Content uploaded by Shreeji Kumawat

Author content

All content in this area was uploaded by Shreeji Kumawat on Oct 17, 2022

Content may be subject to copyright.

Article

1

Analyzing the candidacy of gravitino as constituent particles of

2

dark matter

3

Shreeji Kumawat 1

4

1 Physics student at St. Xavier’s School, India

5

shreejikumawat@gmail.com

6

Abstract: Dark matter makes up about 85% of our universe, a fact supported by both theoretical

7

and observational evidences, yet we are still searching for theories to accurately describe its com-

8

position. Unlike regular baryonic matter, dark matter refuses to interact with electromagnetic

9

radiations, making it more evasive to detect. Several theories have been proposed to explain it’s

10

origin, most popular ones suggesting WIMPs, Light bosons and neutrinos as constituting parti-

11

cles, although none of them have been successfully proven. In this paper we focus on establish-

12

ing gravitino, a type of hypothetical particle in quantum theory of gravity, as a viable candidate

13

for constituting dark matter. We begin by highlighting the basic assumptions and principles re-

14

lied upon, such as the model & Effective Field Theory (EFT), and then proceeding to

15

gradually point towards the proposed relation between dark matter and its decay widths to

16

standard model particles, particularly photons. In a key part of this paper, attempts have been

17

made to indicate that the proposed energy possessed by photons after gravitino decay in our

18

model actually coincides well with the real time energies of gamma ray bursts GRBs) observed

19

just after the detection of gravitational waves. In depth analysis of data made publicly available

20

by LIGO and Fermi GRB observatory is done in order to arrive at this result. In conclusion, we

21

calculated and demonstrated a close relationship between the observed and theoretical value of

22

GRB energies, thus implying that gravitino could well be considered as a constituent for dark

23

matter.

24

25

Keywords: (Gravitons) (Dark Matter) (Quantum) (Gravity) (Gamma Ray Bursts) (Gravitational

26

Waves)

27

28

1. Introduction

29

Dark matter is a hypothetical form of matter which makes up to 85% of all matter in

30

our universe, according to the standard LambdaCDM model [1]. It does not absorb, emit

31

or reflect any kind of radiation and does not even loose kinetic energy to heat except in

32

one inefficient process of gravitational ejection. This makes it difficult to detect with tra-

33

ditional techniques and instruments, forcing us to rely on advanced theoretical ap-

34

proaches for its detection, majority of whom rely on measuring its gravitational effect on

35

nearby objects. This is synonymous to locating a black hole by observing the behavior of

36

baryonic matter around it. Even though its detection has not yet been achieved, we have

37

good reasons to believe it exists. Our currently accepted theory of gravity gives almost

38

nonsensical answers to various astrophysical equations unless an invisible yet large form

39

of matter if accounted for.

40

41

As such, the need for DM was recognized and it was defined to be ‘matter that is not

42

visible but whose energy density scales with the inverse cube of a scale factor’. This im-

43

plies the simple relation:

44

45

2 of 11

where is energy density and α is a scale factor [2]. Having sufficiently introduced

46

DM, we now proceed onto listing the various reasons and methods that prove its undeni-

47

able existence and imply its crucial role in the formation and evolution of our universe.

48

49

50

1.1. Theoretical Evidences

51

52

The first hint towards existence of DM was provided in 1933 when Fritz Zwicky

53

analyzed the Coma Cluster of galaxies with Virial Theorem [3]. The theorem can

54

be stated as a relation between a system’s total kinetic energy and its virial:

55

56

57

58

Modifying the theorem to include gravitationally interacting masses as shown

59

below

60

61

62

63

allowed Zwicky to estimate the mass of the cluster in relation to motion of its

64

galaxies near the edge. But when he calculated the mass again using luminosity

65

and the number of galaxies, he found the two values differed by 400 times, show-

66

ing that some large amount of matter that wasn’t contributing to luminosity was

67

present and affecting the overall cluster motion.

68

69

Rotation curves for the andromeda galaxy were observed by W. Babcock in 1939

70

soon after, reporting mass-luminosity ratio of 50:1 and a peculiar rapid rotation

71

motion in outskirts [4]. This result was later used by Jan Oort to hypothesis ex-

72

istence of DM that caused invisible attraction. Vera Rubin and Kent Ford also

73

measured the velocity curves of spiral galaxies with spectrographs to a great pre-

74

cision, proposing that most galaxies contain DM about 6 times greater than reg-

75

ular matter [5].

