Stochastic gravitational wave background from the collisions of dark matter halos

Recently, the gravitational wave (GW) detecting technology has been developing rapidly. In 2015, the detection of binary black holes merger GW150914 by the LIGO experimental cooperation signaled the first detection of gravitational waves LIGOScientific:2016aoc , while in 2017, the joint detection of GW170817 LIGOScientific:2017vwq and GRB170817A Goldstein:2017mmi opened the new era of multi-messenger astronomy Addazi:2021xuf . In general, with the increasing amount of detected gravitational wave events LIGOScientific:2020ibl one has improved statistics that allows to track the history of the universe Ezquiaga:2017ekz ; Zhao:2017cbb and impose bounds on various cosmological parameters LIGOScientific:2017zic ; LIGOScientific:2017adf , as well as constrain various theories of gravity Baker:2017hug ; Farrugia:2018gyz ; Cai:2018rzd ; Hohmann:2018wxu ; CANTATA:2021ktz . Moreover, for different frequencies and types of gravitational wave sources, various detection means have been designed and implemented. Besides ground-based laser interferometers such as LIGO, Virgo and KAGRA, which probe high frequency bands ($10-10^{4}$ Hz), space-based laser interferometers such as LISA Amaro-Seoane:2012vvq ; LISA:2017pwj ; Papanikolaou:2022chm ; Domenech:2020ssp for intermediate frequency gravitational waves ($10^{-4}-1$ Hz), and the pulsar timing array (PTA) Lentati:2015qwp ; NANOGrav:2020bcs ; Chen:2022ooe ; Chen:2018yje ; EPTA:2023sfo ; Antoniadis:2023xlr for lower frequency bands ($10^{-9}-10^{-6}$ Hz), are also raised. These observational avenues allow us to acquire rich information from GWs of different types and sources, among which stochastic gravitational wave background is attracting increasing interest.

Stochastic gravitational wave background (GWB) is a type of random background signal that exists in an analogous way to the cosmic microwave background. The contribution of GWB can be roughly divided into cosmological sources and astrophysical sources Christensen:2018iqi . Astrophysical originated GWB contains all types of unresolved GW emitting events, including binary black hole mergers LIGOScientific:2016fpe ; Owen:1998xg ; Jaffe:2002rt ; Stergioulas:2011gd ; Clark:2015zxa ; DeLillo:2022blw ; Agarwal:2022lvk . These signals can provide information about astrophysical source populations and processes over the history of the universe Wang:2016tbj ; Flauger:2020qyi ; Bellomo:2021mer ; Zhao:2022cnn . On the other hand, cosmological originated GWB mainly involves primordial gravitational perturbations during the inflation epoch Maggiore:1999vm ; Ananda:2006af ; Caprini:2018mtu , or perturbations arising from primordial black holes fluctuations Musco:2012au ; Papanikolaou:2021uhe ; Papanikolaou:2022hkg ; Banerjee:2022xft ; Papanikolaou:2020qtd . GW signals typically remain unaffected during their propagation, and thus they can provide valuable information about the very early stages of the universe. For instance, different inflationary models can lead to different predictions for the GWB spectrum Dufaux:2007pt ; Cai:2007xr ; Kuroyanagi:2008ye ; Mavromatos:2020kzj ; Zhou:2020kkf ; Achucarro:2022qrl ; Zhu:2021whu ; Cai:2022erk ; Franciolini:2022tfm ; Addazi:2022ukh ; Mavromatos:2022yql ; Kazempour:2022xzy ; Papanikolaou:2022did , and thus GWB can be used as a probe of this primordial universe epoch. Since GWB can provide us with important astrophysical and cosmological probes, it is crucial to understand its composition and properties Allen:1997ad ; Capozziello:2008fn ; Sesana:2008mz ; Romano:2016dpx ; Huang:2016cjm ; Cai:2017aea ; Cai:2019jah ; Regimbau:2022mdu ; NANOGRAV:2018hou ; Speri:2022kaq ; Georgousi:2022uyt ; NANOGrav:2023hfp .

