Possible Explanation of the Geograv Detector Signal during the Explosion of SN 1987A in Modified Gravity Models

TL;DR

This paper proposes that a modified gravity model could explain the Geograv detector signal during SN 1987A, suggesting an abrupt metric change caused by neutrino flux, with implications for current gravitational wave detectors.

Contribution

It introduces a novel explanation for the SN 1987A signal based on extended scalar-tensor theories of gravity affecting detector responses during stellar collapse.

Findings

The initial neutrino pulse could produce a detectable metric change in modified gravity models.

The second, broader neutrino pulse's influence is exponentially suppressed in the detector.

This mechanism explains the observed 1.4s delay between Geograv and LSD signals.

Abstract

A change in gravity law in some regimes is predicted in the modified gravity models that are actively discussed at present. In this paper, we consider a possibility that the signal recorded by the Geograv resonant gravitational-wave detector in 1987 during the explosion of SN 1987A was produced by an abrupt change in the metric during the passage of a strong neutrino flux through the detector. Such an impact on the detector is possible, in particular, in extended scalar-tensor theories in which the local matter density gradient affects the gravitational force. The first short neutrino pulse emitted at the initial stage of stellar core collapse before the onset of neutrino opacity could exert a major influence on the detector by exiting the detector response at the main resonance frequency. In contrast, the influence of the subsequent broad pulse (with a duration of several seconds) in the resonant detector is exponentially suppressed, despite the fact that the second pulse carries an order-of-magnitude more neutrino energy, and it could generate a signal in the LSD neutrino detector. This explains the time delay of 1.4s between the Geograv and LSD signals. The consequences of this effect of modified gravity for LIGO/Virgo observations are discussed.

Equations

$${\rho }_{\nu }(t,r)=\frac{M_{\nu }}{4\pi r^2}f(r-ct),$$

$$\int f(x)dx=1,$$

$$f(x)=\frac{1}{\sqrt{2\pi }{H}_{\nu }}{e}^{-x^2/(2H_{\nu }^2)}.$$

$$\delta t=\frac{r}{2c}{\left(\frac{m_{\nu }{c}^2}{E_{\nu }}\right)}^2.$$

$$\delta t=2.6\times 10^{-4}\left(\frac{r}{50\text{mbox}\text{leavevmode}kpc}\right){\left(\frac{E_{\nu }}{10\text{leavevmode}\text{mbox}MeV}\right)}^{-2}{\left(\frac{m_{\nu }}{0.1\text{leavevmode}\text{mbox}eV}\right)}^2\text{mbox}\text{leavevmode}s$$

$$\frac{d\varphi }{dr}=G\left(\frac{M(r)}{r^2}-\tilde{\epsilon }\frac{d^{2}M(r)}{d{r}^2}\right),$$

$$M(r)=4\pi \underset{0}{\overset{r}{\int }}{r}^{\prime 2}\rho (r^{\prime })d{r}^{\prime },$$

$$\frac{d\Delta \varphi }{dr}=-\epsilon G\frac{d^{2}M(r)}{d{r}^2},$$

$$M(t,r)=4\pi \underset{0}{\overset{r}{\int }}{r}^{{}^{\prime }2}{\rho }_{\nu }(t,r^{\prime })d{r}^{\prime },$$

$$\frac{\partial M(r)}{\partial r}=4\pi r^2{\rho }_{\nu }(t,r)=M_{\nu }f(t,r),$$

$$\frac{{\partial }^{2}M(r)}{\partial r^2}=M_{\nu }\frac{\partial f(t,r)}{\partial r}\sim \frac{M_{\nu }}{H_{\nu }},$$

$$g_t=\frac{\partial \varphi (t,r)}{\partial r}z\sin \theta ,$$

$$g_t=-z\sin \theta \epsilon G{M}_{\nu }\frac{{\partial }^{2}f(t,r)}{\partial r^2}.$$

