CMB Power Spectrum from Noncommutative Spacetime

Qing Guo Huang and Miao Li

Institute of Theoretical Physics

Academia Sinica, P.O. Box 2735

Beijing 100080

Department of Physics

National Taiwan University

Taipei 10764, Taiwan

Very recent CMB data of WMAP offers an opportunity to test inflation models, in particular, the running of spectral index is quite new and can be used to rule out some models. We show that an noncommutative spacetime inflation model gives a good explanation of these new results. In fitting the data, we also obtain a relationship between the noncommutative parameter (string scale) and the ending time of inflation.

April, 2003

Recent results from the Wilkinson Microwave Anisotropy Probe [1-3] give further determination of cosmological parameters and constrain properties of inflationary models tightly. Some analysis was done in [3], and in particular, the new data on the running of the spectral index of the power spectrum of CMB provide good constraints on inflationary models, and some models were analyzed in [4]. In general, models satisfying the slow roll conditions naturally predict too small running of the spectral index. In this note we analyze the model of [5] in which spacetime noncommutativity is taken into account, and find that it is rather easy to accommodate WMAP data in this model. It is quite remarkable that the two parameters in this model can be used to fit four data of the spectral index. We determine these two parameters, and the string scale is in turn determined once the termination time of inflation is known. For a typical termination time of inflation [7], the string scale is cm, or Gev.

Spacetime in general is noncommutative in string theory, this manifests in a general relation [8]. However it is hard to formulate string theory in a fashion to incorporate this relation directly. On a general ground, if inflation reflects physics at a scale close to string scale or a related scale, one expects that spacetime uncertainty must have effects in the CMB power spectrum, observable or to be observable (implications of space-space noncommutativity for inflation were considered in [6]). Lacking a manifest formulation of noncommutativity, we shall be content with a scheme proposed in [5], in which spacetime noncommutativity is put in by hand for a scalar field (inflaton). Skipping the detailed discussions leading to their action, we copy the free scalar action here

where

where is the string length scale, is the scale factor in the metric, and time is defined such that

Unlike in the commuting spacetime, the all important scale factor in our case depends on the comoving wave number .

We shall consider the power-law inflation, and this can be generated by a scalar potential assuming an exponential form. Let be the cosmological time, so , . In terms of , with

To simplify relations that will follow, we define a scale so that and . We shall estimate the relationship between the scale and the termination time later.

The scalar power spectrum can be computed in the usual way, and is given by

and is determined by the condition

Two extremal limits were considered in [5], namely the UV limit and the IR limit. However, the observed results must lie in between, so we need to calculate the power spectrum again. The interested region is actually UV in terms of the time when the perturbation is created, but this was not discussed in [5]. What we need to compute is

Define

Naively, is determined by two microscopic scales and , one would expect that is also at the microscopic scale. This is wrong, due to the large exponent . To achieve sufficient number of e-folds, ought to be large enough, and indeed we will determine it be around 13. Thus, if is smaller than by a few order, the characteristic wave length is macroscopic. The interesting range of is between , and we assume these length scales be smaller than , so . In solving the crossing horizon condition (6), we will expand everything in .

Without the string spacetime uncertainty, or let , the crossing horizon time is

the dependence on is fictitious, since a factor in cancels . Since string uncertainty introduces a uncertainty in time , this ought to be smaller than the time when the perturbation is created, namely, must be small. This ratio is nothing but

For a large enough , is small enough so that the above assumption will be valid.

In solving (6), not only we need to modify according to (2), but also we need to modify the definition of the conformal time according to (2). For instance

where we have truncated to the first nontrivial order in . After a lengthy but straightforward calculation, we find

Solving (6) using (12), we obtain

The power spectrum is readily calculated to be

Denote the coefficient in the front of in the above formula by (the positive one), we compute the spectral index and the running of the index

Since is positive, is larger for smaller , the expected behavior. Now, use the WMAP data

to see whether the main formulas in (15) can fit these. There are only two parameters in (15), namely and , but there are four independent results from the WMAP. We will adopt the following strategy: we first use the two results at to determine the two parameters, and make a prediction about and its running at . Later, we use the results about at two different scales to make a prediction about the running of at the two scales.

Using the results at , we find

Indeed, is a small number, and is also small enough so the approximation used in deriving (15) is valid. These parameters together with (15) predict

The central value of the predicted is quite good compared to that in (16), the central value of the predicted running is larger than that in (16), but within the error bar, we notice that the tendency of the predicted running is good, it is larger at a larger scale. For , it is possible that higher orders in are not negligible, and may improve our result. We also notice that in [3] the central value of the running is estimated to be , much closer to our prediction.

Next, using the values of at two different scales in (16), we determine

and the running of the spectral index

We show the spectral index and the running as functions of in fig.1 and fig.2.

We have used the WMAP data to determine two parameters and in the noncommutative inflation model. Next, we want to determine the relation between the ending time of inflation and the string scale. To do so, note that the comoving wave numbers are measured at the present time, so the normalization of the scale factor at the present is 1. Since the termination of inflation, our universe underwent a radiation dominated epoch and matter dominated epoch (for simplicity we ignore the observed dark energy), so the scale factor was enhanced twice:

where is the termination time of inflation and its unit is second. Using , we find the parameter

is larger than . Next, using the definition of in (8), we get

In the end, and are expressed in terms of

where in defining and , we have turned and in the unit cm.

For parameters in (17), we have

Thus for s, cm, cm. is larger than the Planck scale by 8 orders. For s, the earliest possibility [7], cm, larger than the Planck scale only by two orders. For parameters in (19),

not much different from those in (25).

Figure 1. as a function of . Parameters are determined using data at .

Figure 2. as a function of . Parameters are determined using data at .

To summarize, we have found that the noncommutative inflation model accommodates the recent WMAP data nicely, and if this model possesses any truth about string theory, string theory can already be tested in the observational cosmology, this is very exciting. Although the WMAP results on the spectral index and its running are obtained by combining other experiments and are not yet very conclusive, we believe that future refined results will offer better opportunity for testing whether spacetime uncertainty is a viable physical model for inflation.

Acknowledgments. This work was supported by a grant of NSC, and by a “Hundred People Project” grant of Academia Sinica and an outstanding young investigator award of NSF of China. This work was done during a string workshop organized by ICTS (Interdisciplinary Center of Theoretical Sciences), its hospitality is gratefully acknowledged.

References

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