* Byrd et al. [1995] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu.  A limited memory algorithm for bound constrained optimization.  _SIAM Journal on Scientific Computing_ , 16(5):1190–1208, 1995.
  * Cecchetto [2009] B. T. Cecchetto.  Correction of chromatic aberration from a single image using keypoints, 2009.  URL http://www.cs.ubc.ca/~bentc/downloads/CPSC525.pdf.
  * Chang et al. [1999] E. Chang, S. Cheung, and D. Pan.  Color filter array recovery using a threshold-based variable number of gradients.  _Proceedings of SPIE_ , 14:36–43, 1999.
  * Chen et al. [2000] Y.-S. Chen, S.-W. Shih, Y.-P. Hung, and C.-S. Fuh.  Camera calibration with a motorized zoom lens.  _International Conference on Pattern Recognition_ , 2000.
  * Cicala [2008] R. Cicala.  “This lens is soft” and other myths, 2008.  URL https://www.lensrentals.com/news/2008.12.22/this-lens-is-soft-and-other-myths.
  * CIE [1926] CIE.  Commission internationale de l’Eclairage proceedings: The CIE1924 photopic luminosity function.  Technical report, CIE, 1926.
  * Clark and Evans [1954] P. J. Clark and F. C. Evans.  Distance to nearest neighbour as a measure of spatial relationships in populations.  _Ecology_ , 25(4):445–453, October 1954.
  * Cochran et al. [1967] W. Cochran, J. Cooley, D. Favin, H. Helms, R. Kaenel, W. Lang, G. M. Jr., D. Nelson, C. Rader, and P. Welch.  What is the Fast Fourier Transform?  _IEEE Transactions on Audio and Electroacoustics_ , 15:45–55, June 1967.
  * Coffin [2006] D. Coffin.  Dcraw, 2006.  URL http://www.cybercom.net/~dcoffin/dcraw/.
  * Corporation [2008] N. Corporation, 2008.  URL http://imaging.nikon.com/products/imaging/technology/d-technology/expeed/chromatic/index.htm.
  * Cronk et al. [2006] S. Cronk, C. S. Fraser, and H. B. Hanley.  Automatic calibration of colour digital cameras.  _Photogrammetric Record_ , 21(116):355–372, 2006\.
  * Delaunay [1934] D. Delaunay.  Sur la sphère vide.  _Izvestia Akademii Nauk SSSR, Otdelenie Matematicheskikh i Estestvennykh Nauk_ , 7:793–800, 1934.
  * Dersch [2003] H. Dersch.  Panotools, 2003.  URL http://webuser.hs-furtwangen.de/~dersch/.
  * Electronics and Assoc. [2002] J. Electronics and I. T. I. Assoc.  Digital still camera image file format standard (EXIF) version 2.2, 2002\.  URL http://exif.org/Exif2-2.PDF.
  * Fairchild [2005] M. D. Fairchild.  _Color Apprearance Models_.  Chichester, UK: John Wiley and Sons, Ltd, second edition, 2005.
  * Fang et al. [2007] Y. Fang, B. Wu, and T. Liu.  Eliminating lateral color aberration of a high-resolution digital projection lens using a novel genetic algorithm.  _Optical Engineering_ , 46, 2007.
  * Ferriere and Goedgebuer [1982] R. Ferriere and J. P. Goedgebuer.  A spatially coherent achromatic fourier transformer.  _Optics Communications_ , 42:223–225, 1982.
  * Fischer et al. [2008] R. E. Fischer, B. Tadic-Galeb, and P. R. Yoder.  _Optical system design_.  McGraw-Hill Professional, 2008.
  * Fletcher [1970] R. Fletcher.  A new approach to variable metric algorithms.  _The Computer Journal_ , 13(3):317, 1970.
  * Fryer and Brown [1986] J. G. Fryer and D. C. Brown.  Lens distortion for close-range photogrammetry.  _Photogrammetric Engineering and Remote Sensing_ , 52(1):51–58, 1986.
  * Gamal [2002] A. E. Gamal.  Trends in cmos image sensor technology and design.  In _Electron Devices Meeting, 2002. IEDM ’02. Digest. International_ , pages 805–808, 2002.
  * Goldfarb [1970] D. Goldfarb.  A family of variable-metric methods derived by variational means.  _Mathematics of Computation_ , 24(109):23–26, 1970.
  * Green [1989] W. B. Green.  _Digital Image Processing: A Systems Approach_.  Van Nostrand Reinhold Co., 1989.
  * Grey [1962] D. S. Grey.  Aberration theories for semiautomatic lens design by electronic computers.  _Journal of the Optical Society of America_ , 1962.
  * Gross et al. [2005] H. Gross, W. Singer, and M. Totzeck.  _Handbook of Optical Systems_.  Wiley-VCH, 2005.
  * Group [1992] J. P. E. Group.  JPEG standard (JPEG ISO/IEC 10918-1 ITU-T recommendation T.81), 1992.  URL http://www.w3.org/Graphics/JPEG/itu-t81.pdf.
  * Hallyn [1994] F. Hallyn.  Kepler, Snell and the law of refraction.  _Med. Konink. Acad. Wetensch. Belgie_ , 56(2):119–134, 1994.
  * Hamming [1998] R. W. Hamming.  _Digital filters_.  Courier Dover Publications, 1998.
  * Hay et al. [1963] J. C. Hay, J. Herbert L. Pick, and E. Rosser.  _Adaptation to Chromatic Abarration by the Human Visual System_.  American Association for the Advancement of Science, 1963.
  * Hirakawa and Parks [2005] K. Hirakawa and T. W. Parks.  Adaptive homogeneity-directed demosaicing algorithm.  _IEEE Transactions Image Processing_ , 14:360–369, 2005\.
  * Hornbeck [1987] L. Hornbeck, 1987.
  * Hua et al. [2004] X. S. Hua, L. Lu, and H. J. Zhang.  Optimization-based automated home video editing system.  _IEEE Transactions on TCSVT_ , 5(15), May 2004.
  * Inc. [2009] A. Inc.  Compensate for chromatic aberration in camera raw, 2009.  URL http://help.adobe.com/en_US/Photoshop/11.0/WS0B9A44BF-63F1-498c-936F-3CB7B4DE0FEB.html.
  * Inc. [2000] C. Inc.  Canon multi-layer diffractive optical element, 2000.  URL http://www.canon.com/camera-museum/tech/report/200106/report.html.
  * Inc. [2007] N. Inc.  Nikon D300 chromatic aberration reduction, 2007.  URL http://imaging.nikon.com/products/imaging/lineup/d300/artful/page1.htm.
  * International Telecommunication Union [2002] International Telecommunication Union, 2002.  URL http://www.itu.int/rec/R-REC-BT.709/en.
  * Jacobson [2000] R. E. Jacobson.  _The manual of photography_.  Focal Press, 2000.
  * Jacobson et al. [2000] R. E. Jacobson, S. Ray, G. G. Attridge, and N. Aford.  _The Manual of Photography_.  Focal Press, ninth edition, 2000.
  * Jr. et al. [2010] R. A. M. Jr., S. S. Young, J. Reynolds, K. Krapels, J. Fanning, and T. Corbin.  Review of bayer pattern color filter array (cfa) demosaicing with new quality assessment algorithms.  Technical report, United States Army Research Laboratory, 2010.
  * Karras et al. [1998] G. E. Karras, G. Mountrakis, P. Patias, and E. Petsa.  Modeling distortion of super-wide-angle lenses in architectural and archaeological applications, 1998.
  * Kaufmann and Ladstädter [2005] V. Kaufmann and R. Ladstädter.  Elimination of color fringes in digital photographs caused by lateral chromatic aberration.  _International CIPA Symposium_ , 1:403–408, September 2005\.
  * Kidger [2001] M. J. Kidger.  _Fundamental Optical Design_.  SPIE, 2001.
  * Kingslake [1978] R. Kingslake.  _Lens Design Fundamentals_.  Academic Press, 1978.
  * Koren [2008] N. Koren.  Imatest - RAW files, 2008.  URL http://www.imatest.com/docs/print_raw.html.
  * Kraus [2004] E. Kraus.  Determination of chromatic aberration correction parameters, 2004.  URL http://www.erik-krause.de/ca/index.htm.
  * Laikin [1991] M. Laikin.  _Lens Design_.  Marcel Dekker, New York, 1991.
  * Lanczos [1950] C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators.  _J. Research of the National Bureau of Standard_ , 45:255–282, 1950.
  * Lee and Song [2007] T. Lee and W. Song.  Chromatic aberration correction using iterative edge restoration.  _Signal Processing and Communication Applications_ , 20(15), 2007.
  * Lowe [2004] D. G. Lowe.  Distinctive image features from scale-invariant keypoints.  _International Journal of Computer Vision_ , 60(2):91–110, 2004.  ISSN 0920–5691.
  * Luhmann [2006] T. Luhmann.  Modelling of chromatic aberration for high precision photogrammetry.  _IAPRS_ , XXXVI, 2006.
  * Mallon and Whelan [2007] J. Mallon and P. F. Whelan.  Calibration and removal of lateral chromatic aberration in images.  _Pattern Recogn. Lett._ , 28(1):125–135, 2007\.  ISSN 0167-8655.
  * Marquardt [1963] D. W. Marquardt.  An algorithm for least-squares estimation of nonlinear parameters.  _SIAM Journal on Applied Mathematics_ , 11(2):431–441, 1963.
  * Melchior [2007] P. Melchior.  Simple vs conjugate gradient minimization in n dimensions, 2007.  URL http://www.ita.uni-heidelberg.de/~pmelchior/talks/multimin_110607.pdf.
  * Mielenz [1999] K. D. Mielenz.  On the diffraction limit for lensless imaging.  _Journal of Research of the National Institute of Standards and Technology_ , 104(5), 1999.
  * Nelder and Mead [1965] J. A. Nelder and R. Mead.  A simplex method for function minimization.  _Computer Journal_ , 7:308–313, 1965.
  * Neville [1934] E. H. Neville.  Iterative interpolation.  _Journal of Indian Mathematical Society_ , 20:87–120, 1934\.
  * Newton [1704] I. Newton.  _Opticks_.  William Innys, St. Paul’s, 1704.
  * Ohba et al. [1980] S. Ohba, M. Nakai, H. Ando, S. Hanamura, S. Shimada, K. Satoh, K. Takahashi, M. Kubo, and T. Fujita.  MOS area sensor: Part II - low-noise MOS area sensor with antiblooming photodiodes.  _IEEE Transactions on Electron Devices_ , 27(8):1682–1687, 1980.
  * Optics [2002] S. Optics.  Oslo, 2002.  URL http://www.sinopt.com/.
  * Park and Hong [2000] S. Park and K. Hong.  Practical ways to calculate camera lens distortion for real-time camera calibration.  _Pattern Recognition_ , 34(6):1199–1206, 2000\.
  * Peres [2007] M. R. Peres.  _The Focal Encyclopedia of Photography_.  Focal Press, fourth edition, 2007.
  * Piqueres [2004] J. V. Piqueres.  The office, 2004.  URL http://www.ignorancia.org/uploads/zips/office.zip.
  * Pomaska [2001] G. Pomaska.  Image acquisition for digital photogrammetry using of the shelf and metric cameras.  _The international committee for architectural photogrammetry_ , 2001\.
  * Powell [1981] I. Powell.  Lenses for correcting chromatic aberration of the eye.  _Applied Optics_ , 20(24):4152–4155, 1981.
  * Powell [1964] M. J. D. Powell.  An efficient method for finding the minimum of a function of several variables without calculating derivatives.  _The Computer Journal_ , 7(2):155–162, 1964.
  * Remondino and Fraser [2006] F. Remondino and C. Fraser.  Digital camera calibration methods: considerations and comparisons.  In ISPRS, editor, _International Archives of Photogrammetry, Remote Sensing and Spatial Information Sciences,_ , volume XXXVI, Dresden, Germany, 2006. Institute of Geodesy and Photogrammetry, ETH Zürich.
  * Rifman and McKinnon [1974] S. S. Rifman and D. M. McKinnon.  Evaluation of digital correction techiques for erts images.  Technical report, TRW Systems, Redondo Beach, California, 1974.
  * Scherzer [1947] O. Scherzer.  Sphärische und chromatische korrektur von elektronenlinsen.  _Optik_ , 2:114–132, 1947.
  * Seong and Greivenkamp [2008] K. Seong and J. E. Greivenkamp.  Chromatic aberration measurement for transmission interferometric testing.  _Applied Optics_ , 47(35), 2008.
  * Shack [1971] R. V. Shack.  Production and use of a lenticular hartmann screen.  _Journal of the Optical Society of America_ , 61:656, 1971\.
  * Shanno [1970] D. F. Shanno.  Conditioning of quasi-newton methods for function minimization.  _Mathematics of Computation_ , 24(111):647–656, 1970.
  * Shannon [1949] C. E. Shannon.  Communication in the presence of noise.  _Proceedings of the Institute of Radio Engineers_ , 37(1):10–21, January 1949.
  * Sibson [1981] R. Sibson.  _Interpolating Multivariate Data_ , chapter A brief description of natural neighbor interpolation, pages 21–36.  John Wiley, 1981.
  * Smith [1927] T. T. Smith.  The color correction of an achromatic doublet II.  _Journal of the Optical Society of America_ , 15:249, 1927\.
  * Smith [2008] W. J. Smith.  _Modern optical engineering: the design of optical systems_.  McGraw-Hill Professional, 2008.
  * Thibos [1987] L. N. Thibos.  Calculation of the influence of lateral chromatic aberration on image quality across the visual field.  _Optical Society of America_ , 4(8), 1987.
  * Thibos et al. [1992] L. N. Thibos, M. Ye, and X. X. Zhang.  The chromatic eye: a new reduced-eye model of ocular chromatic aberration in humans.  _Applied Optics_ , 31:3594–3600, 1992.
  * Tsai [1987] R. Y. Tsai.  A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lense.  _IEEE Journal of Robotics and Automation_ , 3(4):323–344, 1987.
  * Tucker et al. [1998] S. C. Tucker, W. T. Cathey, and E. R. Dowski.  Extended depth of field and aberration control for inexpensive digital microscope systems.  _Opticals Express_ , 4:467–474, 1998.
  * Tygier [2005] S. Tygier.  Correcting chromatic aberrations using a diffraction grating in a camera lens.  Technical report, Department of Physics and Astronomy, Manchester University, 2005.
  * van den Heuvel et al. [2006] F. A. van den Heuvel, R. Verwaal, and B. Beers.  Calibration of fisheye camera systems and the reduction of chromatic aberration.  In _ISPRS Image Engineering and Vision Metrology_ , page 157. CycloMedia Technology B.V., 2006.
  * van Walree [1997] P. van Walree.  Chromatic aberrations, 1997.  URL http://www.vanwalree.com/optics/chromatic.html.
  * von Seidel [1856] L. von Seidel.  Astronomische nachrichten, 1856.
  * Watters [2004] J. Watters.  Correcting chromatic aberration with panotools radial shift, 2004.  URL http://photocreations.ca/radial_distortion/index.html.
  * Welford [1986] W. T. Welford.  _Aberrations of Optical Systems_.  CRC Press, 1986.
  * Willson [1994] R. G. Willson.  Modeling and calibration of automated zoom lenses.  Technical report, Carnegie Mellon University, 1994.
  * Willson and Shafer [1991] R. G. Willson and S. Shafer.  Active lens control for high precision computer imaging.  In _Proceedings IEEE International Conference on Robotics and Automation (ICRA 1991)_ , volume 3, pages 2063–2070. Robotics Institute, Carnegie Mellon University, April 1991.
  * Willson and Shafer [1993] R. G. Willson and S. A. Shafer.  A perspective projection camera model for zoom lenses.  _Proceedings of the Conference on Optical 3-D Measurement Techniques_ , 1993.
  * Wolberg and Boult [1992] G. Wolberg and T. E. Boult.  Correcting chromatic aberrations using image warping.  In _DARPA Image Understanding 1992 Workshop_ , pages 363–377. Department of Computer Science, Columbia University, NYC, 1992.
  * Wyszecki and Stiles [1982] G. Wyszecki and W. S. Stiles.  _Color Science: concepts and methods, quantitative data and formulae_.  Chichester, UK: John Wiley and Sons, Ltd, second edition, 1982.
  * Zabolotny [2008] A. Zabolotny.  Lensfun, 2008.  URL http://lensfun.berlios.de/.
  * Zeng et al. [2001] Y. Zeng, L. Cheng, G. Bi, and A. C. Kot.  Integer DCTs and fast algorithms.  _IEEE Transactions on Signal Processing_ , 49:2774–2782, November 2001.
  * Zhang et al. [2010] M. Zhang, Z.-Q. Wang, Y. Wang, and T. Zuo.  The study of correction for chromatic aberrations of human eye with diffractive–refractive hybrid elements.  _International Journal for Light and Electron Optic_ , 2010.
  * Zhu et al. [1994] C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal.  L-BFGS-B – Fortran subroutines for large-scale bound constrained optimization.  Technical report, ACM Transactions on Mathematical Software, 1994.


