^{3}

It is often said that detecting a spectrum of primordial gravitational waves via observing B-mode polarization of the Cosmic Microwave Background is the “Holy Grail” of inflation. The purpose of this short note is to point out that it is indeed of immense scientific interest to search for a signal of gravitational waves in B-mode polarization. However, rather than proving that inflation is the right paradigm of early universe cosmology, a positive signal of direct primordial B-mode polarization might well be due to other sources than inflation. In fact, a careful characterization of the spectrum of B-mode polarization might even falsify the inflationary paradigm.

At the present time, the Cosmic Microwave Background (CMB) is our most accurate tool to study the structure of the universe. CMB temperature maps allow us to study the structure of the universe at the time

There is, however, more information in the CMB than simple temperature maps reveal: CMB radiation is polarized, and the polarization carries a lot of important information about cosmology. CMB polarization can be decomposed into E-mode and B-mode polarization. The E-mode polarization power spectrum has now been observed, as has the cross correlation power spectrum between temperature and E-mode. However, to date no B-mode polarization has been detected.

Primordial adiabatic density fluctuations only lead to E-mode polarization, whereas gravity waves will lead to a signal in B-mode polarization. Thus, the search for B-mode CMB polarization is one of the ways to detect a spectrum of primordial gravitational radiation on cosmological scales.

The inflationary scenario [

It is currently often said that the measurement of gravitational radiation through the detection of primordial B-mode polarization is the “Holy Grail” needed to confirm that inflation is the correct theory of the very early universe (see e.g. [

Here, I point out that IF there is a measurable spectrum of gravitational waves on cosmological scales, these waves could well be due to other sources than inflation. Whereas inflation typically produces a small amplitude of gravitational waves, the other sources which I will mention can produce a larger amplitude. Secondly, I point out that there are other cosmological sources of B-mode polarization. Hence, a positive detection of B-mode polarization might NOT be due to gravitational waves at all.

This Note is intended mainly for experimentalists. I hope to argue here that it is even more important to search for B-mode polarization than if gravitational waves from inflation were the only possible source. In fact, I will show that it is possible that B-mode polarization results could falsify the inflationary paradigm.

The scenario of cosmological inflation [

Space-time sketch of inflationary cosmology. The vertical axis is time, the horizontal axis corresponds to physical distance. The period of inflation lasts from

A space-time sketch of inflationary cosmology is shown in Fig.

It is important to realize that a scale-invariant spectrum of adiabatic fluctuations as the origin of structure in the universe was already postulated a decade before the development of inflationary cosmology [

Part of the motivation for looking beyond inflationary cosmology is the fact that (at least current realizations of) inflation suffer from various conceptual problems (see e.g. [

There are, however, more serious issues. One of them is the “trans-Planckian issue” for fluctuations [

A space-time sketch of inflationary cosmology. As in Fig.

If inflation is obtained by coupling classical matter fields to Einstein gravity, then a cosmological singularity in the past is as unavoidable [

In large field toy models of inflation, the energy scale at which the inflationary phase takes place is typically of the order

The above conceptual problems of inflation motivate the search for alternatives. As will be shown below, there are in fact alternative cosmological scenarios, and some of them yield an almost scale-invariant spectrum of gravitational waves like inflationary cosmology does. In fact, in at least one of these alternative approaches, the amplitude of the gravitational wave spectrum is larger than it is in simple inflationary models.

In this section I will briefly review three alternative scenarios to inflation. This list is not meant to be complete! The three scenarios I will mention are an “emergent stringy universe” [

In the “emergent universe” scenario [

Time evolution of the scale factor (size of space) in the emergent universe scenario. The time axis is the horizontal axis, the vertical axis gives the value of the scale factor. The time

Space-time sketch of the emergent universe scenario. As in Fig.

