Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves.

Transcription

1 Gravitational Wave Astronomy With LISA Rajesh Kumble Nayak, IISER-Kolkata

2 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves.

3 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves. The mission is expected to take-off in year 2020.

4 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves. The mission is expected to take-off in year Motivation for detector in space.

5 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves. The mission is expected to take-off in year Motivation for detector in space. Types of sources in the LISA band.

6 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves. The mission is expected to take-off in year Motivation for detector in space. Types of sources in the LISA band. Brief overview of the LISA mission.

7 Over View LISA - Laser Interferometer Space Antenna, is a ESA- NASA joint space mission to detect low-frequency gravitational waves. The mission is expected to take-off in year Motivation for detector in space. Types of sources in the LISA band. Brief overview of the LISA mission. Time-Delay interferometer.

8 Ext. low FrequencyVery low Frequency Below Hz Hz Low Frequency Hz High Frequency 10 - Few KHz

9 Coalescing neutron Stellar mass Blackhole mergers Rotating Neutron/Compact Stars Supernova Ext. low FrequencyVery low Frequency Below Hz Hz Low Frequency Hz High Frequency 10 - Few KHz Ground Based Detectors Such as LIGO, VIRGO etc

10 Coalescing SMB < 10 7 M s Galactic Binaries EMRI s Coalescing neutron Stellar mass Blackhole mergers Rotating Neutron/Compact Stars Supernova Ext. low FrequencyVery low Frequency Below Hz Hz Low Frequency Hz High Frequency 10 - Few KHz LISA Ground Based Detectors Such as LIGO, VIRGO etc

11 Coalescing SMB upto Coalescing M SMB < 10 7 s M s Stochastic GW background Galactic Binaries Binary stars EMRI s Coalescing neutron Stellar mass Blackhole mergers Rotating Neutron/Compact Stars Supernova Ext. low FrequencyVery low Frequency Below Hz Hz Low Frequency Hz High Frequency 10 - Few KHz Pulsar in Binary (Taylor Pulsar) LISA Pulsar Timing Doppler tracking of inter-planetary spacecrafts Ground Based Detectors Such as LIGO, VIRGO etc

12 Universe Coalescing SMB upto Coalescing M SMB < 10 7 s M s Stochastic GW background Galactic Binaries Binary stars EMRI s Coalescing neutron Stellar mass Blackhole mergers Rotating Neutron/Compact Stars Supernova Ext. low FrequencyVery low Frequency Below Hz Hz Low Frequency Hz High Frequency 10 - Few KHz CMB Pulsar in Binary (Taylor Pulsar) LISA Pulsar Timing Doppler tracking of inter-planetary spacecrafts Ground Based Detectors Such as LIGO, VIRGO etc

13 Sensitivity Curve LISA is sensitive to gravitational waves in the frequency range 10 4 Hz to 1 Hz.

14 Galactic Binaries Binary stars emit gravitation wave a frequency twice that of their orbital frequency.

15 Galactic Binaries Binary stars emit gravitation wave a frequency twice that of their orbital frequency. Lowest of LISA frequency is Hz, orbital frequency must be Hz or orbital period 5.5 Hrs.

16 Galactic Binaries Binary stars emit gravitation wave a frequency twice that of their orbital frequency. Lowest of LISA frequency is Hz, orbital frequency must be Hz or orbital period 5.5 Hrs. Consider two 1 M stars, their orbital separation r o mts, while solar radius is mts

17 Galactic Binaries Binary stars emit gravitation wave a frequency twice that of their orbital frequency. Lowest of LISA frequency is Hz, orbital frequency must be Hz or orbital period 5.5 Hrs. Consider two 1 M stars, their orbital separation r o mts, while solar radius is mts White dwarfs have mass about M while their radius is only 1000 KM.

18 Galactic Binaries Binary stars emit gravitation wave a frequency twice that of their orbital frequency. Lowest of LISA frequency is Hz, orbital frequency must be Hz or orbital period 5.5 Hrs. Consider two 1 M stars, their orbital separation r o mts, while solar radius is mts White dwarfs have mass about M while their radius is only 1000 KM. In our Galaxy there stars and about 50% of them are binary system.