# Possibility of the Traversable Wormholes in the Galactic Halos within $4D$
Einstein-Gauss-Bonnet Gravity

Zinnat Hassan 0000-0002-6608-2075<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

Recently, there has been significant interest regarding the regularization of
a $D\rightarrow 4$ limit of Einstein-Gauss-Bonnet (EGB) gravity. This
regularization involves re-scaling the Gauss-Bonnet (GB) coupling constant as
$\alpha/(D-4)$, which bypasses Lovelock’s theorem and avoids Ostrogradsky
instability. A noteworthy observation is that the maximally or spherically
symmetric solutions for all the regularized gravities coincide in the $4D$
scenario. Considering this, we investigate the wormhole solutions in the
galactic halos based on three different choices of dark matter (DM) profiles,
such as Universal Rotation Curve, Navarro-Frenk-White, and Scalar Field Dark
Matter with the framework of $4D$ EGB gravity. Also, the Karmarkar condition
was used to find the exact solutions for the shape functions under different
non-constant redshift functions. We discussed the energy conditions for each
DM profile and noticed the influence of GB coefficient $\alpha$ in violating
energy conditions, especially null energy conditions. Further, some physical
features of wormholes, viz. complexity factor, active gravitational mass,
total gravitational energy, and embedding diagrams, have been explored.