A possible realization of the emergent universe scenario arises in the context of “String Gas Cosmology” [

Evolution of the temperature (vertical axis) as a function of the logarithm of the radius (horizontal axis). Two possible evolutionary tracks are shown which differ in the amount of entropy which they contain. If we begin the evolution at large radii, then initially the temperature rises as space contracts as for point particle matter. However, once the temperature approaches the limiting Hagedorn temperature

In string gas cosmology, matter in the early universe is a gas of strings in thermal equilibrium. Hence, the thermal fluctuations dominate over the vacuum ones – in contrast to inflationary cosmology where it is assumed that fluctuations arise from a vacuum state. It was realized in [

A second alternative to inflation is the “matter bounce” scenario. If there were a non-singular bouncing cosmology, then there would obviously be no singularity problem. Penrose and Hawking a long time ago taught us that in order to obtain such a non-singular bounce, one needs to go beyond General Relativity as the theory of space-time, or else invoke matter which violates some of the usual energy conditions. There is a long history of such attempts (see [

Space-time sketch of the matter bounce scenario. The vertical axis is conformal time

Given a non-singular cosmology, it is most logical to assume that the contracting phase begins with a matter-dominated phase, then makes a transition to a radiation-dominated phase before undergoing the bounce and re-emerging as an expanding Standard cosmology. Figure

A while back it was realized [

The major problem of the “matter bounce” scenario and of other bouncing cosmologies is the “anisotropy problem”: during the phase of contraction, the energy density in anisotropies grows faster than the energy density in the isotropic matter components. Hence, unless the initial anisotropies are tuned to be extremely small, no smooth cosmological bounce can occur.

The Ekpyrotic scenario [

The Ekpyrotic scenario assumes that the attractive potential

In inflationary cosmology, a scale-invariant spectrum of gravitational waves is generated from initial quantum vacuum fluctuations of the canonical fields which represent each polarization state. We begin with the metric

Inserting the gravitational wave ansatz (

Up to this point, the analysis has been completely general. Let us now consider the application to inflationary cosmology. In this case, the perturbations originate as quantum vacuum fluctuations. Since the amplitude of the quantum vacuum perturbations is given by the Hubble constant

Note that if inflation is realized in the context of Einstein gravity and is generated by matter fields obeying the “Null Energy Condition”, then the Hubble constant must be a decreasing function of time and hence

The rate at which the gravitational waves are squeezed on super-Hubble scales is the same as the rate at which a test scalar field on a fixed background metric is squeezed. If the equation of state of the background is independent of time, the scalar metric fluctuations (the “cosmological perturbations”) are squeezed at the same rate (this is true e.g. in the matter bounce scenario during the relevant time intervals), but it is NOT true in inflationary cosmology where the cosmological perturbations are amplified more during the reheating period than the gravitational waves, which leads to the small value of the tensor to scalar ratio

Before discussing what kinds of spectra of gravitational waves the alternative cosmological scenarios mentioned in Sect.

Cosmic strings are linear topological defects predicted to form in a wide range of phase transitions of matter in the early universe. A subset of grand unified field theories leads to cosmic strings, in particular in many supergravity models [

In models which admit topologically stable cosmic strings, a network of such strings will inevitably form in the early universe (see [

Due to the fact that the distribution of strings is scaling, a scale-invariant spectrum of gravitational waves emerges. In turn, this produces a tensor contribution to the microwave anisotropies, as first calculated in [

Based on the scaling solution of the network of infinite strings, one can calculate the distribution of cosmic string loop sizes. The number density

Based on this scaling distribution of string loops, the energy density distribution in gravitational waves can be calculated, and the resulting CMB temperature anisotropies can be inferred [

The above result for the amplitude of the tensor mode should be compared with the observed amplitude of the scalar mode which is

Although cosmic strings predict a spectrum of gravitational waves which is scale-invariant on large angular scales, the model predicts specific non-Gaussian signatures in position space. In CMB temperature maps the signal is a distribution of edges across which the temperature jumps [

As pointed out in [

Since the polarization signal of cosmic strings also has special non-Gaussian features, it is important to search for them in position space through the use of statistics such as the edge detection algorithms mentioned above.

As initially pointed out in [

After this discussion of gravitational waves and direct B-mode polarization produced by a source which might be present in Standard Cosmology, let us now turn to the predictions for the spectrum of the stochastic background of gravitational radiation from the alternative cosmological models introduced earlier.

We have seen in Sect.

