Inflationary predictions of doublewell, ColemanWeinberg, and hilltop potentials with nonminimal coupling
Abstract
We discuss how the nonminimal coupling between the inflaton and the Ricci scalar affects the predictions of single field inflation models where the inflaton has a nonzero vacuum expectation value (VEV) after inflation. We show that, for inflaton values both above the VEV and below the VEV during inflation, under certain conditions the inflationary predictions become approximately the same as the predictions of the Starobinsky model. We then analyze inflation with doublewell and ColemanWeinberg potentials in detail, displaying the regions in the  plane for which the spectral index and the tensortoscalar ratio values are compatible with the current observations. is always larger than in these regions. Finally, we consider the effect of on small field inflation (hilltop) potentials.
*]and Vedat Nefer Şenoğuz\note[*]Email: nefer.senoguz@msgsu.edu.tr \affiliationDepartment of Physics, Mimar Sinan Fine Arts University, 34380 Şişli, İstanbul, Turkey \keywordsphysics of the early universe, inflation
1 Introduction
The hypothesis of cosmic inflation [1, 2, 3, 4] provides a plausible explanation of the large scale homogeneity of the universe and, more importantly, of the primordial density perturbations that evolve into cosmic structure. A simple way inflation can occur is based on a slowrolling scalar field called the inflaton. Once the Lagrangian for the inflaton field and also the thermal history of the universe after inflation is specified, values for observational parameters can be calculated and compared with constraints coming from measurements of the cosmic microwave background (CMB) anisotropies [5, 6].
The observational parameters, in particular the scalar spectral index and the tensortoscalar ratio , have been calculated for various inflationary potentials (see [7] for a comprehensive subset). An assumption often made in the calculations is that the inflaton is minimally coupled. On the other hand, a renormalizable scalar field theory in curved spacetime also requires the nonminimal coupling between the inflaton and the Ricci scalar [8, 9, 10]. For a given potential, depending on the value of the nonminimal coupling parameter , inflationary predictions and even whether inflation occurs or not can change [11, 12, 13, 14, 15, 16, 17, 18, 19].
Here we will investigate how the value of the nonminimal coupling parameter affects the inflationary predictions for potentials where the inflaton has a nonzero vacuum expectation value (VEV) after inflation. In terms of the redefined field , the nonminimal coupling in the Lagrangian includes a linear term in as well as a quadratic term. Under some conditions on and that are discussed in section 2, this leads to inflationary predictions that approach those of the Starobinsky ( inflation) model [20], which is in good agreement with the current observations [6]. The Starobinskylike behaviour is obtained not just for the wellknown nonminimally coupled quartic potential case but also when inflation occurs near the quadratic minimum of the potential, for inflaton values above (below) the VEV and ().
A reason for considering a nonzero VEV after inflation is that such potentials can be associated with symmetry breaking in the early universe. After a general discussion of inflation with nonminimal coupling for such potentials, we then analyze in detail two archetypal symmetry breaking potentials, namely the doublewell potential (section 3) and the ColemanWeinberg potential (section 5). Although both potentials with nonminimal coupling were previously considered, there are some gaps and disagreements in the literature which we address in these sections. For each potential, we display the observational parameter values as functions of for selected values as well as the regions in the  plane for which the spectral index and tensortoscalar ratio values are compatible with the current observations. Section 4 suggests modifying the doublewell potential to obtain a small field inflation (hilltop) potential, which unlike the other two potentials can fit observations for inflaton values below the VEV and . Finally, section 6 concludes the paper with a summary of our results and a remark on perturbative unitarity violation.
2 Inflation with nonminimal coupling
Suppose we have a nonminimally coupled scalar field with a canonical kinetic term and a potential :
(1) 
where the subscript indicates that the Lagrangian is specified in a Jordan frame. Here, for we will be considering symmetrybreaking type of potentials where the inflaton takes positive values and has a nonzero vacuum expectation value (VEV) after inflation.