$$\rho \frac{{\partial }^2{u}_z}{\partial t^2}=E\frac{{\partial }^2{u}_z}{\partial z^2}+\left(\frac{4}{3}\eta +\zeta \right)\frac{\partial }{\partial t}\frac{{\partial }^2{u}_z}{\partial z^2}+\rho g_t,$$

$$\rho g_t=\frac{\rho }{2}z\frac{{\partial }^2}{\partial t^2}\left[2\epsilon \sin \theta \frac{G{M}_{\nu }}{c^2}f\right].$$

$$h\to 2\epsilon \sin \theta \frac{G{M}_{\nu }}{c^2}f.$$

$$H(\omega )=2\epsilon \sin \theta \frac{G{M}_{\nu }}{c^3}{e}^{-i\omega r/c}{e}^{-H_{\nu }^2{\omega }^2/(2c^2)}.$$

$$u_z(t,z)=\frac{1}{2\pi }\underset{-\infty }{\overset{\infty }{\int }}T(z,\omega )H(\omega )e^{i\omega t}d\omega ,$$

$$T(\omega )=\frac{2L}{{\pi }^2}\frac{{\omega }^2}{{\omega }^2-{\omega }_0^2-i\omega /{\tau }_0},$$

$$\xi (t)$$

$$E_{\max }=\frac{1}{4}M{\omega }_0^2{\xi }_{\max }^2\le \frac{k_B{T}^{\ast }}{2},$$

$$\epsilon \le \frac{{\pi }^2(k_B{T}^{\ast }{)}^{1/2}{c}^3}{2^{3/2}G{M}_{\nu }{\omega }_0^2{M}^{1/2}L\sin \theta }{e}^{H_{\nu }^2{\omega }_0^2/(2c^2)}.$$

$$a\sim \frac{d^2\varphi }{d{r}^2}L\sim G\epsilon L\frac{M_{\nu }}{H_{\nu }^3}$$

$$h=\frac{\delta L}{L}\sim \frac{G{M}_{\nu }\epsilon }{2H_{\nu }{c}^2}\sim \frac{\epsilon }{4}\frac{r_g}{H_{\nu }},$$

$$h\simeq 5\times 10^{-6}\left(\frac{r_g}{0.02\times 3\text{mbox}\text{leavevmode}km}\right){\left(\frac{\Delta t}{10^{-2}\text{mbox}\text{leavevmode}s}\right)}^{-1}\epsilon$$

$$\Delta t=\max \left\{H_{\nu },\frac{r}{2}{\left(\frac{m_{\nu }{c}^2}{E_{\nu }}\right)}^2\right\}.$$

$$\nu \sim \frac{c}{H_{\nu }}\sim \frac{1}{\Delta t}=10^2{\left(\frac{\Delta t}{10^{-2}\text{mbox}\text{leavevmode}s}\right)}^{-1}\text{mbox}\text{leavevmode}Hz$$

$$a_{\mathrm{N}}\sim 2\pi G\frac{M_{\nu }}{4\pi r^2}\frac{L}{H_{\nu }}$$

$$h$$

Full text

I INTRODUCTION

On February 23, 1987, the explosion of a core-collapse supernova (SN) was observed in the Large Magellanic Cloud at a distance $r\simeq 52$ kpc from the Earth (for a review and detailed discussion, see Bah93 ). A statistically significant neutrino signal from the explosion was recorded in the Mont Blanc LSD detector (Italy, USSR) Dadetal87 , and a second neutrino signal was recorded by the Kamiokande II (Japan), IMB (USA), and Baksan Neutrino Observatory (USSR) detectors 4.7 h later. Such a double signal on a time scale of several hours is puzzling, since the neutrino emission lasts only a few seconds in a single collapse. A two-stage collapse may in principle explain this puzzle SteTre87 ; Hiletal87 ; Ruj87 ; Beretal88 ; DadZatRya89 ; ImsNad88 ; ImsRya04 ; BisMoiArd18 . Within the improved rotational mechanism described in ImsRya04 , instability develops and two neutron stars revolving in an orbit are formed due to the presence of a large angular momentum in the core of the original star as it collapses. Losing the energy of orbital motion through gravitational radiation, the binary components approach each other for 4.7 h. Matter from the less massive neutron star is transferred through the Roche point to the more massive neutron star. As a result, the smaller (in mass) neutron star explodes, while the larger one collapses into a black hole with the emission of a second neutrino signal. The presence or absence of recording of neutrino signals from different stages of collapse by different groups of detectors can be explained by a difference between the energy spectra of the neutrinos generated at different evolutionary stages of the collapsar and the detector characteristics Beretal88 (for a review of various possibilities, see also Bah93 , Vis15 ).