76

77

A majority of proofs that followed later focused on galaxical rotation curves, in-

78

sisting that from Kepler’s second law it is expected that rotation velocities de-

79

crease with distance from center. But observations show a flat curve (Fig. 1)

80

meaning that mass distribution must not be regular. If Kepler’s laws are correct

81

then a non-luminous matter must be present in outskirts of such galaxies. Fried-

82

mann solutions for structural formation of our universe also show that if there

83

were only observable matter, galaxies and clusters would not have evolved due

84

to slow density perturbations [6]. All these rationales shown above are used to

85

accept DM existence based on theoretical analysis of known laws.

86

87

3 of 11

88

Figure 1. This shows the galaxy rotation curve as expected vs. its observed value.

89

90

91

1.2. Observational Evidences

92

93

Our conceptual reasons for DM presence are also supported well by numerous

94

strong observational verifications. Radio astronomy showed early on that half-

95

dozen galaxies spun too fast in outer regions, meaning some sort of gravitational

96

attraction was there to keep their stars in orbit [7]. There are several methods to

97

measure a galaxy’s mass, using either scatter in radial velocities or X-ray spec-

98

trum from hot centers to estimate density and pressure. Gravitational lensing is

99

a method where galaxies lying between distant quasars and observer act as a lens

100

bending light. The magnification observed gives us an estimate of mass causing

101

distortions. All above mentioned methods arrive at a common agreement that

102

DM exceeds visible matter in abundance by 5 times [8].

103

104

Cosmic Microwave Background is also by affected by DM in terms of its gravita-

105

tional potential and effect on velocity & density of ordinary matter, thus imply-

106

ing evolution of matter perturbations with time on CMB. Using sky map aniso-

107

tropies, an angular power spectrum can be formed to show densities of baryonic

108

and DM [9]. This spectrum fits perfectly with results from LambdaCDM model

109

but proves difficult to explain with Newtonian mechanics alone.

110

111

Lastly, in a long list of experimental conformations, we lay emphasis on the Bul-

112

let Cluster, which was recently formed as a result of a two-galaxy collision. Its

113

apparent center of mass differs vastly from baryonic COM, an effect lucidly ex-

114

plained by DM models but almost unaccounted for if one relies solely on modi-

115

fied gravity [10].

116

117

2. Quantum Gravity

118

119

The foundation idea of Quantum Gravity is that gravity arises from quanta scaled

120

packets of energy called ‘gravitons’ which do not exist in space, but are space themselves

121

[11]. These mediate the force of gravitation between particles and are responsible for the

122

bend in space-time in presence of mass. They are the carries of hypothesized gravitational

123

fields, are massless and electronically neutral, much like the analogous photons that me-

124

diate electromagnetic interactions. In the standard model of particles, it is theorized to be

125

a spin 2 boson particle because its fields travel at speed of light and are effective at long

126

4 of 11

range. If in the near future the vacancy of a massless spin 2 particles is filled, it is expected

127

to a graviton [12].

128

129

But the candidate we propose for DM is gravitino, a close partner of graviton in the-

130

ory of supergravity. This is a theory that deals with the combination of general relativity

131

and supersymmetry in areas where quantum effects of gravity become considerable on

132

relativistic scale. Supersymmetry proposes a space time symmetry between two classes of

133

particles – bosons and their superpartners – fermions. Considering this, it also implies that

134

boson natured gravitons have a gauge fermion superpartner with spin

named

135

Gravitino, which mediate supergravity interactions [13]. Supersymmetric theories make

136

gravity arise in a natural way and in coming sections we focus on establishing its applica-

137

tion as a good component of DM.

138

139

2.1. Effective field theory (EFT)

140

141

This is another theory working under quantum gravity (QG) and it forms our base

142

for describing dark matter and gravitino relations. EFT accounts for gravity as a well-

143

defined series dependent on higher powers of R, the curvature of space time, as [14].

144

145

This is in contrast with Einstein’s series of gravity which is a linear function of R. We

146

consider higher orders in QG based on several parameters, mass of scalar field being

147

one with most consequences. Generally, values of ‘m’ are kept heavy so as to avoid the

148

effects of relativity breaking under plank length, but in a recent study Jose Cembranos

149

found out that making ‘m’ lighter automatically accounts for dark matter. The particle

150

introduced would be a gravitino with the ability to be casted as a scalar field that could

151

clump just like real matter [15]. This is an especially elegant result as it does not require

152

us to introduce a new particle to the standard model and merely points towards a poten-

153

tial vacancy.

154

155

In another study conducted, this result was used to construct a model of our universe

156

that tracked its evolution through time, assuming gravity to be quantum in nature and

157

caused by gravitinos. The model showed hints of accelerated expansion at the exact point

158

of time we are in today and matched the rate very closely with present observational evi-

159

dence [16]. This new perspective of thinking is advantageous to us, as it starts of with

160

general framework of gravity at quantum level rather the restricting itself to cosmological

161

contexts from the very start.

