Swampland, Axions and Minimal Warm Inflation

Warm inflation has been noted previously as a possible way to implement inflationary models compatible with the dS swampland bounds. But often in these discussions the heat bath dynamics is kept largely unspecified. We point out that the recently introduced Minimal Warm Inflation of arXiv:1910.07525, where an axionic coupling of the inflaton leads to an explicit model for the thermal bath, yields models of inflation that can easily fit cosmological observations while satisfying dS swampland bounds, as well as the swampland distance bound and trans-Planckian censorship.


I. INTRODUCTION
Which low energy effective theories can arise from a UV complete theory of quantum gravity (such as string theory), is a question of both theoretical and phenomenological interest [1][2][3][4]. In particular, inspired by the difficulty of realizing inflation and/or de-Sitter vacua in string theory, it has recently been conjectured that scalar potentials whose potential slow-roll parameters are small, cannot be realised in (asymptotic regimes of) string theory [5][6][7]. Since the slow roll conditions are crucial for conventional models of inflation, if one wants to have inflation in such regimes, one must explore alternative models. One simple way to achieve sufficient amounts of inflation, even for steep potentials, is to employ the "warm inflation" mechanism in the strongly dissipative regime (see [8,9] for some early papers and [10,11] for review). In the warm inflation paradigm, the inflaton loses its energy to a thermal bath. Its utility for swampland purposes has been noted previously [12][13][14][15][16][17] On the other hand, while warm inflation has been studied for a long time as a possibility, realising it in concrete models has been a challenge (see e.g. the discussion in [18,19] and references therein). In particular, any endeavour to realise warm inflation in a strongly dissipative regime has difficulties because the strong dissipation typically destabilizes the inflationary potential.
Very recently however, a class of concrete models ("Minimal Warm Inflation") that realize warm inflation in the strongly dissipative regime, have been put forward [20] (see also [19]). Minimal Warm Inflation gives the inflaton an axionic coupling to non-Abelian gauge fields. This provides a very simple and possibly viable model of the thermal bath. Since the inflaton is an axion, its shift symmetry will protect it from any perturbative backreaction and hence from acquiring a large thermal mass. On the other hand, because it is coupled to the gauge field and since at sufficiently high temperature there are sphaleron transitions between gauge vacua, there is friction. The corresponding axion friction coefficient, Υ, * gaurav.goswami@ahduni.edu.in † chethan.krishnan@gmail.com turns out to be [20] (see e.g. section 9.5 of [21] and also [22][23][24][25]) where, T is the temperature of the bath, Γ sp (T ) is the sphaleron rate, f is the axion decay constant, α g = g 2 /(4π), g being the Yang-Mills gauge coupling, and κ is a dimensionless quantity which depends on the dimension of the gauge group (N c ), the representation of fermions (N f ) if any, and on the gauge coupling. In addition to this, the axion has a UV potential that is responsible for inflation, which softly breaks the shift symmetry without causing too much backreaction [20]. Since axions and gauge fields are ubiquitous in string theory, the mechanism of [20] has ingredients which may be realizable in string theory. But, for many stringy solutions (near the boundary of the landscape), we also know that the scalar potential violates potential slowroll [5][6][7] as was first noted in the example of [26]. This raises the following question: could the ingredients used in [20], which lead to inflation in the specific models studied there, lead to inflation when one is dealing with the kind of steep runaway potentials that are ubiquitous at the boundary of the string landscape?
In this short note, we would like to point out that a simple model in which the inflaton is an axion and its UV potential is of the exponential run-away form is a viable model to achieve warm inflation in the strongly dissipative regime. We will show that (a) CMB observational constraints are easily satisfied, (b) the dS swampland bounds are satisfied by construction, (c) the field excursion can be sub-Planckian so that the requirement of swampland distance conjecture [27,28] is satisfied, and (d) the energy scale of inflation is low enough so that the recently proposed trans-Planckian censorship conjecture [29,30] holds. In the following, we will elaborate on these claims.
In this paper, our primary focus will be on showing that enough inflation to simultaneously satisfy observational data and swampland constraints is possible. In a concluding section, we will comment on what it takes for our scenario to be turned into a full cosmological model.

