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Article
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Analyzing the candidacy of gravitino as constituent particles of
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dark matter
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Shreeji Kumawat 1
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1 Physics student at St. Xavier’s School, India
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shreejikumawat@gmail.com
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Abstract: Dark matter makes up about 85% of our universe, a fact supported by both theoretical
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and observational evidences, yet we are still searching for theories to accurately describe its com-
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position. Unlike regular baryonic matter, dark matter refuses to interact with electromagnetic
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radiations, making it more evasive to detect. Several theories have been proposed to explain it’s
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origin, most popular ones suggesting WIMPs, Light bosons and neutrinos as constituting parti-
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cles, although none of them have been successfully proven. In this paper we focus on establish-
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ing gravitino, a type of hypothetical particle in quantum theory of gravity, as a viable candidate
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for constituting dark matter. We begin by highlighting the basic assumptions and principles re-
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lied upon, such as the model & Effective Field Theory (EFT), and then proceeding to
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gradually point towards the proposed relation between dark matter and its decay widths to
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standard model particles, particularly photons. In a key part of this paper, attempts have been
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made to indicate that the proposed energy possessed by photons after gravitino decay in our
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model actually coincides well with the real time energies of gamma ray bursts GRBs) observed
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just after the detection of gravitational waves. In depth analysis of data made publicly available
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by LIGO and Fermi GRB observatory is done in order to arrive at this result. In conclusion, we
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calculated and demonstrated a close relationship between the observed and theoretical value of
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GRB energies, thus implying that gravitino could well be considered as a constituent for dark
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matter.
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Keywords: (Gravitons) (Dark Matter) (Quantum) (Gravity) (Gamma Ray Bursts) (Gravitational
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Waves)
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1. Introduction
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Dark matter is a hypothetical form of matter which makes up to 85% of all matter in
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our universe, according to the standard LambdaCDM model [1]. It does not absorb, emit
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or reflect any kind of radiation and does not even loose kinetic energy to heat except in
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one inefficient process of gravitational ejection. This makes it difficult to detect with tra-
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ditional techniques and instruments, forcing us to rely on advanced theoretical ap-
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proaches for its detection, majority of whom rely on measuring its gravitational effect on
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nearby objects. This is synonymous to locating a black hole by observing the behavior of
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baryonic matter around it. Even though its detection has not yet been achieved, we have
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good reasons to believe it exists. Our currently accepted theory of gravity gives almost
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nonsensical answers to various astrophysical equations unless an invisible yet large form
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of matter if accounted for.
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As such, the need for DM was recognized and it was defined to be ‘matter that is not
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visible but whose energy density scales with the inverse cube of a scale factor’. This im-
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plies the simple relation:
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where is energy density and α is a scale factor [2]. Having sufficiently introduced
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DM, we now proceed onto listing the various reasons and methods that prove its undeni-
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able existence and imply its crucial role in the formation and evolution of our universe.
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1.1. Theoretical Evidences
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The first hint towards existence of DM was provided in 1933 when Fritz Zwicky
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analyzed the Coma Cluster of galaxies with Virial Theorem [3]. The theorem can
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be stated as a relation between a system’s total kinetic energy and its virial:
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Modifying the theorem to include gravitationally interacting masses as shown
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below
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allowed Zwicky to estimate the mass of the cluster in relation to motion of its
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galaxies near the edge. But when he calculated the mass again using luminosity
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and the number of galaxies, he found the two values differed by 400 times, show-
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ing that some large amount of matter that wasn’t contributing to luminosity was
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present and affecting the overall cluster motion.
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Rotation curves for the andromeda galaxy were observed by W. Babcock in 1939
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soon after, reporting mass-luminosity ratio of 50:1 and a peculiar rapid rotation
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motion in outskirts [4]. This result was later used by Jan Oort to hypothesis ex-
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istence of DM that caused invisible attraction. Vera Rubin and Kent Ford also
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measured the velocity curves of spiral galaxies with spectrographs to a great pre-
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cision, proposing that most galaxies contain DM about 6 times greater than reg-
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ular matter [5].
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A majority of proofs that followed later focused on galaxical rotation curves, in-
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sisting that from Kepler’s second law it is expected that rotation velocities de-
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crease with distance from center. But observations show a flat curve (Fig. 1)
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meaning that mass distribution must not be regular. If Kepler’s laws are correct
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then a non-luminous matter must be present in outskirts of such galaxies. Fried-
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mann solutions for structural formation of our universe also show that if there
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were only observable matter, galaxies and clusters would not have evolved due
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to slow density perturbations [6]. All these rationales shown above are used to
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accept DM existence based on theoretical analysis of known laws.