Keywords: Galactic halos, dark matter, wormhole solutions, $4D$ EGB gravity,
energy conditions.

##  I Introduction

In 1915, Albert Einstein published his seminal work, which revolutionized our
understanding of the fundamental nature of gravity by linking it to the
geometry of space-time. Over the subsequent years, extensive experimental and
observational validation [1] has solidified the significance of this theory,
making it an indispensable tool for researchers worldwide in their pursuit of
a deeper comprehension of the cosmos. This groundbreaking theory has given
rise to many extraordinary phenomena, including gravitational lensing, the
formation of black holes, the detection of gravitational waves, and the
concept of wormholes. One of the particular interests among these phenomena is
the theoretical existence of wormholes, an alternative solution within
Einstein’s Field Equations (EFE) that conceivably bridges the gap between
distinct regions of space-time.
Interest in wormhole space-times traces its origins back to 1916 when Flamm
[2] initially postulated the existence of a tunnel structure within the
Schwarzschild solution, suggesting it to be a potential wormhole. Later,
Einstein and Rosen [3] introduced the concept of a “bridge structure” that
links two exterior regions of a Schwarzschild black hole space-time, thereby
giving rise to an inter-universe connection. Wheeler [4, 5] coined the term
“wormhole” to describe microscopic charge-carrying wormholes. They
demonstrated that if a wormhole somehow attempted to open, it would close so
rapidly that even a single photon might travel through it, thereby sustaining
Einsteinian causality. A detailed discussion of the possibility of a wormhole
was presented in [6]. The potential existence of wormholes in the galactic
halo region was discussed in [7], and gravitational lensing by wormholes was
explored in [8, 9, 10, 11, 12, 13]. For further interesting works on wormhole
geometry, one may check some Refs. [14, 15, 16, 17, 18].
In General Relativity (GR), a key element in understanding wormholes involves
violating energy conditions [19], and the matter associated with violating
energy conditions is termed as exotic matter. Since minimizing the use of
exotic matter poses a primary challenge in wormhole theory, various approaches
have been explored in the literature. These include the cut-and-paste method
[20, 21] and the investigation of wormholes with variable equations of state
(EoS) [22, 23]. Simultaneously, exploring modified theories of gravity has
provided a fresh perspective on wormhole studies. Numerous modified gravity
theories have been employed to delve into wormhole theory, where the
additional terms in the modified field equations facilitate solutions aligning
with the satisfaction of energy conditions by ordinary matter. Readers can
check some works on wormhole geometry in different modified theories include
$f(R)$ gravity [24, 25, 26, 27], $f(R,T)$ gravity [28, 29, 30], braneworld
scenarios [31, 32, 33], Rastall theory [34, 35, 36], $f(Q)$ gravity [37, 38,
39, 40], and $f(Q,T)$ gravity [41, 42, 43].
In string theories, particularly in their low-energy limits, effective field
theories of gravity emerge. In these theories, the Lagrangian incorporates
terms of quadratic and higher orders in the curvature alongside the
conventional scalar curvature term [44, 45, 46]. Notably, the gravitational
action can undergo modification to encompass quadratic curvature correction
terms while maintaining the equations of motion in the second order. This
modification is permissible as long as the quadratic terms manifest in
specific combinations corresponding to the GB invariants denoted by [47]

$\mathcal{G}=R^{2}-4R_{\alpha\nu}\,R^{\alpha\nu}+R_{\alpha\nu\delta\eta}\,R^{\alpha\nu\delta\eta}.$
(1)

Such a gravitational theory is designated as $D\geq 5$ dimensional EGB gravity
theory, with $(D-4)$ extra dimensions playing a pivotal role. The EGB gravity
constitutes a specific instance within the broader framework of the Lovelock
theory of gravitation [48]. Significantly, the equations of motion associated
with EGB gravity exhibit no more than two metric derivatives, ensuring the
absence of ghost-related issues [49]. However, it is crucial to note that the
GB term (1) behaves as a topological invariant in four dimensions,
contributing proportionally to $(D-4)$ in all components of Einstein’s
equations. Consequently, this term does not contribute to the equations of
motion, necessitating $D\geq 5$ to manifest non-trivial gravitational
dynamics.
Nevertheless, Glavan and Lin [50] demonstrated a significant impact on
gravitational dynamics in $D=4$ could be achieved by re-scaling the GB
coupling constant in the EGB gravity action, expressed as

$\alpha\rightarrow\frac{\alpha}{D-4}.$ (2)

A consistent theory of $D\rightarrow 4$ EGB gravity, maintaining invariance
under spatial diffeomorphism while breaking the time diffeomorphism, has been
discussed in [51, 52]. The 4D EGB gravity has garnered substantial attention
and is undergoing extensive examination [53, 54, 55, 56]. A cloud of strings
in 4D EGB was also explored in [57]. The exploration of non-static Vaidya-like
spherical radiating black hole solutions in 4D EGB gravity has been reported
[58], and further black hole solutions can be found in [59, 60, 61, 62, 63].
Furthermore, in Ref. [64], the authors discussed the photon geodesics and the
influence of the GB coupling parameter on the shadow of the 4D EGB black hole.
A detailed review of this topic has been discussed in [65]. Wormhole solutions
were also investigated using the same formalism in $4D$ EGB gravity. Jusufi et
al. [66] discussed wormhole solutions in $4D$ EGB gravity for isotropic and
anisotropic matter sources. Moreover, the stability of thin-shell wormholes
[67] and Yukawa-Casimir wormholes [68] have been investigated. However,
wormholes in the region of galactic dark matter halos have yet to be explored,
which motivates us to study this particular gap.
Recent revelations about the composition of the Universe have brought to light
the fact that a mere 5% of the total matter and energy consists of the visible
planets, stars, and ordinary matter, with the remainder being dominated by
dark matter and dark energy of which 27% is attributed to dark matter. Unlike
regular baryonic matter, this dark matter does not interact with photons but
exerts its influence solely through gravitational forces. In 1933, astronomer
Zwicky was the first to ascertain the presence of dark matter within a distant
galaxy cluster [69, 70]. Oort [71, 72] initially proposed the existence of
dark matter in the Milky Way galaxy and subsequently confirmed through
observational evidence by Diemand and Springel [73, 74]. Dark matter plays a
crucial role in the hypothetical construction of wormholes, with the density
profile of dark matter proposed by researchers playing a pivotal role in the
theoretical development of a fully stable and traversable wormhole. In Ref.
[75], the authors have provided evidence indicating the role of DM in
sustaining the wormhole geometry within the outer reaches of the galactic
halo. Sarkar et al. [76] have demonstrated the presence of wormholes within
the isothermal galactic halo, supported by the presence of DM. Additionally,
Kuhfitting [77] has examined the gravitational lensing phenomena associated
with wormholes within the galactic halo region.
In recent years, there has been a remarkable surge in the number of exact
solutions describing compact objects within classical GR and modified gravity
theories. This surge has pushed the search for solutions to the field
equations into the forefront of mainstream astrophysics, particularly
evidenced in the expansion of realistic stellar models over the last decade.
Utilization of the Karmarkar condition has played a pivotal role in generating
compact objects, enabling the differentiation between the radial and
transverse stresses at various points within the stellar fluid [78, 79]. In
the context of classical GR, the Karmarkar condition is readily integrated to
establish a relationship between the two metric potentials, effectively
simplifying the task of finding exact solutions to the field equations into a
single generating function [80, 81]. The authors of Ref. [82] presented a
convincing discussion regarding the nonexistence of conformally flat-charged
isotropic fluid spheres falling under the category of embedding Class One.
Also, the Karmarkar condition has been extended to encompass time-dependent
systems, enabling its utilization in scenarios involving the modeling of
shear-free, dissipative collapse [83, 84, 85]. Moreover, in [86], anisotropic
stars have been studied via Karmarkar condition in Brans-Dicke gravity and
confirm that their solutions describe compact objects such as PSR J1903+327;
Cen X-3; EXO 1785-248 and LMC X-4 to an excellent approximation. Furthermore,
the Karmarkar condition has been established to be a valuable tool in
exploring other compact objects, including wormholes.
Motivated by the above discussions, we have investigated wormhole solutions
under different DM halo models by employing the Karmarkar condition in the
context of $4D$ Gauss-Bonet gravity. The structure of this paper is outlined
as follows: In section II, we have provided the basic formalism of $4D$ EGB
gravity and corresponding wormhole field equations under this gravity. The
Karmarkar condition and the DM halo profiles have been extensively discussed
in section III and IV, respectively. Moreover, section V discusses wormhole
solutions for different redshift functions under each DM profile. Further,
some physical features of wormholes have been studied in section VI. Finally,
we conclude our finding in section VII.

