Figure 10.6: Laser noise residual in dif-ferent Michelson X generations.
10−4 10−3 10−2 10−1 100
10−22 10−16 10−10 10−4 102
Fourier frequency in Hz ASDinHz/√ Hz
X0 X0model X1 X1model X2 X2model X3 X3model 1pm equiv.
laser noise residual2 far below the other variables. The sharp increase of X3 close to 1 Hz can be explained by the interpolation errors discussed in section12.3.2.2.
10.3 the algebra of commutative and non-commutative delay operators
It can be useful in practice to study and describe TDI combinations from an algebraic perspective. In fact, the problem of up to1.5th generation TDI can be exactly solved using algebraic methods, see [83]. It turns out that the entire space of1.5th generation variables can be generated from just6fundamental combinations [69], while for1st generation, even4combinations are sufficient [32].
To the knowledge of the author, the algebraic problem of second generation TDI is to date still unsolved [83], and no finite set of generators is known.
This opens up the question which, and how many, second generation TDI combinations one should compute to extract all available information. We will briefly explore this question at the end of chapter11.
Regardless, it is useful to realize that once2nd generation TDI combinations
Cf. chapter11for a review of possible second generation combinations.
are found, they can be manipulated algebraically.
Formally, we observe the following properties of the delay operators of first and second generation TDI:
• Each delay operatorDij has a left and right inverseDji, which exactly cancels the applied time shift (see chapter11for more information, in particular eq. (11.2)).
2 Note that we neglected terms proportional to ˙˙˙dAin the computation of∆τin section10.2.5, which might cause a residual laser noise at a higher level than estimated here.
118 introduction to time-delay interferometry (tdi)
• Delay operators are linear operators, in the sense that Dij(x(t) +y(t)) =Dijx(t) +Dijy(t),
Dij(ax(t)) =aDijx(t), (10.31) for any constanta∈Rand any time dependent functions x(t),y(t).
• Delay operators of first generation are commutative, while those of second generation are non-commutative.
Using that delay operators are linear, we can factorize any given TDI combi-nation as
TDI=
∑
ij∈I2
Pijηij, (10.32)
where the set of index touplesI2 ={ij|i,j∈ {1, 2, 3},i6= j}has six elements and eachPij is a polynomial3of delay operators.
For example, the first generation Michelson combination can be factorized as
X1= η12+D12η21+D121η13+D1213η31
−[η13+D13η31+D131η12+D1312η21]
= (1−D131)(η12+D12η21)−(1−D121)(η13+D13η31),
(10.33)
which means its defining polynomialsPij are
P12 = (1−D131), P23 =0 , P31=−(1−D121)D13, P21 = (1−D131)D12, P32 =0 , P13=−(1−D121).
We know from the previous sections that the laser noise appearing in each ηij is strongly suppressed in the full TDI combination. However, this is not the case for most other noises. For the purpose of estimating the coupling of non-suppressed effects, like gravitational waves and most noise sources, it is therefore often sufficient to analyze eq. (10.32) under the assumption that all delays appearing in thePij are constant, or even equal to one average delay. Therefore, any result valid for0th,1st or1.5th generation variables are applicable to the non-suppressed quantities in the second generation variables as well, with increasing level of accuracy.
For example, assuming that all delays are equal, we can writeX1in terms of the simple MichelsonX0,
X1= (1−D131)(η12+D12η21) + (1−D121)(η13+D13η31)
≈ (1−D2) (η12+Dη21−η13−Dη31)
| {z }
X0
, (10.34)
3 Note that these operator-valued polynomials don’t obey the same rules valid for real-valued polynomials. In particular, a polynomial of delay operators has in general no multiplicative inverse. In addition, the delay operators of2nd generation TDI are non-commutative, such that the order of operators must be preserved when factorizing these polynomials.
10.3 the algebra of commutative and non-commutative delay operators 119 where we usedDas the unique equal delay operator of0th generation TDI.
Likewise, we can write X2in terms ofX1 assuming that delays commute, or even in terms ofX0 if they are all assumed equal:
X2=(1−D131−D13121+D1213131)(η12+D12η21)
−(1−D121−D12131+D1312121)(η13+D13η31)
≈(1−D12131)X1.
≈(1−D4)(1−D2)X0.
(10.35)
These properties are very useful in practice. For example, the relationship between X2 and X1 was used in [86] to generalize an already known clock correction algorithm from first to second generation, and it is also regularly used in the LISA performance model group to relate noise residuals for second generation TDI to those of first generation TDI.
We will also use a similar reasoning to relate all second generation combina-tions found in chapter11and presented in table11.3to the four generators of first generation TDI, cf. section11.5.1.
T D I C O M B I N AT I O N S
11
We presented in chapter10how TDI combinations can be built out of the one-way measurements to construct arbitrary two-beam interferometers.
We will extend this approach to construct more general multi-beam interfer-ometers, following the ideas described by [92,63], in section11.1. Here, we also introduce a concise notation for identifying TDI combinations.
