# Short Distance Physics and the Consistency Relation for Scalar and Tensor Fluctuations in the Inflationary Universe

###### Abstract

Recent discussions suggest the possibility that short distance physics can significantly modify the behavior of quantum fluctuations in the inflationary universe, and alter the standard large scale structure predictions. Such modifications can be viewed as due to a different choice of the vacuum state. We show that such changes generally lead to violations of the well-known consistency relation between the scalar to tensor ratio and the tensor spectral index. Vacuum effects can introduce an observable modulation to the usual predictions for the scalar and tensor power spectra. 98.80.Bp; 98.80.Cq; 98.65.Dx; 04.62.+v

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## I Introduction

One of the key predictions of inflation [1] is a spectrum of scalar and tensor fluctuations, which are quantum in origin and are stretched from very small wavelengths to cosmological scales [2]. Recently, there has been much discussion of whether short distance physics might modify the nature of the fluctuations. While our current ignorance about physics at very high energy makes a definite prediction difficult, examples have been put forward where the standard predictions for the amplitude and spectrum of fluctuations are modified [3]. Most focus on scalar fluctuations [4], whose amplitude and spectrum are also determined by the choice of the inflaton potential in the standard theory. Modifications in the scalar fluctuations due to short distance physics might therefore be difficult to tell apart observationally from those due to a different choice of the inflaton potential.

This motivates us to examine the relation between scalar and tensor fluctuations generated during inflation. It is well known that for the simplest inflation models, single-field slow-roll inflation, a certain consistency relation exists between the ratio of tensor to scalar amplitude and the tensor spectral index [5]. It has been argued that such a relation, if observationally verified, would offer strong support for the idea of inflation. Here, we demonstrate that modifications of short distance physics generally lead to violations of this consistency relation (and its generalization in multiple-field models). This opens up the exciting possibility of probing new physics at high energies if tensor modes are detected [6].

To make our discussion general, we choose not to focus on particular modifications of short distance physics, but instead view such modifications as equivalent to a different choice of the vacuum state. Indeed, if new physics modifies the dynamics of fluctuations only on some (proper) scales much smaller than the Hubble radius during inflation, as far as the late-time evolution of a given comoving wave-mode is concerned, the only difference from the standard theory is that the initial condition or vacuum state has been modified [7]. The standard choice is known as the Bunch-Davies vacuum, which corresponds to the Minkowski vacuum in the small wavelength limit [8]. This is in some sense the most natural vacuum. For instance, it respects the symmetry of the de Sitter group. On the other hand, inflation generally does not correspond to exact de Sitter expansion. It is conceivable that physics at very short distances effectively forces another choice of the vacuum. Such examples have been constructed, and it would be useful to investigate the generic observational consequences.

Our discussion is organized as follows. In §II, we derive the scalar and tensor power spectra, allowing for a more general vacuum. We then derive the tensor to scalar ratio and their respective spectral indices in §III, and show how the consistency relation can be broken. There is naturally the concern that a different choice of the vacuum might give rise to a particle background which contributes significantly to the total energy density, hence violating the condition for inflation [7]. This, and multiple-field inflation, will be discussed in §III as well.

## Ii Scalar and Tensor Power Spectra and the Choice of the Vacuum

Here, we follow closely the notation of [9]. Much of the material here has been derived before assuming the Bunch-Davies vacuum. We repeat the essential steps, showing where and how the choice of the vacuum affects the final predictions [10]. The metric can be written as

where is the Hubble scale factor as a function of conformal time , , , and are the scalar perturbations (with comma denoting ordinary spatial derivative, and Latin indices denoting spatial coordinates, which are raised or lowered with the unit matrix), and is the tensor fluctuation satisfying , . The relevant action for a minimally coupled inflaton is

(2) |

where is the Ricci scalar, and is some potential. Decomposing into a homogeneous part and fluctuation , we can expand the above action keeping all (metric and inflaton) perturbations up to second order. It cleanly separates into two parts, containing only scalar () and only tensor fluctuations () respectively [11]:

(3) |

(4) |

where denotes derivative with respect to , and is the Hubble parameter. The scalar fluctuation and the tensor fluctuation are defined as

(5) |

They are chosen so that the corresponding action resembles that for a scalar field with variable mass, which can be readily quantized. The quantity can be viewed as a gauge invariant form of the inflaton fluctuation.