##  II Basic formalisms of $4D$ EGB gravity and wormhole field equations

###  II.1 Basic formalisms of $4D$ EGB gravity

In the $D$ dimensions, the EGB gravity action incorporating a rescaled GB
coupling $\alpha/(D-4)$ to restore dimensional regularization results in the
expression [50]

$\mathcal{I}=\frac{1}{16\pi}\int
d^{D}\,x\sqrt{-g}\left(R+\frac{\alpha}{D-4}\mathcal{G}\right)+\mathcal{S}_{matter},$
(3)

where $R$ represents the Ricci scalar, and $\alpha$ is the GB coefficient with
dimension $[\textit{Length}]^{2}$. The GB invariant is described by
$\mathcal{G}$, and its expression is given in Eq. (1). The $4D$ EGB theory is
defined by considering the limit $D\rightarrow 4$ at the level of equations of
motion rather than in the action. Consequently, in this framework, the GB term
plays a non-trivial role in gravitational dynamics [50].
Now, by varying the action (3) w.r.t the metric tensor, one can find the field
equations

$G_{\mu\nu}+\frac{\alpha}{D-4}\mathcal{H}_{\mu\nu}=8\pi\mathcal{T}_{\mu\nu},$
(4)

where $G_{\alpha\nu}$, $\mathcal{T}_{\alpha\nu}$ and $\mathcal{H}_{\alpha\nu}$
are the Einstein tensor, energy-momentum tensor and the GB tensor,
respectively which are defined by

$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu},$ (5)
$\mathcal{T}_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}\,\mathcal{S}_{m}\right)}{\delta
g^{\mu\nu}},$ (6)

and

$\mathcal{H}_{\mu\nu}=-\frac{1}{2}\mathcal{G}g_{\mu\nu}+2\left[R\,R_{\mu\nu}-2R_{\mu\sigma}R^{\sigma}_{\,\,\nu}-2R_{\mu\rho\nu\sigma}\,R^{\rho\sigma}\right.\\\
\left.-2R_{\mu\sigma bc}\,R^{\sigma bc}_{\,\,\,\,\,\,\,\,\nu}\right],$ (7)

where $R_{\mu\nu}$ represents the Ricci tensor and $R_{\mu\rho\nu\sigma}$ is
the Riemann tensor. The GB terms are generally characterized as total
derivatives in $4D$ space-time, thereby not contributing to the field
equations. Nevertheless, when the coupling constant $\alpha/(D-4)$ undergoes
re-scaling and maximally symmetric space-times with a curvature scale
$\mathcal{K}$, we obtain

$\frac{g_{\mu\sigma}}{\sqrt{-g}}\frac{\partial\mathcal{G}}{\partial
g_{\nu\sigma}}=\frac{\alpha(D-2)(D-3)}{2(D-1)}\mathcal{K}^{2}\delta^{\nu}_{\mu}.$
(8)

Evidently, the GB action variation does not vanish in $D=4$ due to the
influence of the re-scaled coupling constant [50].
To generate wormhole solutions within the framework of $4D$ EGB, we employ the
regularization process outlined in [50]. Notably, the $4D$ spherical solutions
derived in [50] align with those obtained in various other regularized
theories [87, 88, 89].

###  II.2 Wormhole field equations

In this analysis, we focus on a $D$-dimensional spacetime where the two-sphere
[19] is substituted with an $(D-2)$ sphere, as described by the following line
element

$ds^{2}=-e^{2\phi(r)}dt^{2}+\left(1-\frac{b(r)}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}_{D-2},$
(9)

where

$d\Omega^{2}_{D-2}=d\theta_{1}^{2}+\sum\limits_{i=2}^{D-2}\prod\limits_{j=1}^{i-2}\sin^{2}\theta_{j}d\theta_{i}^{2}.$

$\phi(r)$ represents the redshift function, and to bypass the event horizon
issue, the redshift function must be finite everywhere. $b(r)$ is the shape
function; it determines the shape of the wormholes. It is crucial to note that
$b(r)$ must adhere to the boundary condition $b(r=r_{0})=r_{0}$ at the throat,
where $r_{0}$ is such that $r_{0}\leq r\leq\infty$. To assure the
traversability of a wormhole, b(r) should meet the flaring-out condition,
which is deduced from the embedding calculation and expressed as
$(b-b^{\prime}r)/b^{2}>0$ [19]. Alternatively, this condition can be compactly
stated as $b(r_{0})<1$ at the throat, where $r=r_{0}$. Besides, the condition
$1-\frac{b(r)}{r}>0$ is enforced.
Further, we consider the diagonal energy-momentum tensor for an anisotropic
fluid, which can be read as

$\mathcal{T}_{\mu}^{\nu}=\text{diag}[-\rho(r),p_{r}(r),p_{t}(r),p_{t}(r)],$
(10)

where $\rho(r)$ is the energy density. $p_{r}(r)$ and $p_{t}(r)$ represent the
radial pressure and tangential pressure, respectively.
Thus, the gravitational field equation (4) for the metric (9) in the limit
$D\rightarrow 4$ provides the following relations [66]

$8\pi\rho(r)=\frac{\alpha
b(r)}{r^{6}}\left(2rb^{\prime}(r)-3b(r)\right)+\frac{b^{\prime}(r)}{r^{2}},$
(11) $8\pi p_{r}(r)=\frac{\alpha
b(r)}{r^{6}}\left[4r\phi^{\prime}(r)\left(r-b(r)\right)+b(r)\right]\\\
+\frac{2\phi^{\prime}(r)\left(r-b(r)\right)}{r^{2}}-\frac{b(r)}{r^{3}},$ (12)
$8\pi
p_{t}(r)=\left(1-\frac{b(r)}{r}\right)\left[\left(\phi^{\prime\prime}(r)+(\phi^{\prime}(r))^{2}\right)\right.\\\
\left.\times\left(1+\frac{4\alpha
b(r)}{r^{3}}\right)+\frac{1}{r}\left(\phi^{\prime}(r)-\frac{rb^{\prime}(r)-b(r)}{2r(r-b(r)}\right)\right.\\\
\left.\times\left(1-\frac{2\alpha
b(r)}{r^{3}}\right)-\left(\frac{\left(rb^{\prime}(r)-b(r)\right)\phi^{\prime}(r)}{2r(r-b(r)}\right)\right.\\\
\left.\times\left(1-\frac{8\alpha}{r^{2}}+\frac{12\alpha
b(r)}{r^{3}}\right)\right]-\frac{2\alpha b^{2}(r)}{r^{6}},$ (13)

where the prime signifies a derivative w.r.t. radial coordinate $r$.
Let me briefly elaborate on classical energy conditions derived from the
Raychaudhuri equations. Given our focus on anisotropic fluid matter
distribution, the energy conditions derived from standard General Relativity
take the form
$\bullet$ Null energy condition (NEC) if $\rho+P_{r}\geq 0$, $\rho+P_{t}\geq
0$.
$\bullet$ Weak energy conditions (WEC) if $\rho\geq 0$, $\rho+P_{r}\geq 0$,
$\rho+P_{t}\geq 0$.
$\bullet$ Dominant energy conditions (DEC) if $\rho-\mathinner{\\!\left\lvert
p_{r}\right\rvert}\geq 0$, $\rho-\mathinner{\\!\left\lvert
p_{t}\right\rvert}\geq 0$.
$\bullet$ Strong energy conditions (SEC) if $\rho+P_{r}+2P_{t}\geq 0$.
Considering these, we will examine these energy conditions by defining
$NEC1=\rho+P_{r}$, $NEC2=\rho+P_{t}$, $DEC1=\rho-\mid P_{r}\mid$,
$DEC2=\rho-\mid P_{t}\mid$, $SEC=\rho+P_{r}+2P_{t}$
for this investigation.