Besides neutrinos, gravitational waves can also be emitted during SN explosions. Within general relativity, non-spherical explosion is a necessary condition for the emission of a gravitational wave. A possibility of non-sphericity during an explosion leading to the emission of gravitational waves was pointed out by L.M. Ozernoi in 1964. The change in the quadrupole moment can occur if the collapsing core is pear-shaped. In this paper we will discuss a possibility of recording a gravitational signal from SN 1987A while considering the gravitational perturbation in terms of the modified gravity theory rather than general relativity. First, let us briefly outline the history of the question.

Several gravitational-wave signals from the mergers of black holes and neutron stars in binary systems have been reliably recorded with the LIGO/Virgo laser interferometers since 2016 LIGOVirgo17-1 . However, the attempts to record gravitational waves were made previously. After the theoretical works by H. Bondi and J. Weber, who developed the method of recording gravitational waves by solid-state detectors, Weber constructed such detectors in the shape of cylinders whose oscillations were picked up with piezoelectric sensors. A solid-state detector can record a gravitational wave if the latter contains Fourier components close to the resonance frequencies of the cylinder. In 1969 Weber reported the recording of signals that could be gravitational waves, but this result was not confirmed in independent experiments. Nevertheless, the designs of solid-state detectors continued to improve, while their sensitivity increased.

Two solid-state gravitational-wave detectors operated on February 23, 1987. A distinct signal was recorded by the Geograv detector in Rome Amaetal87 , Agletal91 . The probability of an accidental coincidence was estimated in Amaetal87 to be 3% and in a more detailed analysis Agletal91 to be 0.001%. This signal preceded the cluster of neutrino pulses in the LSD detector by $1.4\pm 0.5$ s. The recording of such a signal, unless it was a statistical fluctuation, looks puzzling, because a classical gravitational wave could produce this signal only if an energy of $\sim 2.4\times 10^{3}M_{\odot}$ was released into gravitational waves Amaetal87 , while the entire mass of the collapsed and exploded star, a blue supergiant, was $\sim 16M_{\odot}$.

Therefore it is impossible to explain the signal in the Geograv detector in terms of general relativity. It is thus interesting to consider a possibility of explaining the signal in terms of modifications of general relativity. Some steps have already been taken in this direction. For example, the effect of scalar gravitational waves on the Geograv detector in the field theory of gravitation was investigated in Bar97 .

In this paper we will consider the effect of a neutrino envelope passing through the Earth on the Geograv detector. We carry out our study in the framework of modified gravity, in particular, as a test model we assume beyond Horndeski theory Crisostomi:2016czh ; Achour:2016rkg . In this theory the gravitational potential contains an additional term proportional to the local matter density gradient111Note that such theories satisfy the local gravitational observations (for example, in the Solar system) thanks to the Vainshtein mechanism Vainshtein (for a review, see Babichev:2013usa )..

Thus, the Geograv detector could be affected by the density gradient of the neutrino envelope passing through it. As we will discuss in more detail below, the signal in the Geograv detector could be produced only by the short (with a duration of less than $<10^{-2}$ s) neutrino burst emitted at the initial stage of collapse, while the main signal in the neutrino detectors should occur about $\sim 1$ s later, which gives a natural explanation for the observed earlier signal in Geograv.