We then generalize the conditions for laser noise cancellation in the differ-ent TDI generations in section 11.2, and discuss different symmetries the combinations can obey in section11.3.
Finally, we use these formulations in section11.4to perform an independent replication of the search method outlined in [92,63], finding additional14link combinations which were previously missed. A full summary of the found combinations is given in table11.3. We then relate these combinations to the generators of first generation TDI in section11.5, and show simulation results indicating that just four combinations are sufficient to recover all information encoded in the2nd generation TDI combinations in section11.5.1. Finally, we show how these decompositions can be applied to actual simulated data in section11.5.2
The results reported in this chapter where produced independently by the author, but with regular exchange and comparison of results with the LISA group in Trento, in particular M. Muratore, D. Vetrugno and S. Vitale.
11.1 laser noise cancellation in multi-beam interferometers
We have discussed how to construct two-beam interferometers allowing laser noise cancellation in chapter 10. This approach can be generalized to construct TDI combinations utilizing an arbitrary (even) number of beams, by combining multiple two-beam interferometers.
To understand this, it is useful to consider the spacetime events involved in each measurement ηij. Each ηij represents a one-way measurement, where incoming light from the distant spacecraft is interfered with the local laser beam. The resulting measurement is the difference of the phase of the local laser at the event of reception of the distant beam and that of the distant laser at the event of its emission. We used this property in section10.1.2to combine multipleηij’s such that the laser entering at the event of reception in one of the ηij exactly cancels with the same laser entering at the event of emission of anotherηij.
121
122 tdi combinations of events in the TDI combination, starting at one of the emission events. The colors indi-cate which spacecraft is visited in each link.
The y-axis denotes mean-ing that the first and last event in the chain happen
This allowed construction of single-beam interferometers, where one of the beams is emitted at an emission event, relayed in an arbitrary path along the constellation, and then recombined with a non-delayed version at a re-ception event. Furthermore, in section10.1.3, we combined two such beams which share a common reception event to construct a virtual interferometer, cancelling laser noise if the two respective emission events are simultane-ous.
This means so far, we used two different laser noise cancellation mechanisms when combining measurements:
• Cancellation at a simultaneous reception and emission event, and
• Cancellation at two simultaneous reception events.
Obviously, there is a third option to combine theηij’s which we didn’t use so far, which is cancellation at two simultaneousemissionevents.
As an example, fig. 11.1 shows a schematic overview of the emission and reception events involved in the first generation Michelson combination. The x-axis is a simple counter indexing these events, while the y-axis shows the coordinate time at which the beam arrives at a certain spacecraft.
In the classical TDI description outlined in section10.1.3, the two beams are combined at a reception event. Therefore, the figure is to be read starting from the event labelled #4, where two beams are received simultaneously.
Each of the two beams can then be tracked back to their inital event of
11.1 laser noise cancellation in multi-beam interferometers 123
emission by going through the sequence either descending (from #4 to #0) or ascending (from #4 to #8). Laser noise is cancelled if the two emission events are simultaneous1.
However, we can just as well time shift our whole combination to start at the emission time from the first spacecraft, labelled #0 in the figure. We then track a beam which is emitted at time t = 0 forwards through time, computing it’s reception times going through events #1 to #4. At event #4, it is combined with the second beam, and we change direction. We then compute the emission times of the second beam while going through the rest of the sequence, from events #4 to #8. The condition for laser noise cancellation has not changed, the first and last event have to be simultaneous. The advantage of going through the sequence in this order is that it is easy to generalize it to include more than two beams, or equivalently more than one measurement event. Figure11.2shows an example of such a combination, which uses two measurement events/interferometers. In this example, the two measurement events on spacecraft3have to be aligned in time in such a way that the shared emission event on spacecraft 2 (# 6 in the figure) is simultaneous for both interferometers. We still have the same condition for laser noise cancellation, i.e., the wholephoton path has to be closed, which means the first and last event in the chain have to be in the same spacecraft and simultaneous.
1 ’Simultaneous’ meaning here that the time difference between the two events is sufficient small that the residual laser noise is below the required level, as discussed in chapter10.
124 tdi combinations
In practice, constructing TDI variables starting not from a reception event but from an emission event requires knowledge of not only the delaysDij, which give the emission time of a beam on spacecraftjgiven a reception time i, but also the advancementsDij, which give the reception time of a beam on spacecraftjgiven an emission time on spacecrafti.
Formally, if the coordinate times of the events of emission and reception of a beam emitted from spacecraftiand received by spacecraft jare denotedtei andtrj, respectively, we have
Djix(trj) =x(tei), Dijx(tei) = x(trj), (11.1) which immediately yields the identity
DijDjix(t) =DjiDijx(t) =x(t). (11.2) In LISA, we only directly measureDij, cf. section 6.3. The advancements can be calculated out of these using an iterative procedure on eq. (11.2), see appendix B.4.