The scalar fluctuation obeys:

(6) |

Expanding in plane waves, we have

(7) |

where and are the annihilation and creation operators. Bold faced symbols here denote (3D) spatial vectors. Imposing the commutation relations and implies that the mode functions obey

(8) |

From eq. (6), it is clear that at early times, when , is oscillatory in behavior (here, ). The Bunch-Davies vacuum corresponds to the choice of at early times. The associated operator annihilates the Bunch-Davies vacuum: . More generally, one can choose a different set of mode functions and equivalently a different set of creation and annihilation operators:

(9) | |||

The operator annihilates a different vacuum . We will evaluate all expectation values below using this vacuum, noting that the Bunch-Davies vacuum is recovered if and . Note that the analog of condition (8) for implies the coefficients obey

(10) |

Note that the above choice of the vacuum is not the most general possible [12]. It does describe some of the recent constructions in the literature [13], and is adopted here for simplicity. It is straightforward to generalize our calculation to a wider class of vacua.

We should emphasize here that, if , the state defined above is no longer the ground state within the context of the low energy theory (i.e. the usual field theory calculation). As we have explained in the introduction, this is a deliberate choice to model the effects of short distance physics – from the point of view of the late time evolution of wave-modes, the only effects that short distance physics has are through the definition of the initial state. We refer to this state as the vacuum (or effective vacuum) here, but the reader should keep in mind this is not the vacuum in the usual sense of being the ground state in the low energy effective theory unless .

Eq. (6) tells us that at late times, where is some time independent function of . So, if we define the quantity , its two-point correlation has the following late time asymptote

(11) |

The definition of here agrees with that of [14] in the long wavelength limit. It equals the intrinsic curvature fluctuations on comoving hypersurfaces i.e. quantifying the local departure from flatness. This variable is conveniently frozen when the proper wavelength greatly exceeds the Hubble radius. We define the power spectrum of curvature fluctuations to be [15].

While eq. (6) can be solved exactly by expanding in slow-roll parameters, it is sufficient for our purpose here to obtain by simply matching the early and late time solutions (for the Fourier mode ) at Hubble radius crossing (i.e. ). This yields

(12) | |||||

where the subscript ’cross’ refers to evaluation at Hubble radius crossing, and is the first slow-roll parameter and is defined as . The factor of quantifies the deviation from the standard prediction (corresponding to ) due to the choice of vacuum.

The tensor fluctuations can be derived in an analogous fashion. They obey an equation of motion of the form:

(13) |

Expanding in Fourier modes gives

where is the polarization index, and is the polarization matrix which obeys , and . We have used and to denote the mode function and annihilation operators for gravitons – the extra argument distinguishes them from the corresponding quantities for scalar fluctuations. As before, we can in general choose to have the following behavior at early times:

(15) |

(16) |

Eq. (13) implies that behaves at late times as where is some time independent function of and . From the definition of in eq. (5), it is therefore clear that each Fourier mode of approaches a constant at late times. In other words, the two point correlation function of the tensor fluctuations has the following future asymptote:

(17) | |||||

Similar to the scalar case, we define the power spectrum of tensor fluctuations to be . Matching the solutions for at early and late times at Hubble radius crossing , we obtain [16]

(18) | |||||

Note that we allow here, for the sake of generality, the possibility that i.e. the departure from the Bunch-Davies vacuum can be different for tensor and scalar modes. As we will see, even in the simplest case where , the consistency relation between scalar and tensor modes is generally broken.