On the other hand, according to observations, dark matter (DM) constitutes a significant fraction of the energy density of the universe Planck:2018nkj ; Planck:2018vyg ; Planck:2013pxb . Its microphysical nature and possible interactions remain unknown Bertone:2004pz ; Jungman:1995df ; Preskill:1982cy ; Arkani-Hamed:2008hhe , nevertheless we do know unambiguously that DM takes part in gravitational interaction Rubin:1980zd ; Clowe:2006eq . Current theory predicts that the main part of DM is concentrated in dark halos, which coincide in position with galaxy or galaxy clusters Navarro:1995iw . These galaxies and galaxy clusters, and thus dark halos too, are typically accelerating and merging through their mutual attraction Davis:1985rj ; White:1977jf ; Bullock:1999he . Such processes can in principle release GW signal through gravitational bremsstrahlung Peters:1970mx ; Crowley:1977us ; Smarr:1977fy ; Kovacs:1978eu ; Farris:2011vx ; Mougiakakos:2021ckm ; Herrmann:2021lqe ; Mougiakakos:2022sic ; Jakobsen:2021lvp ; Gondan:2021fpr ; Riva:2022fru ; Inagaki:2010nu ; Inagaki:2012th . This process can be approximately described as an elastic collision between two particles. The approximate calculation results of the gravitational waves released during the collision process have been obtained previously in some literature 2010gfe..book…..M . However, it is important to note that simply considering the gravitational waves released during the elastic collision process is the ideal hypothesis because the release of gravitational waves takes away the system’s energy, making the collision no longer elastic. Therefore, the gravitational wave spectrum calculated in this way can only be applicable below a certain cutoff frequency.

In this work, we are interested in investigating for the first time the possible GW signals that could be emitted through bremsstrahlung during dark halo merger and collisions, and their contribution to the stochastic GWB. In particular, we will first consider a single event of two DM halos collision, and we will calculate the emitted GW signal. Then, we will calculate the energy spectrum contribution to the stochastic GWB, taking the DM halo collision rate into consideration. The structure of the article is as follows. In Section 2 we analyze the GW emitted during the collision of two galaxies or two galaxy clusters. In Section 3 we integrate over redshift and DM halos parameters to extract the contribution to stochastic GWB. Finally, in Section 4 we conclude and discuss our results.

In this section, we aim at estimating the gravitational waves emitted during a single collision event. In particular, we calculate the GW radiated by the collision of two DM halos, which corresponds to the collision of two galaxies or two galaxy clusters.

2010gfe..book…..M considered the gravitational waves released during the elastic collision between two particles. In section 4.4.1 of 2010gfe..book…..M , the energy-momentum tensor was approximated directly using the 4-momentum of the particles before and after the collision without considering collision process.

In the case where the collision velocity is much lower than the speed of light $c$, let A and B be the two particles participating in the collision, $m_{Q}$, $p_{Q}^{\mu},{p_{{}^{\prime}}}_{Q}^{\mu}$ and $\vec{v}_{Q},\vec{v}_{Q}^{{}^{\prime}}$ be the mass, 4-momenta and velocities of particle Q before and after the collision. Then, before the collision occurs, the energy-momentum tensor of the two particles is given by $\sum_{Q=A,B}\frac{p_{Q}^{\mu}p_{Q}^{\nu}}{m_{Q}}\delta^{(3)}(\vec{x}_{Q}-\vec{v}_{Q}t)$. Treating the collision process as instantaneous, the collision occurs at $t=0$. Afterwards, the energy-momentum tensor of the two particles is given by $\sum_{Q=A,B}\frac{{p^{{}^{\prime}}}_{Q}^{\mu}{p^{{}^{\prime}}}_{Q}^{\nu}}{m_{Q}}\delta^{(3)}(\vec{x}_{Q}-\vec{v}^{{}^{\prime}}_{Q}t)$. Let $\theta(t)$ be the unit step function. In this way, the energy-momentum tensor of the particles throughout the entire process can be written as

Based on this energy-momentum tensor, the gravitational wave energy spectrum can be further calculated. In the next section, we will see the energy spectrum given in this article is consistent with that in 2010gfe..book…..M at relatively high frequencies. However, since we give the precise calculation of the particle motion during the entire collision process, in our calculation we also give the contribution of the lower-frequency gravitational waves compared to the results given in 2010gfe..book…..M .

Besides, at very high frequencies, the physical processes corresponding to the emission of high-frequency gravitational waves are not reflected in either analytical method and cannot be dealt with. As a result, the gravitational wave spectra obtained from both methods can only be applicable below a certain cut-off frequency. In fact, without using numerical relativity for high precision calculations, any gravitational wave spectra obtained through analytical methods require an artificially estimated cut-off frequency. In the next section, we will see that the spectra obtained from both methods diverge when integrated to arbitrarily high frequencies, thus necessitating the establishment of a cut-off frequency.

