In the previous sections, we introduced a concise notation for defining TDI combinations, and reviewed an algorithm based on [92] for how to construct the corresponding interferometers out of the six one-way measurements.
However, there are multiple symmetry operations we can apply to a TDI string without changing the resultant combination in a meaningful way. Therefore, each TDI string should be seen as a representative of an equivalence class of distinct TDI combinations.
Furthermore, some combinations are related by simple permutations of the spacecraft indices used in them. Thus, each string can also be seen as repre-sentative of another equivalence class of core TDI combinations, from which all distinct combinations can be generated by index permutations.
We will use a sorting criterion to pick one unique representative for each of these equivalence classes.
Note that such symmetries are also discussed in [92,63]. We list them here for completeness, and to discuss how they are applied in our notation.
11.3.1 String reversal
We constructed our algorithm to use the first event in the string as starting point for the combination. However, we could have just as well started from the opposite end. Practically, this means that the order of visited spacecraft is reversed, while links which previously where interpreted as delays are now advancements, and vice versa. This has two minor impacts which should not lead us to consider these as different combinations.
For one, the sign will be flipped, since we assigned a positive sign to advance-ments and a negative sign to delays in our algorithm. This sign assignment was in itself arbitrary, such that this has no physical meaning. Secondly, the reversed combination will have a small time shift, since the starting time is now defined at the last spacecraft, not the first. As before, an overall timeshift does not distinguish a combination. In addition, the first and last event of the chain can be seen as almost simultaneous within the assumptions of a given TDI generation, such that this timeshift is indeed irrelevant in most situations.
Thus, two strings can be seen as equivalent if one can be produced from the other by reversing the order and changing all signs.
11.3.2 Cyclic string shifts
Applying a cyclic shift of all string indices, e.g.,
"12 21 13 31 -12 -21 -13 -31"→"21 13 31 -12 -21 -13 -31 12"
11.3 symmetries of tdi strings and string normal form 131
(11.25) only has a very subtle impact on the resultant combination. If the photon path described by the combination is perfectly closed, the first and last event are simultaneous. This means that the cyclically shifted combination contains exactly the same events, with exactly the same laser noise cancellations at simultaneous events, and the only change is an overall shift in the starting time.
In reality, the photon path is not perfectly closed, such that the first and last event are not simultaneous. We therefore have the additional effect that the
’open’ end of the path is at a different spacecraft. Non-closedness of the path represents an armlength mismatch of the synthesized interferometer, which will cause a small residual laser noise, as discussed in section12.3.1.
This means we can in principle control which laser noise remains unsup-pressed via cyclic string shifts. This might be exploitable if the lasers have different noise characteristics, such as is the case if they are all locked to one master laser in the constellation, and this noise appears in them with different delays.
Regardless of this minor technical detail, we will consider two combinations equivalent if one can be produced by the other via a cyclic string shift.
11.3.3 Equivalent strings and normal representative
Two strings are seen as equivalent if either of them can be produced from the other one by any combination of string reversals or cyclic shifts.
Since the cyclic permutation group of a list ofNelements has itselfNelements, and there are two versions of each string due to string reversal, we have a total of 2N members of the equivalence class for each string of lengthN.
We will choose a unique representative for each equivalence class via the following criteria:
• Favor strings which start with the longest list of consecutive advance-ments.
• Out of those, favor strings for which the starting indices are ascending ("12","23","31").
• Out of those, prefer the lowest numerical ordering of the spacecraft appearing in the sequence (e.g.,"121..."is favored over"231..."or
"123...").
11.3.4 Permutation of spacecraft indices
LISA obeys the symmetries of an equilateral triangle. In total, there are six possible permutations of the set of spacecraft indices. These can all be written
132 tdi combinations
as multiple applications of the cyclic permutations, 1→2, 2→3, 3→1, and the mirror symmetry along the separating axis between spacecraft2 and3, i.e., by exchanging indices 2↔3, as described in appendix A.
This implies that for any TDI string of a given TDI generation there exist6 versions which are all of the same generation. They differ by the starting spacecraft and by exchanging the order in which the other two spacecraft appear.
Contrary to the previous two kinds of symmetries, some of these6versions can be seen as distinct combinations. We previously discarded combinations as equivalent if the same signals enter in them, up to an overall time-shift or a timing mismatch to be neglected at given TDI generation. This is not the case when changing the starting spacecraft or the order in which spacecraft are visited. For example, the Michelson combinationXdoes not use any measurements between spacecraft2 and3, while its equivalentsY and Z constructed by cyclically shifting all spacecraft indices do use these measurements. They therefore contain different signals, and are seen as distinct combinations.
Some combinations are mirror symmetric. For example, exchanging the role of spacecraft 2 and 3 in the Michelson X combination is equivalent to a reversal of the string, which we previously identified to be equivalent to just a sign flip.
11.3.5 Time reversal symmetry
Reversing just the time direction of all links in a TDI string yields another TDI string of the same generation as the original6. In many cases, this time-reversed string is truly equivalent to the original, in the sense of section11.3.3, or it is equivalent to one of the permutations described in section11.3.4.
However, there are a few cases in which it is not equivalant to either. As an example, we show in fig. 11.3 the time-reversed version of the Beacon combination we originally discussed in fig.11.2. In the original version of the variable, both spacecraft1and2emit two beams each, while all measurements are performed on spacecraft3.
In the time-reversed version, on the other hand, all beams are emitted from spacecraft 3, while they are interfered and measured on spacecraft 1 and 2. This change in topology of the combination cannot be produced by a simple exchange of spacecraft labels, such that both the original and the time reversed version should be seen as distinct combinations.
6 We thank S. Vitale for pointing out this additional symmetry!
11.3 symmetries of tdi strings and string normal form 133 Con-trary to fig.11.2, all beams are now
Time-reversed8link Beacon combination
11.3.6 Core combinations and equivalence
We call adistinct combination the equivalence class of strings which can be produced from each other just by string reversal or cyclic string shifts. There-fore, the three Michelson combinations X1,Y1 and Z1 are distinct combina-tions.
We call a core combination the set of strings which can be produced from each other by either string reversal, cyclic string shifts, permutations of the spacecraft indices or time reversal. For example, X1,Y1 and Z1 are distinct combinations, but represent the same core combination.
Note that for time-symmetric combinations, each core combination represents 3 or 6 distinct combinations, depending on if the combination is mirror symmetric or not. Likewise, for non time-symmetric combinations, there are 6or12 versions, again depending on the mirror symmetry.
To summarize, depending on its symmetries, each core TDI combination represents3,6or12distinct combination.
Since each distinct combination of length N itself represents 2N possible strings (cf. section11.3.3), the equivalence class of one core TDI combination of length Nhas either 6N, 12Nor 24Nmembers.
134 tdi combinations