Our choice for consists of a constant term and a nonminimal coupling between the inflaton and the Ricci scalar. The constant term is familiar from the EinsteinHilbert action, and the term is required in a renormalizable scalar field theory in curved spacetime [8, 9, 10]. We are using units where the reduced Planck scale is set equal to unity, so we require after inflation. Therefore taking , we have . The term in can be neglected in some specific models such as when the standard model Higgs is the inflaton [21, 22], but may well play an important role in other models. For instance, if inflation is associated with symmetry breaking at or near the grand unified theory scale , is possible for values of similar to values required for standard model Higgs inflation.
As we will see, as cosmological scales exit the horizon in the observationally favored region of parameters, and the Starobinskylike regime corresponds to . The effective gravitational constant remains positive throughout the evolution of the field. Indeed, if we switch to the Einstein frame and make a field redefinition so that the kinetic term is again canonical, we see that is only reached at infinite values of the field (see section 2.1 and ref. [23]). This implies, in particular, that if there can be no transition from the symmetric () phase to the brokensymmetry () phase. Nevertheless, we include this case in our investigations as the field evolution does not have to start from the symmetric phase, and could for example start from values above the VEV as would be expected for chaotic initial conditions [4].
It has been appreciated [24, 16, 25, 23, 26, 27, 28, 29, 30, 31] that different choices for and can share the same attractor point with the Starobinsky ( inflation) model [20, 32], which predicts
(2) 
to leading order in the number of efolds , where is the running of the spectral index. An example relevant to our discussion is
(3) 
which is a special case of the strong coupling attractor model discussed in [28] (see also [25]).
In terms of the redefined field so that after inflation, includes a linear term in as well as a quadratic term. If () as cosmological scales exit the horizon, it means the inflaton is away from the minimum and . Then eq. (3) is satisfied for , the nonminimally coupled quartic model wellknown since the late eighties [15, 16, 33, 34]. On the other hand, if () as cosmological scales exit the horizon, so that eq. (3) is satisfied for , with () for inflaton values above (below) the VEV during inflation.
Since a generic potential will be quadratic close enough to its minimum, it seems that a generic can share the predictions of the Starobinsky model up to leading order in the number of efolds . For this to happen, and values should satisfy some constraints which we discuss in section 2.2. Before that, we briefly review how to calculate the observational parameters for inflation with nonminimal coupling.
2.1 Calculating the observational parameters
For calculating the observational parameters given eq. (1), it is convenient to switch to the Einstein () frame by applying a Weyl rescaling , so that the Lagrangian density takes the form [35]
(4) 
where
(5) 
and . If we make a field redefinition
(6) 
we obtain the Lagrangian density for a minimally coupled scalar field with a canonical kinetic term.
For , eq. (5) gives
(7) 
It will be useful to consider some simplifying cases of this expression:
 [style=multiline,font=]
 1.

Weak coupling limit
If and , and . (Provided , these conditions will be satisfied when for inflation below the VEV, and for inflation above the VEV.) Then, the inflationary predictions are approximately the same as for minimal coupling in general. Note, however, that if is very flat as cosmological scales exit the horizon, then even a small correction in the potential can significantly alter the inflationary predictions, as we will discuss in section 4.  2.
 3.

Strong coupling limit
If , we have(9) Using eq. (6), we obtain where is positive during inflation. This exponential behaviour in terms of the canonical field makes it difficult to satisfy observations except for the special cases discussed in section 2.2 where the Einstein frame potential has a plateau due to cancellations between and .
Once the Einstein frame potential is expressed in terms of the canonical field, the observational parameters can be calculated using the slowroll parameters (see ref. [37] for a review and references):
(10) 
where ’s in the subscript denote derivatives. The spectral index , the tensortoscalar ratio and the running of the spectral index are given in the slowroll approximation by
(11) 
The amplitude of the curvature perturbation is given by
(12) 
which should satisfy from the Planck measurement [5] with the pivot scale chosen at Mpc. The number of efolds is given by
(13) 
where the subscript “” denotes quantities when the scale corresponding to exited the horizon, and is the inflaton value at the end of inflation, which we estimate by .