##  III Embedding class-1 space-time

In the present work, our main focus is to develop a wormhole shape function
using the Karmarkar conditions that describe wormhole geometry. For this
purpose, we assume static spherically symmetric space-time defined as:

$ds^{2}=-e^{\xi(r)}dt^{2}+e^{\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (14)

where $\xi$ and $\lambda$ are function of radial coordinate $r$ only.
According to the above line element, the non-vanishing covariant Riemannian
curvature components are

$\,\,\,\,\,\,\,\,R_{1212}=R_{2121},\,\,\,R_{1221}=R_{2112},\,\,\,R_{1313}=R_{3131},\\\
R_{1331}=R_{3113},\,\,\,R_{2323}=R_{3232},\,\,\,R_{2332}=R_{3223},\\\
R_{1414}=R_{4141},\,\,\,R_{4114}=R_{1441},\,\,\,R_{4242}=R_{2424},\\\
R_{4224}=R_{2442},\,\,\,R_{4343}=R_{3434},\,\,\,R_{4334}=R_{3443}.$ (15)

The concept of embedding $n$-dimensional space $V_{n}$ within a pseudo-
Euclidean space $E_{n}$ has been a subject of significant interest, as
highlighted in the works of Eisland [90] and Eisenhart [91]. When an
$n$-dimensional space $V_{n}$ can be isometrically immersed in an
$(n+m)$-dimensional space, where $m$ represents the minimum number of
additional dimensions, $V_{n}$ is considered to possess $m$-Class embedding,
in that context, metric described in (14) typically yields a four-dimensional
spherically symmetric space-time, placing it within Class two, denoting $m=2$
and its embedding in a six-dimensional pseudo-Euclidean space. On a different
note, it’s worth mentioning that it is possible to devise a parametrization
that allows the space-time outlined in equation (14) to be incorporated into a
five-dimensional pseudo-Euclidean space, corresponding to Class $m=1$,
commonly referred to as embedding Class one [90, 91, 92]. In order for a
spherically symmetric space-time, whether static or non-static, to be
classified as Class-one, the system must adhere to the following requisite and
appropriate conditions:

  * •

the Gauss equation:

$\displaystyle\mathcal{R}_{mnpq}=2\,\epsilon\,{b_{m\,[p}}{b_{q]n}}\,,$ (16)

  * •

the Codazzi equation:

$\displaystyle
b_{m\left[n;p\right]}-{\Gamma}^{q}_{\left[n\,p\right]}\,b_{mq}+{{\Gamma}^{q}_{m}}\,{}_{[n}\,b_{p]q}=0.$
(17)

Here, we consider the case where $\epsilon=\pm 1$ and square brackets denote
antisymmetrization. The coefficients of the second differential form are
represented by $b_{mn}$. By utilizing Eqs. (16) and (17) and applying the
prescribed mathematical procedure, we can calculate the Karmarkar condition as
follows:

$R_{1414}=\frac{R_{1224}R_{1334}+R_{1212}R_{3434}}{R_{2323}},$ (18)

with $R_{2323}\neq 0$ [93]. The form of space-time satisfying the Karmarkar
condition is known as embedding class-I. By substituting the non-zero
components of Riemann curvature (15) in the Karmarkar relation (18), we get
the following differential equation:

$\frac{\xi^{\prime}(r)\lambda^{\prime}(r)}{1-e^{\lambda(r)}}=\xi^{\prime}(r)\lambda^{\prime}(r)-\xi^{\prime
2}(r)-2\xi^{{}^{\prime\prime}},$ (19)

where $e^{\lambda(r)}\neq 1$. On solving the above differential equation, one
can find

$e^{\lambda(r)}=1+c_{1}e^{\xi(r)}\xi^{{}^{\prime}2}(r),$ (20)

where $c_{1}$ is the constant of integration. Using the above expression (20),
we shall obtain wormhole shape functions for different redshift functions in
section-V.

##  IV Dark matter halos

Several inquiries are highlighted herein in the DM halos context, particularly
those involving galactic wormholes, primarily deduced from rotation curves
[94, 95]. It is essential to acknowledge that the postulation of “dark matter”
arose from the review of Oort constants [96] and investigations into the
masses of galaxy clusters [69, 70], aiming to explain the observed flat
rotation curves. To address this phenomenon, the Navarro-Frenk-White density
profile function [97, 98] and, alternatively, the Einasto profile was
suggested, with the latter demonstrating improved alignment with specific dark
matter halo simulations [99, 100]. Readers interested in galactic wormholes
are encouraged to refer to some of the recent relevant literature [101, 102,
103]. Generally, the wormhole throat is surrounded by varying types of DM
halos. The observed deviation from expected energy conditions substantiates
the presence of these dark halos. This study explores the representations
associated with different kinds of dark matter halos, incorporating wormhole
solutions within various DM halo profiles in this section.

###  IV.1 Universal Rotation Curve (URC) dark matter profile

For the first case, we consider the URC DM halo profile, which is defined by
[104]

$\rho(r)=\frac{\rho_{s}\,r_{s}^{3}}{(r+r_{s})(r^{2}+r_{s}^{2})},$ (21)

where $r_{s}$ and $\rho_{s}$ denote the core radius and central density,
respectively. The subsequent aspect under consideration is the rotation curve,
a parameter for which the analysis is widely regarded as a significant pillar
supporting the presence of dark matter in galaxies. Persic et al. [105], in
their investigation utilizing $H_{\alpha}$ data and incorporating certain
radio rotation curves, have observed an extensive array of rotation curves.
Their findings suggest that rotation curves can be effectively accommodated
for various luminosities and across various galaxy types, encompassing
spirals, low-surface-brightness ellipticals, and dwarf-irregular galaxies.
Consequently, they introduced the term “universal rotation curve” to replace
the conventional term “rotation curves,” reflecting the broad applicability of
the phenomenon.

###  IV.2 Navarro-Frenk-White (NFW) dark matter profile

An approximate analytical formulation of the NFW density profile is
established by drawing upon the Cold Dark Matter ($\Lambda$CDM) theory and
numerical simulations [106, 97, 98]. In the context of galaxies and clusters,
the dark matter halo can be characterized through the NFW density profile,
expressed as

$\rho(r)=\frac{\rho_{s}}{(r/r_{s})(1+r/r_{s})^{2}},$ (22)

where $\rho_{s}$ denotes the dark matter density during the collapse of the
dark matter halo, and $r_{s}$ represents the scale radius. It is widely
recognized that the NFW density profile encompasses a diverse range of dark
matter models characterized by minimal collision effects between dark matter
particles.

###  IV.3 Scalar Field Dark Matter (SFDM) profile

The SFDM model contains two distinct density profiles, namely the Bose-
Einstein Condensate (BEC) profile and the finite BEC profile [107, 108]. For
the sake of simplicity, we concentrate on the BEC profile. This particular
profile aligns with the static solution of the Klein-Gordon equation and
involves a quadratic potential governing the scalar field. The corresponding
SFDM profile [109, 110] of DM halo is given by

$\rho(r)=\frac{\rho_{s}\sin(\pi r/r_{s})}{\pi r/r_{s}},$ (23)

where $r_{s}$ denotes the radius at which the pressure and density are zero,
and $\rho_{s}$ is the central density.

##  V Wormhole solutions for different redshift functions

In this section, we will explore wormhole solutions with non-zero tidal force
solutions, where $\phi(r)\neq 0$. A non-constant redshift function can improve
the stability of a wormhole and enhance its traversability. By appropriately
choosing the redshift function, one can control the gravitational tidal forces
experienced by travelers. For instance, a well-chosen redshift function can
ensure that these tidal forces remain weak enough for safe passage through the
wormhole, which is essential for the traversability condition. However, many
researchers have focused on the zero tidal force (ZTF) model, where $\phi(r)$
is kept constant, to simplify the calculations of the energy conditions.
Maintaining a constant redshift may not be realistic from a physics
perspective. Therefore, removing this assumption and allowing for a non-
constant redshift function is crucial. In [111], the authors confirm that the
ZTF condition does not enable wormholes to exist with isotropic pressure. The
physical properties of traversable wormholes under a phantom energy state were
investigated in [112], and it was discussed that a non-constant redshift
function could help minimize the amount of exotic matter needed to sustain the
wormhole. Additionally, the impact of redshift functions on the WEC within the
framework of $f(R)$ gravity has been discussed in [113]. Thus, a non-constant
redshift function offers numerous theoretical advantages in wormhole geometry,
thereby improving their feasibility and versatility as solutions within GR and
beyond.
Motivated by the above lines, we shall consider two different non-constant
redshift functions to study the properties of wormhole solutions. Now, to
compute the shape functions, we compare the coefficients of the metrics (14)
and (9), and we obtain

$\xi(r)=2\phi(r)\,\,\,\,\,\,\,\,\,\text{and}\,\,\,\,\,\,\,\,e^{\lambda(r)}=\left(1-\frac{b(r)}{r}\right)^{-1}.$
(24)

Now, with the above relations, we could able to obtain the shape function from
Eq. (20) under different redshift functions.

