andδτˆi2(τ)are assumed to be zero-mean noise processes, we can now apply a linear fit to our raw data to remove the term ντ, and get the de-trended phase variable,
φτˆi(τ)≈ ϕ(τ)−ν·δτˆi2(τ). (13.9) Any errors in a time shift applied toφτˆi will now couple scaled3by ˙φτˆi ≈50 Hz, such that the precision requirement on these time shifts is relaxed by 5-6 orders of magnitude.
In addition, each individual delayed term used in the construction of the TDI variable will be much smaller, and therefore introduce significantly less numerical noise. The quantization noise present in the original variable before detrending is still there, but it will be suppressed towards low frequencies by the TDI transfer function.
The downside of detrending is that the clock noise termνδτˆi2(τ)can no longer be compensated by time shifting the data. Instead, it remains as a new noise term, scaled4by the beatnote frequencyν.
The final TDI combination will contain many of these noise terms from all measurements entering into it. They will appear with different time shifts applied to them, and scaled by different beatnote frequencies.
As we will show in section 13.3, these noise terms can be compensated by subtracting a combination of the clock noise measured in the sideband interferometers in an additional processing step after the TDI variable is constructed.
13.3 clock noise correction in tdi
This section is based on a publication developed in close collaboration with J.-B. Bayle [43]. A previous version of this article was also included in [20]. It is a continuation of the work published with M. Tinto in [86].
We described in section13.2that one way of doing the INREP processing is to subtract any large trends from the phase or frequency data before constructing the TDI variables. As mentioned there, this approach requires an additional clock noise correction step.
As described in chapter 6, the GHz clock sideband modulations allow an independent measurement of the differential clock jitter, which can be used for clock-noise reduction algorithms. A first version of such an algorithm was presented in [51], which perfectly cancels clock noise assuming constant armlengths. In [86], it was shown that this algorithm can be extended to
3 This is a rough estimate from integrating the laser frequency noise of 30 Hz/√
Hz up to the Nyquist frequency of 2 Hz.
4 Any large deterministic offset inδτˆi2(τ)would be absorbed into the fit, such that the measured average beatnote frequency will be different from the actual frequency.
172 time synchronization and tdi
linearly time-varying armlengths and still reduces clock noise below require-ments. The correction algorithms given in these references are specific to be applied to either the Michelson or Sagnac combinations. We present here a general clock-noise reduction algorithm which can be directly applied to all second-generation TDI combinations presented in section11.4.
We study in section13.3.1how clock noise enters in the standard TDI combi-nations, and propose a general algorithm to remove it. In section13.3.2, we present numerical simulations, and discuss the main results in section13.3.3.
In particular, we give models for the limiting effects and compare the perfor-mance of our algorithm against existing clock-noise reduction schemes.
13.3.1 Clock-noise reduction
As a starting point, we use a simplified version of the phasemeter equations presented in chapter 7. We disregard all noises expect clock noise qei and modulation noiseMij, and consider neither the filter, the timestamping oper-ator nor laser locking5. In addition, we assume that large trends have been removed from the data, such that we can directly use the equations describing the fluctuations. For legibility, we will drop the explicit( )e for all quantities, e.g., we writeqi ≡qei in this section.
These simplifications give
iscij,c= −q˙iaij, (13.10a)
iscij,sb= νmjiD˙ij(q˙j+M˙ji)−νijm(q˙i+M˙ij)
−q˙i(aij+D˙ijνmji −νijm), (13.10b) for the ISC beatnote fluctuations, and
refij,c =−q˙ibij, (13.11a)
refij,sb=νikm(q˙i+M˙ik)−νijm(q˙i+M˙ij)
−q˙i(bij+νikm−νijm), (13.11b) for the reference interferometers.
Under these assumptions, the test-mass (TM) beatnote frequency fluctuations are identical to the reference (REF) ones.
13.3.1.1 Intermediary variables
Inserting eqs. (13.10a) and (13.11a) in the expressions given for the intermedi-ary variables in section12.1, we find how clock noise enter intoηij,
ηij =D˙ijbjkq˙j−aijq˙i, (13.12)
ηik =−(bij+aik)q˙i. (13.13)
5 We included all of these effects in the INREP pipeline results shown in chapter9, showing that the reduction is effective in any case. We also include them in the dedicated numerical simulations shown in section13.4.