II. MINIMAL WARM INFLATION WITH A RUNAWAY POTENTIAL
In this section, we will carefully analyse a simple model of inflation which has the following features: (a) its scalar potential is consistent with dS swampland conjecture (i.e. it has a steep potential), (b) the inflaton field excursion required to achieve sufficient number of e-foldings of inflation is sub-Planckian (as expected from swampland distance conjecture), (c) the energy scale of inflation (and the corresponding number of e-folds) is consistent with the trans-Planckian censorship conjecture, (d) it is based on warm inflation realised in a strongly dissipative regime, (e) the model is a Minimal Warm Inflation model and thus, there is a clear understanding of the thermal bath in terms of axionic couplings of the inflaton, and, finally, (f) the model is consistent with cosmological observations.

A. Equations and approximations
For warm inflation, at the background level, one is interested in the dynamics of the homogeneous inflaton field φ(t) and temperature of the bath T (t). The evolution equations arë where Υ(T ) is the axion friction coefficient, which, for our purpose, is given by Eq (1) and ρ R =g * T 4 whereg * = π 2 g * 30 , all the other symbols have their usual meaning. It is useful to to work with the dimensionless quantity Q defined by In the following, we follow the convention of the literature on cold inflation and define the potential slow roll parameters in the usual following way, In the literature on warm inflation, it is usual to define another set of slow roll parameters, for which we use the following notation: in addition, one can have the usual Hubble slow-roll parameters. Before proceeding, let us note the following important points: (a) During warm inflation, friction due to the thermal bath ensures that the inflaton slow rolls even when the potential is steep, this means that theφ term in Eq (2) can be ignored, (b) We want the Universe to inflate, so we need V (φ) to be larger thanφ 2 /2 and ρ R (energy density of radiation). Thus, in Eq (3), the only term in bracket which is relevant is V (φ), (c) We want to deal with warm inflation in strongly dissipative regime, this corresponds to 3H Υ(T ) in Eq (2) i.e. the condition (d) It can be shown that, when w 1 and η w 1, the first term in Eq (4), i.e.ρ R can be ignored. Notice that in strongly dissipative regime with Q 1, one can have w and η w too small even if V and η V are O(1); (e) Finally, let us note that warm inflation requires that T > H and when this does not hold good, we are dealing with cold inflation.
With all the above approximations, Eq (2), Eq (3) and Eq (4) take the form: notice that we are not assuming that V and η V are small.

B. Potential and constraints
In the following, we work with a typical run away potential In cold models, this gives a power law expansion a ∼ t q with q = 2/α 2 . For such a potential, one finds that Similarly, for this model, η w = 2 w . Note that while V and η V are fixed quantities, since Q is in general temperature dependent, w and η w will change as inflation proceeds.

dS swampland constraint
The de-Sitter swampland bounds [5][6][7] (see also [31,32]) dictate that at least one of the potential slow roll parameters among V or η V must be an O(1) number in Planck units. More specifically, For concreteness we will consider values of α belonging to the range (0.2, 5) which corresponds to V in the range (0.02, 12.5) and η V in the range (0.04, 25). If we are within this range, we will say that we have satisfied the dS swampland bound -even though it is clear that at the (lower) boundaries of this range, we are already in some tension. We will see however that in many cases we will be able to satisfy the bound comfortably, so this should not concern us unduly.

Constraint from swampland distance conjecture
Swampland distance conjecture [27,28] states that as we explore distances comparable to M pl in scalar field space, towers of states become exponentially light. Thus, a potential obtained from low energy effective field theory can only be trustworthy for sub-Planckian field excursions. We will try to determine whether there is any allowed range of parameters which give us the condition ∆φ M pl . In practice we will demand the field range to be an order of magnitude lower than M pl . Before proceeding, let us note that larger values of α give us a potential which is more steep: as we shall see, this will correspond to a smaller field excursion.