## Iii Discussion

Eq. (12) and (18) are the main results of the last section. The effect of the vacuum choice is encapsulated in the two quantities and defined in the two equations. Note that the coefficients , , and , which control and and define the vacuum, are not completely arbitrary because they have to satisfy normalization conditions eq. (10) & (16). Let us now compute the quantities of interest: the scalar and tensor spectral indices and the tensor to scalar amplitude ratio. For historical reasons, the scalar and tensor power spectra are often rescaled and expressed as

(19) | |||

The symbol represents some fiducial wave number. Therefore, the tensor to scalar ratio is

(20) |

and the scalar and tensor spectral indices are

(21) | |||||

where is the first slow-roll parameter defined after eq. (12), and is the second slow-roll parameter, , where is proper time. The above can be derived by noting that in the slow-roll approximation, . Eq. (20) & (21) agree exactly with the standard results [17] if .

Clearly, the usual consistency relation [5],

(22) |

is broken in general unless a special relationship between the coefficients , and is satisfied:

(23) |

Even in the simplest case where (i.e. the modifications to the vacuum for gravitons and for scalar fluctuations are identical), the consistency relation is not satisfied unless is independent of as well [18].

Of course, for arbitrary and , the spectrum can display strong deviations from the power-law form, in which case the spectral index is a poor parametrization of the shape of the power spectrum. However, as pointed out by [7], one expects (and ) to be close to unity so that the particle background does not dominate over the potential energy of the inflaton i.e. the vacuum is ’close’ to the Bunch-Davies form. To be concrete, suppose the vacuum is such that , and (eq. 9, 15), where the S and T labels have been dropped since the same argument applies to both scalar and tensor fluctuations. It can be shown that (eq. 10, 16). Putting the above into eq. (12) and (18), it can be seen that the scalar and tensor power spectra take the form of a weakly modulated power-law , where in the last equality we have kept terms up to first order, and where we have used

(24) | |||

where and , both of which are small. The description of the spectrum in terms of a running power law is sensible, especially if the variation of and with is sufficiently slow. As we have remarked before, even in the simplest case where , the consistency condition is broken unless is -independent (eq. 23).

The arguments of [7] show that , where is the mass scale at which new physics kicks in. If , can be safely ignored for inflation at e.g. GUT scale [19]. However, it is quite possible in which case, , though small, can conceivably be comparable to (or even larger than) the slow roll parameters . In this case, while the tensor to scalar ratio is essentially unchanged i.e. (since is small, modifications due to the vacuum are higher order; eq. 20), the power spectrum shapes could be modified by a non-negligible amount compared to the departure from scale invariance due to (eq. 21, 24). Since upcoming precision measurements promise to constrain the spectral index (at least for scalar modes) to percent-level accuracy, vacuum effects (possibly as large as ) should not be ignored a priori.

We end this discussion by noting that the consistency condition (eq. 22) can also be broken in multiple-field models of inflation, without resorting to a novel vacuum. However, even for multiple-field models (with the standard vacuum), the consistency condition is not arbitrarily broken, but weakens to an inequality[20] . On the other hand, even this weakened consistency condition can be violated by a suitable choice of the vacuum in the single-field model we have considered i.e. by choosing .

Ultimately, to make progress, we need a better understanding of what forms and might take – this would help us decide whether resorting to multiple-field models or modifying the vacuum (or even developing alternatives to inflation) is a more plausible option in the event that a violation of the consistency relation (eq. 22) is indeed observed. Perhaps the most important (and in retrospect not surprising) lesson of our exercise is that the key predictions of inflation for the scalar and tensor fluctuations depend critically on the choice of the vacuum state. While the Bunch-Davies vacuum is perhaps the most natural one, fundamentally it is not obvious whether short-distance physics will indeed pick out this vacuum. This is an important question which deserves further investigation. Our main result is that a specific condition on the vacuum (eq. 23) has to be obeyed for the well-known consistency relation between scalar and tensor modes to be respected.

###### Acknowledgements.

We thank Robert Brandenberger, Richard Easther, Brian Greene, Alan Guth, Gary Shiu and Ed Witten for useful discussions, and Scott Dodelson for his excellent notes. LH and WHK are supported respectively by the DOE Outstanding Junior Investigator Program, and the Columbia University Academic Quality Fund.## References

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