On the other hand, if the signal in the Geograv detector on February 23, 1987, was a random fluctuation, then it can serve to obtain upper bounds on the parameters of the gravity theory. Such a bound is possible even in the absence of a signal at the detector noise level from the condition that the signal does not exceed the noise level.

II THE SOURCE OF THE SIGNAL

During the gravitational collapse of a stellar core a strong neutrino flux comes at two stages: at the stage of initial stellar core collapse before the onset of neutrino opacity, when bulk neutrino emission occurs, and at the later neutrino-opaque stage with surface neutrino emission (for core-collapse supernovae, see ZasPos06 , Bah93 ). Initially, the stellar core collapses to a density $\rho_{s}\simeq 2.8\times 10^{14}$ g cm*-3* during the time $\Delta t\sim 0.001-0.01$ s with energy release $M_{\nu}\sim 0.01M_{\odot}$ in the form of neutrinos. The second stage lasts much longer, of the order of several seconds, with the release of energy $M_{\nu}\sim 0.1M_{\odot}$ in the form of neutrinos. The neutrinosphere inevitably becomes non-spherical due to the presence of a magnetic field and presupernova core rotation. As a result, the duration of the initial and subsequent stages of neutrino emission in different directions is different. Therefore, when the signal from the SN is recorded, there is an additional factor which depends on the angle and that we, however, disregard here.

As we show below, the typical time scale of the neutrino pulse plays a crucial role in the possibility of gravitational signal detection. For this reason, the second stage was unobservable for the Geograv detector. Conversely, a low neutrino flux at the first stage makes it unobservable for neutrino telescopes. Thus, Geograv and LSD could record the neutrino signal from the first and second stages, respectively. This gives a natural explanation for the fact that Geograv recorded the signal 1.4 s earlier than did the neutrino detector.

Neutrinos are emitted in the form of a spherical envelope with a characteristic thickness $H_{\nu}\sim c\Delta t$, see Fig. 1. The envelope density in the laboratory frame (in the detector frame) reads

[TABLE]

where $f(x)$ is the normalized envelope density profile at fixed $t$ (integration is along the radial direction),

[TABLE]

and $M_{\nu}c^{2}$ is the energy of the envelope in the laboratory frame. For simplicity we will assume that the function $f(x)$ has the form of a Gaussian with the center at an envelope radius $x=r(t)$ and a characteristic width $H_{\nu}$:

[TABLE]

The real distribution can have a more complex form with different growth and decay time scales.

Since the neutrinos have a small, but nonzero rest mass, these particles move with a speed slightly lower than the speed of light. As was first pointed out by Zatsepin in Zat68 , this allows to put an upper bound on the neutrino mass from the observations of SN explosions. The neutrino signal delay compared to the speed of light is

[TABLE]

Numerically, the quantity

[TABLE]

is equal in order of magnitude to the neutrino envelope broadening if we set the typical width of the energy spectrum equal to $\delta E_{\nu}\sim E_{\nu}$. Up-to-date data on the neutrino masses are given in GorRub16 . The direct experimental constraint on the electron neutrino mass from the $\beta$-decay of tritium is $m_{\nu}\leq 2$ eV (for Majorana neutrinos $m_{\nu}\leq 0.3$ eV). A constraint of the same order of magnitude $m_{\nu}\leq 0.2-1$ eV is obtained from the cosmological data on the cosmic microwave background and the formation of large-scale structures. The lower bound on the electron neutrino mass follows from the constraints on the difference of the squares of the masses of mass states, which, in turn, follow from the observations of neutrino oscillations. In the case of a direct hierarchy of masses without degeneracy, the electron neutrino mass is $m_{\nu}\geq 0.009$ eV, but may also be $m_{\nu}\geq 0.05$ eV. At a mass $m_{\nu}\leq 0.1$ eV the neutrino envelope at a distance $r\sim 50$ kpc will not spread out, and its width depends only on the emission process. This may not the case for neutrinos from other galaxies. The spreading can become significant starting from some distance, which affects the detection efficiency. The detectors may not record the neutrino envelopes from SN explosions in distant galaxies due to this effect.