19 There millions of binary stars in the LISA band especially between 0.1 mhz to 3 mhz, and individual sources can not be resolved. These stars generate a stochastic noise the detector called white dwarf noise.

20 There millions of binary stars in the LISA band especially between 0.1 mhz to 3 mhz, and individual sources can not be resolved. These stars generate a stochastic noise the detector called white dwarf noise. Sensitivity curve for binary confusion noise

21 We have catalog of about 400 such objects from optical astronomy. There are about 5 of them known in the LISA.

22 We have catalog of about 400 such objects from optical astronomy. There are about 5 of them known in the LISA. LISA observation can give distribution of them in our Galaxy and one can constrain the Galactic models based on LISA observations.

23 Super massive blackhole binaries There are strong evidence of existence of super massive blackhole of mass 10 5 to 10 8 M at the center of most of galaxies.

24 Super massive blackhole binaries There are strong evidence of existence of super massive blackhole of mass 10 5 to 10 8 M at the center of most of galaxies. It is seen that Galaxies collide,

25 Binary blackhole s are one of promising sources in the LISA band, can seen almost any where in the Universe.

26 Sensitivity curve for blackhole binaries:

27 Super-massive blackhole binaries are very important sources in the LISA band. The binary blackholes can be used to make an independent estimate of Cosmological parameter.

28 Super-massive blackhole binaries are very important sources in the LISA band. The binary blackholes can be used to make an independent estimate of Cosmological parameter. They serve as test for Einstein theory in the strong field limit.

29 Super-massive blackhole binaries are very important sources in the LISA band. The binary blackholes can be used to make an independent estimate of Cosmological parameter. They serve as test for Einstein theory in the strong field limit. Can be used to test the alternative theory of gravity.

30 Extreme mass ratio insprial(emri s) Galactic centers, such as our Galaxy hosts blackhole of mass 10 5 M. Some of the stars orbiting around them can fall in to blackholes emitting gravitational waves:

31 Extreme mass ratio insprial(emri s) Galactic centers, such as our Galaxy hosts blackhole of mass 10 5 M. Some of the stars orbiting around them can fall in to blackholes emitting gravitational waves: Or one can have blackholes of mass 1 10 M falling into central blackhole

32 We have small objects of mass 1 to 50 M falling in to blackholes of mass 10 6 M is called EMRI

33 Sensitivity curve for EMRI:

34 In order to observer astrophysically important sources such as these, one needs a space mission such as LISA.

35 Mission overview

36 The mission LISA consists of three identical spacecrafts orbiting around the Sun forming a giant laser interferometer of arm length 10 6 KM Mission overview

37 Mission overview The mission LISA consists of three identical spacecrafts orbiting around the Sun forming a giant laser interferometer of arm length 10 6 KM Six laser beams are exchanged between these three spacecraft to detect gravitational wave signal in the band Hz with peak sensitivity of h

38 Each spacecraft hosts, two test masses and optical system to communicate with the distance spacecraft

39 LISA orbits

40 LISA orbits Each of the LISA spacecraft go around Sun, in elliptical orbits such that entire formation remain almost equilateral triangle. The plane of the LISA triangle makes an angle of 60 o with the ecliptic plane. The center of this LISA constellation moves around the Sun in an earth-like orbit (R = 1AU), 20 o behind the Earth. The triangular constellation revolves once around its center as it completes one orbit in the course of one year.

41 A choice of orbits A choice of orbits are given by, X 1 = R(cosψ 1 + e)cosε, Y 1 = R 1 e 2 sinψ 1, Z S ε Y X SC3 R R(1+e) l y z 60 o O SC1 l/2 x Z 1 = R(cosψ 1 + e)sinε. SC2 tanε = e = α 1 + α/ 3, α = l 2R ( α + 4 ) 1/2 3 3 α2 1 ψ 1 + esinψ 1 = Ωt Orbits for other two spacecraft can obtained by rotating this by 180 o about z axis

42 CW-Frame and Equations Clohessy and Wiltshire or Hill s equations are linearised dynamical equations for test-particles in the neighborhood of reference point, in our case the LISA centroid. These equations are written in a frame which has its origin on the reference orbit and also rotates with angular velocity Ω. The equation for a free test particle are given by, ẍ 2Ωẏ 3Ω 2 x = 0, ÿ + 2Ωẋ = 0, z + Ω 2 z = 0.

