Discussion: Theoretical signal-to-noise ratio for the stochastic background

In order to reliably detect a stochastic background of gravitational radiation, we will need to be able to say (with a certain level of confidence) that an observed positive mean value for the cross-correlation signal measurements is not the result of detector noise alone, but rather is the result of an incident stochastic background. This leads us natually to consider the signal-to-noise ratio, since the larger its value, the more confident we will be in saying that the observed mean value of our measurements is a valid estimate of the true mean value of the stochastic background signal. Thus, an interesting question to ask in regard to stochastic background searches is: ``What is the theroretically predicted signal-to-noise ratio after a total observation time , for a given pair of detectors, and for a given strength of the stochastic background?'' In this section, we derive the mathematical equations that we need to answer this question. Numerical results will be calculated by example programs in Secs. and .

To answer the above question, we will need to evaluate both the mean value

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As described in Sec. , if the magnitude
of the noise intrinsic to the detectors is much larger than the
magnitude of the signal due to the stochastic background, then

where is an arbitrary filter function. The choice

maximizes the signal-to-noise ratio (). It is the

it is convenient to choose the normalization constant so that

For such a ,

which leads to the

This is equation (3.33) in Ref. [36].

But suppose that we do *not* assume that the noise intrinsic to
the detectors is much larger in magnitude than that of the stochastic
background.
Then Eq. () for needs to be modified to
take into account the non-negligible contributions to the variance
brought in by the stochastic background signal.
(Equation () for is unaffected.)
This change in implies that Eq. () for
is no longer optimal.
But to simplify matters, we will leave as is.
Although such a no longer maximizes the signal-to-noise
ratio, it at least has the nice property that, for a stochastic background
having a constant frequency spectrum, the normalization constant
can be chosen so that is independent of .
The expression for the actual optimal filter function, on the other
hand, would depend on .

So keeping Eq. () for , let us consider a
stochastic background having a constant frequency spectrum as described
above.
Then we can still choose so that

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The new squared signal-to-noise ratio is divided by the above expression for .

Note the three additional terms that contribute to the variance
.
Roughly speaking, they can be thought of as two ``signal+noise''
cross-terms and one ``pure signal'' variance term.
These are the terms proportional to and ,
respectively.
When is small, the above expression for
reduces to the pure noise variance term ().
This is what we expect to be the case in practice.
But for the question that we posed at the beginning of the section,
where no assumption is made about the relative strength of the
stochastic background and detector noise signals, the more complicated
expression () for should be used.
The function `calculate_var()`, which is defined in the
following section, calculates the variance using this equation.