III AN ADDITIONAL CONTRIBUTION TO THE GRAVITATIONAL POTENTIAL

In this paper we make use a prediction of beyond Horndeski’ theory for Geograv detector. The beyond Horndeski’ theory is a general scalar-–tensor theory containing one additional scalar degree of freedom (in addition to graviton). General relativity is a special case of this theory. In the spherically symmetric case and in a quasi-static approximation the gradient of the gravitational potential ’in beyond Horndeski theory reads KobWatYam15 :

[TABLE]

where

[TABLE]

and $\tilde{\varepsilon}$ is a dimensionless parameter expressed via the fundamental parameters of the theory and the cosmological solution222Note that in the Horndeski theory Hor74 the second term in (6) is absent.. There are, in particular, constraints on $\tilde{\varepsilon}\leq 1$ based on the stability of neutron stars Saietal15 . The derivative $d^{2}M(r)/dr^{2}$ has a contribution from the density gradient. Note that no similar expression has been found for the case of a fast moving medium. Therefore, strictly speaking, we cannot apply (6) to the neutrino envelope.

Instead we will use a phenomenological approach and interpret (6) as the result of gravity modification that is also valid for moving matter. More specifically, we write the additional contribution to the derivative of the gravitational potential in the detector rest frame as

[TABLE]

with the parameter $\varepsilon$ being arbitrary and, generally speaking, unrelated to any specific modified gravity model. As we will see below, the signal in the Geograv detector can be explained by the presence of the term (8). On the other hand, we will obtain an independent constraint for the parameter $\varepsilon$.

For further purposes we will need an estimate of $d^{2}M(r)/dr^{2}$ in (8). For the neutrino envelope considered above we have

[TABLE]

[TABLE]

[TABLE]

since the function $f(t,r)$ changes on the length scale $H_{\nu}$.

IV THE SIGNAL IN THE DETECTOR

Consider a solid-state detector of a cylindrical shape of length $L$ and whose axis and the the plane of the incoming neutrino envelope make an angle $\theta$. The basics of operation of such detectors is described in detail in AstKvaTeo82 . A gravitational wave or other action on the cylinder causes its oscillations that are recorded with the piezoelectric elements fixed to the detector. We will place the origin of coordinates at the cylinder center; the “z” axis is directed along the cylinder. The tidal acceleration acting on a cylinder mass element with coordinate $z$ reads

[TABLE]

where $\theta$ was $30^{\circ}$ at the time of observation of the first neutrino signal from SN 1987A. The extra contribution to the gravitational potential (8) is given by

[TABLE]

Note that the ordinary Newtonian force contains $1/r^{2}$ factor, where $r=52$ kpc, and, therefore, its contribution to the signal is negligible with respect to the noise level.

The equation for longitudinal cylinder oscillations with a tidal force reads LLUprug

[TABLE]

where $u_{z}$ is a vector component, $\rho$ is the density, $E$ is Young’’s modulus, $\eta$ and $\zeta$ are the viscosity coefficients of the cylinder material defining the damping time of its oscillations. Note that for $f(t,r)=f(r-ct)$ we have $\partial^{2}f/\partial r^{2}=(1/c^{2})\partial^{2}f/\partial t^{2}$. Therefore, given the above relations, the term $\rho g_{t}$ on the right-hand-side of (14) can be written as

[TABLE]

We see that Eq. (4.5) from AstKvaTeo82 coincides with (14) after the substitution

[TABLE]

Thus, the response in the detector can be taken into account by the same method AstKvaTeo82 that was applied in the case of gravitational waves in general relativity, but with the substitution of the combination of quantities (16) for the wave amplitude. The detector response is expressed via the signal spectral density close to resonances. Since the signal frequency is lower than the first resonance frequency of the Geograv detector in the case under consideration, the region near the first resonance frequency $\omega_{0}$ plays a major role. Close to the resonance a cylindrical detector is equivalent in response to a system of two weights connected by a spring. When choosing (3), the Fourier transform of the quantity (16) is

[TABLE]

The cylinder oscillations resulting from the envelope passage are found by the convolution

[TABLE]

where $T(z,\omega)$ is the response function obtained by the Fourier transform of Eq. (14).