13.3 clock noise correction in tdi 173 Note thatbij =−bik, since the two reference interferometers on one spacecraft
use the same lasers. In the following, we choose to only use reference beatnote frequenciesbij from left MOSAs.
13.3.1.2 Clock-noise residuals
From these intermediary variables, we can build laser noise-free TDI combi-nations. They can be expressed as polynomials of delay operatorsPij, in the form
TDI=
∑
i,j∈I2
Pijηij. (13.14)
whereI2 = {(1, 2),(2, 3),(3, 1),(1, 3),(3, 2),(2, 1)}is the set of the6MOSA index pairs.
Inserting eqs. (13.12) and (13.13), we find that clock noise enters in the TDI combination as
TDIq=
∑
i,j,k∈I3+
[PkiD˙ki−Pik]bijq˙i−
∑
i,j∈I2
Pijaijq˙i, (13.15) withI3+ ={(1, 2, 3),(2, 3, 1),(3, 1, 2)}as the set of triplets of spacecraft indices in ascending order.
To estimate the contribution of clock noise before any correction, we as-sume that all light travel times are constant and equal to L, such that we can commute delay operators. As shown in [24], these commutators only yield multiplicative terms 1. Also, we suppose that all clock noises are uncorrelated but have the same PSD Sq˙(ω). Lastly, we assume that beatnote frequency offsets are constant. The clock noise residual PSD then reads
STDIq(ω)≈
∑
i,j,k∈I3+
aijPeij(ω) +ajiPeji(ω)
−bij[ePik(ω)−Peki(ω)De˙ ki(ω)]2Sq˙(ω),
(13.16)
Here,De˙ij andPeij are the Fourier transforms of delay operators and polynomi-als thereof, see [24] for further information.
As an example, we can use eq. (13.16) to work out clock noise residuals in the second-generation Michelson combinationX2,
X2= (1−D˙121−D˙12131+D˙1312121)(η13+D˙13η31)
−(1−D˙131−D˙13121+D˙1213131)(η12+D˙ 12η21). (13.17) We find that
SXq
2(ω)≈16 sin2(2ωL)sin2(ωL)AX2(ω)Sq˙(ω), (13.18) withAX2(ω)is a scaling factor that depends only on the beatnote frequencies,
AX2(ω) = (a12−a13)2+a221+a231
−4b12(a12−a13−b12)sin2(ωL). (13.19)
174 time synchronization and tdi
Figure 13.1: Compar-ison of the residual clock noise in second-generation Michelson X2combination, and the usual LISA 1 pm noise allocation curve.
We assumed here a state-of-the-art space-qualified USO and a realistic set of beatnote frequency offsets.
10−4 10−3 10−2 10−1 100
10−10 10−9 10−8 10−7 10−6 10−5
Fourier frequency in Hz ASDinHz/√ Hz
Clock-noise residuals 1-pm noise allocation
Figure13.1 shows this clock noise residual assuming the same USO model presented in chapter 6, with Sq˙(f) = 4×10−27f−1 in fractional frequency fluctuations, and a realistic set of beatnote frequency offsets. We compared it to a typical 1 pm LISA noise allocation for a single noise source, given by dividing eq. (8.1) by a factor10.
We see that below 0.2 Hz, clock noise significantly violates this requirement and must be suppressed.
13.3.1.3 Building correcting expression
Inspecting eq. (13.15), we observe that clock noise enters in our TDI combina-tion coupled with delay polynomials. We can rearrange it to get
TDIq=
∑
i,j,k∈I3+
h
Pij(bjk−aij)q˙i−Pik(bij+aik)q˙i +Pij(D˙ ijbjkq˙j−bjkq˙i)i.
(13.20)
Following chapter7, the beatnote frequency offsetsaij, bij are time-dependent.
As such, commuting them with a delay operator yields an error term. This effect will be studied in section13.3.3.1. We neglect it for now, and write the previous equation as
TDIq≈
∑
i,j,k∈I3+
h(bjk−aij)Pijq˙i−(bij+aik)Pikq˙i
+bjkPij(D˙ijq˙j−q˙i)i.
(13.21)
We observe that clock noise appears on the right side of thePij, eitherdirectly as ˙qi, or as adifferential termD˙ ijq˙j−q˙i.