- freely available
Universe 2019, 5(9), 192; https://doi.org/10.3390/universe5090192
2. Numerical Methods
2.2. Time Variables
2.4. Equations of Motion
2.5. Integrating and Evaluating the Background
2.5.1. Integration Strategies
- simplicity, one needs to integrate a system of variables with respect to time just once;
- less integration overhead, the integration software itself has a computational cost. Each time it is used, there are the costs of initialization, step computation and destruction.
- step sizes smaller than necessary. When performing the integration, the Ordinary Differential Equations (ODE) solver evaluates the steps of all components being integrated and adapts the time step such that the error bounds are respected by all components. For this reason, including the ‘C’ set in the integration can result in a larger set of steps for all quantities ‘B + C’;
- difficult to efficiently modularize, in some analysis, just a few (or just one) observables from ‘C’ are necessary. When performing all integration at once, you have two options, integrate everything every time, even when they are not all necessary, or to create a set of flags that control which quantities should be integrated by branching (e.g., if-statements). Naturally, the if-clauses create an unnecessary overhead if they are placed inside the integration loop. Alternatively, if one decides to create a loop for every combination to avoid this overhead, then he/she will end-up with different loops (where n is the number of variables in ‘C’) which creates new problems such as code repetition and harder code maintenance.
2.5.2. Evaluating the Background
3. Linear Perturbations
4. Cancellation Errors
Conflicts of Interest
|SNeIa||Type Ia Supernovae|
|ODE||Ordinary Differential Equation|
|CMB||Cosmic Microwave Background|
|LSS||Large Scale Structure|
|CAMB||Code for Anisotropies in the Microwave Background|
|CLASS||The Cosmic Linear Anisotropy Solving System|
|MCMC||Markov Chain Monte Carlo|
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The observed anisotropies in the CMB are very small (), and, for this reason, they are within the regime of validity of the first order perturbation theory.
The relation between z and is monotonic since .
The ± sign in the definition of is included in order to have it monotonically increasing with t or in the expansion/contraction, respectively.
Note that here we are defining as a constant evaluated today, in the literature, the definition as a function of z is also frequently found, i.e., , where .
It would be necessary to save/read the last evaluation interval in a per-thread basis.
When are not positive definite one can use a similar algorithm with and , where the is a scale s controls the transition between linear and log scale.
Usually, the relative tolerance controls the final error of the code while the absolute tolerance is application specific and, in the few cases where it is used, it serves to avoid excess in the tolerance. For example, if a given function f is zero at a point x and we want to compute its approximation p, the error control is
Thus, without the absolute tolerance, the error control would never accept the approximation, unless it was a perfect approximation .
We can write and , for an eigenfunction of with eigenvalue , i.e., .
It is for this reason that some authors use the expressions super-/sub-Hubble scales, scales much larger/smaller than . We can also find the expressions super-/sub-horizon to refer to the same scales. Nevertheless, this nomenclature is based on the fact that, for some simple models, the horizon is also proportional to , which can be wrong and counter-intuitive in many cases and as such should be avoided.
For a complete exposition about this subject see .
The precision is usually defined in binary basis, since it is that which are used in the computer representation. This precision does not translate to a fixed number of decimal places. For instance, in the IEEE 754 standard, the common used double precision FP number has a 53 bits base and 11 bits exponent; this basis roughly translates to decimal places.
However, in practice, we have the addition of positive and negative numbers. In this case, the error sign of the individual operations are mixed and, consequently, some cancel out. As a rule of thumb, some authors suggest the use of the square root of the number of operations to estimate the roundoff error, see  for more details about the statistical analysis of the accumulated roundoff error.
With the exception of computations that involves a very large number of operations, n must be of the order of trillions () to produce a error.
In many cases, the order of the operations follow a restrict precedence rule dependent on the compiler/specification.
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