential. Of course, just analyzing the present data without this powerful physical theory insight, does not allow to discriminate between classes of models, and so, very superficially and incompletely, it would seem that almost all the potentials are still at the same footing, waiting for the new data to discriminate them.
In the Ginsburg-Landau spirit the potential is a polynomial in the field starting by a constant term (4). Linear terms can always be eliminated by a constant shift of the inflaton field. The quadratic term can have a positive or a negative sign associated to unbroken symmetry (chaotic inflation) or to broken symmetry (new inflation), respectively.
As shown in refs. (6); (7) a negative quadratic term and a negligible cubic term in new inflation provides a very good fit to the CMB+LSS data, (the inflaton starts at or very close to the false vacuum ). The analysis in refs.(6); (7) showed that chaotic inflation is clearly disfavoured compared with new inflation. Namely, inflaton potentials with are favoured with the inflaton starting to evolve at .
We can therefore ignore the linear and cubic terms in . I f we restrict ourselves for the moment to fourth order polynomial potentials, eq.(6) and imply that the inflaton potential is a double well (broken symmetric) with the following form:
The mass term and the coupling are naturally expressed in terms of the two energy scales which are relevant in this context: the energy scale of inflation and the Planck mass GeV,
Here is the quartic coupling.
The MCMC analysis of the CMB+LSS data combined with the theoretical input above yields the value for the coupling (6); (7). turns out to be order one consistent with the Ginsburg-Landau formulation of the theory of inflation (7).
According to the current CMB+LSS data, this fourth order double–well potential of new inflation yields as most probable values: (6); (7). This value for is within reach of forthcoming CMB observations (21). For the best fit value , the inflaton field exits the horizon in the negative concavity region intrinsic to new inflation. We find for the best fit (6); (7),
It must be stressed that in our approach the amplitude of scalar fluctuations allows us to completely determine the energy scale of inflation which turns out to coincide with the Grand Unification energy scale (well below the Planck energy scale). Namely, we succeed to derive the energy scale of inflation without the knowledge of the value of from observations. guarantees the validity of the effective theory approach to inflation. The fact that the inflaton mass is implies the appearance of infrared phenomenon as the quasi-scale invariance of the primordial power.
Since the inflaton potential must be bounded from below , the highest degree term must be even and with a positive coefficient. Hence, we consider polynomial potentials of degree where .
The request of renormalizability restricts the degree of the inflaton potential to four. However, since the theory of inflation is an effective theory, potentials of degrees higher than four are in principle acceptable.
A given Ginsburg-Landau potential will be reliable provided it is stable under the addition to the potential of terms of higher order. Namely, adding to the th order potential further terms of order and should only produce small changes in the observables. Otherwise, the description obtained could not be trusted. Since, the highest degree term must be even and positive, this implies that all even terms of order higher or equal than four should be positive.
Moreover, when expressed in terms of the appropriate dimensionless variables, a relevant dimensionless coupling constant can be defined by rescaling the inflaton field. This coupling turns out to be of order where is the number of efolds since the cosmologically relevant modes exit the horizon till the end of inflation showing that the slow-roll approximation is in fact an expansion in (5). It is then natural to introduce as coupling constant . This is consistent with the stability of the results in the above sense. Generally speaking, the Ginsburg-Landau approach makes sense for small or moderate coupling.
Odd terms in the inflaton field are allowed in in the effective theory of inflation. Choosing an even function of implies that is a symmetry of the inflaton potential. At the moment, as stated in (12); (7), we do not see reasons based on fundamental physics to choose a zero or a nonzero cubic term, which is the first non-trivial odd term. Only the phenomenology, that is the fit to the CMB+LSS data, decides on the value of the cubic and the higher order odd terms. The MCMC analysis of the WMAP plus LSS data shows that the cubic term is negligible and therefore can be ignored for new inflation (6); (7). CMB data have also been analyzed at the light of slow-roll inflation in refs. (8).
In the present paper we systematically study the effects produced by higher order terms () in the inflationary potential on the observables and .
We show in this paper that all curves for a large class of double–well potentials of arbitrary high order in new inflation fall inside the universal banana region depicted in fig. 10. Moreover, we find that the curves for even double–well potentials with arbitrarily positive higher order terms lie inside the universal banana region [fig. 10]. This is true for arbitrarily large values of the coefficients in the potential.
