Dark Matter Gravity Waves at LIGO and LISA

TL;DR

This paper investigates the potential of gravitational wave detectors like LIGO and LISA to detect primordial intermediate mass black holes as dark matter, concluding current observations are unlikely to be from such dark matter PIMBHs.

Contribution

It analyzes the merger rates of dark matter PIMBHs and discusses the detector sensitivities needed to observe their gravitational waves, proposing future detector requirements.

Findings

LIGO's observed black holes are likely from stellar collapse, not dark matter PIMBHs.

Detecting dark matter PIMBHs requires detectors sensitive below 10Hz.

A second space-based detector beyond LISA is necessary for comprehensive dark matter studies.

Abstract

We study the merger rate of dark matter PIMBHs(=Primordial Intermediate Mass Black Holes). We conclude that the black holes observed by LIGO in GW150914 and later events were probably not dark matter PIMBHs but rather the result of gravitational collapse of very massive stars. To study the PIMBHs by gravitational radiation will require a detector sensitive to frequencies below 10Hz and otherwise more sensitive than LIGO. The LISA detector, expected to come online in 2034, will be useful at frequencies below 1Hz but a second space-based detector beyond LISA, sensitive up to 10Hz, will be necessary fully to study dark matter.

Equations

$$\tau ={\left(\frac{M_{MACHO}}{M_{\odot }}\right)}^{1/2}(0.2M_{\odot })$$

$$10M_{\odot }<M_{PIMBH}<625M_{\odot }$$

$$\tau =8.3\times 10^{-12}{\eta }^4\mathrm{years}$$

$$\sigma =\pi {\left(\frac{85\pi }{3}\right)}^{\frac{2}{7}}{R}_s^2{\left(\frac{v}{c}\right)}^{-\frac{18}{7}}(pc{)}^2$$

$$\sigma \simeq 1.4\times 10^{-14}{M}_{30}^2(v{)}^{-\frac{18}{7}}(pc{)}^2$$

$$V=\left(\frac{M}{\rho }\right)\simeq (5\times 10^{14})\left(\frac{M_{12}}{{\rho }_{.002}}\right)(pc{)}^3$$

$$N$$

$${\rho }_{DM}\simeq 3.6\times 10^{10}{M}_{\odot }/(Mpc{)}^3\equiv 2.3\times 10^{-29}g/(cm{)}^3$$

$$n=\left(\frac{{\rho }_{DM}}{10^{12}{M}_{\odot }}\right)\simeq 0.036(Mpc{)}^{-3}.$$

$$\Gamma =nN\simeq 2.4\times 10^{-4}\left[v^{-\frac{11}{7}}\rho \right]/(Gpc{)}^3/yr$$

$${\Gamma }_{theory}\simeq nN\simeq 2.4\times 10^{-4}/(Gpc{)}^3/yr$$

$$2/(Gpc{)}^3/yr<{\Gamma }_{expt}<53/(Gpc{)}^3/yr$$

$${\Gamma }_{theory}{}_{\sim }^{<}1.9\times 10^{-3}/(Gpc{)}^3/yr$$

Full text

December 2018

**Dark Matter Gravity Waves

at LIGO and LISA**

Paul H. Frampton*paul.h.frampton@gmail.com**

*Department of Mathematics and Physics “Ennio de Giorgi”,

University of Salento, Lecce, Italy.*

We study the merger rate of dark matter PIMBHs(=Primordial Intermediate Mass Black Holes). We conclude that the black holes observed by LIGO in GW150914 and later events were probably not dark matter PIMBHs but rather the result of gravitational collapse of very massive stars. To study the PIMBHs by gravitational radiation will require a detector sensitive to frequencies below 10Hz and otherwise more sensitive than LIGO. The LISA detector, expected to come online in 2034, will be useful at frequencies below 1Hz but a second space-based detector beyond LISA, sensitive up to 10Hz, will be necessary fully to study dark matter.

The first direct detection by LIGO [1] of gravitational waves, identified as originating from a merger of two black holes with masses $\sim 36M_{\odot}$ and $\sim 29M_{\odot}$ which occurred at a red shift $Z\sim 0.09$, was observed on September 14, 2015 and, after detailed analysis ensured the data were ironclad, was first publicly announced on February 11, 2016. This represents the most important cosmological discovery so far in the 21st century, opening up an entirely new window to observe our Universe. At last, the direct detection of gravity waves has confirmed the indirect detection deduced from a binary pulsar by Hulse and Taylor[2, 3, 4, 5] and totally vindicates the prediction of gravity waves made one hundred years ago by Einstein[6].