Unfortunately, for general values of and , it is difficult and inconvenient to express the potential in terms of the canonical field . We therefore rewrite these slowroll expressions in terms of the original field for the numerical calculations, following the approach in ref. [23]. Using eq. (6), eq. (10) can be written as
(14) 
where we defined
(15) 
Similarly, eq. (12) and eq. (13) can be written as
(16)  
(17) 
To calculate the numerical values of , and we also need a numerical value of . Assuming a standard thermal history after inflation,
(18) 
Here is the
energy density at the end of inflation, is the energy density at the
end of reheating and is the equation of state parameter during
reheating, which we take to be constant.
(19) 
To represent a plausible range of , we can consider three cases: In
the high case is taken to be 1/3, which is equivalent to
assuming instant reheating. In the middle case we take
and the reheat temperature GeV, calculating using the
standard model value for the number of relativistic degrees of freedom ().
In the low case we take GeV (again with
).
2.2 The Starobinsky conditions
As mentioned in the beginning of section 2, for a potential which is quartic away from the minimum or quadratic close to the minimum, if some conditions on and values are satisfied, predictions approach the Starobinsky point given by eq. (2) on the  plane. Following the discussion in ref. [31], we will now derive these conditions using the relation of the Starobinsky point with the order and residue of the leading pole in the kinetic term.
Let’s write the Einstein frame Lagrangian density in terms of :
(20) 
Suppose is given by a Laurent series with a leading pole located at whereas is given by a Taylor series starting from a nonvanishing constant term as cosmological scales exit the horizon:
(21) 
In analogy with motion of a particle for and , slowroll inflation occurs for . We can calculate the inflationary predictions using the usual slowroll expressions (see section 2.1) and , obtaining [31]
(22) 
For a standard thermal history after inflation, the current data [5, 6] favors , which corresponds to the case . Note that for this case does not depend on , which is to be expected since the kinetic term is invariant under . The Starobinsky model predictions given by eq. (2) correspond to and .
Now consider inflation with , so that corresponds to . This implies that we can look for Starobinskylike solutions, with inflaton values above (below) the VEV if (). Using eq. (5), we obtain
(23) 
Note that this equation differs from eq. (22) of ref. [31] due to the term in . As a consequence, the kinetic term can remain positive after as well as during inflation for both signs of .
First, consider above VEV solutions satisfying as cosmological scales exit the horizon. In this case
(24) 
Note that from eq. (22), with , so corresponds to . Also, the assumption corresponds to . When these conditions are satisfied, the leading order inflationary predictions coincide with those of the –attractor models [39, 40, 31], namely,
(25) 
Second, let’s assume that as cosmological scales exit the horizon. Further assuming , eq. (23) simplifies to , that is, and . Therefore the Starobinsky model predictions given by eq. (2) are obtained for inflaton values both above and below the VEV whenever these two assumptions are satisfied. Using , the two assumptions correspond to and , respectively.
As long as these conditions are satisfied and is given by eq. (21), the inflationary predictions will match the Starobinsky model predictions up to leading order in . From eq. (21) we obtain
(26) 
which implies that regardless of the value of (as long as it is ), Starobinskylike solutions are obtained if is approximately given by the double well potential for which , as cosmological scales exit the horizon. Therefore in terms of , the potentials satisfying eq. (21) can be written as
(27) 
This again shows that the Starobinsky point given by eq. (2) is obtained for the potential away from the minimum (), and the potential near the minimum (). From (for ), the value of as cosmological scales exit the horizon is and for these two cases, respectively.
We can summarize the Starobinsky conditions as follows. The inflationary predictions for coincide with the Starobinsky model predictions up to leading order in if:
 [style=multiline,font=]
 1.

The inflaton is above the VEV, is quartic for , and . (On the other hand, from eq. (24), for .)
 2.

The potential is quadratic around the minimum for and
(28)
where () for inflaton values above (below) the VEV. These conditions satisfy the strong coupling limit eq. (9), so that a plateau type Einstein frame potential is obtained during inflation in terms of the canonical scalar field:
(29) 
Both 1. and 2. are special cases of the strong coupling attractor model eq. (3), with and , respectively. The case is discussed in ref. [29] and also belongs to the class of the induced inflation models discussed in ref. [30].
