Minimizing the tensor-to-scalar ratio in single-field inflation models

Inflation is currently the best theory we have to explain and describe the initial state of the early universe [1, 2, 3, 4, 5, 6, 7]. The theory of inflation proposes that the early universe experienced a period of accelerated, quasi-exponential expansion. During this period, inflationary dynamics produced a nearly flat and homogeneous universe, while quantum fluctuations in the inflaton field itself imprinted the universe with a characteristic spectrum of scalar density fluctuations/inhomogeneities that would seed the large-scale structure of the late-time universe. The statistical features of this spectrum, observed in the Cosmic Microwave Background (CMB), largely align with generic inflationary predictions that suggest that these fluctuations should be adiabatic, Gaussian, and nearly scale-invariant [8, 9, 10].

While the inflationary paradigm has been a spectacular success, seen a number of novel predictions confirmed, and justifiably become the dominant paradigm for modeling the early universe [11, 12, 13, 14, 15, 16], the enthusiasm for inflation has been moderately tempered in recent years. This is no doubt partially due to the failure to detect observable B-modes, which have long hailed as a “smoking-gun” for inflation (see e.g. [17, 18]; c.f. [19]) because they would inform us about another key prediction of inflation models: the tensor/scalar ratio $r$. This observable has been one of the primary parameters used to constrain inflationary models because it gives us direct information concerning the microphysics of inflation; i.e., its energy scale and the form of its potential. Furthermore, it has traditionally been held that standard, “simple” single-field inflationary models typically produce large, or at least observable, tensor/scalar ratios [20]. Consequently, the null results have eliminated many of the simplest inflation models and inspired some to explore alternative approaches with renewed attention such as bouncing and string gas cosmologies (although inflation remains the dominant paradigm) [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 15, 12, 11, 31].

However, the answer to the question of what we can infer about the status of simple inflation models based on the detection or non-detection of $r$ is a subtle business. This of course depends upon one’s definition of “simple”, which, similarly to [32], we take to roughly be that the model consists of a single, canonical scalar field with a potential that can be approximated by an effective operator expansion (c.f. [33, 34] for further discussions on possible definitions of “simple” in this context). To this point, the authors of [32] provide a straightforward counter-example to the claim that simple single-field inflation models cannot produce a small tensor/scalar ratio: they construct a hilltop model with a potential described by a leading order quadratic term, and show that subleading order operators in the potential can induce an earlier end to inflation, and in doing so, lower $r$ “arbitrarily”.

This result intersects with some more general observations, especially emphasized by [35] and further explored in [36, 37], that:

1. 1.

   There are a number of inflation models in the literature with unregularized potentials that are unbounded from below.

2. 2.

   Producing a physically viable inflation model depends upon stabilizing, or “correcting”, the potential. Otherwise, the universe would immediately re-collapse.

3. 3.

   The observable predictions of inflation models can depend sensitively on the nature of the correction terms that ensure inflation ends smoothly.

In light of these observations, it remains to be seen whether and to what extent an arbitrarily small tensor/scalar ratio can be realized within the construction of [32].

In this paper, we answer this question by explicitly constructing models of inflation that can realize a vanishingly small $r$. To do so, we explore corrections to the quadratic hilltop model as suggested by [32] to show how a small $r$ can be realized within this construction. Furthermore, while we find that these models can easily produce values for $r$ well below even the most optimistic observational sensitivities, we also find that $r$ can essentially be lowered arbitrarily, at least until its value to asymptote around $r\sim 10^{-11}$.

The paper proceeds as follows. Sec. (II) describes the theory of inflation, slow-roll dynamics, and the observables associated with inflation models. Sec. (III) discusses the necessity of correcting, or regularizing, unbounded inflationary potentials, and explores some of the ways that this has been done in the literature. This section also shows how the standard quadratic hilltop model can be corrected by considering the behavior of terms in the generic Taylor expansion of the potential. Sec. (IV) explores the observational predictions for $r$ and $n_{s}$ for this general model, and shows how $r$ can be lowered in a nearly arbitrary way before hitting a lower bound. Sec. (V) concludes.

We begin with a discussion of the basic inflation scenario: a single, canonical scalar field $\phi$ minimally coupled to gravity in a homogeneous and isotropic FLRW background. The scale factor of the universe $a(t)$ evolves according to the Friedmann equations

	$\displaystyle H^{2}$	$\displaystyle=\frac{1}{3M_{\mathrm{Pl}}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right),$		(1)
	$\displaystyle\dot{H}$	$\displaystyle=\frac{1}{2}\frac{\dot{\phi}^{2}}{M_{\mathrm{Pl}}^{2}},$		(2)

where $H=\dot{a}/a$ and ${M_{\mathrm{Pl}}}$ is the Planck mass; and the scalar field $\phi$ evolves according to the Klein-Gordon equation in a Friedmann background

$$\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0,$$

(3)

where $V^{\prime}=dV/d\phi$.