It has always been a productive research directions to bridge between different subfields. In the case of the present Letter, we consider a possible bridge between the detection of gravity waves[1] and a theory of dark matter [7, 8, 9, 10]. In one of these dark matter papers [7] the underpinning of the idea that DM(= Dark Matter) $\equiv$ PIMBHs(=Primordial Intermediate Mass Black Holes) is discussed thoroughly and compared to the competing theories where the dark matter constituents are elementary particles. It is argued (see especially the Discussion section of [7]) that, based on the entropy of the universe, DM$\equiv$PIMBHs is strongly favoured over particle theory alternatives. Therefore, in this Letter, we assume it to be correct.

As discussed in the first paper#2#2#2An important precursor was by Chapline[9] who assumed PBHs lighter than the Sun. suggesting DM$\equiv$PIMBHs [8], published before the LIGO announcement, the optimal experiment for detecting PIMBHs employs microlensing. This technique dates back to Einstein[11] and Paczynski[12]. Its usefulness was amply demonstrated by the MACHO Collaboration[13] who observed 17 microlensing events which lensed stars in the Magellanic Clouds and used a telescope at the Mount Stromlo Observatory in Australia. Their microlensing events included light curves with duration $\tau$ ranging from 34 days to 230 days. From Paczynski’s formula[12]

[TABLE]

this range in $\tau$ translates to $0.2M_{\odot}\mathop{}_{\textstyle\sim}^{\textstyle<}M_{MACHO}\mathop{}_{\textstyle\sim}^{\textstyle<}10M_{\odot}$ for the MACHO mass. The MACHO Collaboration found insufficient mass in MACHOs to constitute all the dark matter, actually less than $20\%$ of the required amount.

According to the theory of dark matter proposed in [7, 8, 10] the remaining $80\%$ resides in PIMBHs with masses $M_{PIMBH}>10M_{\odot}$. If we restrict our attention to masses readily accessible to microlensing within five years, i.e. $\tau\mathop{}_{\textstyle\sim}^{\textstyle<}5y$ then the new masses in play are

[TABLE]

The mass range in Eq.(2) includes the mass values $\sim 36M_{\odot}$ and $\sim 29M_{\odot}$ involved in the first black hole merger, dubbed GW150914, discovered by LIGO[1]. This does not necessarily imply that the theoretical idea $DM\equiv PIMBHs$ has been either confirmed or refuted by LIGO. In the present notes, this question will be discussed in some detail and our preliminary conclusion will be that the LIGO black holes are probably not part of the dark matter, but are more likely not primordial, rather the end products from the gravitational collapse of very massive stars.

Let us anticipate the result that the GW150914 black holes are non-primordial and retrace the cosmological origin of the binary. The merger itself happened [1] at $Z=0.09$ which corresponds to cosmic time $t=12.6Gy$ and is $1.2Gy$ before the present $t_{0}=13.8Gy$. The capture and inspiral of two black holes can take gigayears, although we assume this is less that 7.3Gy which means we do not study cosmic times less than $t(Z=1)=5.3Gy$. For later use we note that the average cosmological matter density at $Z=1$, being proportional to $(1+Z)^{3}$, is $8$ times that at the present time $Z=0$.

To justify the choice of $Z\leq 1$ for the lifetime of the GW150914 binary, we use Eq. (5.10) in Peters’ 1964 paper[14]. Inserting $m_{1}=m_{2}=30M_{\odot}$ and initial separation $a_{0}=\eta R_{S}$ where $R_{S}=90$km is the Schwarzschild radius, we find a lifetime

[TABLE]

which is below $7.3$Gy as long as $\eta<1.7\times 10^{5}$. But this implies the huge initial separation $a_{0}=1.5\times 10^{7}$ km $\simeq 20R_{\odot}$. This makes our $Z\leq 1$ assumption very robust. This calculation was done for a circular orbit but, as can be seen from Fig. 2 of [14], for an eccentric orbit the lifetime $\tau$ in Eq.(3) becomes even shorter.