Trans-Planckian censorship
The Trans-Planckian Censorship Conjecture [29,30] states that the cosmological evolution in effective theories consistent with quantum gravity must be such that quantum fluctuations at sub-Planckian length scales must never become classical. This requirement imposes strong constraints on cosmic inflation. In particular, it is argued in [30] that this will imply that the potential energy during inflation, V , must satisfy the inequality (see also [17])
In the rest of this work, we shall work with the following values: A s = 2.099 × 10 −9 , n s = 0.965 and r < 0.07.

Free parameters and dependent quantities
Before proceeding, let us note that in this model, at this stage, there are 6 free parameters: α, V 0 , g * (the number of relativistic species in the radiation bath with which the inflaton is coupled), κ (which depends on the Yang Mills gauge group that the axion couples to and the representation of matter fields), α g and the axion decay constant f . As is apparent from Eq (1), the last three parameters always turn up in the combination κα 5 note thatc has mass dimension −2 so, we shall often work with the parameter M 2 plc . Thus, for a fixed value Other quantities would be determined in terms of these parameters. Note that, for time dependent quantities, one constrains their values at the time t * , when the pivot scale crossed the Hubble radius during inflation: this epoch will be denoted with a subscript * (except for those quantities for which there is already a subscript, for these, there will be a superscript * e.g. * w ). Thus, some of the dependent variables are T * , H * , V * , Q * etc. Just to avoid confusion, let us emphasize the obvious point that the subscript * in g * andg * has nothing to do with this epoch.
The inflaton field would have a specific value at the epoch t * i.e. φ * = φ(t * ). In cold inflation, one determine φ * by the following procedure: first find the field value at the end of inflation: i.e. first find φ end such that V (φ end ) = 1. Then, use the fact that the pivot scale crossed the Hubble radius N cmb e-foldings before the end of inflation: in the equation the last factor on the right hand side will, in general, depend on the field value as well as the parameters in the potential; for every choice of such parameters and N cmb , one can determine φ * from the above equation. A similar procedure can be followed in warm inflation, by using Eq (10) and Eq (11) it is easy to see that one gets If one wishes to use Eq (19) to relate N cmb and φ * , one needs to start from φ end . In this paper we will not be concerned with the mechanism to end inflation 1 , thus we can not determine φ end at this stage and hence we shall not have an equation which connects N cmb and φ * . Thus, the value of φ * will be left undetermined in what follows.
Whenever a mechanism to end inflation is introduced in the model of our interest, one knows the field value at which inflation ends and then the above steps can be followed to find φ * .

Strategy
From the discussion in the last section it is clear that we can have six free parameters: V 0 , α,g * , M pl 2c , φ * and N cmb , out of which, as we argued, φ * will be undetermined. The expression for V * is where it is worth emphasising that, at this stage, V * itself depends on, not only φ * , but also α and V 0 . But, the parameters α, φ * and V 0 only turn up in this combination, thus, we could work with V * as our parameter 2 . The possible values of N cmb we will be interested in are 40, 50 and 60; once one of these values is chosen, we have the freedom to vary the rest of the four parameters. Thus, for a chosen value of N cmb , we have four parameters left to be determined: We have two observables whose values have been observationally fixed: A s and n s . Imposing the requirement of getting the correct values of these observables fixes two of the parameters, say, α and V * and then we have the two parametersg * and M 2 plc which will be left free. All the other dependent quantities (such as Q * , H * , T * etc) will be determined in terms of A s , n s ,g * and c. Additional constraints e.g. r < 0.07, ∆φ < M pl , 0.2 < α < 5, Q * 1 etc will restrict the allowed regions ing * −c parameter space. 1 Note that as a model of cold inflation, the potential given by Eq (13) is also incomplete in the sense that it also does not have a natural end point to its evolution. But for cold inflation, even if one assumes that such a mechanism exists and does not disrupt the predictions for cosmological perturbations, this model leads to r = 8(1−ns) which, for the measured value of ns gives a value of r which is ruled out. 2 Another way to say the same thing is that one could always readjust either α or V 0 (or both) to ensure that φ * = 0, this will imply that V * = V 0 .