3 Doublewell potential
In this section we analyze the prototypical symmetry breaking potential [41]
(30) 
referred to as the doublewell potential, the Higgs potential or the LandauGinzburg potential. First, let us briefly review inflation with this potential for the minimal coupling case, which was analyzed in several papers, see e.g. refs. [42, 43, 44, 45, 46, 47, 7, 48]. When inflation occurs near the minimum, in terms of , the potential is approximately quadratic: . Since for quadratic inflation, the observable part of inflation occurs near the minimum for . Then the quadratic potential predictions of
(31) 
are obtained for inflation both below the VEV and above the VEV.
For inflation above the VEV, if then we have quartic inflation with and . The predictions interpolate between the quadratic and quartic limits for , remaining out of the 95% CL Planck contour (Planck TT+lowP+BKP+lensing+ext) [5] for all . Whereas for inflation below the VEV, if then as cosmological scales exit the horizon, so the potential is effectively of the new inflation (small field or hilltop inflation) type
(32) 
which implies a strongly red tilted spectrum with suppressed . As a result, although both the and limits are ruled out, the  values are in the 68% CL Planck contour (Planck TT+lowP+BKP+lensing+ext) [5] for a narrow range around (specifically, between and 25 for the high case), see figures 2 and 3. Note that all the figures in this section are obtained for the high case, using the equations given in section 2.1. In particular from eq. (17) we obtain:
From the discussions in section 2, we expect that we need () to improve the fit to the observations for inflation below (above) the VEV. Indeed, figures 2 and 3 show that for inflation below the VEV, the predictions move out of the 95% CL Planck contour for . For inflation above the VEV, the case interpolating between quadratic and quartic inflation is already out of the Planck range, as is the case which leads to an even redder spectrum and larger .
It was discussed in section 2 that if certain constraints on and values are satisfied, a potential quadratic near its minimum or a potential quartic away from it correspond to special cases of the strong coupling attractor model of ref. [28]. The doublewell potential satisfies both conditions. The Einstein frame potential can be written as
(33) 
Therefore the discussion in section 2.2 is directly applicable. In particular, eq. (28) implies that for inflation below the VEV, as is increased for a given , the predictions eventually approach the Starobinsky point given by eq. (2). As can be seen from figure 2, this transition is rather abrupt for and occurs near . This means that if the Starobinsky point is excluded by future observations, the entire parameter space will be ruled out for below VEV inflation with this potential.
For above VEV inflation with (which includes the induced gravity case ) and , the inflationary predictions are given by eq. (25). Thus, the  values are in the 95% (68%) CL contours for (0.007) for the high case, see figures 2 and 4. Note that for very low reheat temperatures the  values can move out of the 68% CL contour, see figure 1. For negligible values of , as is the case in standard model Higgs inflation [21], the model is reduced to the nonminimally coupled quartic inflation model. Our results in this limit agree with previous results [15, 16, 33, 34].
Combining eq. (25) with eq. (28) we see that for above VEV inflation the Starobinsky point is obtained when:
(34) 
In the induced gravity limit, using eq. (8), the Einstein frame potential can be written as
(35) 
coinciding with the  model of refs. [39, 40]. Thus the inflationary predictions approach the quadratic potential predictions given by eq. (31) for , and eq. (25) for larger . The doublewell potential in the induced gravity limit was previously considered for inflation in refs. [49, 13, 50, 51]. Ref. [50] also calculated for values between 0 and 1. Our results agree with ref. [50] for but not for the induced gravity limit . For the latter case our results agree with ref. [51]. Finally, ref. [23] analyzed the doublewell potential with nonminimal coupling in detail. The difference between our work and theirs is that we take as explained in section 2, whereas they take . As a consequence, although the predictions on the  plane look generally similar, there are a few differences between our and their results. Namely, for inflation below the VEV their predictions approach the Starobinsky point given by eq. (2) when , whereas our predictions approach it when eq. (28) is satisfied. For above VEV inflation, their predictions approach eq. (2) only for large values of , whereas our predictions approach it when eq. (34) is satisfied.
4 Small field inflation potentials
Consider new inflation type models where the inflaton is below the VEV during inflation. For the doublewell potential, we see that consistency with observations require so that very large values are needed for subPlanckian values of the VEV . On the other hand, a potential which is flatter near the origin could be compatible with observations even if . As an example we take a simple generalization of the doublewell potential:
(36) 
For the weak coupling limit and , we have . If then during inflation, and the Einstein frame potential can be written as
(37) 
where we have defined .
