# The effective Planck mass and the scale of inflation

## Abstract

Observable quantities in cosmology are dimensionless, and therefore independent of the units in which they are measured. This is true of all physical quantities associated with the primordial perturbations that source cosmic microwave background anisotropies such as their amplitude and spectral properties. However, if one were to try and *infer* an absolute energy scale for inflation—a priori, one of the more immediate corollaries of detecting primordial tensor modes—one necessarily makes reference to a particular choice of units, the natural choice for which is Planck units. In this note, we discuss various aspects of how inferring the energy scale of inflation is complicated by the fact that the effective strength of gravity as seen by inflationary quanta necessarily differs from that seen by gravitational experiments at presently accessible scales. The uncertainty in the former relative to the latter has to do with the unknown spectrum of universally coupled particles between laboratory scales and the putative scale of inflation. These intermediate particles could be in hidden as well as visible sectors or could also be associated with Kaluza–Klein resonances associated with a compactification scale *below* the scale of inflation. We discuss various implications for cosmological observables.

## Keywords

Extra Dimension Hide Sector Visible Sector Strong Gravity Inflaton Field## 1 Preliminaries

The strength of the gravitational force depends on the scale at which it is measured.^{1} At laboratory scales, the strength of gravity is characterized by the reduced Planck mass \(M_\mathrm{pl}= 2.435\times 10^{18}\) GeV which determines Newton’s constant \(G_N = M_\mathrm{pl}^{-2}\). However, like all other interactions, quantum corrections effect the effective strength of gravity depending on the characteristic energy of the process probing it.^{2}

Massive particles are particularly interesting for the threshold effects they impart once we start to probe energies above their mass \(M\), i.e. at distances below \(M^{-1}\). This can be understood via a simple thought experiment [5]: consider scattering a test particle off a very heavy point mass. The inverse Fourier transform of the scattering amplitude yields the gravitational potential generated by the source. Once the inter-particle separation approaches \(\Delta x \sim M^{-1}\), \(M\) being the mass of some heavy particle, virtual pairs of these particles are created, the positive/ negative energy virtual quanta of which are attracted/ repelled by the source, creating a gravitational dipole distribution that effectively anti-screens the source, strengthening its gravitational field. Therefore, the strength of gravity is increased by this effective ‘vacuum polarization’ far enough away from the threshold induced by a particle of mass \(M_j\) that couples to gravity,^{3} i.e. as we probe increasingly shorter distances \(\Delta x \ll M^{-1}_j\).

*total*energy momentum tensor of the theory. We further consider the limit where the external momentum satisfies \(p^2 \gg M^2\) where \(M^2\) is the mass of the heaviest particle that can run through the loops. In this limit, the theory becomes conformal, which fixes the finite part of the loop integral to be

*That is,*\(M_{**}\)

*is the effective cut-off of gravity at short distances*. However, one must take care to distinguish between the scale of strong gravity \(M_{**}\) from the strength of gravity at a particular energy scale, which we denote \(M_*\). Whereas the former sets the scale at which unitarity starts to break down in the effective theory,

^{4}the latter determines the strength of gravitationally mediated processes at any particular scale

*below*\(M_{**}\). As detailed in Appendix B, although every massive species contributes to lowering the scale at which strong gravity effects become important, one has to distinguish between species that universally couple directly to the matter energy-momentum tensor at tree level [such as massive Kaluza–Klein (KK) gravitons, non-minimally coupled scalars, and \(U(1)\) gauge fields] from ordinary four-dimensional fields that couple at one loop, in terms of their effects on the strength of gravity

*as one crosses the threshold set by the mass*\(M\)

*of the species,*but are still far below the scale \(M_{**}\). Whereas the former immediately affect the strength of gravity, the latter do not make their effects known until very close to \(M_{**}\).

^{5}Therefore, for the rest of our discussion we denote use \(N\) as a shorthand for the weighted index that effectively counts the number of universally coupled degrees of freedom below the energy scale of interest corresponding to the generalization of (3), such that the strength of gravity at that scale, henceforth taken to be the scale of inflation, is given by

^{6}As is to be expected, all dimensionless observables, such as the amplitude and spectral properties of the perturbations, are unaffected by the changing strength of gravity at inflationary energies. However, when one tries to

*infer*an absolute energy scale for inflation, one finds that it is undetermined commensurate with (5) up to the unknown spectrum of universally coupled species between laboratory scales and the inflationary scale, the details of which we elaborate upon in the following.

## 2 The scale of inflation

^{7}sourced by the comoving curvature perturbation \(\mathcal{R}\), defined as the conformal factor of the 3-metric \(h_{ij}\) in comoving gauge

^{8}:

*up to our ignorance of the effective strength of gravity at the scale*\(H_*\), given by

^{9}species up to the scale \(H_*\)—whether they exist in the visible sector or in any hidden sector. Presuming \(r_* = 0.1\), Eq. (11) implies an energy scale for inflation of \(V_*^{1/4} = 7.6\times 10^{-3}M_*\).

In order to keep track of concepts in the discussion to follow, we distinguish between what we henceforth refer to as *the scale of inflation*—defined as \(H_*\) during inflation—and the *energy scale of inflation*, defined as \(V_*^{1/4}\). The reason for this distinction is that \(H_*\) defines (among other things) the scale above or below which massive particles respond to the background expansion irrespective of any direct couplings to the inflaton^{10} whereas \(V^{1/4}_{*}\) defines the physical energy density in the inflaton field as seen by particles that couple to it, such as all decay products produced in (p)reheating. We take this distinction for granted in what follows.