D. Relevant quantities in terms of free parameters
Let us now find the expressions of all the quantities in terms of A s , n s ,g * andc.
For the scenario studied in [20], with the friction coefficient given by Eq (1), the general expressions for T (φ) as well as Q(φ) for any potential have been given in [20]. One can use Eq (1), Eq (5), Eq (10), Eq (11), Eq (12) and the expression for ρ R (t) to derive those expressions. We make use of the expressions for T (φ) and Q(φ) from [20] for the potential of our interest: Eq (13). Thus the expression for Q(φ) turns out to be 3 Similarly, from the expression for T (φ) for [20], one can show for our case that

Scalar amplitude
We now make use of the above expressions to write down various observable quantities in terms of the free parameters and observables. The expression for the amplitude of scalar perturbations for warm inflation in strongly dissipative regime is available in the literature (see e.g. Eq (75) of [34]) and it can be written in the following form convenient for our purpose 4 where, c = (4π 3/2 Q 9 3 ) −1 = 7.626 × 10 −10 and the above expression is sufficiently accurate only in the limit Thus, from the value of A s , we find the constraint where, the numerical value of the constant on the RHS is 2.752 and, if one prefers, one could replace V * by V 0 in accordance with Eq (21).

Scalar spectral index
The expression for scalar spectral index in Minimal Warm Inflation turns out to have the following form [20]: Using the fact that in the model of our interest, η w = 2 w and using the definition of w , one can see that 1 − n s = 33 14 using the fact that Q * 1 and the observed value of n s , one finds that

Tensor to scalar ratio
On the other hand, as long as T < M pl , the tensor spectral amplitude is given by the same expression as in cold inflation i.e., thus, using Eq (11), Eq (27) (and using the value of c ), the tensor to scalar ratio (in the limit Q 1) can be written as

Constraints
As discussed, in order to implement dS swampland bounds, the following inequality must hold good 0.2 < α < 5 .
Now we turn to the field excursion of inflaton, the minimum number of e-foldings of inflation required will be given by Note that, for the potential give by Eq (13), V /V = −M pl /α so that the only field dependence in the above integral is from Q(φ). One could try to estimate the value of the integral for N cmb by regarding Q(φ) as being a constant i.e. by replacing Q(φ) by Q * , one gets, One can now use Eq (30) and hence get where, the inequality must hold good if swampland distance bound is to be satisfied and the possible values of N cmb of interest are 40, 50 and 60.

Summary: all variables in terms ofg * andc
As we saw, the variables of interest are V * , α,g * ,c, and N cmb . Here, N cmb ∈ {40, 50, 60}. Let us see how we can relate all the relevant variables, we follow the following steps: (a) Eq (30) can be used to express α in terms of Q * : with c 1 = 0.122.
(c) Now, in Eq (24), we can use the above expression for V * and find T * in terms of Q * ,g * andc: here, c 3 = (d) Now we can make use of Eq (27), and get an expression for Q * in terms ofc. We get where, For any chosen value ofcM 2 pl andg * , we can use Eq (40) to find Q * , then use Eq (37) to find α, then use Eq (38) to find V * and finally, use Eq (39) to find T * .