Furthermore, the inflaton field may be a condensate of fermion-antifermion pairs in a grand unified theory (GUT) in the inflationary background. In this paper we explicitly write down in closed form the inflaton potential dynamically generated as the effective potential of fermions in the inflationary universe. This inflaton potential turns out to belong to the Ginsburg-Landau class of potentials considered in this paper. We find that the corresponding curves lie inside the universal banana region provided the one-loop part of the inflaton potential is at most of the same order as the tree level piece. Therefore, a lower bound for the ratio tensor/scalar fluctuations is present for all potentials above mentioned. For the current best value of the spectral index (7); (1) the lower bound turns out to be .
Namely, the shape of the banana region fig. 10 combined with the value for the spectral index yields the lower bound . If one consider low enough values for (in disagreement with observations) can be arbitrarily small within the GL class of inflaton potentials.
The upper border of the universal region tells us that for . Therefore, we have inside the region within the large class of potentials considered here
Interestingly enough is within reach, although borderline for the Planck satellite (21).
Among the simplest potentials in the Ginsburg-Landau class, the one that best reproduces the present CMB+LSS data, is the fourth order double–well potential eq.(7), yielding as most probable values: . Our work here shows that adding higher order terms to the inflaton potential does not really improve the data description in spite of the addition of new free parameters. Therefore, the fourth order double–well potential gives a robust and stable description of the present CMB/LSS data and provides clear predictions to be contrasted with the forthcoming CMB observations (21).
The question on whether a lower bound for is found or not depends on whether the Ginsburg-Landau (G-L) effective field approach to inflation is used or not. Namely, within the G-L approach, the new inflation double well potential determines a banana shaped relationship which for the observed value determines a lower bound on . The analysis of the CMB+LSS data within the G-L approach which we performed in refs. (6); (7) shows that new inflation is preferred by the data with respect to chaotic inflation for fourth degree potentials, and that the lower bound on is then present. Without using the powerful physical G-L framework such discrimination between the two classes of inflation models is not possible and the lower bound for does not emerge. Other references in the field (i. e. (13); (19); (20)) do not work within the Ginsburg-Landau framework, do not find lower bounds for and cannot exclude arbitrarily small values for , much smaller than our lower bound .
This paper is organized as follows: in section II we present in general inflaton potentials of arbitrary high degree, specializing then to fourth and sixth–order polynomial potentials and displaying their corresponding curves. Sec. III contains the th order double–well polynomial inflaton potentials with arbitrary random coefficients and their curves. Sec. IV presents the limits of these polynomial potentials and we present in sec. V the exponential potential and its infinite coupling limit. In sec. V we compute the inflaton potential from dynamically generated fermion condensates in a de Sitter space-time displaying their curves. Finally, we present and discuss the universal banana region in sec. VII together with our conclusions.
Ii Physical parametrization for inflaton potentials
We start by writing the inflaton potential in dimensionless variables as (12)
where is the energy scale of inflation and is a dimensionless function of the dimensionless field argument . Without loss of generality we can set . Moreover, provided we can choose without loss of generality .
In the slow-roll regime, higher time derivatives in the equations of motion can be neglected with the final well known result for the number of efolds
where is the inflaton field at horizon exit. To leading order in we can take to be the value at which attains its absolute minimum , which must be zero since inflation must stop after a finite number of efolds (7).
Then, in chaotic inflation we have , with for , while in new inflation we have with for . We consider potentials that can be expanded in Taylor series around , with a non-vanishing quadratic (mass) term.
It is convenient to rescale the inflaton field in order to conveniently parametrize the higher order potential. We define a coupling parameter by rescaling the inflaton and its potential keeping invariant the quadratic term, that is
For a potential expanded in power series around we write:
The positive sign in the quadratic term corresponds to chaotic inflation (in which case ), while the negative sign corresponds to new inflation (in which case is chosen such that vanishes at its absolute minimum).
Clearly plus the set of coefficients provide an overcomplete parametrization of the inflaton potential which we will now reduce. In the case of chaotic inflation a convenient choice is , so that
which represents a generic higher order perturbation of the trinomial chaotic inflation studied in refs. (6).
In the case of new inflation, where , it is more convenient to set without loss of generality that . In order to have appropriate inflation, must be the absolute minimum of and the closest one to the origin on the positive semi–axis. That is,
and then fixes from eq.(12) the constant term in the potential
We thus get for the inflaton potential
For the coupling and the field using eq.(14),
From eq.(11) it now follows that the parameter can be expressed as the integral
with for chaotic inflation and for new inflation. Eq.(22) can be regarded as a parametrization of and in terms of the rescaled exit field . Clearly, as a function of , is uniformly of order . is numerically of order as long as is of order one. As we shall see, typically both at horizon exit and are of order one. We have for new inflation and for chaotic inflation.