For , this small field inflation potential (also called hilltop potential) appears often in the literature, see for example refs. [58, 37, 7] and references therein. Using eqs. 10, 11 and 13, we obtain
(38) 
which shows that is suppressed and tends to be smaller than the range favored by observations. To be more specific, let’s consider the most optimistic high case, where using eqs. 18, 19, and , we have
(39) 
Note that the energy scale during inflation is lower for lower values, which correspond to lower , and therefore values as well. For , can be inside the 95% CL contour given by the Planck collaboration (Planck TT+lowP+BKP+lensing+ext) [5] only for , see figure 5. If , is outside the 95% CL contour for any value.
Repeating the calculation for the potential given by eq. (37), we obtain
(40) 
These expressions are in excellent agreement with the numerical results given in figure 5, which were calculated using the Jordan frame potential given by eq. (36). They show that values increase and the fit to observational data improves provided . In particular, can be inside the 95% CL contour for much smaller VEVs, namely for , , for , 8, 10 respectively.
5 ColemanWeinberg potential
Symmetry breaking due to the ColemanWeinberg mechanism [59] was associated with inflation since the early eighties when the first new inflation models were proposed [2, 3, 60]. In these models the effective potential can be written as [61, 62]
(41) 
For a minimally coupled scalar, the inflationary predictions of this potential were analyzed in ref. [63] (see also refs. [46, 47, 7, 64, 48, 65]). They are generally similar to the predictions of the doublewell potential: Again, for inflation occurs around the quadratic minimum , leading to eq. (31). For inflation above the VEV, the predictions again interpolate between the quadratic and quartic limits, remaining out of the 95% CL Planck contour. Whereas for inflation below the VEV, if then as cosmological scales exit the horizon, so the potential is effectively of the new inflation (small field or hilltop inflation) type
(42) 
which predicts , and a tiny as given by eq. (38). Comparing with eq. (41), the parameter in eq. (42) is given by
(43) 
where using eq. (13) we obtain
(44) 
Here, is a branch of the Lambert function satisfying .
Similarly to the doublewell potential, although both the and limits are ruled out for inflation below the VEV, the  values are in the 68% CL Planck contour (Planck TT+lowP+BKP+lensing+ext) [5] for a narrow range around (specifically, between and 38 for the high case), see figures 6 and 7. Note that all the figures in this section are also obtained for the high case.
From figure 6 we see that below VEV inflation predictions are incompatible with the observational data for and . This is expected from the discussion in section 4 since under these conditions the ColemanWeinberg potential is approximately given by eq. (37) with during inflation. As is clear from figure 5 and eq. (40), a small nonminimal coupling cannot bring into agreement with observations. For instance remains for even under the most favorable instant reheating assumption. This result agrees with ref. [66], but disagrees with ref. [67] where the quartic term in eq. (37) is erroneously neglected.
The effect of the nonminimal coupling on the predictions of inflation below the VEV is similar for ColemanWeinberg potential to the doublewell potential. Both potentials are quadratic near their minima, so that the Starobinsky point given by eq. (2) is obtained when eq. (28) holds. Numerically, approaching the Starobinsky point requires even larger for the ColemanWeinberg potential as can be seen from figure 6. The predictions move out of the 95% CL Planck contour for .
For inflation above the VEV, the case interpolating between quadratic and quartic inflation is already out of the Planck range, as is the case which leads to an even redder spectrum and larger (see figure 8). The case is more subtle since the ColemanWeinberg potential in the Jordan frame is not simply quartic away from the minimum but also contains a logarithmic factor. There is similarly a logarithmic factor in the Einstein frame potential written in terms of :
(45) 
Thus, the predictions of eq. (25) are approached only when the logarithmic factor can be treated as constant, that is, when the contribution from its derivative can be neglected. Taking derivative of eq. (45) and using (see section 2.2), we find that this requires
(46) 
Numerically,  values are in the 95% (68%) CL contours for (0.008), assuming the high case and , see figures 6 and 8. The Starobinsky point given by eq. (2) is obtained both for and for with extremely small values of , whereas the predictions move out of the observationally favored region in the  plane as approaches (see figure 9).