E. Analysis
In this section, we find the possible values which the various quantities can take in this scenario. Before proceeding let us note that in the following, we can treat the requirement of obtaining the correct central value of the amplitude of the scalar perturbations (A s ) and that of scalar spectral index (n s ) as already built in. Thus, for all the possible values of the various parameters in the following, we get the correct values of A s and n s : given this, if we also wish to impose other requirements e.g. Let us recall that the inflaton in this model has an axionic coupling to a non-Abelian gauge theory and the sphaleron transitions between gauge vacua, existing at sufficiently high temperatures, provide the friction necessary for warm inflation. If the corresponding non-Abelian gauge theory has gauge group SU (3), there will be 8 gauge bosons, each of which will contribute two relativistic degrees of freedom, and so, including the inflaton itself, there will be 17 relativistic degrees of freedom. In most of the rest of this section, we shall present the results for the case for which g * = 17 and find what happens when we changec (which is the only other free parameter). In the last column of table (I), we have also included the results for the case g * = 199 corresponding to the gauge group SU (10). The temperature dependence of sphaleron rate, on which Eq (1) is based, is best known for pure SU (2) and SU (3) theories, but the results for the case of SU (10) in the last column of of table (I), are presented assuming the same form of Υ as in Eq (1).
The results of the last section indicate that all the quantities (e.g. α, Q * , T * , V * etc) are power functions of the parameterscM 2 pl andg * with different powers and pre-factors. By using the equations of the last section, one can easily arrive at the corresponding powers and prefactors for all the quantities. Thus, if one plots the logarithm of any of the dependent variables as a function of the logarithm of any of the independent variables i.e. From Eq (38) and Eq (39), it can be seen that the ratio of energy density of radiation to that of potential energy density of inflaton is    (3)) and the fourth case has g * = 199 (corresponding to the gauge group SU (10)). To find ∆φ i.e. entries in row 10, we set N cmb = 50. For the last two rows, the value of f is calculated by assuming that κ = 100 and g = 10 −1 . First case is for c =cmin, for second case, the value ofc is chosen such that V 1/4 * turns out to be 10 −10 M pl (to satisfy trans-Planckian censorship conjecture) while for the third case, the value ofc is chosen such that ∆φ turns out to be 0.1M pl . For the last case, for g * = 199, the value ofc is chosen such that ∆φ turns out to be 0.1M pl .
When the Universe is inflating, this ratio is expected to be small as compared to 1: from the values of c 2 and c 3 given in the last section, this is easy to see that its numerical value is 0.0037. Thus, this ratio doesn't depend on eitherc M 2 pl org * : this is easily seen from row 8 of table (I). This justifies the use of Eq (11) instead of Eq (3).
We now letc change over a large range of values and note what happens. What should be the range over which we varyc? In Eq (1), κ is an O(100) number (see e.g. [20]), while we expect that the axion decay constant would be sub-Planckian. This implies that unless α g is too small, in Eq (16), the quantityc M 2 pl will be a large quantity. This fits well with the fact that in Eq (40), Q * is required to be a large quantity since we work in the strongly dissipative regime. But, in Eq (40), the power ofc M 2 pl is so small that one needs very large values ofc M 2 pl to get reasonably large values of Q * . We thus letc M 2 pl change over a huge range: 10 7 to 10 21 . In fig (1), we show how the parameter in our potential α and the potential slow roll parameter V change as we changec M 2 pl over its range. In order to satisfy Eq (26) if we want the minimum value of Q * to be, say, Q min * For a fixed g * , this will imply a corresponding lower limit or upper limit on the various quantities. The entries in the first column of table (I) are for the casec =c min : sincec is expected to be greater thanc min the entries in this column are lower limits for possible values for Q * , α, , and upper limits for V * , T * , H * , r, ∆φ and f . The entry in row 4 of this column indicates that the lower limit of V is 0.32. Thus, if we wish to obtain the observed values of n s and A s and we wish to be in a strongly dissipative regime, we can not have too small values of V i.e. we shall need to have a steep potential (which is -happilyrequired by dS swampland conjectures). In fig (1), the range of values ofc which give a value of Q * smaller than Q min * is denoted by the shaded region. Another quantity of interest is the inflaton field excursion which depends additionally on N cmb . In fig (2), for every choice of N cmb , the value ofc M 2 pl smaller than the one at which the slanted lines intersect the horizontal line corresponding to ∆φ = M pl gets ruled out if we wish to demand that ∆φ be less than M pl . Once one chooses a value ofc which satisfies this requirement, all the other ingredients of the model fall in place. It may not be satisfactory to only have ∆φ < M pl , we may wish to have ∆φ hierarchically smaller than M pl . In the last column of table (I), i.e. for third case, the value ofc is chosen such that ∆φ turns out to be 0.1M pl (i.e. one order of magnitude below the Planck scale). Note that the corresponding value of H * is well above the value of H during big bang nucleosynthesis as we want. The corresponding value of V is quite large, i.e. in this model, to make field excursion hierarchically small, we need to take the potential to be very steep. Also, note the various labelled vertical lines in fig (1) and fig (2): the line labelled "c min " corresponds to case 1 in table (I), the line labelled "TCC" corresponds to case 2 in table (I) while the line labelled "∆φ = 0.1M pl " corresponds to case 3 in table (I). The last column in table (I) provides the details for the case in which ∆φ = 0.1M pl for the case g * = 199.
For g * = 17, fig (3) shows the dependence of a few quantities on the value ofcM 2 pl . Note that, for the entire range, T * stays greater than H * : as required by warm inflation. Also, it is worth noting that the tensor to scalar ratio, r is unobservably small for the entire range of values.
Finally, we may wish to satisfy the requirement of trans-Planckian censorship conjecture. In the second column of table (I), the value ofc is chosen such that V 1/4 * turns out to be 10 −10 M pl (to satisfy the trans-Planckian censorship conjecture). Note that: (a) the corresponding value of H * is well above the value of H during big bang nucleosynthesis, and (b) the corresponding ∆φ is sub-Planckian, but is not (significantly) hierarchically smaller.
In Eq (1), if we set the values of κ(≈ O(100)) and g, then, for every chosen value ofc, we can find the corresponding value of the axion decay constant, f , Following [20] we demand that f V 1/4 * . If we set κ = 100 and g = 0.1, the corresponding values of f are shown in row 11 of table (I). From the last row of the table, it is easy to see that for this reasonably small choice of the value of gauge coupling (g = 0.1), the axion decay is several orders of magnitude smaller than V 1/4 * , as desired.
Thus, it is clear that in this model, one can successfully obtain sufficient number of e-foldings of inflation even if we wish to keep the potential to be steep and hence slow roll parameters to be O(1) (i.e. dS swampland requirements get satisfied). Furthermore, one can get sub-Planckian field excursion (as required by swampland distance conjecture) and a successful realisation of warm inflation in a strongly dissipative regime. Additionally, there is a large region in the parameter space of the model such that one can satisfy the requirements of trans-Planckian censorship conjecture.

