Imprints of Short Distance Physics On Inflationary Cosmology
Abstract
We analyze the impact of certain modifications to short distance physics on the inflationary perturbation spectrum. For the specific case of powerlaw inflation, we find distinctive – and possibly observable – effects on the spectrum of density perturbations.
pacs:
PACS numbers: 98.80.Cq,12.90.+b,11.25.wInflation stretches quantum fluctuations to astrophysical scales, providing a microscopic mechanism for the formation of galaxies structure . Most models of inflation yield far more quasiexponential expansion than the 60 efoldings required to solve the difficulties faced by the standard model of the hot big bang. Consequently, astrophysical scales in the present universe map to physical distances in the primordial universe that are exponentially smaller than any conceivable fundamental length. As explained in brand1 ; BM ; N , this introduces an implicit assumption into the perturbation spectrum calculation: that spacetime physics and quantum mechanics can be extrapolated to arbitrarily small physical lengths, independently of any fundamental length scale. A fundamental length is predicted by almost all attempted unifications of general relativity with the other fundamental forces of nature, and also by quantum theories of gravity. Naïve dimensional arguments identify this scale with the Planck length but, for example, the additional scale introduced by string theory, the string length, can easily be one or two orders of magnitude larger than the Planck length stringscale , if not more adds . Thus we ask two important questions: can a fundamental length change the predicted perturbation spectrum and, if so, are these differences detectable observationally? If these questions are answered in the affirmative, they may provide astrophysical tests for theories of nature containing a fundamental length scale, including string theory.
Initial investigations BM ; N relied on assessing the robustness of the usual inflationary spectra to changes in the perturbations’ dispersion relations. A different approach was initiated in Kempf ; KN ; EGKS where a nominally string inspired fundamental length appears in the uncertainty relations and provides a short distance cutoff. We showed that the de Sitter space power spectrum predicted for this model is rescaled by a multiplicative constant, observable only if one has independent knowledge of the Hubble constant during inflation EGKS . We emphasize that the calculations of EGKS and those presented here are not performed within string theory, but instead make use of a standard field theory modified on short scales in a manner inspired by string theory. Our aim is to determine if such short distance scale modifications can yield astrophysically observable signatures.
In almost all non de Sitter inflationary backgrounds the expansion rate is slower than exponential, and the physical horizon size increases with time. We predicted that the spectrum will be changed more dramatically at long wavelengths than at short wavelengths (larger ), since the impact of the fundamental length increases with its ratio to the physical horizon size EGKS . Since the resulting change to the spectrum is more complicated than a simple rescaling, it is – in principle – observable.
This letter discusses the perturbation spectrum generated by power law inflation when a fundamental length is inserted into the uncertainty relations. The shape of the spectrum does change, in line with our initial expectation. To this extent, we agree with KN which presents a qualitative analysis of the same problem. However, the magnitude of the effect drops more slowly as the fundamental length decreases than the analysis of KN suggests, with important consequences for the possibility of observing this signal.
In broad outline, the procedure for determining the spectrum is not changed by the introduction of a fundamental length. We begin with appropriately normalized field oscillations, which are quantum in origin but obey the classical equations of motion, and extract the spectrum from their asymptotic amplitudes. However, the fundamental length modifies both the evolution equation and the explicit form of the normalization condition.
For wavelengths much greater than the fundamental length, tensor modes obey
(1) 
with . Scalar modes obey a similar equation, with replaced by . Here is the field responsible for inflation, and . For the special case of powerlaw inflation, , so scalar and tensor modes obey identical equations, although their power spectra differ in their normalizations. Since the two types of modes have the same equations of motion at long wavelength, we assume that they also obey the same equations of motion at short wavelength, where the influence of shortdistance physics is important, and that any modulations of the power spectrum due to shortdistance physics apply identically to tensor and scalar modes. With this assumption, modifications to short distance physics can result in violations of the socalled “consistency condition” for inflationhui01 ; Shiu:2001sy . While this is reasonable, it is by no means guaranteed: scalar modes are a mix of field and metric fluctuations in an arbitrary gauge MFB , where as the tensor modes are purely metric. It is not clear that a shortdistance cutoff will affect fluctuations of the metric in the same way as fluctuations in an arbitrary scalar field. However, general coordinate invariance implies that we can transform (for example) to a gauge in which even the scalar fluctuations are purely “metric”, and if this property is preserved at short distances then the effect of new physics on the scalars and tensors should be identical. Even in the existing literature, metric (gravitational wave) fluctuations can be treated as a generic scalar field at short distances. This is also reasonable, but not inevitable.
