# Sensitivity Studies for Third-Generation Gravitational Wave Observatories

###### Abstract

Advanced gravitational wave detectors, currently under construction, are expected to directly observe gravitational wave signals of astrophysical origin. The Einstein Telescope, a third-generation gravitational wave detector, has been proposed in order to fully open up the emerging field of gravitational wave astronomy. In this article we describe sensitivity models for the Einstein Telescope and investigate potential limits imposed by fundamental noise sources. A special focus is set on evaluating the frequency band below 10 Hz where a complex mixture of seismic, gravity gradient, suspension thermal and radiation pressure noise dominates. We develop the most accurate sensitivity model, referred to as ET-D, for a third-generation detector so far, including the most relevant fundamental noise contributions.

###### pacs:

04.80.Nn, 95.75.Kk## 1 Introduction

The currently operating Gravitational Wave (GW) detectors LIGO [1], Virgo [2], GEO 600 [3] and TAMA [4] are based on extremely sensitive Michelson interferometers. While the sensitivity achieved by these first generation detectors is mainly limited by shot noise, mirror thermal noise and seismic noise, for the second generation of instruments, such as Advanced LIGO [5], Advanced Virgo [6], GEO-HF [7] and LCGT [8], additional fundamental noise sources will start to play a role towards the low-frequency end of the detection band: Thermal noise of the test mass suspension, photon radiation pressure noise and seismically driven gravity gradients acting on the test masses. These three sources of noise will become even more important for third-generation GW observatories such as the Einstein Telescope (ET) [9], [10], as these detectors aim to significantly increase the detection band towards frequencies as low as a few Hz [11], [13]. Therefore, major parts of the ET design are driven by exactly these noise sources. An overview of the importance of the sub-10 Hz band for astrophysical and cosmological analyses can be found in [9].

In this article we will give an overview of the currently ongoing ET design activities with a special focus on the modelling of the achievable sensitivity taking the most important fundamental noise sources into account. The first sensitivity estimate for a third-generation interferometer was described in [11], [12] and was based on a single interferometer covering the full frequency range from about 1 Hz to 10 kHz. In the following we will refer to this sensitivity curve as ET-B. Subsequently we developed a more realistic design, taking cross-compatibility aspects of the various involved technologies into account. This led to the so-called xylophone-design, in which one GW detector is composed of two individual interferometers: A low-power, cryogenic low-frequency interferometer and a high-power, room-temperature high-frequency interferometer. A detailed description of this xylophone detector sensitivity, in the following referred to as ET-C, can be found in [13]. The ET-C configuration will serve as a starting point for the investigations described in this article. We improved the sensitivity models for ET by including additional new noise sources as well as by amending and updating noise contributions already previously included. These improvements, which led to a new sensitivity estimate, referred to as ET-D, will be presented and discussed in this article.

In section 2 we discuss seismic and gravity gradient noise, followed by the quantum noise contribution in section 3. Thermal noise of the suspensions and test masses will be presented in section 4. An improved noise budget for the Einstein Telescope is then given in section 5. We conclude with a brief overview of the configuration of a full third-generation observatory, consisting of several GW detectors.

## 2 Seismic Isolation and Gravity Gradient Noise

Seismic noise couples into the differential arm length of a GW detector
via two main paths. First of all, seismic excitation can mechanically couple through
the suspension and seismic isolation systems. Secondly, seismic noise excites density
fluctuations in the environment of the GW detector, which couple via
gravitational attraction to the test mass position. In the following we will refer to these two noise
sources as seismic noise and gravity gradient noise, respectively.
The main difference between these two noise sources is that while seismic
noise can be reduced by application of complex seismic isolation systems,
the only guaranteed way to reduce the gravity gradient noise is to
reduce the initial seismic excitation.^{1}^{1}1Many promising gravity gradient noise subtraction schemes have
been suggested in the literature [19]. However, as none of these schemes has been demonstrated
so far, we do not consider them in this article. Therefore, third-generation GW detectors are
proposed to be built in quiet underground locations.

The seismic noise contribution of the low-frequency interferometer of ET-C was based on a seismic excitation of (where is the frequency in Hz) and a generic 50 m tall seismic isolation system consisting of 5 passive pendulum stages, each of 10 m height. A more realistic seismic isolation design, based on the Virgo super attenuator concept [14], [15], has been developed recently [16]. To achieve a lower cut-off frequency the height of the individual pendulum stages of the super attenuator will be extended to 2 m per stage. The overall isolation of the proposed modified super attenuator, consisting of 6 pendulum stages (each stage providing horizontal as well as vertical isolation) and a total height of 17 m, is shown in the center panel of Figure 1. Using the seismic excitation level, measured in an underground facility of the Black Forest Observatory (BFO) [17], [18], shown in the left panel of Figure 1, we can derive the expected seismic noise contribution to the ET noise budget. The result is shown in the right hand panel of Figure 1. Reducing the height of the seismic isolation system from 50 to 17 m increases the cut-off frequency only slightly from about 1.2 to 1.7 Hz.

