Tensor mode fluctuations

Inflation theory proposed by Guth[6] set out to solve the problem of the apparent flatness and isotropy of the Universe on scales that were not causally connected by invoking a short period of rapid exponential expansion driven by symmetry-breaking of a scalar field. A bonus of this theory is a natural way of seeding the Universe with the necessary perturbations by stretching quantum field fluctuations from the sub-atomic level to cosmological scales. Another consequence is the possible generation at the same time of a similar stochastic background of gravitational waves. Whether or not significant significant tensor perturbations are produced depends on the rate of change of the inflaton field. In inflation theories where there is a slow roll-over such as Linde[13] scaler modes will likely dominate as the anisotropy they cause is approximately $(\Delta T/T)_S\approx\frac{1}{5}H\delta\phi/\dot{\phi}$ as compared to those produced by tensor modes $(\Delta T/T)_T\approx \frac{1}{2}(8\pi G)^{1/2}\delta\phi$, where H is the Hubble constant during inflation, $\delta\phi$ is the amplitude of fluctuations in the inflaton field $\phi$ and $\dot{\phi}$ is the differential with respect to conformal time. In models of power-law inflaton (eg. Abbott and Wise[7]) accelerated expansion can occur implying a large $\dot{\phi}$ and so reduced scalar perturbations.

The predicted spectra indices of perturbations of scaler modes is $n_S\approx 1$ and for tensor modes $n_T \approx 0$, but in the above case of significant gravitational wave production the scalar mode spectra index drops resulting in the so-called ``tilted'' spectrum inflationary models proposed to answer the problem of over-production of small scale perturbations in inflationary CDM models. These ``tilted'' models can be tested for by comparing the results from CMB experiments working on different angular scales such as COBE and Tenerife.

  

Figure: Relation between the ratio r of tensor to scalar contributions to the CMB quadrapole on the scalar mode spectral index.

Steinhardt[14] has considered the equation-of-state expected during inflation in order to quantify the relative contributions on tensor and scale modes to the CMB anisotropy. By taking the ratio rbetween the scalar C₂^S and tensor C₂^T contributions to the expected CMB quadrupole anisotropy C₂ (ie. r = C₂^T/C₂^S) for different scaler mode spectra indices he finds the relation $r \approx 7(1-n_S)$. Figure shows this implications of relation. Due to the relatively steep slope of this line significant (r<1) tensor modes are found if the scalar spectral drops below 0.87. The thick grey lines signify limits consistent with inflation.

  

Figure: The expected CMB angular power spectrum in the case of equal contributions of scalar and tensor modes to the quadrupole.

Coming back to the observable effects on the CMB angular power, Figure  shows the scalar and tensor mode CMB angular power spectra and their sum for the critical case $r\approx 1$. As can be seen the Doppler peak is half the size as the equivalent purely scalar CDM model shown in figure  due to the reduced scalar spectral index of n=0.85 in this case. The relative height to the ``Sachs-Wolfe plateau'' is further reduced, as the gravitational waves raise the plateau but not the peak because they red-shift quickly away once within the horizon scale at $\ell\approx 100$. This will made the tensor mode signal easy to check for once clear observations are made covering angular scales up to and including the peak (ie mapping on degree scales).