162

163

2.2. Gravitational Fields

164

165

The LIGO (Laser Interferometer Gravitational Waves Observatory) has already de-

166

tected many gravitational waves which are produced by all matter that accelerates. These

167

wave like ripples in space time travel at the speed of light and transport energy as gravi-

168

tational radiation. A gravitational field (gravitino field) is model used to explain the im-

169

pact of gravity on other objects and is observed in form of gravitational waves [17]. It is

170

massless in nature and contains a spectrum of two massive fields of spin 2 and 0 whose

171

properties can be derived with help of quantum gravity [18]. They have the ability to ide-

172

ally explain DM that is only gravitationally coupled to normal matter while postulating

173

its existence to be light in nature. The waves would also carry information about DM and

174

be directly related to its abundance.

175

176

2.3. Decay widths

177

178

5 of 11

The intriguing property of such massless gravitational fields is that they decay both

179

gravitationally and to standard model particles [19]. In this section, we shall attempt to

180

show a concise and brief version of the original derivation wherein the different decay

181

widths of these gravitino fields to various particles are calculated.

182

183

We start with general relativity and integrate the fluctuations due to gravitinos to get

184

the classical effective action at second order curvature.

185

186

187

188

Here R is Ricci scalar. Many complex summations of are included in later terms

189

which depend on the cosmological constant, which is the renormalization scale,

190

the Lagrangian and M as the total energy scale up to which we can trust the EFT. The

191

green’s function poles can be used to identify new degrees of freedom with respect to a

192

matrix. Two pairs of complex poles are found for each spin field which are then renormal-

193

ized till reduced plank scale. This allows us to obtain the gravitational decay width in

194

terms of mass of the field as:

195

196

197

198

199

200

201

202

203

204

Here is the reduced plank mass equal to and is the decay

205

width due to gravitation. Using this equation (Fig 2.) we can now calculate the decay

206

widths of these fields into various standard model particles. The result of such calcula-

207

tions is that the fields do not decay to gauge bosons and the charged leptons of the stand-

208

ard model, but they do decay to gluons such as photons and neutrinos with low enough

209

mass [20]. Decay width to photons is given by:

210

211

212

213

214

The fact that gravitino fields decay to photons is especially emphasized as their ob-

215

servability during gamma ray bursts is of astronomical importance to us. The Unruh –

216

DeWitt detector implies a decay of gravitons into photons using this pathway [21]:

217

218

or

219

220

The author will use the information provided until now to make the hypotheses that

221

follow below and these are subject to human error due to negligence/ignorance.

222

6 of 11

223

224

Figure 2. The complete and general equation used to derive decay widths of gravitinos to standard model particles.

225

226

3. Approximate equivalence in Gamma Ray Flux

227

228

We illustrated in the above sections that gravity, when quantized as gravitino, could

229

account for dark matter and a field of such particles would decay into photons with a

230

considerable width. Using this result, it is also possible to calculate the flux of a gamma

231

ray produced upon decay of gravitinos. The in-depth calculations of the flux are shown

232

in this paper [22] for further reference but maybe be briefly summed up as:

233

234

235

236

where is the energy possessed by gamma rays that are produced by decay of dark

237

matter particle of mass . The flux of such a monochromatic gamma ray is given by

238

line of sight integral over dark matter distribution and the explicit component is:

239

240

241

242

where is the decay width and

is the milky way’s dark matter halo

243

density profile given by:

244

245

246

247

Using and we get local dark matter density of 0.4

248

.

The flux calculated now, assuming gravitons to have mass 200 GeV and lifetime

249

seconds is approximately in the range of units. This is also shown in

250

the graph (Fig 3.).

251

252

7 of 11

253

Figure 3. This shows the predicted gamma ray flux from gravitino decay [22].

254

255

3.1. Methodology and Principle

256

257

It is known that neutron-neutron star collisions and black hole neutron star collisions

258

release huge gravitational waves which are detected by LIGO. The data observed from

259

gravitational waves publicly is available on the GW Open Science Center and after deep

260

query analysis of it using the fact that the theoretical mass range of neutron stars is gen-

261

erally between 1.174 – 2.116 solar masses, we list 5 GW events that have the possibility of

262

being emitted from neutron star mergers. The code used by the author during analysis is

263

listed in section 3.2 for reference.