While these are the full dynamics, in order to produce the needed accelerated expansion, inflation occurs when the potential $V(\phi)$ is the primary driver of the dynamics. This permits us to work in the slow-roll approximation. In this approximation, the kinetic term and the field acceleration are both vanishingly small $\dot{\phi}^{2}=\ddot{\phi}\simeq 0$. This leads to the following system of simpler equations:

	$\displaystyle H^{2}$	$\displaystyle\simeq\frac{V(\phi)}{3M_{\mathrm{Pl}}^{2}},$		(4)
	$\displaystyle 3H\dot{\phi}$	$\displaystyle\simeq-V^{\prime}(\phi).$		(5)

The slow-roll approximation can be conveniently parameterized in terms of the slow-roll parameters $\epsilon$ and $\eta$.

	$\displaystyle\epsilon$	$\displaystyle\equiv\frac{M_{\mathrm{Pl}}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2},$		(6)
	$\displaystyle\eta$	$\displaystyle\equiv M_{\mathrm{Pl}}^{2}\frac{V^{\prime\prime}}{V}.$		(7)

The accelerated expansion that inflation induces can only be sustained when these parameters are small, i.e., $\epsilon\ll 1$ and $\eta\ll 1$, and inflation ends when $\epsilon(\phi_{E})\simeq 1$.

We can also use this to calculate the number of e-folds $N(\phi)$ between the end of inflation $E$ and some earlier initial time $I$ when the fluctuations exit the horizon and freeze.

$$N(\phi)=\log(\frac{a_{E}}{a_{I}})=\int_{\phi_{\mathrm{I}}}^{\phi_{\mathrm{E}}}\frac{d\phi}{M_{\mathrm{Pl}}}\frac{1}{\sqrt{2\epsilon(\phi)}}.$$

(8)

Inflation generally needs $N\simeq 60$ e-folds to successfully resolve the aforementioned fine-tuning puzzles as well as to be compatible with observations.

This theory provides an explanatorily satisfying solution to the fine-tuning puzzles that inspired its development and elegantly produces a flat, homogeneized universe. Where inflation really shines; however, is that the theory also explains and predicts how the universe generates the tiny inhomogeneities observed in the CMB that would later grow into the large-scale cosmic structure that we see today. Tiny quantum fluctuations in the inflaton field itself $\delta\phi$ produce density perturbations in the metric that inflationary dynamics then stretch and amplify over cosmological scales. Scalar density perturbations are produced with a power spectrum

$\displaystyle\mathcal{P}_{\phi}=\left\langle|\delta\phi(k)|^{2}\right\rangle\propto A_{s}(k)^{n_{s}-1},$

(9)

and amplitude

$\displaystyle A_{s}=\frac{1}{8\pi^{2}}\frac{1}{\epsilon}\frac{H^{2}}{M_{\mathrm{Pl}}},$

(10)

where $k$ refers to the scale/wavenumber of the mode and $n_{s}$ is the so-called scalar spectrum index—a key observable that characterizes the scale-dependence of the fluctuation power spectrum which is defined as:

$$n_{s}(k)-1=\frac{d\ln\mathcal{P}_{\phi}}{d\ln k}.$$

(11)

Additionally, inflation produces primordial gravitational waves, which are tensor perturbations that produce a distinctive B-mode polarization pattern. The amplitude of tensor perturbations is given by

$$A_{t}=\frac{2}{\pi^{2}}\frac{H^{2}}{M_{\mathrm{Pl}}^{2}},$$

(12)

which allows us to define the scalar/tensor ratio $r$ as

$$r=\frac{A_{s}}{A_{t}}.$$

(13)

The quantities $r$ and $n_{s}$ are the two primary observables used to constrain inflationary models. Furthermore, they can be directly calculated from the slow-roll parameters, leading to

	$\displaystyle n_{s}-1$	$\displaystyle=-6\epsilon\left(\phi_{I}\right)+2\eta\left(\phi_{I}\right),$		(14)
	$\displaystyle r$	$\displaystyle=16\epsilon\left(\phi_{I}\right),$		(15)

where quantities are evaluated when the modes exit the horizon at $\phi_{I}$.

While the above are options are straightforward ways one could think of to stabilise the potential, they are all somewhat ad-hoc. As any analytic potential admits of a Taylor expansion,

$$V=V_{0}+\left.\frac{dV}{d\phi}\right|_{\phi=0}\phi+\left.\frac{1}{2}\frac{d^{2}V}{d\phi^{2}}\right|_{\phi=0}\phi^{2}+\left.\frac{1}{6}\frac{d^{3}V}{d\phi^{3}}\right|_{\phi=0}\phi^{3}+\cdots,$$

(21)

there are a huge number of possibilities that could show up at higher orders. Thinking of this as an effective field theory [51], the relevant symmetries and energy scale cut-off will determine which of the terms show up and at what order the expansion terminates. There are many potentials whose leading order terms are quadratic and thus would be well represented at leading order by the quadratic hilltop model, but with a number of higher order terms that could perhaps serve our purposes to both stabilise the potential, as well as modify observable predictions for $r$ or $n_{s}$.