43 The solutions: x(t) = y(t) = + ( ẋ0 Ω sinωt 3x 0 + 2ẏ ) 0 cosωt + 2 Ω ( 6x 0 + 4ẏ ) 0 Ω ( y 0 2ẋ ) 0 Ω z(t) = z 0 cosωt + ż0 Ω sinωt ( ) 2x 0 + ẏ0 Ω sinωt + 2ẋ 0 Ω cosωt 3(2Ωx 0 + ẏ 0 )t Ignoring runaway solutions and offset solution the condition for stable configuration is given by z 0 = µ 3x 0 and ż 0 Ω = 1 2 µ 3y 0, µ = ±1

44 The solutions are given by, x(t) = 1 2 ρ 0 cos(ωt φ 0 ), y(t) = ρ 0 sin(ωt φ 0 ), z(t) = µρ cos(ωt φ 0), where ρ 0 = 4x y2 0 tanφ 0 = y 0 2x 0

45 The orbits given earlier when when approximated to first order in α = l/(2r) and transformed to CW frame we get: x k y k z k [ = ercos Ωt (k 1) 2π ] 3 [ = 2eRsin Ωt (k 1) 2π ] 3 = [ 3eRcos Ωt (k 1) 2π ] 3 we identify ρ 0 = 2eR and φ 0 = 2π(k 1)/3

46 The general result is In the CW frame there are just two planes which make angles of ±π/3 with the (x-y) plane, in which test particles obeying CW equations perform rigid rotations about the origin with angular velocity Ω. S. V. Dhurandhar et. al, Fundamentals of the LISA Stable Flight Formation, Class. Quantum Grav (2005).

47 Time-Delay interferometer There are various difficulties in building conventional interferometer in space.

48 Time-Delay interferometer There are various difficulties in building conventional interferometer in space. LISA uses 1 Watt laser, after traveling for 10 9 mts power of laser is reduced to few pico watts. It is not practical to reflect this back to from the interferometer.

49 Time-Delay interferometer There are various difficulties in building conventional interferometer in space. LISA uses 1 Watt laser, after traveling for 10 9 mts power of laser is reduced to few pico watts. It is not practical to reflect this back to from the interferometer. For locking Michelson like interferometer, distance between the end points must be integral multiple of λ 2, which is impossible in space.

50 Time-Delay interferometer There are various difficulties in building conventional interferometer in space. LISA uses 1 Watt laser, after traveling for 10 9 mts power of laser is reduced to few pico watts. It is not practical to reflect this back to from the interferometer. For locking Michelson like interferometer, distance between the end points must be integral multiple of λ 2, which is impossible in space. Because different lasers are used at eand points, laser noise play important role.

51 Lasers typically have a frequency fluctuation of the order: ν 0 ν

52 Lasers typically have a frequency fluctuation of the order: ν 0 ν h In the case of ground based detectors, where both arms are exactly of same length (integral multiple of λ 2 )and the laser noise cancels out

53 Lasers typically have a frequency fluctuation of the order: ν 0 ν h In the case of ground based detectors, where both arms are exactly of same length (integral multiple of λ 2 )and the laser noise cancels out. Time-Delay interferometry is one of novel technique suggested by Armstrong et. al (J.W. Armstrong, F.B Estabrook and M. Tinto, Astrophys. J. 527, 814(1999).) for constructing unequal arm interferometer in space In this method the individual beams are combined off-line after introducing suitable time delay corresponding to the light travel time across the arms to simulate the interferometer.

54 Unequal Arm Interferometer

55 Unequal Arm Interferometer Let Φ(t) be the Phase fluctuation of laser

56 Unequal Arm Interferometer Let Φ(t) be the Phase fluctuation of laser If L 1 L 2 Φ 1 (t) = Φ(t 2L 1 ) Φ(t) Φ 2 (t) = Φ(t 2L 2 ) Φ(t)

57 Unequal Arm Interferometer Let Φ(t) be the Phase fluctuation of laser If L 1 L 2 Φ 1 (t) = Φ(t 2L 1 ) Φ(t) Φ 2 (t) = Φ(t 2L 2 ) Φ(t) Clearly, Φ 1 (t) Φ 2 (t) 0

58 We can still cancel-out the laser phase noise by taking the combination,

59 We can still cancel-out the laser phase noise by taking the combination, X 1 = [Φ 1 (t 2L 2 ) Φ 1 (t)] X 2 = [Φ 2 (t 2L 1 ) Φ 2 (t)]

60 We can still cancel-out the laser phase noise by taking the combination, X 1 = [Φ 1 (t 2L 2 ) Φ 1 (t)] X 2 = [Φ 2 (t 2L 1 ) Φ 2 (t)] Clearly, X 1 X 2 = 0!