264

265

A release of gamma ray burst should happen soon after a collision happens and this

266

is detected by observatories on earth usually within a short time interval of GW trigger

267

time. The public data on Heasarc NASA by the Fermi Gamma Ray Observatory is ana-

268

lyzed for listed 5 GW events to extract GRB trigger, time interval and fluence. Flux is then

269

calculated separately for each event using queried data. The time interval between GW

270

and GRB triggers is also mentioned for reader reference.

271

272

The author theorizes that since neutron star collisions release a huge number of grav-

273

itational waves and thus by implication gravitino fields in our model, some of these

274

gravitinos must have decayed into gamma rays. If energy/flux of these gamma rays is

275

calculated, suitable comparisons can be made between the computed value and the theo-

276

retical value of energy/flux hypothesized to be possessed by gamma rays after decay from

277

gravitino fields. Equal values must increase the probability that dark matter is indeed

278

made up of gravitinos for future conceptual investigations in this matter. Accordingly,

279

highly unequal values must also eliminate our idea to constitute DM with gravitinos.

280

281

282

3.2. Working

283

The python code used by the author to query and analyze GW event catalogue is:

284

285

! pip install -q 'gwpy==2.0.2'

286

import gwpy

287

print(gwpy.__version__)

288

289

from gwosc.datasets import find_datasets

290

8 of 11

from gwosc import datasets

291

292

print("List of available catalogs")

293

print(find_datasets(type="catalog"))

294

print("")

295

List of available catalogs

296

['GWTC-1-confident', 'GWTC-1-marginal', 'GWTC-2', 'GWTC-2.1-auxiliary', 'GWTC-2.1-confident',

297

'GWTC-2.1-marginal', 'GWTC-3-confident', 'GWTC-3-marginal', 'Initial_LIGO_Virgo', 'O1_O2-Pre-

298

liminary', 'O3_Discovery_Papers', 'O3_IMBH_marginal']

299

300

gwtc1 = datasets.find_datasets(type='events', catalog='all')

301

print('All events:', events)

302

print("")

303

print(datasets.find_datasets(type='events', catalog='any', detector="any", segment=(second-

304

ary_mass)))

305

from gwosc.datasets import event_secondarymass

306

result = event_secondarymass(1.174, 2.116)

307

print ('list of events:', result)

308

309

list of events: GW170817, GW190425, GW190917_114630, GW200105_162426,

310

GW200115_042309

311

312

313

Thus, GW170817, GW190425, GW190917 ,GW200105 and GW200115 are the events

314

which include a possible neutron star collision. The data for bursts observed on these

315

dates is extracted from HEARSEC and the fluence and time period is noted as shown in

316

following table.

317

318

Table 1. This shows the data for gamma ray bursts observed near given gravitational wave events.

319

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:

320

321

The plot of observed flux as function of energy for each event is shown along with the

322

straight line of predicted flux (Fig 4.).

323

324

9 of 11

325

Figure 4. Flux values of GRBs from neutron star mergers plotted along with predicted value of flux

326

from gamma ray after gravitino decay

327

328

3.3. Results

329

We infer from the plot graph that the flux of most events aligns approximately with the

330

theorized value. Minor time differences are noticed between GW triggers and GRB trig-

331

gers, implying the good possibility of both events originating from a common source.

332

The author attributes minor gaps between predicted and actual flux values to plausible

333

technological, source, numerical and human errors.

334

4. Conclusion

335

In this paper we have shown that the existence of dark matter is supported by ample

336

number of evidences and that its absence indeed causes serious issues to remain unex-

337

plained by our exclusive models. Its origin and composition are much contemplated

338

over by defining a quantum theory of gravity and the various implications it would

339

have. Gravitinos, the mediating particle of gravitational forces in supergravity, are

340

thought to constitute dark matter and in such a scenario, it is predicted that the gravita-

341

tional field would decay due to standard model particles. Its decay to photons would

342

eventually give rise to gamma rays with predicted values of flux. The author endeavors

343

to analyze gravitational wave and gamma ray bursts data from neutron star collisions in

344

order to obtain experimental flux observed. Both these values are clearly shown to be

345

approximately equal, implying that according to our model gravitinos must constitute

346

dark matter and have predictably decayed into gamma rays. Although there is a long

347

way to go before it can be said for certain what dark matter is composed of, this conclu-

348

sion helps us evaluate that the scope of quantum gravity. The result proceeds to further

349

cement our belief that gravitinos have a good candidacy to be considered as comprising

350

particles for dark matter.

351

352

353

354

355

356

10 of 11

357

358

359

360

361

Funding: This research received no external funding or grants.