Thus, given our ignorance regarding inflationary energy scales and a lack of any especially compelling theoretical motivations to consider stabilising correction terms of any particular type, we will work very generally and consider models given by a polynomial expansion

$$V(\phi)=V_{0}+\sum_{n=2}^{n=q}\frac{a_{n}}{n!}\left(\frac{\phi}{\mu}\right)^{n},$$

(22)

where the various $a_{n}$’s represent expansion coefficients of approximately $\mathcal{O}(1)$, and from now on we choose $\mu=M_{Pl}$ so that $\phi$ is scaled as $\phi\rightarrow\phi/M_{Pl}$. As depicted in Eq. (23), we restrict $a_{2}<0$ in order to ensure that our leading order term is described by the quadratic hilltop model, while $a_{n}$, where $2<n<q$, can take positive or negative values, and $a_{q}$ is the order at which the expansion terminates and must be positive to ensure that the potential stabilises.

$$\begin{array}[]{|c|c|}\text{ coefficient }&\text{ range }\\ \hline\cr\hline\cr a_{2}&\in[-1.0,-.01]\\ a_{2<n<q}&\in[-1.0,1.0]\\ a_{q}&\in[.01,1.0]\end{array}$$

(23)

So, for example, going out to $q=6$ would be given by the following:

$$V(\phi)=V_{0}+\frac{a_{2}}{2!}\phi^{2}+\frac{a_{3}}{3!}\phi^{3}+\frac{a_{4}}{4!}\phi^{4}+\frac{a_{5}}{5!}\phi^{5}+\frac{a_{6}}{6!}\phi^{6},$$

(24)

where $a_{6}$ is required to be positive, $a_{2}$ is required to be negative, and all the $a_{n}$’s in between are bounded as indicated above in Eq. (23). Of course, there is a tremendous amount of freedom in an expansion of this type, and the resulting potentials can take on a number of shapes, possibly having multiple minima. Thus, when we generate a potential, we then span a wide range of $\phi$ to find the global minimum and adjust $V_{0}$ such that $V(\phi_{vev})=0$ at this point (see [52] for a similar set-up with more general polynomials that include linear terms). Here we choose to span from $\phi\in[0,20M_{Pl}]$ in search of stabilising minima.

Given the high dimensionality of the parameter space and potential degeneracies between the parameters, we will pursue two strategies to explore it and assess the observational predictions of a single-field model given by an expansion of the form Eqs. (22-23). In Sect. IV.1, we will pursue a Markov chain Monte Carlo (MCMC) strategy to sample values from the parameter space to ascertain some generic features of this model. In Sect. IV.2, we will pursue minimization strategies to see how far we truly can push $r$ with this general set up.

Throughout the literature on inflation there has been much discussion concerning what the inflationary paradigm generically predicts for crucial observables such as $r$ and $n_{s}$. While many of the simplest and most commonly explored models do predict a relatively high $r$ (and some have been eliminated as a result), there are now a number of counter examples that show that single-field inflation can also produce an undetectably negligible tensor/scalar ratio [32, 54, 35]. Even going beyond our most optimistic scenarios for observational sensitivities though, studies such as this one which suggests that single-field inflation can produce tensor/scalar ratios as low as $r\sim 10^{-11}$, [55] which investigates a two-field model of natural inflation that can produce tensor/scalar ratios as low as $r\sim 10^{-15}$, or [56] which studies the mechanisms by which additional fields can suppress $r$, underscore the almost infinite flexibility of the inflation paradigm.

Here, we explored modifications to the quadratic hilltop model and explicitly showed that introducing possible correction terms can radically alter the observable predictions, where the form of the correction terms we considered was motivated by a widely applicable effective field theory operator expansion. Such higher order correction terms are both necessary to stabilise the potential in order to ensure a smooth end to inflation, and also represent additional degrees of freedom that can lower $r$ significantly. We showed this to be the case both by randomly sampling many possible viable models described by such an expansion, as well as by optimizing to find the minimum allowed values of $r$.

Where does this leave us? A null detection of $r$ seemingly will never totally rule out single-field inflation. Yet, as these examples suggest, the further down $r$ goes the more structurally complicated the models need to become [34, 33]. While such models may indeed be simple in the sense that they can be described by a single canonical scalar field minimally coupled to gravity, pushing $r$ lower does indeed seem to require more structural complexity. Like dark energy [41], early universe models suffer from surprisingly stubborn underdetermination problems; however, future observations will no doubt continue to offer valuable information concerning the physical processes underlying cosmological phenomena.

I am very grateful to Pedro Ferreira and David Sloan for many illuminating discussions. I am also grateful for financial support from St. Cross College, Oxford.