61 We can still cancel-out the laser phase noise by taking the combination, X 1 = [Φ 1 (t 2L 2 ) Φ 1 (t)] X 2 = [Φ 2 (t 2L 1 ) Φ 2 (t)] Clearly, X 1 X 2 = 0! LISA is a more complex system. There are three arms and six beams.

62 Delay Operator S. V. Dhurandhar et. al, Algebraic approach to time-delay data analysis for LISA, Phs Rev. D65, (2002), gr-qc/ Let a(t) be any arbitrary function of time and L k be the length of the kth arm, then we define time delay operator for the arm k as, D k a(t) = a(t L k ) c = 1 and all the distances are measured in the unit of time.

63 Delay Operator S. V. Dhurandhar et. al, Algebraic approach to time-delay data analysis for LISA, Phs Rev. D65, (2002), gr-qc/ Let a(t) be any arbitrary function of time and L k be the length of the kth arm, then we define time delay operator for the arm k as, D k a(t) = a(t L k ) c = 1 and all the distances are measured in the unit of time. Some Properties of Delay Operator: the delay corresponding to the length of ll 1 + ml 2 + nl 3 is, D l 1 Dm 2 Dn 3 This is equivalent to successively applying the delay operator

64 Delay Operator S. V. Dhurandhar et. al, Algebraic approach to time-delay data analysis for LISA, Phs Rev. D65, (2002), gr-qc/ Let a(t) be any arbitrary function of time and L k be the length of the kth arm, then we define time delay operator for the arm k as, D k a(t) = a(t L k ) c = 1 and all the distances are measured in the unit of time. Some Properties of Delay Operator: the delay corresponding to the length of ll 1 + ml 2 + nl 3 is, D l 1 Dm 2 Dn 3 This is equivalent to successively applying the delay operator They commute: D k 1 Dl 2 = Dl 2 Dk 1

65 Back to LISA geometry and beams 1 L 3 L 2 2 L 1 3

66 Back to LISA geometry and beams U 1 = D 2 C 3 C 1 1 L 3 L 2 U 1 2 L 1 3

67 Back to LISA geometry and beams U 1 = D 2 C 3 C 1 1 -V L 3 1 L 2 U 1 2 V 1 = C 1 D 3 C 2 L 1 3

68 Back to LISA geometry and beams Ui s in +ve direction U 1 = D 2 C 3 C 1 U 2 = D 3 C 1 C 2 1 L 3 -V 1 U 3 = D 1 C 2 C 3 U 2 L 2 U 1 2 U 3 V 1 = C 1 D 3 C 2 L 1 3

69 Back to LISA geometry and beams Ui s in +ve direction U 1 = D 2 C 3 C 1 U 2 = D 3 C 1 C 2 1 L 3 -V 1 U 3 = D 1 C 2 C 3 U 2 Vi s in -ve direction -V 3 L 2 U 1 U 3 2 V 1 = C 1 D 3 C 2 L 1 V 2 = C 2 D 1 C 3 V 3 = C 3 D 2 C 1 3 -V 2

70 Back to LISA geometry and beams Ui s in +ve direction U 1 = D 2 C 3 C 1 U 2 = D 3 C 1 C 2 1 L 3 -V 1 There are six beams U 3 = D 1 C 2 C 3 U 2 Vi s in -ve direction -V 3 L 2 U 1 U 3 2 V 1 = C 1 D 3 C 2 V 2 = C 2 D 1 C 3 V 3 = C 3 D 2 C 1 3 L 1 -V 2 Redundancy in Data