The following is a more detailed and accurate calculation process we carried out. According to observations, such a collision typically has a huge duration, which in turn implies that the energy radiated through GWs per unit time is not very large, and thus we can safely use linear perturbation theory in the involved calculations. Specifically, we use Maggiore:2007ulw

where $G$ is the gravitational constant, $c$ is the speed of light, and $r$ is the distance from us to the center of mass of the two galaxies or galaxy clusters. Moreover, $I_{ij}$ is the quadruple moment

where $T^{\mu\nu}$ is energy-momentum tensor, $\rho$ is energy density, and $y^{i}$ is the spatial coordinate. Since the goal of our calculation is to acquire an estimation of the order of the magnitude of the resulting signal, we can consider these two DM halos as mass points, with mass $M_{a}$ and position $\mathbf{y}_{(a)}(t)$ at time $t$. Hence, the density $\rho$ can be written as

while the quadruple moment $I_{ij}(t)$ becomes

Finally, since the relative speed of two galaxies or galaxy clusters is much smaller than the speed of light, we can use Newtonian mechanics to handle their dynamics.

For simplicity, we write the equations in the center-of-mass frame of these two mass points. By definition, we have

where $M_{A}$, $M_{B}$ are the masses of the mass points A and B, with $\mathbf{r_{A}}$, $\mathbf{r_{B}}$ their position vectors. From Newtonian mechanics we have

which using (7) gives

where we have defined $\mu_{B}\equiv\frac{GM_{B}}{(1+\frac{M_{A}}{M_{B}})^{2}}$. Additionally, we assume that the two points are initially at infinite distance, their relative speed is $v_{\infty}={v_{A}}_{\infty}+{v_{B}}_{\infty}$, and the impact parameter is $b=b_{A}+b_{B}$. From Newtonian mechanics we know that the trajectory of each point is a hyperbola and the two points are moving in a plane (we set this plane as $z=0$ plane, and thus $\mathbf{r}_{A}=(x_{A},y_{A},0)$), while the total energy of the system is positive. Additionally, the mass center of these two DM halos will not follow a hyperbolic trajectory at all times, in order to acquire a collision. In Fig. 1 we depict an illustrative representation of the initial conditions of the collision.

Let us start with the beginning of the collision, when the two DM halos start moving towards each other. For point A we have

where

We proceed by defining $\lambda_{A}$ through

hence

Note that $t=0$ corresponds to the time when the two mass points have the shortest distance.

In order to obtain the GW amplitude $h_{ij}$, we proceed to the calculation of the quadrupole moment $I_{ij}(t)$ and its second time derivative. We have

From (15), (16) we find

and thus inserting into (18) we extract all the second time derivatives of the quadrupole moment $I_{ij}(t)$, namely

For simplicity, we define $M\equiv M_{A}+M_{B}$, and the mass ratio $x\equiv M_{A}/M_{B}$. Noting that we have $M_{A}b_{A}=M_{B}b_{B}$ and $M_{A}v_{A\infty}=M_{B}v_{B\infty}$, so that $1/x=b_{A}/b_{B}=v_{A\infty}/v_{B\infty}$.

As a result, we get $M_{A}=\frac{M}{(1+1/x)},M_{B}=\frac{M}{1+x},b_{A}=\frac{b}{1+x},b_{B}=\frac{b}{1+1/x},v_{A\infty}=\frac{v_{\infty}}{1+x},v_{B\infty}=\frac{v_{\infty}}{1+1/x}$. Substitute the above equations into the formula of $\mu_{B},a_{A},e_{A}$,we have

We note that $e_{A}$ is independent from mass ratio $x$, so $e_{A}=e_{B}$. Besides, $v_{A\infty}/a_{A}=v^{3}_{\infty}/GM$ is also independent from mass ratio $x$, so the $\lambda_{A}$ is just equal to $\lambda_{B}$.In short, the mass ratio $x$ only affects $\ddot{I}_{ij}$ by factors like $\mu_{B}M_{A}/a_{A}=Mv^{2}_{\infty}x/(1+x)^{3}$.Therefore, we can get the results in any mass ratio from the equal mass result by the following equation,