Consider two PIMBHs approaching each other with some impact parameter and relative velocity $v$ (which will be written in units of $200km/s$). In approach they produce a time-varying quadrupole and hence gravity wave (GW) emission. The cross-section was derived by Quinlan and Shapiro[15], and confirmed by Mouri and Taniguchi[16], to be given by

[TABLE]

where $R_{s}$ is the Schwarzschild radius given by $R_{s}=3\eta$ km for $M_{PIMBH}=\eta M_{\odot}$. Let us define $M_{30}$ by $M_{PIMBH}=M_{30}(30M_{\odot})$ and, as stated above, $v\equiv v_{PIMBH}/(200km/s)$. With these definitions, we find from Eq.(4) that

[TABLE]

Using Eq.(5), we can calculate the rate of mergers per unit volume in the dark halo. Let us write the halo mass as $M_{Halo}=M_{12}(10^{12}M_{\odot})$ which defines $M_{12}$. We also define $\rho_{0.002}$ by assuming a uniform dark matter density $\rho=\rho_{0.002}(0.002M_{\odot})/(pc)^{3}$. The halo volume $V$ is

[TABLE]

The rate of mergers in the halo is

[TABLE]

To estimate the merger rate per unit volume in the visible universe we need the mean density $\rho_{DM}$ of dark matter

[TABLE]

and the density $n$ of halos which is

[TABLE]

The mean rate of PIMBH mergers $\Gamma$ in the visible universe is

[TABLE]

Taking red shift $Z=0$ values $v\simeq\rho\simeq 1$ the merger rate is estimated as

[TABLE]

which is to be compared with the LIGO estimate[1]

[TABLE]

There is significant disagreement between Eq.(11) and Eq.(12). However, the first LIGO event occurred not at $Z=0$ but at $Z\simeq 0.09$ [1]. As already mentioned, the beginning of the binary capture could not be earlier than $Z=1$. This provides an enhancement due to the factor in square brackets in Eq.(10) where, although the velocity $v$ does not change significantly, the density $\rho$ is increased by a factor $(1+Z)^{3}\mathop{}_{\textstyle\sim}^{\textstyle<}8$ which changes the estimate in Eq.(11) to

[TABLE]

Nevertheless, the marked disagreement between Eq.(12) and Eq.(13) leads us to the conclusion that the black holes involved in the merger which generated the gravity waves observed by LIGO in [1] are probably not part of the dark matter but are more likely to be non-primordial and result from gravitational collapse of very massive stars#3#3#3We note that in [17] which assumed the start of the merger process was at a larger redshift an enhancement in the factor $[\left[(v)^{-\frac{11}{7}}\rho_{0.002}\right]$ by $6\times 10^{6}$ led to a conclusion that ”the possibility that LIGO saw dark matter cannot be immediately excluded”..

An interesting question is can LIGO or any of its upgrades detect the dark matter predicted by the $DM\equiv PIMBHs$ theory?

Two observations can be made immediately: (i) The merger rate calculated in Eq.(7) is independent of the PIMBH mass so increasing this mass has no significant effect on the rate. (ii) Increasing the PIMBH mass does effect the frequency of the gravity waves as the frequency is inversely proportional to the mass. The frequencies involved in the LIGO chirp ranged from 35Hz to 250Hz. The lowest frequency#4#4#4This lowest-frequency sensitivity limit is principally a result of microseismicity. to which LIGO is sensitive is [18] 10Hz.

By scaling the chirp frequencies from those of GW150914, we can check that when the black hole mass exceeds $105M_{\odot}$ the low frequency beginning of the chirp is undetectable by LIGO. For black hole mass $M>750M_{\odot}$, LIGO misses the chirp entirely. For the detection of gravity waves from dark matter, we may thus regard LIGO as a high (audible) frequency detector.

There is a second gravity wave detector, LISA(=Laser Interferometer Space Antenna) [19, 20] which will be space-based and expected to begin taking data in $2034$. Its frequency $f$ sensitivity will be $0.1mHz<f<1Hz$ so that its frequency ceiling is significantly below LIGO’s frequency floor. This is unfortunate for the merger of black holes with $750M_{\odot}$, for example, where the chirp frequency ranges from 1.4 Hz to 10Hz.

The LIGO and LISA detectors of gravitational radiation were designed well before the origin and nature of dark matter were understood at the level of [7]. It is clear that at least one additional gravity-wave detector with greater reach than LIGO, and sensitive to the crucial frequency range $1Hz<f<10Hz$, will be necessary to investigate the properties of dark matter. It seems that this additional detector must, like LISA, be space-based because any ground-based detector, even underground, cannot be sufficiently sensitive at such low frequencies.

Acknowledgement

We thank Steven Balbus and Selma E. de Mink for useful discussions.