Conditions for laser noise cancellation

We can evaluate if a string corresponds to a combination which is TDI gen-eration 1, 1.5 or 2 using modified versions of the rules presented in sec-tion10.2.

Combinations satisfying these conditions can then be found numerically, by simply generating and testing all possible TDI strings up to a given length.

11.2 conditions for laser noise cancellation 127

11.2.1 1st Generation

If all armlengths are constant and symmetric, it is sufficient if the same spacecraft are visited on the forward and backwards paths of the combination.

Formally, we have to count the total number of occurences of each spacecraft in all positive substrings, which has to be equal to the number of occurences of each spacecraft in all negative substrings.

For example, the Sagnac combination is given by the string "1321 -1321", which fulfills this condition.

11.2.2 1.5th Generation

As before, we now have to consider the direction of links. This requires to expand the photon path into a list of delays, and then count that the same links are used in forward and backwards direction.

Since all delays are still constant, we can freely commute them in this list. Us-ing eq. (11.2), we see that each advancementD_ij cancels with a corresponding delay D_ji.

In summary, the algorithm to check if a string is TDI1.5is:

• Expand the TDI string into list of advancements and delays. For exam-ple:

"13121 -13121"→[13, 31, 12, 21,−^13,−^31,−^12,−²¹] (11.11)

• Count the number of each advancementij, which has to be equal to the number of each corresponding delay−^ji.

11.2.3 2nd Generation

The algorithm described in section11.1.1 For this study, it is more useful to not apply the optional last step of the algorithm, such that all timeshifts are shifted to the second laser noise term.

always yields an expression of the form

TDI= (1−^Di1...in)Φ_in, (11.12)

with the chain of delays in the second term corresponding exactly to the indices of the TDI string. We have to check if the overall nested time shift in that term evaluates to zero when performing a first order approximation in the arm lengths.

11.2.3.1 Forward time shifts for linear delays

We first need to determine how the forward timeshiftsD_Aare related to the delaysD_Ain the linear approximation. We have again

D_Ax(t) =x(t−^dA−^d˙At), (11.13)

128 tdi combinations

withd_A and ˙d_A assumed constant for the delays.

We want to model our advancements in exactly the same form, which will allow us to reuse eq. (10.26). So we use³

D_Ax(t) =x(t−^a_A−^a˙_A^t). (11.14) Utilizing eq. (11.2), we then see that

x(t) =D_AD_Ax(t)

=D_Ax(t−^dA−^d˙At)

= x(t−^a_A−^a˙_A^t−^dA−^d˙A(t−^a_A−^a˙_A^t))

= x(t−^a_A−^dA+d^˙_Aa_A+ (a˙_Ad˙_A−^a˙_A−^d˙A)t)

(11.15)

For the equation to hold, the terms with and without a time-dependence both need to cancel independently, yielding the two solutions

a_A =−₁ ^dA

−^d˙A, (11.16)

˙

a_A =− ^d˙A

1−^d˙A. (11.17)

We can expand both expressions to first order in ˙dA to get

a_A ≈ −^dA(1+d^˙_A), (11.18)

˙

a_A ≈ −^d˙A. (11.19)

11.2.3.2 Laser noise cancellation to first order inL˙

We can reuse the result and reasoning we had in section10.2to show that the overall nested time shift is structurally identical to eq. (10.26),

DA˜1... ˜Anx(t) =x The difference is that we now allow each ˜A_i to be either an advancement or a delay, replacingTA˜i →^dAi, ˙TA˜i →^d˙Ai for a delay, andTA˜i → −^d_Ai(1+d^˙_Ai), T˙A˜i → −^d˙_Ai for an advancement. The bar onA_i acts as a reminder that when we use this formula in the two index convention, we need to reverse the order of indices for the advancements to get the correct inverse operator.

Using these replacement rules, we see immediately that the sum ∑ⁿ_k=1T˙A˜k

cancels as long as the same arms are transversed an equal number of times in forward and backwards direction, which is exactly the condition for TDI generation1.5.

3 Note that in the two index convention, we haveD_ijas the inverse of the delayD_ji, with the indices switched. In our one-index notation for delays, we will useD_Aas the inverse ofD_A.

11.2 conditions for laser noise cancellation 129 In addition, we need to check if all delays and advancements in the last two

sums cancel, i.e., that

at least to first order in ˙L. The first order approximation means here that we can use the simplified substitutionTA˜i → −^L_Ai for the advancements in the left side of the equation, while we need to use the full expression for the right side of the equation.

Equation (11.21) can be checked by either evaluating it symbolically, or by implementing an appropriate counting algorithm on the links appearing in the nested delay, as described in [92]. We chose the former option for convenience, using the computer algebra softwareMathematica.

11.2.3.3 Laser noise cancellation expressed in the Fermi frame

It was argued in [63] that the search for TDI combinations can in principle be performed in any reference frame, since the simultaneity of two events (here the emission and reception events of the beams) remains true in any reference frame⁴.

Furthermore, it was shown that there exists a reference frame in which the terms for Land ˙Lhave certain symmetries, and can be expressed in terms of the relative coordinate velocity⁵of the spacecraft:

L_ij = L_0,ij

Using these conditions and neglecting effects second order inv_ijin expressions equivalent to eq. (11.21) led to the discovery of some new 2nd generation TDI combinations in [63] which were previously discarded by the search algorithm described in [92]. Notably, it was shown that several new 12link combinations suppress laser noise to the same level as the previously known 16link combinations.

4 In reality, the recombination events are not exactly simultaneous, but occur with a small time offset on the order of ps when expressed in the TCB [63]. It might be possible to construct a coordinate system with a coordinate singularity which blows up this ps time interval to an arbitrarily large number, which conversely would mean that finding a ’small’ overall light travel time in such a coordinate system would not translate to an equally small time interval in the TCB. This is not the case for the systems considered here, for which the relative scaling of these time intervals due to coordinate transformations is small enough to not affect the result.

5 Note that in these equations (taken directly from [63], see there for more information), the armlength are expressed in meters, not seconds, explaining the occurence of the speed of light c, which is absent in all our equations.

130 tdi combinations