In what follows we therefore use instead of as a coupling constant and make contact with eq.(10) by setting
We can easily read from this equation the order of magnitude of and since is given by eq.(9) and and are of order one. Hence, and .
As we will see below, the coupling (or ) is the most relevant coupling since it is related to the inflaton rescaling: the tensor–scalar ratio and the spectral index vary in a more relevant manner with than with the rest of the parameters in the potential eq.(15).
By construction the function has the following properties
for in chaotic inflation;
for in new inflation;
as in chaotic inflation;
as in new inflation.
In terms of this parametrization and to leading order in , the tensor to scalar ratio and the spectral index read:
Notice that both and are of order for generic inflation potentials in this Ginsburg-Landau framework as we see from eq.(25). Moreover, the running of the scalar spectral index from eq.(24) and its slow-roll expression turn out to be of order
Since can be inverted for any , these two relations can also be regarded as parametrizations and in terms of the coupling constant .
We are interested in the region of the plane obtained from eq.(25) by varying (or ) and the other parameters in the inflaton potential. We call this region.
From now on, we will restrict to new inflation.
For a generic [with the required global properties described above] we can determine the asymptotic of , since they follow from the weak coupling limit and from the strong coupling limit . When , then and from the property above,
When we have in new inflation and then,
We see that in the strong coupling regime becomes very small and becomes well below unity. However, the slow-roll approximation is valid for and in any case, the WMAP+LSS results exclude (1). Therefore, the strong coupling limit is ruled out.
Since may be small only in case is large (the logarithmic derivative of has a milder effect for large .) Therefore, we only find in a strong coupling regime. Notice that is much smaller than in the strong coupling regime [eq.(21)].
Let us now study large classes of physically meaningful inflaton potentials in order to provide generic bounds on the region of the plane within an interval of surely compatible with the WMAP+LSS data for , namely . To gain insight into the problem, we consider first the cases amenable to an analytic treatment, leaving the generic cases to a numerical investigation. As we will see below, the boundaries of the region turn out to be described parametrically by the analytic formulas (32) and (50).
ii.1 The fourth degree double–well inflaton potential
The case when the is the standard double–well quartic polynomial
By explicitly evaluating the integral in eq. (22) one obtains
and then, from eq. (25)
where . As required by the general arguments above, is a monotonically decreasing function of , ranging from till when increases from till . In particular, when vanishes quadratically as,
The concavity of the potential eq.(30) for the inflaton field at horizon crossing takes the value
We see that vanishes at , that is at . (This is usually called the spinodal point (17)). Therefore,
The negative concavity case for is specific to new inflation eq.(30). can be expressed as a linear combination of the observables and as
As expected in the general framework presented above, the limit implies weak coupling , that is, the potential is quadratic around the absolute minimum and we find,
which coincide with and for the monomial quadratic potential in chaotic inflation.
In the limit which implies (strong coupling), we have
Notice that the slow-roll approximation is no longer valid when the coefficient of becomes much larger than unity. Hence, the results in eq.(35) are valid for . We see that in this strong coupling regime (see fig. 1), becomes very small and becomes well below unity. However, the WMAP+LSS results exclude (1). Therefore, this strong coupling limit is ruled out.
For the fourth order double–well inflaton potential, the relation defined by eq.(32) is a single curve depicted with dotted lines in fig. 1. It represents the upper border of the banana shaped region in fig. 1.
Notice that there is here a maximum value for , namely with (7). The curve has here two branches: the lower branch in which increases with increasing and the upper branch in which decreases with increasing .
ii.2 The sixth–order double–well inflaton potential
where for stability we assume . Moreover, if we regard this case as a higher order correction to the quartic double–well potential, then is positive.
It is convenient to use as free parameter so that and . Thus,
where in order to ensure that .
The integral in eq. (22) can be explicitly evaluated with the result
According to the general arguments presented above [see the lines below eq. (22)] one can verify that is a monotonically decreasing function of for , where .
The scalar index and the tensor–scalar ratio are evaluated from eq. (25) as
Various curves are plotted in fig. 1 for several values of in the interval sweeping the region . We see that for increasing [namely, for increasing sextic coupling and decreasing quartic coupling, see eq.(38)] the curves move down and right, sweeping the banana-shape region depicted on fig. 1.
Clearly, is a variable more relevant than . Changing moves and in the whole available range of values, while changing only amounts to displacements transverse to the banana region in the plane. In particular, for a given becomes smaller for increasing .
We see in fig. 1 two important limiting curves: the and the curves. When the function reduces to the fourth order double-well potential eq.(30) and we recover its characteristic curve . When the potential has no quartic term and reduces to the quadratic plus sixth order potential:
In summary, the quadratic plus quartic broken–symmetry potential describes the upper/left border of the banana–shaped region of fig. 1, while the quadratic plus sextic broken–symmetry potential describes its lower/right border.