The response function of the equivalent spring detector near the first resonance can be approximately given as AstKvaTeo82

[TABLE]

where $\tau_{0}=Q/\omega_{0}$ is the oscillation damping time and $Q$ is the detector $Q$-factor. We obtain the motion for a weight of the equivalent spring detector at $t>r/c$ by calculating the integral (18):

[TABLE]

The energy released in the detector does not exceed the spike in the effective temperature recorded on February 23, 1987:

[TABLE]

where $M$ is the cylinder mass (the mass of each of the two weights of the equivalent detector is equal to $M/2$) and $T^{*}\sim 135$ K is the temperature spike (at a noise level of 29 K); the equality in (21) will hold if the spike in temperature is explained by the passage of the neutrino envelope. Hence we obtain

[TABLE]

This constraint is shown in Fig. 2.

The envelope width $H_{\nu}=c\Delta t$ dependent on the neutrino signal duration $\Delta t$ plays a crucial role in exponential factor in (22). The duration of the first neutrino emission stage coincides in order of magnitude with the free-fall time $t_{\rm ff}\sim(G\rho_{s})^{-1/2}\sim 4\times 10^{-3}$ s at $\rho_{s}\sim 10^{12}$ g cm*-3*. Numerical SN explosion simulations Nad78 show a neutrino burst on a characteristic time scale $\Delta t\sim 10^{-3}$ s due to the sharp growth in plasma temperature during the collapse. This primary burst continues until the onset of neutrino opacity, the optical depth for neutrinos $\tau_{\nu}=1$. It can be seen from Fig. 2 that the signal in the Geograv detector can be generated for $\Delta t\ll 10^{-2}$ s. The most conservative constraint $\tilde{\varepsilon}\leq 1$ Saietal15 based on the stability of neutron stars at $\varepsilon\sim\tilde{\varepsilon}$ would rule out a duration $\Delta t\geq 10^{-2}$ s.

The second neutrino burst (at the first stage of two-stage collapse) after the onset of neutrino opacity lasts much longer, of the order of several seconds, and, therefore, its influence on the detector is exponentially suppressed. The neutrino luminosity in the second burst probably decreases exponentially and not as a Gaussian, but only the characteristic decline time is important for a qualitative estimate. Meanwhile, during the second burst an order of magnitude greater energy, $6.3\times 10^{53}$ erg$=0.35M_{\odot}c^{2}$ Beretal88 , is released into neutrinos and, therefore, the signals in the neutrino detectors are due to to the second burst. This explains the delay of 1.4 s between the signal recordings by the Geograv detector and the LSD neutrino detector. However, it should be noted that the time in the neutrino detector is measured from the first neutrino event and, therefore, this delay could be slightly different in view of the statistical fluctuations.