Gravity gradient noise has been described in detail [20, 21, 22]. In our simulations we estimate the power spectral density of the gravity gradient noise contribution as:

(1) |

where is the gravitational constant, the density of the rock around the GW detector, the arm length of the interferometer, the frequency and the power spectral density of the ground motion. accounts for the actual coupling transfer function from seismic excitation to the differential arm length noise and depends for instance on the wave type of the seismic excitation (e.g. ratio of P and S waves) and soil characteristics.

Within the ET design study we carried out a campaign of measuring the seismic noise in various underground locations across Europe. These measurements have indicated that a couple of the test locations show a seismic excitation level similar to or below the BFO measurement [23]. Therefore, we assumed the BFO seismic excitation as a conservative estimate of a potential ET site. The left panel of Figure 2 shows the corresponding gravity gradient noise contribution at the BFO site for different . Since the detailed evaluation of a realistic for potential ET sites is an ongoing activity, we will use , as given in the literature [20, 21], in the following for the ET-D sensitivity. Please note that our models do not take atmospheric newtonian noise into account.

## 3 Shaping of Quantum Noise

Quantum noise, composed of photon shot noise at high-frequencies and photon radiation pressure noise at low frequencies, contributes significantly to the overall sensitivity of ET’s high frequency and low-frequency detectors. The high-frequency interferometers will feature a light power stored in the arm cavities of about 3 MW to reduce shot noise, while the low-frequency interferometers make use of only 18 kW of light power in the arms, in order to reduce the radiation pressure noise. For ET-C the quantum noise contribution was optimised by making use of detuned (low-frequency interferometer) and tuned (high-frequency interferometer) signal recycling [24, 25], together with an assumed generic quantum noise reduction of 10 dB at all frequencies.

Such a broadband quantum noise reduction can in principle be achieved by injecting squeezed light states with a frequency dependent squeezing angle [26] into the output port of the interferometer. Starting from a frequency independent squeezing angle, it is possible to use the dispersion occurring in reflection of a cavity, to create squeezed light states with a frequency dependent squeezing angle. Figure 3 shows a simplified schematic of a dual recycled interferometer with arm cavities, consisting of a power recycling mirror (PRM), the beam splitter (BS), the arm cavity mirrors (IM and EM) and the signal recycling mirror (SRM). In addition the injection of the frequency dependent squeezed light states is also shown: The squeezed light states leave the squeezing source (Squeezer) and are reflected at two filter cavities before they are injected via a Faraday rotator into the interferometer mode. Finally, the interferometer output mode, consisting of the squeezed field and the signal field, is detected on the main photo diode (PD).

In general, for an interferometer with signal recycling two filter cavities are necessary: One for compensating the dispersion of the signal recycling resonance and one to reduce radiation pressure noise. The bandwidth and detuning of the filter cavities depends on the actual optical parameters of the main interferometers (e.g. arm length, SRM reflectivity, tuning of the signal recycling cavity) and need to be matched very accurately to establish the full sensitivity improvement of the squeezed-light injection. For the ET-D high-frequency interferometer we have the special case that only one filter cavity will be required, as it employs tuned signal recycling with a bandwidth significantly larger than the cross-over frequency of radiation pressure and shot noise.

The major loss mechanism for the squeezed light reflected off the filter cavities originates from the fact that for frequencies close to the resonance of the filter cavities, the squeezed states partly enter the filter cavity and experience unavoidable roundtrip losses. We recently performed a detailed analysis of the requirements for the ET filter cavities as well as quantifying the squeezing losses inside the filter cavities [27], [28]. We assumed a squeezing level of 10 dB, an antisqueezing level of 15 dB and a loss of 75 ppm per roundtrip inside a filter cavity. In order to reduce the influence of the intra-cavity losses, we chose a rather long filter cavity length of 10 km, which allows us to keep the filter cavity finesse at a moderate level. The right hand plot in Figure 3 shows the corresponding quantum noise contributions of ET-D for the low frequency interferometer with filter cavity losses included (solid lines).

For the low frequency interferometer it is also interesting to compare different signal recycling options, properly accounting for the squeezing losses inside the filter cavities. The left hand plot of Figure 4 shows the ET-D configuration, employing detuned signal recycling together with two filter cavities, in comparison to tuned signal recycling as well as an configuration without signal recycling. Both of these latter configurations require only a single filter cavity, but this comes at the price of significantly lower sensitivity in the frequency band of interest (4–30 Hz).