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Figure 1. This shows the galaxy rotation curve as expected vs. its observed value.
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1.2. Observational Evidences
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Our conceptual reasons for DM presence are also supported well by numerous
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strong observational verifications. Radio astronomy showed early on that half-
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dozen galaxies spun too fast in outer regions, meaning some sort of gravitational
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attraction was there to keep their stars in orbit [7]. There are several methods to
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measure a galaxy’s mass, using either scatter in radial velocities or X-ray spec-
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trum from hot centers to estimate density and pressure. Gravitational lensing is
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a method where galaxies lying between distant quasars and observer act as a lens
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bending light. The magnification observed gives us an estimate of mass causing
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distortions. All above mentioned methods arrive at a common agreement that
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DM exceeds visible matter in abundance by 5 times [8].
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Cosmic Microwave Background is also by affected by DM in terms of its gravita-
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tional potential and effect on velocity & density of ordinary matter, thus imply-
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ing evolution of matter perturbations with time on CMB. Using sky map aniso-
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tropies, an angular power spectrum can be formed to show densities of baryonic
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and DM [9]. This spectrum fits perfectly with results from LambdaCDM model
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but proves difficult to explain with Newtonian mechanics alone.
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Lastly, in a long list of experimental conformations, we lay emphasis on the Bul-
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let Cluster, which was recently formed as a result of a two-galaxy collision. Its
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apparent center of mass differs vastly from baryonic COM, an effect lucidly ex-
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plained by DM models but almost unaccounted for if one relies solely on modi-
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fied gravity [10].
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2. Quantum Gravity
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The foundation idea of Quantum Gravity is that gravity arises from quanta scaled
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packets of energy called ‘gravitons’ which do not exist in space, but are space themselves
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[11]. These mediate the force of gravitation between particles and are responsible for the
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bend in space-time in presence of mass. They are the carries of hypothesized gravitational
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fields, are massless and electronically neutral, much like the analogous photons that me-
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diate electromagnetic interactions. In the standard model of particles, it is theorized to be
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a spin 2 boson particle because its fields travel at speed of light and are effective at long
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range. If in the near future the vacancy of a massless spin 2 particles is filled, it is expected
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to a graviton [12].
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But the candidate we propose for DM is gravitino, a close partner of graviton in the-
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ory of supergravity. This is a theory that deals with the combination of general relativity
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and supersymmetry in areas where quantum effects of gravity become considerable on
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relativistic scale. Supersymmetry proposes a space time symmetry between two classes of
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particles – bosons and their superpartners – fermions. Considering this, it also implies that
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boson natured gravitons have a gauge fermion superpartner with spin
named
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Gravitino, which mediate supergravity interactions [13]. Supersymmetric theories make
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gravity arise in a natural way and in coming sections we focus on establishing its applica-
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tion as a good component of DM.
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2.1. Effective field theory (EFT)
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This is another theory working under quantum gravity (QG) and it forms our base
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for describing dark matter and gravitino relations. EFT accounts for gravity as a well-
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defined series dependent on higher powers of R, the curvature of space time, as [14].
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This is in contrast with Einstein’s series of gravity which is a linear function of R. We
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consider higher orders in QG based on several parameters, mass of scalar field being
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one with most consequences. Generally, values of ‘m’ are kept heavy so as to avoid the
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effects of relativity breaking under plank length, but in a recent study Jose Cembranos
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found out that making ‘m’ lighter automatically accounts for dark matter. The particle
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introduced would be a gravitino with the ability to be casted as a scalar field that could
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clump just like real matter [15]. This is an especially elegant result as it does not require
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us to introduce a new particle to the standard model and merely points towards a poten-
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tial vacancy.
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In another study conducted, this result was used to construct a model of our universe
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that tracked its evolution through time, assuming gravity to be quantum in nature and
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caused by gravitinos. The model showed hints of accelerated expansion at the exact point
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of time we are in today and matched the rate very closely with present observational evi-
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dence [16]. This new perspective of thinking is advantageous to us, as it starts of with
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general framework of gravity at quantum level rather the restricting itself to cosmological
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contexts from the very start.