We will use the same notation for nested advancements which we used for nested delays. I.e., we have
Di1i2...in =Di1i2Di2i3. . .Din
−1in, (11.3)
which gives the time of reception of a beam on spacecraft in which was emitted from spacecrafti1, being relayed by all the spacecraft corresponding to the indices inbetween, from left to right. Note that eq. (11.2) generalizes to
Di1i2...inDinin−1...i1 = Dinin−1...i1Di1i2...in =1 . (11.4) As an example of how to construct a TDI variable starting at an emission event, let us consider again the1st generation Michelson combination,
X1= η12+D12η21+D121η13+D1213η31
−[η13+D13η31+D131η12+D1312η21]
= (D12131−D13121)Φ1.
(11.5)
This equation is to be evaluated at a reception time on spacecraft1. As argued above, we can simply time-shift the whole combination by applyingD13121, such that it is evaluated at an emission event at one of the beams,
As described in appendix A, we seperate nested delays and advancements in Dby a semicolon, i.e., D13121;13121≡ D13121D13121
which yields
D13121X1 =D13121η12+D13121D12η21+D13121D121η13+D13121D1213η31
−D13121[η13+D13η31+D131η12+D1312η21]
=D13η31+D131η13+D1312η21+D13121η12
−D13121[η13+D13η31+D131η12+D1312η21]
=(1−D13121;13121)Φ1.
(11.6)
We can make the following observations:
11.1 laser noise cancellation in multi-beam interferometers 125
• All delays and advancements in the final result are shifted to the second term, while the first term enters without any timeshift. This leads to the interpretation of this signal in [92] that we have constructed an interferometer where one of the beams travels forwards and backwards in time, and interferes with itself at it’s event of emission.
• The nested delays and advancements in that second term enter exactly in the same order as the visited spacecraft in our visualization, cf. fig.11.1.
• Each ηij is evaluated at such times that the two laser noise contribu-tions in them correspond to the emission and reception events in our visualization. For example:
– The first termD13η31 contains laser noise from spacecraft1at the event of emission at the time origin of the combination (#0), and laser noise of spacecraft3at the event of reception (#1),
– the ’middle’ term D13121η12−D13121η13is the recombination event (#4), at which both beams interfere; here, both η’s are evaluated at the same time, such that the common noise of spacecraft1cancels, – and the final term D13121;1312η21is evaluated at the second to last event (#7). This η21 contains a laser noise term which is then emitted at the last event in our figure (#8).
11.1.1 Algorithm for TDI variable construction
The above observations lead us to a This algorithm was
first described in [92], and updated here to our notation.
general algorithm of how to construct any multi-beam interferometer. We assume that the combination we want to construct is given as a string of visited spacecraft, directly corresponding to the nested delays in eq. (11.6). Paths in forward time direction are indicated with a positive sign, while those with a backward time direction receive a negative sign. For example, the 1st generation Michelson combination described in eq. (11.6) would be encoded as"13121 -13121", but this notation is not restricted to two-beam interferometers.
The full algorithm2 to build a TDI combination given by a photon path is then:
• Expand the nested photon path into a list of single links. E.g.,
"13121 -13121"→[13, 31, 12, 21,−13,−31,−12,−21]. (11.7)
• Create an empty list of delays/advancementsT = [ ], and initialize an expression C =0.
2 The last step is optional. Without it, the resulting TDI combinations usually contain a large number of nested timeshifts, which can become computationally expensive when actually computing the TDI response. It is therefore often advisable to multiply the final combination by the inverse of the first half ofT, such that half of the timeshifts in the combination are cancelled. Physically, this timeshift corresponds to building the combination starting from the central event instead of the first event.
126 tdi combinations
• Iterate through the expanded photon path, doing the following opera-tions, assuming the indices areij:
– If the entry is positive (an advancement), add
C +=TDijηji. (11.8)
– If the entry is negative (a delay), instead subtract
C −=Tηij (11.9)
– Then, append eitherDij orDij to T.
• Apply am overall timeshift to the combination undoing the first half of T to reduce the total number of time shifts (Optional).
We will denote the application of this algorithm by TDI["string"]. For example, we would write for the first generation Michelson
TDI["13121 -13121"] = (1−D131)η12+ (D12−D1312)η21
−(1−D121)η13−(D13−D1213)η31. (11.10) This algorithm is functionally identical to that presented in [92] and simply updated to agree with our notation. We can convert our photon paths to the notation used there (and the other way around) using the following steps:
• Expand nested delays:
"13121 -13121"→[13, 31, 12, 21,−13,−31,−12,−21]
• Replace the two spacecraft indices with corresponding link, so exchange 12↔30, 23↔10, 31↔20,
21 ↔3 , 32↔1 , 13↔2 ,
−12 ↔ −3 , −23↔ −1 , −23↔ −2 ,
−21↔ −30, −32↔ −10, −13↔ −20.
• Collapse the resulting list back into a string, e.g.,
"2 2’ 3’ 3 -2’ -2 -3 -3’".