## 4 Thermal noise contributions

Brownian fluctuations couple into the differential arm length signal as thermal noise of the test mass itself and of its suspension. Both of these noise contributions can be significantly reduced by lowering the temperature of the test masses and the corresponding suspensions. The Japanese CLIO project [29] has successfully demonstrated the operation of a laser interferometer at cryogenic temperatures. The recently funded LCGT detector [8] is expected to transfer cryogenic technologies to a full scale second-generation GW detector.

While for the ET high frequency interferometers even at room temperature the various thermal noise contributions either do not play a significant role or can be sufficiently reduced by increasing the beam size on the test masses and the use of higher-order Laguerre Gauss beam shapes [30] or so-called ‘Mesa’ beams [31], the ET low-frequency interferometers are expected to operate at cryogenic temperature.

In the frequency band from 1 to 10 Hz suspension thermal noise is the dominating thermal noise contribution. When operating the low frequency interferometer at cryogenic temperatures the last stage suspension does not only need to be compliant with the thermal noise requirements, but the actual design is also driven by the requirement to extract any heat (deposited by the laser beams in the test masses) via the suspension. Our model [34] assumes a mirror temperature of 10 K, silicon fibres of 2 m length and 3 mm diameter as well as a temperature of the penultimate mass of 2 K. The righthand plot of Figure 2 shows the simulated suspension thermal noise contribution for the ET low frequency detector [34] using a branched system of multiple oscillators consisting of the main mirror, the penultimate mass and the reaction mass [32], [33].

Fused silica, which is the material of choice for the test masses of all first generation GW detectors, has a high dissipation at low temperatures and therefore cannot be used as substrate material for cryogenic test masses [38], [39]. Sapphire and silicon have been proposed as alternative materials [8], [41] and there are strong R+D efforts to evaluate the optical and mechanical properties of these two candidate materials. In the following we assume silicon as test material for ET, but sapphire would yield similar results. A detailed noise analysis of a cryogenic test mass for ET is given in [40]. The total thermal noise of the test masses is a combination of coating Brownian, substrate Brownian, substrate thermoeleastic and thermo-optic noise. Of these four contributions coating Brownian noise is the most important one because in the frequency range of interest it is a factor of about 5 larger than any of the other thermal noises. The power spectral density of coating brownian noise can be described as

(2) |

where is the frequency, the total thickness of the coating, describes the beam radius, and are the Young’s modulus values for the substrate and coating respectively. and are the mechanical loss values for the coating for strains parallel and perpendicular to the coating surface [35].

Using silicon test masses (62 cm diameter, 30 cm thickness) of 10 K, a Young’s modulus of silicon of 164 GPa,
loss angles of and for the low and high-refraction
coating materials^{2}^{2}2Unfortunately the available measurements indicate higher
loss angles for the coating materials at cryogenic temperatures than at room
temperature [42]. However, since research on cryogenic coatings just started,
we optimistically assumed that by
the time construction of third-generation instruments starts, coatings will be
available featuring the same loss angles as current coatings at room temperature
[36, 37], respectively, and a laser beam radius of 12 cm, we get
a total mirror thermal noise contribution for the low-frequency
detector as indicated by the orange dashed line in the right hand plot of Figure 4.
In this scenario the mirror thermal noise would be at least
a factor of three below the quantum noise for all frequencies. Hence, we could in principle consider
reducing the beam size on the test masses, which could allow for the reduction of
the mirror size. Smaller test masses and smaller laser beams would be
advantageous for many aspects of the observatory design, such as for instance the
total mass of the cryogenic payload or the mode matching into the arm cavities.
However, on the other hand reducing the mirror mass will increase the radiation
pressure noise contribution.

The right hand plot of Figure 4 shows a trade-off analysis of beam size and mirror mass. Starting from a mirror of 62 cm diameter and 30 cm thickness which corresponds to a beam radius of 12 cm, we reduce the mirror diameter to 52 cm and 45 cm, while keeping the aspect ratio of the mirror substrate constant. Using this assumption, already a small reduction in the beam size will increase the radiation pressure noise dramatically and subsequently spoil the sensitivity in the sub-10 Hz band. Therefore, we assume for the low frequency interferometer of ET-D a reduced beam radius of 9 cm, corresponding to an effective test mass diameter of 45 cm, but at the same time keep the overall test mass weight at about 200 kg.

