Expanding PN Tidal Corrections

Chapter IV: Tidal Splicing for BHNS and BNS Systems

4.3 Expanding PN Tidal Corrections

4.3 Expanding PN Tidal Corrections

connection terms are not included in our expansions because of discrepencies in the calculation of the leading order 1.5PN coefficients; a discussion of those coefficients, including how to add them when those discrepencies are resolved, can be found in the Appendix 4.6.

To complete the PN energy and flux, we need to include the dynamics of BBH systems with object spins χA, χB aligned with the orbital angular momentum, and rotationally induced quadrupole moments ¯Q_A, ¯Q_B. (Though ¯Q_{B H} = 1, we allow it to take on arbitrary values in order to extend to general systems). The dimensionless ¯Q_Ais related the dimensionful version by

Q¯_A =− Q_A

m³_Aχ²_A. (4.35)

To match the 2.5PN order of the tidal terms, we include spin-orbit, spin-spin and rotationally in- duced quadrupolar moment effects [45, 47, 48, 51, 64, 71, 102, 105, 123, 157] within both the orbital energy,

E_BBH(v)=− νv² 2

1+ −3 4 − ν

12

!

v²+ 2χ_AX_A 1+ X_A 3

!

−2χ_BX_B 1+ X_B 3

! ! v³ + 1

8 −27+19ν− ν² 3

!

+2χ_Aχ_Bν−( ¯Q_A+1)χ²_AX²_A−( ¯Q_B+1)χ²_BX_B²

! v⁴

+* ,

χ_AX_A* ,

3+5X_A

3 +29X²_A 9 + X³_A

9 + -

+ χ_BX_B* ,

3+5X_B

3 + 29X²_B 9 + X_B³

9 + - + -

v⁵+O(v⁶) , (4.36) and the flux,

F_BBH(v) =32ν²v¹⁰ 5

1+ −1247 336 − 35

12ν

! v² + 4π+ χ_AX_A −5

4 −3X_A 2

!

+ χ_BX_B −5 4 − 3X_B

2

! ! v³ +

−44711

9072 + 9271ν

504 + 65ν² 18

! + 31

8 χ_Aχ_Bν+ 33 16 +2 ¯Q_A

! χ²_AX²_A + 33

16 +2 ¯Q_B

! χ²_BX_B²

! v⁴+

−8191

672 − 583ν 25

! π +χ_AX_A*

,

−13

16 + 63X_A

8 − 73X²_A

36 − 157X³_A 18 +

- +χ_BX_B*

,

−13

16+ 63XB

8 − 73X²_B

36 −157X³_B 18

! + -

v⁵+O(v⁶)

. (4.37)

Therefore the full 2.5PN energy and flux expression are simply E(v) =E_BBH(v)+E_λ¯

2(v)+E_λ¯

3(v), (4.38)

F(v) =F_BBH(v)+F_λ¯

2(v)+F_λ¯

3(v). (4.39)

At this point, we can repeat the expansion procedure from above to generate the series terms for the TaylorT4 and TaylorT2 approximants. Again, we treat terms likeO(1)∼ O( ¯λ2v¹⁰) ∼ O( ¯λ3v¹⁴) as leading order and carry up through 2.5PN order.

Computing the TaylorT4 expansion according to Eq 4.7 where again we can break the series up into the terms corresponding to a BBH system ( ¯λ₂ = λ¯₃ = 0,Q¯ = 1) contained inF_BBH(v), and all of the expansion terms corresponding to the NS inF_Tid(v) where,

F_Tid(v) =32νv⁹ 5M

"

5( ¯Q_A−1)χ²_AX²_A

v⁴+λ¯2AX⁴_Av¹⁰* ,

X5

i=0

F_2A,ivⁱ+ -

+λ¯3AX⁶_Av¹⁴* ,

X5

i=0

F_3A,ivⁱ+ - +(A↔B)

#

. (4.40)

The coefficientsF_2A,iandF_3A,i are given in the Appendix, Eq 4.67.

Again, the ¯λ₂×χ_A,B cross terms appearing in these coefficients are not due to spin-tidal interaction terms in the Hamiltonian, but instead are a consequence of the series expansion power counting.

The ( ¯Q_A−1) in thev⁴term arises from the fact that we have incorporated the BBH part of ¯Q_Ainto F_BBH(v) (for which ¯Q_{B H} =1) and included the remainder here.