Following Kempf , tensor fluctuations obey:
(2) 
where is the scale factor, the prime denotes differentiation with respect to conformal time , while with being the Fourier transform of the physical coordinates , and
(3) 
When evaluating the derivatives of with respect to , we are holding (and not the usual comoving momentum ) fixed with time. It is therefore convenient to express and in terms of by introducing the Lambert function Lambert , which is defined so that :
(4) 
where .
For power law inflation
(5) 
For inflation to occur, we need .
The cutoff is introduced by requiring that , motivated by the notions of a minimum distance in string theory and the socalled “stringy uncertainty principle” sur . Fluctuations with comoving wavenumber reach the cutoff at , where
(6) 
where the implicit definition of comes from setting when , and is the usual physical time.
Writing in order to extract the dependence and abbreviating as , we have
(7) 
In the de Sitter case, can be eliminated from the equation of motion, but this cannot be done here. During powerlaw inflation, different modes sample a different value of the Hubble constant as they cross the horizon, and this will be reflected in the scaledependent modifications to the spectrum we will observe. Despite this, our analysis of the de Sitter case EGKS can easily be adapted to the powerlaw problem.
When , has a branch point Lambert . Physically, this is the moment when () and the fluctuation with wavelength is “created”. As in EGKS , we solve for the leading behavior of by extracting the most singular terms of the equation of motion,
(8) 
where dots denote derivatives with respect to , and
(9) 
The solution to (8) is:
(10) 
Here, is the second Hankel function, and are constants. The first Hankel function is its complex conjugate .
The solution is normalized by the Wronskian condition
(11) 
Using Hankel function identities abramowitz we deduce
(12) 
This, together with equations (10) and (13), gives the general result for all possible boundary conditions. More specifically, though, we must fix and , i.e. we must specify boundary conditions for equation (8) or, equivalently, the form of the vacuum state. Ultimately, a more complete understanding of short scale physics would allow a first principles selection of boundary conditions. Here we simply note that if , the spectrum never approaches the exact powerlaw form, even when the Hubble parameter is arbitrarily small. Consequently, is the only vacuum choice with constant coefficients that reduces to the BunchDavies vacuum at low energies. We thus focus our analysis on the “minimal” choice that for all , but we stress that this choice is not unique: and could depend on , or in a such a way that the constraint of (12) was always satisfied, and that was nonzero at high energies, and vanishing sufficiently rapidly as to avoid any experimental constaints on nonzero . Our results obviously depend on this choice and this issue certainly deserves further investigation. Finally, we point out that that setting in Equation (12) reproduces the de Sitter result, after reconciling the normalization factors.
We solve the mode equations numerically EGKS ; Easther and match the numerical solution to the approximate analytical form, including subleading corrections, near . We obtain the scalar spectrum by solving the mode equations for multiple values of , and then extracting the necessary late time limit to compute
(13) 
where we have obtained the scalar spectrum from the tensor one, as outlined above.
The spectrum is only well defined if the minimum length () is less than the horizon size (), or is less than unity. The critical mode, , that crosses the horizon at the moment when is
(14) 
For large values of , is enormous. This reflects the massive amount of inflation that takes place between the Planck time ( in natural units) and the moment at which ; the numerical value of can always be rescaled by redefining , the value of when .
Fig. 1 shows the spectrum for the longest modes, with . There is a large modulation in the spectrum, corresponding to the slow decrease in as the universe evolves. However, these modes have a much larger amplitude than those contributing to the CMB power spectrum and structure formation. Fig. 2 displays the spectrum with and a “window” of values with amplitudes of the same order as the modes which are the precursors to structure formation. We have not carefully normalized this spectrum (which requires assumptions about the dark matter composition, , etc.), since any signal of transPlanckian physics is much smaller than the uncertainty in currently available data. Instead we assume that , which – given that turner – corresponds to .
The standard powerlaw spectrum is modulated by an “oscillation” whose amplitude and wavelength depend on both the fundamental length and the powerlaw parameter, . The oscillations are attributable to successive modes undergoing increasing numbers of periods between the initial time and horizon exit, with a full extra period corresponding to a single oscillation in the spectrum.