###  V.1 $\phi(r)=-\frac{k}{r}$

For the first case, we consider the redshift function defined by

$\phi(r)=-\frac{k}{r},$ (25)

where $k$ is any positive constant. Note that this form of redshift function
satisfies the asymptotic flatness condition, i.e., $\phi(r)\rightarrow 0$ as
$r\rightarrow\infty$. Kar and Sahdev [114] proposed this specific redshift
function and investigated the nature of matter (WEC) and the embedding of the
space-like slices. They also discussed the human traversability of these space
times under this redshift function. Later, L. A. Anchordoqui et al. [115]
discussed evolving wormholes with this specific redshift function. They
studied some issues related to the WEC violation and human traversability in
these time-dependent geometries. Additionally, with this specific redshift
function, wormhole solutions have been investigated in different modified
theories of gravity, including $f(R)$ gravity [116], $f(R,\Phi)$ gravity
[117], and $f(Q)$ gravity [37].
Using Eqs. (20), (24) and (25), one can obtain the shape function

$b(r)=r-\frac{r^{5}}{r^{4}+4c_{1}k^{2}e^{-2k/r}}.$ (26)

Note that when we impose the throat condition, i.e., $b(r_{0})=r_{0}$, we get
the trivial solution $r_{0}=0$. Thus, to avoid this issue, we add a free
parameter $\delta$, and hence the shape function (26) becomes

$b(r)=r-\frac{r^{5}}{r^{4}+4c_{1}k^{2}e^{-2k/r}}+\delta.$ (27)

Now, to find the integrating constant $c_{1}$, we use the throat condition and
obtain

$c_{1}=\frac{r_{0}^{4}(r_{0}-\delta)}{4\delta k^{2}e^{-2k/r_{0}}}.$ (28)

Substituting the value of $c_{1}$ in Eq. (27), one can find the final version
of the shape function

$b(r)=r-\frac{\delta r^{5}}{\delta
r^{4}+r_{0}^{4}(r_{0}-\delta)e^{-2k(\frac{1}{r}-\frac{1}{r_{0}})}}+\delta,$
(29)

where $0<\delta<r_{0}$. Hence, in this case, the Morris-Throne wormhole metric
(9) can be read as

$ds^{2}=-e^{-\frac{2k}{r}}dt^{2}+\left(\frac{\delta r^{4}}{\delta
r^{4}+r_{0}^{4}(r_{0}-\delta)e^{-2k(\frac{1}{r}-\frac{1}{r_{0}})}}-\frac{\delta}{r}\right)^{-1}\\\
\times dr^{2}+r^{2}d\Omega^{2}_{D-2}.$ (30)

Clearly, in the limit $r\rightarrow\infty$, we obtain

$\lim\limits_{r\rightarrow\infty}\frac{b(r)}{r}\rightarrow 0,$

which confirms that the asymptotically flatness condition is satisfied.
Moreover, the flare-out condition is also satisfied at the throat for
$0<\delta<r_{0}$ as well as for the appropriate choice of the other free
parameters. The graphical behavior of the shape functions can be found in
Figs. 1 and 2.

Figure 1: Asymptotically flatness condition of the wormhole against the radial
distance $r$ for the redshift $\phi(r)=-\frac{k}{r}$. We use $k=0.2$ and
$r_{0}=1$. Figure 2: Flare-out condition of the wormhole against the radial
distance $r$ for the redshift $\phi(r)=-\frac{k}{r}$. We use $k=0.2$ and
$r_{0}=1$.

For the redshift function (25), the radial and tangential pressures can be
read as

$p_{r}=\frac{1}{8\pi}\left[\frac{1}{r^{7}}\left(-\delta
r\mathcal{L}_{2}+\alpha\delta^{2}(r-4k)+\alpha r^{3}-r^{5}\right)\right.\\\
\left.+\alpha\delta^{2}r^{3}(r-4k)\mathcal{L}_{1}^{2}+\frac{\delta\mathcal{L}_{1}}{r^{2}}\left(2k\left(4\alpha\delta+r^{3}+2\alpha
r\right)\right.\right.\\\ \left.\left.+r\left(-2\alpha\delta+r^{3}-2\alpha
r\right)\right)\right],$ (31)
$p_{t}=\frac{1}{8\pi}\left[\frac{\delta}{2r^{8}}\left(-8\alpha\delta
k^{2}+\mathcal{L}_{1}^{2}\left(2\delta
r^{9}\left(k^{2}\left(12\alpha\delta+r^{3}\right.\right.\right.\right.\right.\\\
\left.\left.\left.\left.\left.+12\alpha
r\right)-kr\left(26\alpha\delta+r^{3}+26\alpha
r\right)-2r^{2}\left(-2\alpha\delta+r^{3}\right.\right.\right.\right.\right.\\\
\left.\left.\left.\left.\left.-4\alpha
r\right)\right)\right)+\mathcal{L}_{1}\left(-8\alpha\delta
k^{2}r^{5}-6\alpha\delta r^{7}-8\alpha r^{8}+4r^{10}\right)\right.\right.\\\
\left.\left.-2k^{2}r^{3}-8\alpha k^{2}r+24\alpha
kr^{2}-\mathcal{L}_{1}^{3}\left(4\alpha\delta^{2}r^{14}(r-6k)(2r-k)\right)\right.\right.\\\
\left.\left.+3kr^{4}+32\alpha\delta kr-2\alpha\delta r^{2}-2\alpha
r^{3}+r^{5}\right)-\mathcal{L}_{3}\right],$ (32)

where,
$\mathcal{L}_{1}=\frac{1}{r_{0}^{4}(r_{0}-\delta)e^{2k\left(\frac{1}{r_{0}}-\frac{1}{r}\right)}+\delta
r^{4}}$,
$\mathcal{L}_{2}=\left(2k\left(2\alpha+r^{2}\right)+r^{3}-2\alpha r\right)$,
and $\mathcal{L}_{3}=\frac{2\alpha\left(\delta-\mathcal{L}_{1}\delta
r^{5}+r\right)^{2}}{r^{6}}$.
Now, for the URC model, the NEC can be read at the throat

$NEC1\bigg{|}_{r=r_{0}}=\frac{1}{8\pi
r_{0}^{4}}\left[\alpha-\frac{r_{0}^{2}\left(r_{s}^{3}\left(1-8\pi\rho_{s}r_{0}^{2}\right)+\mathcal{L}_{4}\right)}{(r_{s}+r_{0})\left(r_{s}^{2}+r_{0}^{2}\right)}\right],$
(33)
$NEC2\bigg{|}_{r=r_{0}}=\frac{\rho_{s}r_{s}^{3}}{(r_{s}+r_{0})\left(r_{s}^{2}+r_{0}^{2}\right)}+\frac{1}{16\pi
r_{0}^{8}}\left[2\delta^{2}\right.\\\
\left.\times(k-2r_{0})\left(k\left(4\alpha+r_{0}^{2}\right)+r_{0}^{3}-2\alpha
r_{0}\right)+\delta r_{0}(5r_{0}-2k)\right.\\\
\left.\left(k\left(4\alpha+r_{0}^{2}\right)+r_{0}^{3}-2\alpha
r_{0}\right)-4\alpha r_{0}^{4}\right],$ (34)

where $\mathcal{L}_{4}=r_{s}^{2}r_{0}+r_{s}r_{0}^{2}+r_{0}^{3}$.
Again, for the NFW model, the expression for NEC can read as

$NEC1=\frac{\rho_{s}r_{s}^{3}}{r(r_{s}+r)^{2}}+\frac{1}{8\pi}\left[\frac{1}{r^{7}}\left(-\delta
r\mathcal{L}_{2}+\alpha\delta^{2}(r-4k)\right.\right.\\\ \left.\left.+\alpha
r^{3}-r^{5}\right)+\mathcal{L}_{1}^{2}\alpha\delta^{2}r^{3}(r-4k)+\frac{\delta\mathcal{L}_{1}}{r^{2}}\left(2k\left(4\alpha\delta+r^{3}\right.\right.\right.\\\
\left.\left.\left.+2\alpha r\right)+r\left(-2\alpha\delta+r^{3}-2\alpha
r\right)\right)\right].$ (35)

At wormhole throat $r=r_{0}$, the above expression reduces to

$NEC1\bigg{|}_{r=r_{0}}=\frac{1}{8\pi
r_{0}^{2}}\left[\frac{\alpha}{r_{0}^{2}}-\frac{1}{(r_{s}+r_{0})^{2}}\left(-8\pi\rho_{s}r_{s}^{3}r_{0}+r_{s}^{2}\right.\right.\\\
\left.\left.+2r_{s}r_{0}+r_{0}^{2}\right)\right].$ (36)

Similarly, we could obtain the NEC at the throat for the SFDM model as

$NEC1\bigg{|}_{r=r_{0}}=\frac{\alpha+8\rho_{s}r_{s}r_{0}^{3}\sin\left(\frac{\pi
r_{0}}{r_{s}}\right)-r_{0}^{2}}{8\pi r_{0}^{4}}.$ (37)

Figure 3: The variation of $\rho+p_{r}$ against the radial distance $r$ for
the redshift $\phi(r)=-\frac{k}{r}$. we use $\rho_{s}=0.004\,kpc^{-2}$,
$r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and $r_{0}=1$.

Figure 4: The variation of $\rho+p_{r}+2p_{t}$ against the radial distance $r$
for the redshift $\phi(r)=-\frac{k}{r}$. we use $\rho_{s}=0.004\,kpc^{-2}$,
$r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and $r_{0}=1$.

Figure 5: The variation of $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}$
against the radial distance $r$ for the redshift $\phi(r)=-\frac{k}{r}$. we
use $\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and
$r_{0}=1$.