Parameter | ET-D-HF | ET-D-LF |

Arm length | 10 km | 10 km |

Input power (after IMC) | 500 W | 3 W |

Arm power | 3 MW | 18 kW |

Temperature | 290 K | 10 K |

Mirror material | Fused silica | Silicon |

Mirror diameter / thickness | 62 cm / 30 cm | min 45 cm/ TBD |

Mirror masses | 200 kg | 211 kg |

Laser wavelength | 1064 nm | 1550 nm |

SR-phase | tuned (0.0) | detuned (0.6) |

SR transmittance | 10 % | 20 % |

Quantum noise suppression | freq. dep. squeez. | freq. dep. squeez. |

Filter cavities | km | km |

Squeezing level | 10 dB (effective) | 10 dB (effective) |

Beam shape | LG | TEM |

Beam radius | 7.25 cm | 9 cm |

Scatter loss per surface | 37.5 ppm | 37.5 ppm |

Partial pressurefor HO, H, N | , , Pa | , , Pa |

Seismic isolation | SA, 8 m tall | mod SA, 17 m tall |

Seismic (for Hz) | ||

Gravity gradient subtraction | none | none |

## 5 Overall sensitivity of the Einstein Telescope

Table 1 shows the most important parameters of the ET-D interferometers. The corresponding noise budgets for the high and low-frequency interferometers are shown in figure 5. The sensitivity of the low-frequency detector is limited by seismic noise below 1.7 Hz, while gravity gradient noise directly limits in the frequency band between 1.7 and 6 Hz. For all frequencies above 6 Hz (apart from the violin mode resonances) quantum noise is the limiting noise source.

The crossover frequency of the sensitivities of the low and high-frequency interferometers is at about 35 Hz. Above this frequency the high frequency interferometer is limited only by 2 noise sources: Mirror thermal noise limits the sensitivity between 40 and 200 Hz, while the high-frequency section is limited by quantum noise.

Figure 6 shows the evolution of the sensitivity models for the Einstein Telescope over the past years. The very first strawman design was based on a single cryogenic interferometer covering the full frequency range of interest (ET-B) [11]. The introduction of the xylophone design resulted in the ET-C sensitivity. In this article we significantly refined the xylophone concept and obtained the ET-D sensitivity, which is slightly worse than the ET-C sensitivity, but much more realistic. The loss of sensitivity below 1.7 Hz can be attributed to the application of a more realistic suspension model yielding increased seismic noise. The slightly worse sensitivity in the range between 2 and 8 Hz is the result of the inclusion of suspension thermal noise for the cryogenic interferometer as well as the omission of any potential gravity gradient subtraction. The most significant difference between ET-C and ET-D shows up between 6 and 10 Hz and originates from including realistic squeezing losses experienced inside the filter cavities.

## 6 Building a Full Third-Generation Observatory

As discussed in the previous sections, one ET detector, covering the full detection band will be made of two individual interferometers, one for low frequencies and one for high frequencies. However, the full ET observatory will consist of 3 detectors arranged in triangular shape [43], thus ultimately 6 interferometers will form the whole observatory.

It needs to be pointed out that the sensitivities described in this article refer to a single pair of low and high frequency interferometers of 10 km arm length and an opening angle of (as shown in subplot A of figure 7). The actual effective sensitivity of the full triangular ET observatory depends on the the orientation and polarisation of the source of interest. Let us assume a source directly positioned above the observatory emitting GW of plus polarisation. In case of the configuration A in figure 7, the sensitivity, , is then exactly represented by the ET-D trace in figure 6.

If we now decrease the opening angle of the two interferometers to (see configuration B in figure 7), the effective sensitivity for plus polarised GW is given by

(3) |

which is equivalent to shifting the ET-D curve in figure 6 up by about 15 %.

Finally, if we consider the full triangle for a plus polarised source, we find that the blue and red detectors in configuration C of figure 7 have the same sensitivity, while no signal shows up in the green detector. If we combine the red and blue detectors, the noise of the two needs to be added in quadrature because it is uncorrelated, while the coherent signals need to be added up linearly. Thus the overall sensitivity of the full triangular observatory can be written as

(4) |

which would be equivalent to shifting the ET-D curve in figure 6 down by about 18 %.

Subplot D of figure 7 shows a schematic drawing of the full ET observatory configuration assumed for the ET-D sensitivity. Included are all main mirrors of the 6 interferometers, as well as a total of 9 required filter cavities for the frequency dependent squeezing. In total this sums up to 7 laser beams per tunnel.

## 7 Summary and Future Plans

In this article we described a snapshot of the ongoing sensitivity studies for a European third-generation GW observatory. The ET-D sensitivity represents a much more realistic sensitivity compared to previous models, because we included new noise sources as well as improved the accuracy of several already previously included fundamental noise sources. Key points of the new sensitivity model are the inclusion of suspension thermal noise and a realistic seismic isolation system for the low frequency interferometer and the proper accounting for squeezing losses inside the filter cavities. Finally it needs to be pointed out that the current model does not rely on any subtraction techniques for gravity gradient noise.

In the future, we plan to further refine our sensitivity models by including noise contributions from optical components outside the arm and filter cavities as well as by taking technical contributions such as laser frequency and laser amplitude noise into account.

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