During two-stage collapse one expects additional neutrino signals 4.7 h later that were recorded by the Kamiokande II, IMB, and Baksan detectors. In order to estimate the signal in the gravitational-wave detector from the second stage one needs to know the duration of the neutrino emission that is generated as the neutron star fragments collide Beretal88 or as the neutron star collapses into a black hole. First of all, it can be seen from the observational data (all neutrino events are listed in Bah93 ) that the detectors recorded the neutrino signal for several seconds: up to 12.4 and 5.6 s in the case of the Kamiokande II and IMB detectors, respectively. Thus, the main neutrino flux constituted an extended envelope the signal from which in the Geograv detector is exponentially suppressed, as seen from Fig. 2. The following question remains: could more rapid variations be present in the overall extended signal, as is the case during an ordinary core-collapse SN explosion, when the first short neutrino burst is generated during the primary stellar core collapse? If the second stage is associated with the merger of two neutron stars or a black hole and a neutron star, then the data on short cosmic gamma-ray bursts, which are now known to be generated in such mergers, may turn out to be useful (though no gamma-ray burst was observed during the explosion of SN 1987A). The variability of some short gamma-ray bursts occurs on a time scale of $\sim 10^{-3}$ s. If the neutrino envelopes could have the same time scale, then they could be recorded by the Geograv detector. In this case, the question about the absence of recording of a second gravitational signal remains unresolved. At the same time, other calculations Becetal10 suggest that the typical variability time in the fireball of a gamma-ray burst $\Delta t$ is closer to $\sim 10^{-2}$ s, while many short bursts last up to 3 s, and the signal in the detector will then be exponentially suppressed, Thus, the absence of a second Geograv signal concurrent with Kamiokande II and IMB can be explained either by the absence of rapid variability, i.e., a large $\Delta t\geq 10^{-2}$ s, or by minor energy release into neutrinos in episodes of rapid variability.

Note that the distance to the source does not enter explicitly into the condition (22). Therefore, one would think that the detectors should record the signals from SN explosions in other galaxies very frequently. However, (22) can depend on the distance via $\Delta t$, which increases with distance due to a nonzero neutrino mass, according to Eq. (5). This effect can explain the absence of signals from distant galaxies. At even greater distances, $\sim 1$ Gpc, the influence of a cosmological redshift should manifest itself.

V ESTIMATING THE SIGNAL IN A GRAVITATIONAL INTERFEROMETER

Above we calculated the signal in a resonant solid-state detector. Let us now estimate the signal that could be recorded by gravitational interferometer with free mirrors if a SN exploded during their operation. This question can be of importance for LIGO/Virgo observations and future gravitational detectors, since sooner or later another corecollapse SN explosion will occur in our Galaxy or its surroundings. According to various estimates, such explosions occur once in 20–100 years. Let us consider an interferometric detector in the form of two free masses spaced a distance $L$ apart. The neutrino envelope crosses the detector during a characteristic time $\Delta t\sim H_{\nu}/c$. A tidal acceleration

[TABLE]

acts during this time. Since the envelope moves relativistically, a contribution of not only the energy, but also the momentum of the envelope will be present in the gravitational force. However, this contribution is of the same order of magnitude as (23) and we disregard it in our order-of-magnitude estimate. The change in interferometer arm in the envelope passage time is estimated as $\delta L\sim a(\Delta t)^{2}/2$ and the relative change in the size is

[TABLE]

where the gravitational radius $r_{g}=2GM_{\nu}/c^{2}$. Numerically, we have

[TABLE]

This quantity at $\varepsilon>10^{-15}$ can exceed the effect of the gravitational wave.

If there is a broadening of the neutrino envelope due to a nonzero neutrino mass, according to Eq. (4), then in the above calculation we should substitute the following quantity as $c\Delta t$:

[TABLE]

In our calculation we also assumed the envelope width to be $c\Delta t\geq L$. For future space interferometers like LISA this condition may not be satisfied. In the case of $c\Delta t\ll L$, the tidal acceleration (23) will be gained on a scale of the order of $\sim c\Delta t$ and, therefore, the additional small factor $c\Delta t/L$ will enter into (25).

Each interferometric detector has its sensitivity curve; it can receive signals only in a limited frequency range. Therefore, for an excessively long pulse (low characteristic frequency) the signal will be outside the detection range. The characteristic signal frequency

[TABLE]

at $\Delta t$ specified in the normalization falls into the most sensitive range of the LIGO/Virgo detectors. Therefore, during the explosion of a supernova like SN 1987A it would be possible to obtain a strong constraint on $\varepsilon$.