Now, we will discuss the energy conditions for URC, NFW, and SFDM DM halo
profiles. We consider some particular choice of free parameters
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$ [118], throat radius $r_{0}=1$,
$\delta=0.9$ (as $0<\delta<r_{0}$) and $k=0.2$. Mathematically, one can check
the RHS of Eqs. (33), (36), and (37) by simply putting the values of the
parameters and taking the GB coefficient $\alpha>0$ provides a negative
quantity. This confirms the violation of NEC at the wormhole throat.
Graphically, we have presented the behavior of NEC for each DM profile in Fig.
3. Moreover, we have studied other energy conditions, such as SEC and DEC in
Figs. 4 and 5. SEC is violated near the throat; however, far from the throat,
SEC will be satisfied. Moreover, radial DEC is satisfied at the throat. But
for $\alpha>>0$, DEC will disrespect the energy conditions.

###  V.2 $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$

In this case, we consider the redshift function of the form

$\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}}),$ (38)

where $\eta$ is any positive parameter. Note that this non-constant redshift
function respects the asymptotic condition of a traversable wormhole. With the
above choice of redshift function, Jusufi et al. [119] studied traversable
wormholes supported by Casimir energy in general relativity. Also, in [40],
GUP-Corrected Casimir wormholes have been discussed in modified $f(Q)$ gravity
with this logarithmic redshift function. Motivated by the above articles, we
consider this specific choice of redshift function to check the stability of
wormhole solutions in $4D$ EGB gravity.
Similar to the previous subsection, we could obtain the shape function by
using Eqs. (20), (24) and (38),

$b(r)=r-\frac{r^{5}\left(r^{2}+\eta^{2}\right)}{r^{4}\left(r^{2}+\eta^{2}\right)+4\eta^{2}c_{1}}.$
(39)

It is clear that when we impose the throat condition on the above equation, we
get the trivial solution $r_{0}=0$. Thus, we introduced a free parameter
$\lambda$ to the above shape function, and hence it becomes

$b(r)=r-\frac{r^{5}\left(r^{2}+\eta^{2}\right)}{r^{4}\left(r^{2}+\eta^{2}\right)+4\eta^{2}c_{1}}+\lambda.$
(40)

Now, we impose $b(r_{0})=r_{0}$ to the above expression (40) to obtain $c_{1}$

$c_{1}=\frac{\mathcal{K}_{1}}{4\eta^{2}\lambda},$ (41)

where
$\mathcal{K}_{1}=r_{0}^{4}\left(r_{0}^{2}+\eta^{2}\right)\left(r_{0}-\lambda\right)$.
Substituting the value of $c_{1}$ in Eq. (40), we can obtain the shape
function

$b(r)=r-\frac{r^{5}\left(r^{2}+\eta^{2}\right)\lambda}{\lambda
r^{4}\left(r^{2}+\eta^{2}\right)+\mathcal{K}_{1}}+\lambda.$ (42)

where $0<\lambda<r_{0}$. Hence, the Morris-Throne wormhole metric (9) reduces
to

$ds^{2}=-\left(1+\frac{\eta^{2}}{r^{2}}\right)^{-1}dt^{2}+\\\
\left(\frac{r^{4}\left(r^{2}+\eta^{2}\right)\lambda}{\lambda
r^{4}\left(r^{2}+\eta^{2}\right)+\mathcal{K}_{1}}-\frac{\lambda}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}_{D-2},$
(43)

which is asymptotically flat space-time. In fact, one can smoothly verify that
the case $0<\lambda<r_{0}$ gives the asymptotically flat solution. Also, we
investigate the important criteria of a traversable wormhole is the flare-out
condition, which is also satisfied within $0<\lambda<r_{0}$. In this case, we
consider the throat radius $r_{0}=1$ and $\eta=2$. The graphical behavior of
shape functions can be found in Figs. 6 and 7.

Figure 6: Asymptotically flatness condition of the wormhole against the radial
distance $r$ for the redshift
$\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. We use $\eta=2$ and
$r_{0}=1$. Figure 7: Flare-out condition of the wormhole against the radial
distance $r$ for the redshift
$\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. We use $\eta=2$ and
$r_{0}=1$.

For the given redshift function (38) and shape function (42), we can compute
the components of the energy-momentum tensor. In fact, the radial pressure can
be read as follows

$p_{r}=\frac{1}{8\pi
r^{6}}\left[\alpha\left(\lambda-\frac{4\lambda^{3}\left(\mathcal{K}_{2}-1\right)}{r^{2}\left(\frac{\lambda^{2}}{r^{2}}+1\right)}-\lambda\mathcal{K}_{2}+r\right)\left(\lambda-\lambda\mathcal{K}_{2}\right.\right.\\\
\left.\left.+r\right)-\frac{2\lambda^{3}r\left(\mathcal{K}_{2}-1\right)}{\frac{\lambda^{2}}{r^{2}}+1}-\left(r^{3}\left(\lambda-\lambda\mathcal{K}_{2}+r\right)\right)\right],$
(44)

where $\mathcal{K}_{2}=\frac{r^{5}\left(\eta^{2}+r^{2}\right)}{\lambda
r^{4}\left(\eta^{2}+r^{2}\right)+r_{0}^{4}\left(\eta^{2}+r_{0}^{2}\right)(r_{0}-\lambda)}$.
On the other hand, tangential pressure in this case can be read as

$p_{t}=\frac{1}{8\pi}\left[\frac{1}{r}\left(\lambda\left(\mathcal{K}_{2}-1\right)\left(\frac{\lambda^{2}\left(\mathcal{K}_{3}+1\right)}{2r^{4}\left(\frac{\lambda^{2}}{r^{2}}+1\right)\left(1-\mathcal{K}_{2}\right)}\right.\right.\right.\\\
\left.\left.\left.\times\left(\frac{12\alpha\lambda}{r^{3}}+\frac{4\alpha}{r^{2}}-\frac{12\alpha\lambda\mathcal{K}_{2}}{r^{3}}+1\right)+\frac{\left(2\lambda^{4}+3\lambda^{2}r^{2}\right)}{r^{2}\left(\lambda^{2}+r^{2}\right)^{2}}\right.\right.\right.\\\
\left.\left.\left.\times\left(\frac{4\alpha\left(\lambda-\lambda\mathcal{K}_{2}+r\right)}{r^{3}}+1\right)+\frac{1}{r}\left(\left(-\frac{\lambda^{2}}{r^{3}+\lambda^{2}r}\right.\right.\right.\right.\right.\\\
\left.\left.\left.\left.\left.+\frac{\mathcal{K}_{3}+1}{2r\left(\mathcal{K}_{2}-1\right)}\right)\left(1-\frac{2\alpha\left(\lambda-\lambda\mathcal{K}_{2}+r\right)}{r^{3}}\right)\right)\right)\right)\right.\\\
\left.-\frac{2\alpha\left(\lambda-\lambda\mathcal{K}_{2}+r\right)^{2}}{r^{6}}\right],$
(45)

where
$\mathcal{K}_{3}=\frac{2r^{5}r_{0}^{4}\left(2\eta^{2}+3r^{2}\right)\left(\eta^{2}+r_{0}^{2}\right)(r_{0}-\lambda)}{\left(\lambda
r^{4}\left(\eta^{2}+r^{2}\right)-\eta^{2}\lambda
r_{0}^{4}+\eta^{2}r_{0}^{5}-\lambda r_{0}^{6}+r_{0}^{7}\right)^{2}}$.
Now, with the above pressure components, we can examine the energy conditions
for the URC, NFW, and SFDM dark matter halo profiles and try to generate plots
to evaluate the validity of the energy conditions.
The NEC at wormhole throat ($r=r_{0}$) for each DM halo profile has been
obtained and shown in Eq. (46).

$NEC1\bigg{|}_{r=r_{0}}=\begin{cases}\frac{1}{8\pi
r_{0}^{4}}\left[\alpha-\frac{r_{0}^{2}\left(r_{s}^{3}\left(1-8\pi\rho_{s}r_{0}^{2}\right)+r_{s}^{2}r_{0}+r_{s}r_{0}^{2}+r_{0}^{3}\right)}{(r_{s}+r_{0})\left(r_{s}^{2}+r_{0}^{2}\right)}\right],&\text{(URC
profiles)}\\\ \\\
\frac{\alpha(r_{s}+r_{0})^{2}-r_{0}^{2}\left(-8\pi\rho_{s}r_{s}^{3}r_{0}+r_{s}^{2}+2r_{s}r_{0}+r_{0}^{2}\right)}{8\pi
r_{0}^{4}(r_{s}+r_{0})^{2}},&\text{(NFW profiles)}\\\ \\\
\frac{\alpha+8\rho_{s}r_{s}r_{0}^{3}\sin\left(\frac{\pi
r_{0}}{r_{s}}\right)-r_{0}^{2}}{8\pi r_{0}^{4}},&\text{(SFDM
profiles)}\end{cases}$ (46)

Note that the above expression (46) is independent of the free parameters
$\eta$ and $\lambda$, which confirms no influence of those parameters at the
throat. However, outside the throat, we can see their influences. Also, in
this case, we consider the appropriate choice of free parameters
$\lambda=0.3$, $\eta=2$, $r_{0}=1$ along with the same values of $\rho_{s}$
and $r_{s}$, which we choose in the previous subsection. Numerically, one can
check by putting the above values along with $\alpha>0$, which gives a
negative quantity of NEC. Graphically, we have presented the behavior of NEC
in Fig. 8. Moreover, we studied SEC in Fig. 9 and found that SEC is violated
near the throat for $\alpha>0$. We also noticed that as we increase the values
of $\alpha$, the contribution of violation becomes more. Further, we checked
DEC and observed that DEC is satisfied in the vicinity of the throat (see Fig.
10).
A detailed summary of the energy conditions for URC, NFW, and SFDM galactic DM
profiles against both redshift functions have been calculated and shown in
Table- 1.