Similarly for TaylorT2, the updated time and phase expressions for Eq 4.14, 4.15 are T_Tid(v)=− 5M

256νv⁸

"

−

10( ¯Q_A−1)χ²_AX²_A

v⁴+λ¯_2AX⁴_Av¹⁰* ,

X5

i=0

T_2A,ivⁱ+

- +λ¯_3AX⁶_Av¹⁴*

,

5

X

i=0

T_3A,ivⁱ+ -

+(A↔ B)

#

, (4.41)

P_Tid(v)=− 1 32νv⁵

"

−

25( ¯Q_A−1)χ²_AX²_A

v⁴+λ¯_2AX⁴_Av¹⁰* ,

X5

i=0

P_2A,ivⁱ+

- +λ¯_3AX⁶_Av¹⁴*

,

5

X

i=0

P_3A,ivⁱ+ -

+(A↔B)

#

. (4.42)

The individual coefficientsT_2A,i,T_3A,i,P_2A,i,P_3A,i are given in the Appendix, Eq 4.68, 4.69.

Spinning Dynamical Tides

With our extension into spinning systems, we need to take care to treat the dynamical tides more carefully. Since the dynamical tides is caused by the changing tidal field due to the orbital motion interacting with the internal f−modes of the deformable object, when the object is also spinning, then its internal modes will effectively experience a driving frequency equal to the orbital motion shifted by the object’s spin. We characterize this by making a slight correction to the characteristic parameterγ_`Ain Eq 4.27.

Given an aligned spin of χ_Aand a moment of inertiaI_A, then we can compute the rotation frequency of the deformable object as

ω¯_A = Mω_A = χA

X_AI¯_A, (4.43)

where we have madeI_A dimensionless by

I¯_A = I_A

m³_A. (4.44)

Thus, the effective orbital frequency the resonant modes will experience is simply the difference between the orbital frequency and the rotational frequency of the object. With that in mind, we rewriteγ_`Aas

γ_`A =

`

v³−ω¯_A ω¯f`A

. (4.45)

We take the absolute value because Eq 4.27 is undefined for negative values ofγ_`A, which can occur at low orbital frequencies with aligned spin objects. This corresponds to saying that the resonance modes of the object only care about the magnitude of the frequency of the changing tidal field.

Within Eq 4.30, we also need to make changeω→

v³−ω¯_A

/M.

To show how this affects the profile of the dynamical tides correction, in Fig 4.1 we plot the profile of κ` from Eq 4.28 as a function of orbital frequency. We assume an NS with ¯λ2 ≈ 800 in an equal mass system and use the universal relations (see Table 4.2) to compute the other relevant tidal parameters. We compare the nonspinning NS against both aligned and antialigned spinning NS with magnitude |χNS| = .2, which corresponds to a rotational frequency of f_NS ≈ 312Hz. The upper frequency termination point is at an orbital frequency ofMω_ISCO = 6^−3/2, which has been used as an estimate a BNS inspiral termination criterion [31].

From Fig 4.1 we can see that aligned spins pushed the resonance peak later in the inspiral while antialigned spins move the peak to earlier frequencies. In fact, with large enough aligned spins it is possible that the peak of the resonance is never reached before the system enters the merger/ringdown phase.

At low frequencies, the nonspinning system has the expected behavior of reducing to the simple static tides behavior (κ_` = 1). However, this is not true of the spinning systems, both of which asymptote to a slightly different value. Physically, these differences are due to the deformations of the object experiencing a driving frequency not from the orbital frequency (which is vanishingly small), but its own rotaiton along its axis. We expect this differnece not to be a significant con- tribution to the waveform as the relative size of the tidal effects already vanishes (O(v¹⁰)) at low frequencies anyways. The aligned spin object then passes through a point in the evolution where the orbital frequency matches exactly with its rotational frequency and the effective dynamical tidal field vanishes.

The possible issue is how the antialigned spin object dives through κ_` = 0 and plunges negative.

We recognize there is a caveat with Eq 4.28, which is that this formula only valid up to frequencies shortly after the resonance peak [144]. While for nonspinning systems and spin aligned systems the resonance peak occurs near the end of the inspiral or after merger/ringdown and thus within the

Figure 4.1: The dynamical tide amplification κ<sub>`</sub> as a function of orbital frequency, showing how the resonance profile from Eq 4.28 changes between the nonspinning case (black) and the spin aligned/antialigned (blue/red) cases for both` =2 (solid) and` =3 (dashed) tidal deformabilities.

The vertical lines represent the resonance frequency for both modes in every case. The parameters correspond to an NS with ¯λ₂ ≈ 800 in an equal mass system. For this NS, a spin magnitude of

|χ_NS| =.2 corresponds to a rotational frequency of f_NS ≈312Hz.

range of validity, that condition does not necessarily apply in the antialigned case. If the antialigned spin is large enough, as seems to be the case in Fig 4.1, the resonance frequency occurs early enough in the evolution that this approximate formalism potentially breaks down while still in the inspiral, necessitating a more delicate handling of the dynamical tides.

Until such a formalism is developed for antialigned NS spins, we instead assume that the object is nonspinning (i.e. we setωA = 0) for the purposes of Eq 4.28, 4.30; the aligned spin case will use Eq 4.45 as expected.