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2.2. Gravitational Fields
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The LIGO (Laser Interferometer Gravitational Waves Observatory) has already de-
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tected many gravitational waves which are produced by all matter that accelerates. These
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wave like ripples in space time travel at the speed of light and transport energy as gravi-
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tational radiation. A gravitational field (gravitino field) is model used to explain the im-
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pact of gravity on other objects and is observed in form of gravitational waves [17]. It is
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massless in nature and contains a spectrum of two massive fields of spin 2 and 0 whose
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properties can be derived with help of quantum gravity [18]. They have the ability to ide-
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ally explain DM that is only gravitationally coupled to normal matter while postulating
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its existence to be light in nature. The waves would also carry information about DM and
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be directly related to its abundance.
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2.3. Decay widths
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The intriguing property of such massless gravitational fields is that they decay both
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gravitationally and to standard model particles [19]. In this section, we shall attempt to
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show a concise and brief version of the original derivation wherein the different decay
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widths of these gravitino fields to various particles are calculated.
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We start with general relativity and integrate the fluctuations due to gravitinos to get
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the classical effective action at second order curvature.
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Here R is Ricci scalar. Many complex summations of are included in later terms
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which depend on the cosmological constant, which is the renormalization scale,
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the Lagrangian and M as the total energy scale up to which we can trust the EFT. The
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green’s function poles can be used to identify new degrees of freedom with respect to a
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matrix. Two pairs of complex poles are found for each spin field which are then renormal-
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ized till reduced plank scale. This allows us to obtain the gravitational decay width in
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terms of mass of the field as:
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Here is the reduced plank mass equal to and is the decay
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width due to gravitation. Using this equation (Fig 2.) we can now calculate the decay
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widths of these fields into various standard model particles. The result of such calcula-
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tions is that the fields do not decay to gauge bosons and the charged leptons of the stand-
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ard model, but they do decay to gluons such as photons and neutrinos with low enough
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mass [20]. Decay width to photons is given by:
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The fact that gravitino fields decay to photons is especially emphasized as their ob-
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servability during gamma ray bursts is of astronomical importance to us. The Unruh –
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DeWitt detector implies a decay of gravitons into photons using this pathway [21]:
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or
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The author will use the information provided until now to make the hypotheses that
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follow below and these are subject to human error due to negligence/ignorance.
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Figure 2. The complete and general equation used to derive decay widths of gravitinos to standard model particles.
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3. Approximate equivalence in Gamma Ray Flux
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We illustrated in the above sections that gravity, when quantized as gravitino, could
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account for dark matter and a field of such particles would decay into photons with a
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considerable width. Using this result, it is also possible to calculate the flux of a gamma
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ray produced upon decay of gravitinos. The in-depth calculations of the flux are shown
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in this paper [22] for further reference but maybe be briefly summed up as:
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where is the energy possessed by gamma rays that are produced by decay of dark
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matter particle of mass . The flux of such a monochromatic gamma ray is given by
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line of sight integral over dark matter distribution and the explicit component is:
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where is the decay width and
is the milky way’s dark matter halo
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density profile given by:
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Using and we get local dark matter density of 0.4
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.
The flux calculated now, assuming gravitons to have mass 200 GeV and lifetime
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seconds is approximately in the range of units. This is also shown in
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the graph (Fig 3.).
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Figure 3. This shows the predicted gamma ray flux from gravitino decay [22].
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3.1. Methodology and Principle
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It is known that neutron-neutron star collisions and black hole neutron star collisions
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release huge gravitational waves which are detected by LIGO. The data observed from
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gravitational waves publicly is available on the GW Open Science Center and after deep
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query analysis of it using the fact that the theoretical mass range of neutron stars is gen-
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erally between 1.174 – 2.116 solar masses, we list 5 GW events that have the possibility of
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being emitted from neutron star mergers. The code used by the author during analysis is
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listed in section 3.2 for reference.
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A release of gamma ray burst should happen soon after a collision happens and this
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is detected by observatories on earth usually within a short time interval of GW trigger
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time. The public data on Heasarc NASA by the Fermi Gamma Ray Observatory is ana-
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lyzed for listed 5 GW events to extract GRB trigger, time interval and fluence. Flux is then
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calculated separately for each event using queried data. The time interval between GW
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and GRB triggers is also mentioned for reader reference.
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The author theorizes that since neutron star collisions release a huge number of grav-
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itational waves and thus by implication gravitino fields in our model, some of these
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gravitinos must have decayed into gamma rays. If energy/flux of these gamma rays is
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calculated, suitable comparisons can be made between the computed value and the theo-
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retical value of energy/flux hypothesized to be possessed by gamma rays after decay from
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gravitino fields. Equal values must increase the probability that dark matter is indeed
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made up of gravitinos for future conceptual investigations in this matter. Accordingly,
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highly unequal values must also eliminate our idea to constitute DM with gravitinos.