Iii Higher–order even polynomial double-well inflaton potentials
The generalization of the sixth order inflaton potential with broken symmetry to arbitrarily higher orders is now straightforward:
with the constraint eq.(17)
which guarantees that is an extreme of .
We consider here the case when all higher coefficients are positive or zero :
such that is the unique positive minimum.
We determine the shape of the region for arbitrary positive or zero values of the coefficients [subject to the constraint (43)], performing a large number of simulations with different setups. After producing coefficients we numerically computed the function following eq.(22)
and obtain the curves from eq. (25) by plotting directly vs. .
Uniform distributions of coefficients are obtained by setting
where the numbers are independently and uniformly distributed in the unit interval. We used the parametrization eq.(44) also when the are chosen according to other rules.
For example, in figs. 2-3 we plot the results when , that is for the ten degree polynomial. In this case we let and take independently the values or , for a total of 78 distinct configurations of coefficients. For better clarity, in figs. 2-3 we also include the two border cases and .
For higher values of we extracted the numbers at random within the unit interval. In particular, for the highest case considered, , we used three distributions: in the first, the were all extracted independently and uniformly over the unit interval; in the second we set and extracted the independently and uniformly; in the third we picked at random four freely varying and fixed to 1 the remaining 45 ones (that is we picked at random four possibly non–zero , setting the rest to zero); the values of the four free were chosen at random in the same set of values of the case. The results of these simulations are shown in fig. 5.
As evident from fig. 3, where the curves are split in upper/lower branches with growing/decreasing and especially from fig. 5, the case of the quadratic plus th order polynomial provides a bound to the banana region from below. That is, for any fixed value of , the quadratic plus th order polynomial provides the lowest value for .
One sees from fig. 5 that some blue curves go beyond the slashed red curve for the quadratic plus potential on the right upper border of the banana region . Namely, the right upper border of the region is not given by the quadratic plus potential while this potential provides the lower border of the region.
We performed many other tests with intermediate values of and several other distributions, including other dependent distributions, with characteristic values for growing linearly with or decreasing in a power–like or exponential way. In all cases, the results were consistent with those given above.
It is also important to observe that the class of potentials considered, that is arbitrary even polynomials with positive or zero couplings, is a class of weakly coupled models. This is evident from fig. 4, were is plotted vs. the coupling , which remains of order one when decreases well below the current experimental limits. This weak coupling is the reason why the addition of higher even monomials to these potentials causes only minor quantitative changes to the shape of the curves.
The inflaton potential eq.(42) in the original inflaton field takes therefore the form
Therefore, since the coupling is we have the -th term in the potential suppressed by the -th power of as well as by the factor .
In particular, the quartic term
possesses a very small quartic coupling since . Notice that these suppression factors are natural in the GL approach and come from the ratio of the two relevant energy scales here: the Planck mass and the inflation scale . When the GL approach is not used these suppression factors do not follow in general.
The validity of the GL approach relies on the wide separation between the scale of inflation and the higher energy scale (corresponding to the underlying unknown microscopic theory as discussed in ref. (5).) It is not necessary to require in the GL approach but to impose (5)
This last condition gives an upper bound for the inflaton field depending on the large argument behavior of . We get for example:
The validity of the effective GL theory relies on that separation of scales and the GL approach allows to determine the scale of inflation as GeV (at the GUT scale) and well below the Planck scale using the amplitude of the scalar fluctuations from the CMB data (6); (7).
Inflaton potentials containing terms of arbitrary high order in the inflaton are considered in ref. (13), sec. 25.3.2 without using the GL approach and within the small field hypothesis . Smallness conditions on the expansion coefficients are required in ref. (13). This is actually not needed in the GL approach, whose validity relies only on the wide separation of scales between and , at least in the case of even polynomials with positive coefficients.
Iv The quadratic plus the th order double-well inflaton potential
In order to find the observationally interesting right and down border of the banana we consider the quadratic plus the th order potential for new inflation (10),
This integral can be expressed as a sum of terms including logarithms and arctangents (9).
while in the strong coupling limit at fixed and take the values
in accordance with the general formula eq.(28). In fig. 6 we plot vs. for the potential eq.(45) and the exponents and . We see that for vs. tends towards a limiting curve. For we reach the upper end of the curve [the monomial quadratic potential eq.(47)] while for large the left and lower end of the curve is reached. However, the current CMB–LSS data rule out this strong coupling part of the curve for .