The amplitude and period (in ) of the oscillations are roughly proportional to . In principle is predicted by fundamental theory, but from our perspective here it is a free parameter. If is identified with the string scale, it could conceivably be two orders of magnitude longer than the Planck length, and we use this value in the numerical plots. Observationally, the key parameter is the ratio of the fundamental length to the Hubble radius . In powerlaw inflation (and any other non de Sitter model) is a slowly changing parameter. The observationally relevant range of is fixed by the amplitude of the power spectrum, which is deduced from observations of the CMB and large scale structure.
The rate of change of is determined by , and as increases the wavelength of the fluctuations in the spectrum increases while their amplitude goes down. This accords with our physical understanding of the oscillations: for a fixed value of (and thus ), decreases with increasing . Thus the wavelength of the oscillations (in ) increases with , since at horizon exit changes more slowly with at larger . The variation in the amplitude arises because we are effectively holding fixed at horizon exit, but decreases as is increased. Consequently, the effective value of for a given mode decreases as is increased, which accounts for the dependence of the amplitude of the oscillations. The oscillations do not vanish as becomes arbitrarily large, although their wavelength becomes arbitrarily long, and we approach the de Sitter limit where the spectrum is shifted by a constant multiplicative factor.
In Fig. 2, the spectrum is almost flat, and the values that would be measured by CMB experiments are modified by between and %. A signal of this size lies at the limits of detectability, even with ideal experiments, and would be swamped by cosmic variance at all but the largest values of . Existing constraints on the spectral index put a weak lower bound on of around 20. With this value, the oscillations’ wavelength is so short that the resulting spectrum appears to include a random noise term when plotted over the range of values relevant to structure formation.
Despite the extreme challenge and perhaps near impossibility of detecting an effect of this size for reasonable values of , the conclusions of this letter are still much more optimistic than we might have otherwise expected. First, and in accord with our previous de Sitter calculation, we find that the magnitude of the modification to the spectrum is a function of , where appears to be slightly smaller than . This disagrees with KN , in which it is argued that is roughly unity. However, KN relies on a WKB approximation to the mode equation and chooses the vacuum to be the purely “” WKB solution. We have decomposed our numerical solutions into the two WKB solutions at a time when the WKB approximation holds well, and the actual solution (using the initial conditions described above and also advocated by KN ) contains a mixture of both WKB solutions, where the coefficient on the plus “” scales like with . It is this mode mixing of the standard WKB solutions, with a mixing coefficient of order , that accounts for the magnitude of our results. Thus the pure “” WKB approximation is not consistent with these initial conditions, perhaps explaining why the estimate of KN for the impact of the fundamental length on the spectrum is significantly less than we find here.
We have assumed that the minimum length lies somewhat below the Planck scale. While this is justifiable from a stringy perspective, if we had put the fundamental length equal to the Planck length () the effect we see decreases significantly. Moreover, as mentioned, the equations we solve are not derived from string theory and hence there is no guarantee that they are the ones that quantum gravity will give us. Nevertheless, we find it encouraging that whereas there are 16 orders of magnitude separating the Planck scale from conventional accelarator experiments, the string scale modifications we study here yield cosmological effects that may be only one or two orders of magnitude below the threshold of observability.
In principle existing CMB measurements put experimental restrictions on a portion of the plane. Given the accuracy of current data, the constraints on would be extremely weak, and we have not performed this calculation. As CMB data and surveys of Large Scale Structure (and our ability to work backwards from the observed to the primordial spectrum) improve, it may become possible to place meaningful restrictions on short scale physics using astrophysical and cosmological data.
It is natural to ask whether we can do a full string theoretic version of this calculation. One approach would be to study the full twopoint functions of graviton and inflaton string excitations, either from string field theory or along the lines of Ref. CMNP . These quantities are sensitive to the large number of high energy degrees of freedom found in string theory, another inherently stringy feature.
Acknowledgements.
We thank Robert Brandenberger and Jens Niemeyer for discussions. The work of BG is supported in part by DOE grant DEFG0292ER40699B and the work of GS was supported in part by the DOE grants DEFG0295ER40893, DEEY76023071 and the University of Pennsylvania School of Arts and Sciences Dean’s funds. ISCAP gratefully acknowledges the generous support of the Ohrstrom Foundation.References
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