362

363

Data Availability Statement: The data used in this research for gravitational wave events has been

364

taken from https://www.gw-openscience.org/ and the coding interface used is https://colab.re-

365

search.google.com/github/gw-odw/odw-2022/blob/main/Tutorials/Day_1/Tuto%201.1%20Discov-

366

ering%20Open%20Data.ipynb. The gamma ray burst data has been obtained from https://heas-

367

arc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl.

368

369

370

References

371

372

1. "NASA Science Universe – Dark Energy, Dark Matter". NASA Science. Retrieved 23 May 2021.

373

2. Baumann, Daniel. "Cosmology: Part III" (PDF). Mathematical Tripos. Cambridge University. pp. 21–22. Ar-

374

chived from the original (PDF) on 2 February 2017. Retrieved 24 January 2017

375

3. Zwicky, F. (1937). "On the Masses of Nebulae and of Clusters of Nebulae". The Astrophysical Journal. 86: 217–

376

246. Bibcode:1937ApJ....86..217Z. doi:10.1086/143864.

377

4. Babcock, Horace W. (1939). "The rotation of the Andromeda Nebula". Lick Observatory Bulletin. 19: 41–

378

51. Bibcode:1939LicOB..19...41B. doi:10.5479/ADS/bib/1939LicOB.19.41B.

379

5. Overbye, Dennis (27 December 2016). "Vera Rubin, 88, Dies; Opened Doors in Astronomy, and for

380

Women". The New York Times. Retrieved 27 December 2016

381

6. Jaffe, A.H. "Cosmology 2012: Lecture Notes" (PDF). Archived from the original (PDF) on 17 July 2016.

382

7. "Superstars of Astronomy podcast" (PDF).

383

8. Allen, Steven W.; Evrard, August E.; Mantz, Adam B. (2011). "Cosmological Parameters from Clusters of Gal-

384

axies". Annual Review of Astronomy and Astrophysics. 49 (1): 409–470. arXiv:1103.4829. Bib-

385

code:2011ARA&A..49..409A. doi:10.1146/annurev-astro-081710-102514. S2CID 54922695.

386

9. Ade, P.A.R.; et al. (2016). "Planck 2015 results. XIII. Cosmological parameters". Astron. Astrophys. 594 (13):

387

A13. arXiv:1502.01589. Bibcode:2016A&A...594A..13P. doi:10.1051/0004-6361/201525830. S2CID 119262962.

388

10. Clowe, Douglas; et al. (2006). "A Direct Empirical Proof of the Existence of Dark Matter". The Astrophysical

389

Journal Letters. 648 (2): L109–L113. arXiv:astro-ph/0608407. Bib-

390

code:2006ApJ...648L.109C. doi:10.1086/508162. S2CID 2897407.

391

11. Feynman, R. P.; Morinigo, F. B.; Wagner, W. G.; Hatfield, B. (1995). Feynman Lectures on Gravitation. Addi-

392

son-Wesley. ISBN 0-201-62734-5

393

12. For a comparison of the geometric derivation and the (non-geometric) spin-2 field derivation of general rela-

394

tivity, refer to box 18.1 (and also 17.2.5) of Misner, C. W.; Thorne, K. S.; Wheeler, J. A. (1973). Gravitation. W.

395

H. Freeman. ISBN 0-7167-0344-0.

396

13. de Gouvêa, André; Moroi, Takeo; Murayama, Hitoshi (1997-07-15). "Cosmology of supersymmetric models

397

with low-energy gauge mediation". Physical Review D. 56 (2): 1281–1299. arXiv:hep-ph/9701244. Bib-

398

code:1997PhRvD..56.1281D. doi:10.1103/physrevd.56.1281. ISSN 0556-2821. S2CID 15935292.

399

11 of 11

14. https://doi.org/10.1103/PhysRevLett.102.141301

400

15. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.141301

401

16. https://arxiv.org/pdf/2105.03751.pdf

402

17. Consistent coupling of the gravitino field to a gravitational background with torsion

403

https://www.osti.gov/biblio/6091911

404

18. X. Calmet, Int. J. Mod. Phys. D 22, 1342014 (2013). https://doi.org/10.1142/S0218271813420145. arXiv:1308.6155

405

[gr-qc]

406

19. Dark matter in quantum gravity https://doi.org/10.1140/epjc/s10052-018-6005-8

407

20. Dark matter in quantum gravity https://doi.org/10.1140/epjc/s10052-018-6005-8

408

21. Response of the Unruh-DeWitt detector in a gravitational wave background https://arxiv.org/abs/2109.14183

409

22. Gamma-ray lines from dark matter decay https://iopscience.iop.org/article/10.1088/1742-

410

6596/384/1/012001/pdf

411

412