We can now use (3) in order to obtain the GW signal in the time domain. As typical values we set $M_{A}=M_{B}=10^{9}M_{\odot}$, namely the order of mass of a (dwarf) galaxy, where $M_{\odot}$ is the mass of the Sun, and we use ${v_{A}}_{\infty}={v_{B}}_{\infty}=300km/s$, $b_{A}=b_{B}=10^{4}ly$, which are the typical values for galaxy collisions. Moreover, we assume that the collision happens at a distance of $10^{9}ly$ from the Earth, which is roughly the distance of the source of GW150914. Hence, we can estimate the magnitude of the GW signal. In Fig. 2 we present the obtained dimensionless GW signal $\bar{h}_{ij}$, as a function of time $t$. Since $t=0$ corresponds to the time of shortest distance, the change rate of $\bar{h}_{ij}$ is fastest at this time, as expected. As we observe, the variation of $\bar{h}_{ij}$ is of the order of $5\times 10^{-22}$ during the collision. However, this variation corresponds to a large time scale (about $10^{15}s$), which implies that a single signal of this kind of GW is extremely hard to be detected. Additionally, we can see that the evolution of $\bar{h}_{12}$ is faster than that of $\bar{h}_{11}$, $\bar{h}_{22}$, which implies that $\bar{h}_{12}$ will be dominant in relatively higher frequency than that of $\bar{h}_{11}$, $\bar{h}_{22}$.

The GW amplitude in TT gauge $h^{TT}$ is the traceless version of $\bar{h}_{ij}$.We can project the GW amplitude in TT gauge by

while all other $h^{TT}_{ij}$ are equal to zero.

We proceed by taking the Fourier transformation of $\bar{h}_{ij},h^{TT}_{ij}$, in order to investigate its spectrum. In particular, we use

where $\omega=2\pi f$, with $f$ the frequency. $\tilde{\bar{h}}_{ij}(f)$ obey the power law in a very good approximation for a very wide frequency range. Besides, as $\tilde{\bar{h}}_{11},\tilde{\bar{h}}_{22}\propto 1/f^{2}$, while $\tilde{\bar{h}}_{12}\propto 1/f$, we can infer that $\tilde{\bar{h}}_{11},\tilde{\bar{h}}_{22}$ will be dominant in the low frequency band while $\tilde{\bar{h}}_{12}$ will be dominant in relatively high frequencies. In Fig. 3 we present the dependence of $\tilde{h}^{TT}_{ij}(\omega)(f)$ on $f$.

In this section, we calculate the contribution of the DM halos collisions to the stochastic gravitational wave background. Specifically, we integrate the gravitational wave spectrum of a single collision event over the number density of GW sources.

In principle, in order to compare a theoretical model with observations, one uses both the fractional energy density spectrum $\Omega_{gw}(f)$, as well as the characteristic strain amplitude $h_{c}(f)$ Romano:2016dpx . They are related to the energy spectrum of GWB through the expression

where $f$ is the frequency of GW detected on Earth, and $\rho_{c}\equiv 3c^{2}H_{0}^{2}/8\pi G$ is the critical energy density. The energy spectrum of the stochastic GWB, $\frac{d\rho_{\mathrm{gw}}}{d\ln f}$, can be written as

with $z$ the redshift at the GW emission. Additionally, $\frac{dE(\xi)_{gw}}{d\ln(f_{r})}$ is the energy spectrum of a single GW event, which is calculated through the analysis of subsection 2, and $f_{r}$ is the GW frequency in the rest frame of GW sources, and thus $f_{r}=(1+z)f$.

We mention that we denote the parameters related to the number density of GW sources collectively by $\xi=\left\{\xi_{1},\ldots,\xi_{m}\right\}$, and therefore $\frac{dn}{d\xi_{1}\ldots d\xi_{m}dz}d\xi_{1}\ldots d\xi_{m}dz\equiv\frac{dn}{d\xi dz}d\xi dz$ is the number density of sources in the redshift interval $[z,z+dz]$ and with source parameters in the interval $[\xi,\xi+d\xi]$. Hence, in the simple single event of two DM halos collision of the previous section we have $\xi=\{M,x,v_{\infty},b\}$, where $M=M_{A}+M_{B}$, $x=M_{A}/M_{B}$, $v_{\infty}={v_{A}}_{\infty}+{v_{B}}_{\infty}$ and $b=b_{A}+b_{B}$.

Let us now calculate the full distribution function $\frac{dn}{dzd\xi}=\frac{dn}{dzdMdxdv_{\infty}db}$. As we have checked numerically, the variance of $b,v_{\infty}$ has a minor effect on the final result, not affecting the order of magnitude. Hence, it is a good approximation to omit the change of $b,v_{\infty}$, and consider that $\xi=\{M,x\}$. Hence, we have

where the varying range of $M$ and $x$ is taken from Genel:2010pb .

In the following subsections we will separately calculate the energy spectrum of a single GW event $\frac{dE(\xi)_{\mathrm{gw}}}{d\ln f_{r}}$, and the number density of GW sources $\frac{dn}{dzdMdx}$.