In 2017 the LIGO/Virgo detectors recorded the gravitational-wave burst GW170817 [12]. The short gamma-ray burst GRB 170817A was recorded from its localization region by the Fermi-GBM telescope $1.74\pm 0.05$ s later. This event is most likely associated with the merger of two neutron stars or a neutron star and a black hole with masses of $1.17-1.60M_{\odot}$. A powerful neutrino signal must be generated during such a merger. According to optical observations, the source is located in the galaxy NGC 4993 at a distance of $40\pm 8$ Mpc from the Earth LIGOVirgo17-2 , i.e., farther than SN 1987A approximately by a factor of $\sim 10^{3}$. The absence of anomalies in the LIGO/Virgo observations can be explained by one of the following factors or their combination: a small parameter $\varepsilon\ll 10^{-15}$, a large $\Delta t\gg 10^{-2}$ s or the influence of the neutrino mass and the spreading of the neutrino envelope to large $\Delta t$ as the neutrinos traverse a distance of $40\pm 8$ Mpc (according to Eq. (5)) at $m_{\nu}\geq 0.01$ eV.

Note also that the Newtonian part of the tidal acceleration produced by the passing envelope is

[TABLE]

and produces a single pulse with an amplitude

[TABLE]

whose value is far too small to be detected.

VI CONCLUSIONS

In this paper we showed that the signal probably recorded in the Geograv gravitational-wave detector during the explosion of SN 1987A Amaetal87 could have been produced by a gravitational field perturbation during the passage of a powerful neutrino flux through the detector. The appearance of such a strong signal is possible in modern scalar-–tensor gravity theories, like extensions of Horndeski’ theory, in which the gravitational potential depends not only on the objects mass and distance, but also on the local matter density gradient. This model successfully explains the time delay of 1.4 s between the Geograv and LSD signals. The signal in Geograv is attributable to the neutrinos emitted at the initial stage before the onset of neutrino opacity, while the signal in LSD was produced by the main neutrino flux.

The signal from the neutrino pulse has different polarization compared to a gravitational wave in general relativity, and it produces oscillations in a solid-state resonant detector. Yet another difference is that the additional effect due to the neutrino pulse is not of the form of oscillations, contrary to the case when gravitational waves in general relativity or scalar gravitational waves Bar97 are emitted by an oscillating body.

We also showed that LIGO/Virgo gravitational interferometers would see such signals when the neutrino pulses exploded supernovae were passing through detectors.

It should be stressed that exact solutions for the gravitational potentials for a fast moving medium in beyond Horndeski theory have not been found yet. In this paper we use a phenomenological approach by introducing an unknown parameter $\varepsilon$ that relates the potential gradient to the envelope density gradient in the detector rest frame, by extending the static solution KobWatYam15 ; Saietal15 to the case of moving neutrinos. With this assumptions we demonstrated a possibility to explain the signal in the Geograv detector and we found exponential dependence of the effect on the neutrino signal duration.

It would be of interest to also consider the collapse in modified gravity when simulating the gravitational pre-supernova core collapse, and to calculate the shape of the neutrino signal. This would allow to study self-consistently the problem of the detection of gravitational signals in such theories.

The question of whether an additional contribution in gravitational potential is present when a pulse of gravitational waves passes is also of interest333We are grateful to the referee of this paper who pointed to this possibility.. Such an pulse does not spread out and, therefore, the signal can arrive from great distances. In beyond Horndeski’ theory the matter energy-–momentum tensor makes an additional contribution to the potential gradient. It would be interesting to see whether a gravitational wave, i.e., the energy-momentum pseudo-tensor of the gravitational field, can make a similar contribution.

Acknowledgments

This research was supported by the CNRS/RFBR Cooperation program for 2018-–2020 n. 1985 (France), Russian Foundation for Basic Research (project n. 18-52-15001 CNRS) (Russia) Modified gravity and black holes: consistent models and experimental signatures,” and the research program “Programme national de cosmologie et galaxies” of the CNRS/INSU.