^{11}\(M_* = M_\mathrm{pl}\). However, given our ignorance of particle physics between collider scales and the scale of inflation in addition to all hidden sector physics, \(M_*\) is in general lower than \(M_\mathrm{pl}\) according to (12), where \(N\) represents a model dependent parameter that obscures our ability to infer the actual energy scale of inflation from observations of CMB temperature and polarization anisotropies. That is,

## 3 Extra species as Kaluza–Klein states

^{12}In the former case, the moduli corresponding to the extra dimensions remain fixed at their minima during inflation and we have available the usual relation between the fundamental gravity scale \(M_{**}\)

*below the effective compactification scale*and the long wavelength strength of gravity (the Planck mass):

^{13}\(V_n = M^{-n}\), so that

### 3.1 Extra KK species and the scale of inflation

^{14}\(M_{**}\) (of the same order or an order of magnitude higher for \(2\le n\le 6\)), even though it is always less than the effective cut-off \(M_*\) at the scale \(H_*\) through (11). We note that this is never problematic, even though \(M_{**}\) is the cut-off induced by the underlying UV completion. This is because we remain in the perturbative regime with respect to corrections from the heavy states that UV complete the theory, which relies on derivatives being suppressed relative to this scale i.e. by the ratio \(H_*/M_{**}\), guaranteed to be less than unity by (26).

Furthermore, we stress that although extra dimensions (compactified at a scale below that of inflation) provide a natural context for the appearance of extra massive species, the relation (13) is also valid in a strictly four-dimensional context and illustrates an irreducible uncertainty in our ability to infer a scale for inflation given our lack of knowledge of particle physics from collider energies up to the energy scale of inflation.

### 3.2 Large number of species in string theory

Apart from the possibility of having light KK modes of large extra dimensions, the fundamental gravity scale can be lowered due to a large number of species from hidden sectors (even coupled gravitationally to the Standard Model), or even from string excitations whose number increase exponentially with their mass. In the later case, the effective number of particle species which are not broad resonances, with width less than their mass is \(\widetilde{N}\simeq 1/g_s^2\) [23, 24, 25, 26, 27].

## 4 (P)reheating

^{15}

## 5 Discussion

It is commonly presumed that detection of a primordial tensor mode background would allow us to determine the (energy) scale of inflation in the context of single field inflation. The purpose of this note was to highlight the fact that, instead, one can only *infer* the (energy) scale of inflation from observations up to our ignorance of the scale \(M_* = M_\mathrm{pl}/\sqrt{N}\), the precise value for which depends on the spectrum of all universally coupled species with masses up to \(H_*\), and for which field content of the standard model alone suggests an \(N\) different from one though still of order unity. These observations raise the possibility that the energy scale for inflation can be significantly lowered by the presence of many gravitationally coupled species, an observation that has a particularly natural realization in extra-dimensional scenarios, although is equally pertinent in a four-dimensional context.

## Footnotes

- 1.
E.g. via Cavendish type experiments where we have precise knowledge of two masses (one of which could be a test mass), or equivalently in principle through gravitational scattering experiments.

- 2.
- 3.
This is true regardless of whether these massive particles couple directly to the sector that contains the probe particle (e.g. the Standard Model) or not.

- 4.
And is thus unitarized by the appearance of new degrees of freedom at \(M_{**}\) from some UV completion, such as string theory.

- 5.
We are grateful to Sergey Sibiryakov for discussions concerning this point.

- 6.
- 7.
In what follows, we assume that all of the extra species have sufficiently suppressed couplings to the inflaton during inflation (e.g. either through derivative couplings or as Planck suppressed interactions), so that isocurvature perturbations are not significantly generated. This is trivially true for hidden sector fields.

- 8.
The comoving (or unitary) gauge is defined as the foliation where inflaton field fluctuations have been locally gauged away. In words, it is the time slicing where the inflaton itself is the clock.

- 9.
So that (12) denotes a tree level relation.

- 10.
In addition to quantum corrections to the effective action itself being set by the ratio \(H^2_*/M^2_*\).

- 11.
Note that this would require a desert not only in the sector in which the standard model resides, but in all other hidden sectors as well.

- 12.
We presume for simplicity that there are no further hierarchies between the extra dimensions. Note that \(\mu _c\) can be in general (hierarchically) different from the actual compactification scale associated to their inverse size.

- 13.
This will remain true for more general compactifications (up to factors of order unity) provided again that there are no further hierarchies among the extra dimensions.

- 14.
For a large enough ratio \(\widetilde{N}/N\)—guaranteed for \(n \ge 2\) through the hierarchy implied by (26).

- 15.
- 16.
As we shall see shortly, the bare cosmological constant term is necessitated to ensure the satisfaction of the Slavnov–Taylor identities when expanding around flat space.

- 17.
- 18.
Order unity non-minimal couplings will generically be generated through renormalization group (RG) flow for the singlet component of the Higgs and any other massive scalars present in the early universe [34].

- 19.
In full generality we allow for the generation of a non-trivial the adiabatic sound speed \(c_s < 1\) as well as higher spatial derivative terms from the curvature squared corrections.

- 20.
- 21.
With six extra dimensions for example, this factor equals \(1/12\). One may be tempted to redefine \(\omega \) in (C7) to absorb the factor of \(n(n+2)\), however, this appropriately rescales the arguments of the potential energy contributions such that (C17) also results for the redefined variables.

## Notes

### Acknowledgments

We wish to thank Robert Brandenberger, Cliff Burgess, Sergey Sibiryakov, and Michael Trott for many informative discussions and comments on the draft. During an earlier portion of this collaboration, SP was supported by a Marie Curie Intra-European Fellowship of the European Community’s 7th Framework Program under Contract No. PIEF-GA-2011-302817.

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