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3.2. Working
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The python code used by the author to query and analyze GW event catalogue is:
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! pip install -q 'gwpy==2.0.2'
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import gwpy
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print(gwpy.__version__)
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from gwosc.datasets import find_datasets
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from gwosc import datasets
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print("List of available catalogs")
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print(find_datasets(type="catalog"))
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print("")
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List of available catalogs
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['GWTC-1-confident', 'GWTC-1-marginal', 'GWTC-2', 'GWTC-2.1-auxiliary', 'GWTC-2.1-confident',
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'GWTC-2.1-marginal', 'GWTC-3-confident', 'GWTC-3-marginal', 'Initial_LIGO_Virgo', 'O1_O2-Pre-
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liminary', 'O3_Discovery_Papers', 'O3_IMBH_marginal']
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gwtc1 = datasets.find_datasets(type='events', catalog='all')
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print('All events:', events)
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print("")
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print(datasets.find_datasets(type='events', catalog='any', detector="any", segment=(second-
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ary_mass)))
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from gwosc.datasets import event_secondarymass
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result = event_secondarymass(1.174, 2.116)
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print ('list of events:', result)
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list of events: GW170817, GW190425, GW190917_114630, GW200105_162426,
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GW200115_042309
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Thus, GW170817, GW190425, GW190917 ,GW200105 and GW200115 are the events
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which include a possible neutron star collision. The data for bursts observed on these
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dates is extracted from HEARSEC and the fluence and time period is noted as shown in
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following table.
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Table 1. This shows the data for gamma ray bursts observed near given gravitational wave events.
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GW event name
GRB event name
Trigger time gap
GRB fluence (erg/cm^2)
GRB time interval(sec)
GRB flux(erg/cm^2sec)
GW170817
GRB170817529
00:00:02
2.85e-07
2.6
1.10e-08
GW190425
GRB19042508
06:10:22
4.27e-07
7.6
5.62e-08
GW190917
GRB190916590
02:23:44
2.95e-06
37.37
7.89e-07
GW200105
GRB200105914
05:31:02
3.37e-06
16.89
1.19e-07
GW200115
GRB200114153
00:42:26
5.10e-06
29.6
1.72e-07
Flux is calculated manually using the relation:
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The plot of observed flux as function of energy for each event is shown along with the
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straight line of predicted flux (Fig 4.).
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Figure 4. Flux values of GRBs from neutron star mergers plotted along with predicted value of flux
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from gamma ray after gravitino decay
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3.3. Results
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We infer from the plot graph that the flux of most events aligns approximately with the
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theorized value. Minor time differences are noticed between GW triggers and GRB trig-
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gers, implying the good possibility of both events originating from a common source.
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The author attributes minor gaps between predicted and actual flux values to plausible
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technological, source, numerical and human errors.
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4. Conclusion
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In this paper we have shown that the existence of dark matter is supported by ample
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number of evidences and that its absence indeed causes serious issues to remain unex-
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plained by our exclusive models. Its origin and composition are much contemplated
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over by defining a quantum theory of gravity and the various implications it would
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have. Gravitinos, the mediating particle of gravitational forces in supergravity, are
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thought to constitute dark matter and in such a scenario, it is predicted that the gravita-
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tional field would decay due to standard model particles. Its decay to photons would
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eventually give rise to gamma rays with predicted values of flux. The author endeavors
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to analyze gravitational wave and gamma ray bursts data from neutron star collisions in
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order to obtain experimental flux observed. Both these values are clearly shown to be
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approximately equal, implying that according to our model gravitinos must constitute
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dark matter and have predictably decayed into gamma rays. Although there is a long
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way to go before it can be said for certain what dark matter is composed of, this conclu-
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sion helps us evaluate that the scope of quantum gravity. The result proceeds to further
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cement our belief that gravitinos have a good candidacy to be considered as comprising
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particles for dark matter.
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Funding: This research received no external funding or grants.
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Data Availability Statement: The data used in this research for gravitational wave events has been
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taken from https://www.gw-openscience.org/ and the coding interface used is https://colab.re-
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search.google.com/github/gw-odw/odw-2022/blob/main/Tutorials/Day_1/Tuto%201.1%20Discov-
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ering%20Open%20Data.ipynb. The gamma ray burst data has been obtained from https://heas-
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arc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl.
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