The key difference between the predictions of inflation and string gas cosmology relates to the index of the spectrum. Whereas inflationary models always yield a red spectrum, string gas cosmology generically produces a blue shift (which makes it easier to detect on shorter wavelengths – see [

In the case of the matter bounce scenario, the evolution of the spectrum of gravitational waves and of cosmological perturbations is identical on super–Hubble scales during the contracting phase. If the bounce phase is short, the evolution in this phase will not lead to a change in the amplitude of the fluctuations (neither of cosmological perturbations nor of gravitational waves). Thus, rather generically a matter bounce scenario will lead to a scale-invariant spectrum of gravitational waves with a large amplitude (

The spectrum of fluctuations produced by the matter bounce can be distinguished from that generated by simple single field inflationary models in terms of the induced non-Gaussianities. Specifically, the matter bounce predicts a special shape [

In the case of the Ekpyrotic scenario, the equation of motion of the cosmological perturbations depends on the potential of the scalar field, whereas that of the gravitational waves does not.

If we start out with vacuum fluctuations for both cosmological fluctuation and gravitational wave modes on sub-Hubble scales in the contracting phase, then, unlike in the Matter Bounce scenario, the increase of the amplitude of the gravitational wave modes on super-Hubble scales is insufficient to turn the initial vacuum spectrum into a scale-invariant one. The predicted spectrum of gravitational waves is very blue and hence primordial gravitational waves are negligible on scales of cosmological interest today. In contrast, the cosmological perturbation modes couple to the potential and this allows them to attain a scale-invariant spectrum.

In this Note I have shown that there are many sources of gravitational waves which could lead to a roughly scale-invariant spectrum on cosmological scales. Thus, the detection of relic gravitational radiation via B-mode polarization will NOT prove inflation. Since several of the mechanisms described here predict a spectral amplitude which is larger than that generated in the simplest inflationary models, one may argue that it is more likely that a positive signal will be due to a source different from inflation. Luckily, the different sources of gravitational waves – all of them giving a roughly scale-invariant spectrum – lead to specific predictions with which they can be distinguished. String gas cosmology produces a slight blue tilt of the spectrum, whereas inflationary cosmology always produces a slight red tilt. Cosmic strings lead to specific non-Gaussian signatures which can be identified in position space (see e.g. [

It must also be kept in mind that gravitational radiation is not the only way to generate primordial B-mode polarization. Once again, cosmic strings formed in a phase transition during the early Standard Cosmology phase of the evolution of the universe will generate primordial polarization which is statistically equally distributed between E-mode and B-mode [

The message which this Note is supposed to convey is the following: the search for B-mode polarization is an extremely interesting field, much more interesting than if the only source of such primordial polarization were gravitational waves from inflation. It will be very important to carefully analyze the data without the prejudice that inflation is the only source of a signal. Otherwise, the possible existence of cosmic strings could be missed. More strikingly, if B-mode polarization is found and is shown to be due to gravitational waves, then if the spectrum is slightly blue one would have falsified the inflationary paradigm.

This work is supported by an NSERC Discovery Grant, by the Canada Research Chairs program and by a Killam Research Fellowship. I thank Gil Holder, Matt Dobbs and Xingang Chen for comments on the draft.

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This is a theory paper and it does not present new data.]

The word “approximately” is to be understood slightly differently in the two contexts above. All simple inflationary models lead to a slight deviation from scale-invariance – this explains the meaning of “approximate” in the first case. The simplese inflationary models produce a purely adiabatic spectrum of fluctuations, but a generic more complicated model will produce an iso-curvature component to the fluctuation spectrum – this is the way in which the second use of “approximate” should be understood.

The Hubble radius separates small scales where fluctuations typically oscillate from large scales where the microphysical oscillations are frozen out and the evolution of the perturbations are dominated by the gravitational dynamics of the background. See [

In numbers, more than

The decrease of

As in the case of string gas cosmology, this scenario is free from the trans-Planckian problem for fluctuations provided that the energy scale of the bounce is lower than the Planck scale.

Taking the numbers from the more updated simulations of [

The reason for the blue tilt is easy to understand physically [