71 Back to LISA geometry and beams Ui s in +ve direction U 1 = D 2 C 3 C 1 U 2 = D 3 C 1 C 2 1 L 3 -V 1 There are six beams U 3 = D 1 C 2 C 3 U 2 Vi s in -ve direction -V 3 L 2 U 1 Beams U 3 2 V 1 = C 1 D 3 C 2 V 2 = C 2 D 1 C 3 V 3 = C 3 D 2 C 1 3 L 1 -V 2 Redundancy in Data

72 Known Noise cancellation solution

73 Laser noise Cancellation Data combination A general data combination is given by the combination of U i and V i of the form X = 3 i=1 p i V i + q i U i p i and q i are polynomial in D i, This is clear from the properties of D i

74 Laser noise Cancellation Data combination A general data combination is given by the combination of U i and V i of the form X = 3 i=1 p i V i + q i U i p i and q i are polynomial in D i, This is clear from the properties of D i A noise cancellation Data combination, we need to determine p i and q i such that, 3 i=0 p i V i + q i U i = 0 We need to solve for (p i,q i ) as functions of D i.

75 The solution to this equation is well known in the algebra and forms a module called First Module of Syzygies, over the polynomial ring D i.

76 Vector Space Vs Modules Vector spaces are defined over field F. Modules are defined over ring R.

77 Vector Space Vs Modules Vector spaces are defined over field F. Modules are defined over ring R. Each element of field has multiplicative inverse. The elements of ring in general do not have multiplicative inverse.

78 Vector Space Vs Modules Vector spaces are defined over field F. Modules are defined over ring R. Each element of field has multiplicative inverse. Set of all real numbers R, Complex numbers C form fields. The elements of ring in general do not have multiplicative inverse. Set of polynomials are important example for rings. No inverse for Polynomial!

79 Vector Space Vs Modules Vector spaces are defined over field F. Modules are defined over ring R. Each element of field has multiplicative inverse. Set of all real numbers R, Complex numbers C form fields. Basis can generate the complete vector space by taking the combination a i v i where a i F and v i s are basis elements. The elements of ring in general do not have multiplicative inverse. Set of polynomials are important example for rings. No inverse for Polynomial! They are called generators! and any element of module can be written of the form p i g i where p i R and g i s are the generators.

80 The Generators The solutions are represented by generator with (p i,q i ), X (1) = α = ( ) 1, D 3, D 1 D 3, 1, D 1 D 2, D 2, X (2) = β = ( ) D 1 D 2, 1, D 1, D 3, 1, D 2 D 3, X (3) = γ = ( D 2, D 2 D 3, 1, D 1 D 3, D 1, 1 ), X (4) = ζ = ( ) D 1, D 2, D 3, D 1, D 2, D 3. With these generator any solution can be expressed as: X(p i,q i ) = 4 α (I) X (I) I=1 In general, α (I) can be polynomials in D i

81 Advantages of the Formalism All the Noise cancellation combination can be obtained using a set of generators. Any noise cancellation data combination can be written as, 4 A α A X (A) where α A are polynomials in D i.

82 Advantages of the Formalism All the Noise cancellation combination can be obtained using a set of generators. Any noise cancellation data combination can be written as, 4 α A X (A) A where α A are polynomials in D i. The analysis can be extended to cancel optical bench noise in a straight forward manner.

83 Advantages of the Formalism All the Noise cancellation combination can be obtained using a set of generators. Any noise cancellation data combination can be written as, 4 α A X (A) A where α A are polynomials in D i. The analysis can be extended to cancel optical bench noise in a straight forward manner. Each of the data combination have different signal response.

84 Advantages of the Formalism All the Noise cancellation combination can be obtained using a set of generators. Any noise cancellation data combination can be written as, 4 α A X (A) A where α A are polynomials in D i. The analysis can be extended to cancel optical bench noise in a straight forward manner. Each of the data combination have different signal response. The parameterisation of noise cancellation data combination is useful for application such as maximisation of SNR..etc..

85 Advantages of the Formalism All the Noise cancellation combination can be obtained using a set of generators. Any noise cancellation data combination can be written as, 4 α A X (A) A where α A are polynomials in D i. The analysis can be extended to cancel optical bench noise in a straight forward manner. Each of the data combination have different signal response. The parameterisation of noise cancellation data combination is useful for application such as maximisation of SNR..etc..