III. DISCUSSION
We have showed in this paper that Minimal Warm Inflation [20] provides a viable realization of inflation that can simultaneously meet observational data while satisfying all of the relevant swampland constraints. It gives an explicit scenario to implement a strongly dissipative heat bath for the warm inflation paradigm. While the fact that these results fell out essentially effortlessly is encouraging, in this section, as a counterpoint to our discussions so far, we will point out various features of this approach that needs to be sorted out before it can qualify as a fully satisfactory model of inflation.
• We have not discussed mechanisms for ending infla-tion or for ensuring reheating. These are necessary steps for transitioning to the Big-Bang nucleosynthesis phase and to have a complete cosmological model. We have kept our discussion most basic and general: it seems likely that with a more specific model, these things can be arranged.
• Our model is a small field model. This was more or less inevitable since large field models are more or less in direct tension with swampland constraints. Small field models often require fine tuning in the initial conditions to have enough inflation. It will be interesting to see if this is true in Minimal Warm Inflation. We could draw our conclusions in this paper without explicitly evolving the equations. If instead one decides to study the evolutions in detail, this point should become clearer.
• A somewhat related point is that the field range we find can be made reasonably small, namely about an order of magnitude smaller than Planck scale, but not significantly smaller. Perhaps this is a hint that we can manage to get inflation without too much fine-tuning?
• In some of our discussions of the temperature dependence of the Υ paraamter for various gauge groups, we have extrapolated results known only for small gauge group ranks [24] to higher rank gauge groups. But this is a minor issue, which does not seriously affect our punchlines.
• One of the features of the construction of [20] is that their UV-potential breaks the shift symmetry of the axion completely. Since the axion arises as an angle field, even after symmetry-breaking and non-perturbative effects, we expect it to have a discrete shift symmetry of 2π in suitable units. So the breaking here is an explicit breaking of the 2π periodicity, but it is claimed [20] that it is soft. Questions regarding the breaking of what can be viewed as a discrete gauge symmetry have been discussed previously in a related set up in [36], see also related very recent discussions in [37]. It will also be interesting to consider thermal backreaction questions from this perspective. We will not have much to add to these discussions. A closely related question is that of realizing these in string theory, where a lot has been discussed about (the difficulty with) large axion field ranges [3].
• A related point is that following [20], we have demanded that the scale of the UV-potential be hierarchically above the axion decay constant. It will be good to have a better understanding of the two scales.