Using eq. (8) we can write the Einstein frame potential in the induced gravity limit () as follows [68]:
(47) 
where () during inflation above (below) the VEV. Analysis of this potential [51, 68] shows that inflation below the VEV is not compatible with the current observational data, whereas above VEV inflation predictions interpolate between the linear potential and quadratic potential predictions for and , respectively. The linear potential predictions
(48) 
are in the Planck %95 CL contour, which explains the light green region around the dotted line in figure 6.
6 Conclusion
In this work we discussed the inflationary predictions of models with the Lagrangian
(49) 
where the inflaton has a nonzero VEV after inflation, and . In terms of the redefined field , the nonminimal coupling in the Lagrangian includes a linear term in as well as a quadratic term. This leads to an attractor behaviour where the predictions approach the Starobinsky model predictions, not just for the wellknown nonminimally coupled quartic potential case but also when the inflaton is near the minimum () as cosmological scales exit the horizon.
After discussing the conditions under which Starobinskylike behaviour is obtained in section 2, we analyze two prototypical symmetry breaking potentials: the doublewell potential in section 3 and the ColemanWeinberg potential in section 5. For each potential, we display the regions in the  plane for which the spectral index and the tensortoscalar ratio values are compatible with the current observations.
If () for inflation above (below) the VEV, large portions of the  plane lead to predictions compatible with the current constraints on and , see figures 2 and 6. Most of these portions lead to predictions approaching the Starobinsky model predictions, so the allowed parameter space would shrink drastically if future observations rule out the Starobinsky model. In particular, if the upper bound on becomes , both the doublewell and ColemanWeinberg potentials would be ruled out as inflationary models, for any value of and .
Although we have displayed the inflationary predictions for a wide range of and values, it is questionable whether the entire range can be theoretically justified. In particular can be difficult to realize starting from a fundamental theory [69]. Furthermore, for the wellknown nonminimally coupled quartic potential solution, expanding the action around the vacuum reveals a cutoff scale [70, 25, 71], and requiring this to be higher than the energy scale during inflation corresponds to using eq. (19) and eq. (2). On the other hand, ref. [72] has emphasized that the cutoff scale depends on the background value of the field and can remain above the relevant energy scales during and after inflation. In any case consistency with observations only require for this solution.
Starobinskylike inflationary predictions also arise when eq. (28) is
satisfied. The observable part of inflation then occurs near the minimum
where the potential is , and . This is another special case of the strong coupling attractor
model [28]. Interestingly, even though the
Starobinskylike regime corresponds to (with and
for inflation above and below the VEV, respectively), the cutoff
remains at the Planck scale [29, 30]. Thus,
although consistency with observations require large values of for
subPlanckian VEVs, no perturbative unitarity violation is expected at
scale around the vacuum, unlike the nonminimally coupled quartic
potential case.
Finally, in section 4, we briefly considered a higher order version of the doublewell potential, which for subPlanckian VEVs corresponds to a small field (hilltop) inflation potential, with an additional quadratic term coming from the nonminimal coupling. Unlike the two abovementioned potentials, for this potential inflation below a subPlanckian VEV can be compatible with observations for a positive , and a tiny is predicted. All the considered potentials predict a running of the spectral index that is too small to be observed in the near future, with typically around .
Acknowledgements
VNŞ thanks Diederik Roest for a useful discussion. This work is supported by TÜBİTAK (The Scientific and Technological Research Council of Turkey) project number 116F385.
Footnotes
 For a derivation of eq. (18) see e.g. ref. [38].
 as low as 10 MeV is consistent with big bang nucleosynthesis, however it is difficult to explain how baryogenesis could occur at such low temperatures.
 This potential also arises in some supersymmetric new inflation models [52, 53, 54, 55] and was analyzed in refs. [56, 57].
 The cutoff scale around the vacuum changes when eq. (28) is satisfied since expanding the Einstein frame Lagrangian for small values of , the leading order kinetic term is no longer canonically normalized but instead given by .
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