Figure 8: The variation of $\rho+p_{r}$ against the radial distance $r$ for
the redshift $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. we use
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and
$r_{0}=1$.

Figure 9: The variation of $\rho+p_{r}+2p_{t}$ against the radial distance $r$
for the redshift $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. we use
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and
$r_{0}=1$.

Figure 10: The variation of $\rho-\mathinner{\\!\left\lvert
p_{r}\right\rvert}$ against the radial distance $r$ for the redshift
$\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. we use
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and
$r_{0}=1$. The behavior of the energy conditions around the throat
---
Physical expressions |  $\phi(r)=-\frac{k}{r}$ with $\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and $r_{0}=1$ |  $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$ with $\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and $r_{0}=1$
Energy conditions for URC profile
$\rho$ |  $\rho>0$ |  $\rho>0$
$\rho+p_{r}$ |  $\rho+p_{r}<0$ for $\alpha>0$ |  $\rho+p_{r}<0$ for $\alpha>0$
$\rho+p_{t}$ |  $\rho+p_{t}<0$ for $\alpha>0$ |  $\rho+p_{t}<0$ for $\alpha>0$
$\rho+p_{r}+2p_{t}$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}<0$ for $\alpha>0$
Energy conditions for NFW profile
$\rho$ |  $\rho>0$ |  $\rho>0$
$\rho+p_{r}$ |  $\rho+p_{r}<0$ for $\alpha>0$ |  $\rho+p_{r}<0$ for $\alpha>0$
$\rho+p_{t}$ |  $\rho+p_{t}<0$ for $\alpha>0$ |  $\rho+p_{t}<0$ for $\alpha>0$
$\rho+p_{r}+2p_{t}$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}>0$ for $\alpha>0$
Energy conditions for SFDM profile
$\rho$ |  $\rho>0$ |  $\rho>0$
$\rho+p_{r}$ |  $\rho+p_{r}<0$ for $\alpha>0$ |  $\rho+p_{r}<0$ for $\alpha>0$
$\rho+p_{t}$ |  $\rho+p_{t}<0$ for $\alpha>0$ |  $\rho+p_{t}<0$ for $\alpha>0$
$\rho+p_{r}+2p_{t}$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$ |  $\rho+p_{r}+2p_{t}<0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{r}\right\rvert}>0$ for $\alpha>0$
$\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}>0$ for $\alpha>0$ |  $\rho-\mathinner{\\!\left\lvert p_{t}\right\rvert}>0$ for $\alpha>0$
Table 1: Outlook of the energy conditions

##  VI Some physical features of wormhole solutions

In this particular section, we study some analysis of the obtained wormhole
solutions such as the complexity factor, active gravitational mass, total
gravitational energy, and the embedding diagrams of the wormholes.

###  VI.1 Complexity factor

A recent contribution in Ref. [120] proposed a novel complexity definition for
self-gravitating fluid distributions. This definition centers on the intuitive
concept that the gravitational system’s least complexity should be represented
by a homogeneous energy density distribution accompanied by isotropic
pressure. The work in [120] reveals that in static spherically symmetric
space-times, an associated scalar emerges from the orthogonal splitting of the
Riemann tensor [121, 122]. This scalar, denoted as $\Upsilon_{TF}$, summarizes
the core aspects of complexity, and its expression can be read as

$\Upsilon_{TF}=8\pi\Delta-\dfrac{4\pi}{r^{3}}\int_{0}^{r}\tilde{r}^{3}\rho^{\prime}(r)d\tilde{r}\,,$
(47)

where $\Delta=p_{r}-p_{t}$. Moreover, the above expression (47) allows us to
write the Tolman mass as

$m_{T}=(m_{T})_{\Sigma}\left(\frac{r}{r_{\Sigma}}\right)^{3}+r^{3}\int_{0}^{r_{\Sigma}}\frac{e^{(\xi+\lambda)/2}}{\tilde{r}}\Upsilon_{TF}d\tilde{r},$
(48)

and this serves as a vital justification for defining the complexity factor
using the aforementioned scalar, as it effectively incorporates all
modifications arising from both energy density inhomogeneity and pressure
anisotropy on the active gravitational mass. Notably, the condition of
vanishing complexity $(\Upsilon_{TF}=0)$ can be fulfilled not solely in the
simplest scenario of isotropic and homogeneous systems but also across all
cases where

$\Delta=\dfrac{1}{2r^{3}}\int_{0}^{r}\tilde{r}^{3}\rho^{\prime}(r)d\tilde{r}.$
(49)

In this context, fulfilling the vanishing complexity condition results in a
non-local equation of state, providing an additional condition to close the
system of Einstein’s field equations. This application has been presented in
some recent papers, as seen in Refs. [123, 124, 125].
In this paper, we are interested in studying the complexity factor for the
wormhole solutions under galactic regions. It is known that wormholes are
defined for $r_{0}\leq r<\infty$, in that case the complexity factor (47)
should be modified and can be redefined as

$\Upsilon_{TF}=8\pi\Delta-\dfrac{4\pi}{r^{3}}\int_{r_{0}}^{r}\tilde{r}^{3}\rho^{\prime}(r)d\tilde{r}.$
(50)

Note that the standard definition requires $r_{0}=0$, but we must discard this
case to ensure a finite size of the wormhole throat.

Figure 11: The dynamics of $\Upsilon_{TF}$ against the radial distance $r$ for
the redshift $\phi=\frac{-k}{r}$. we use $\rho_{s}=0.004\,kpc^{-2}$,
$r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and $r_{0}=1$.

Figure 12: The dynamics of $\Upsilon_{TF}$ against the radial distance $r$ for
the redshift $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. we use
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and
$r_{0}=1$.

Now, considering the relevant equations, we have studied the behavior of the
complexity factor (using the above Eq. (50)) for three galactic halos
wormholes with respect to the radial coordinate shown in Figs. 11 and 12. We
noticed that the complexity factor $\Upsilon_{TF}$ tends toward zero as the
radial coordinate approaches infinity $r\rightarrow\infty$ or moves away from
the wormhole throat. As explained in [120], a minimal complexity factor,
corresponds to a configuration characterized by homogeneous energy density and
isotropic pressure. Additionally, a complexity factor of zero suggests the
presence of inhomogeneous energy density and anisotropic pressure, provided
these two effects counterbalance each other in the overall complexity factor.
Consequently, in the vicinity of the wormhole throat, the complexity factor
exhibits a monotonically increasing direction, and as we increase the range of
the radial coordinate, $\Upsilon_{TF}$ approaches zero. Moreover, we noticed
that the above holds for any values of $\alpha>0$. Therefore, the complexity
factor converges to zero for high radial coordinates and $\alpha>0$ in the
context of galactic halo wormholes in EGB gravity. Further, in the dynamics of
the complexity factor, the role of pressure isotropy emerges as more crucial
compared to the homogeneity of energy density. Furthermore, one can read some
recent interesting papers on this topic given in Refs. [126, 127, 128].

###  VI.2 Active gravitational mass

The active mass function for our wormhole within the region from the wormhole
throat $r_{0}$ up to the radius $R$ can be read as

$\mathcal{M}_{\mathcal{A}}=4\pi\int_{r_{0}}^{R}\rho(r)r^{2}dr,$ (51)

where $\mathcal{M}_{\mathcal{A}}$ is the active gravitational mass. Note that
the positive nature of the active gravitational mass indicates that the models
are physically acceptable. Keeping this in mind, we shall investigate three
different types of DM halo models and try to find some conditions for which
one can find positive mass function.
For the URC profile, the expression for $\mathcal{M}_{\mathcal{A}}$ can be
read as

$\mathcal{M}_{\mathcal{A}}=4\pi\rho_{s}r_{s}^{3}\left[\frac{1}{4}\log\left(r_{s}^{2}+r^{2}\right)+\frac{1}{2}\log(r_{s}+r)\right.\\\
\left.-\frac{1}{2}\tan^{-1}\left(\frac{r}{r_{s}}\right)\right]_{r_{0}}^{R}.$
(52)

It is observed from the above expression (52) that the active gravitational
mass $\mathcal{M}_{\mathcal{A}}$ of the wormhole for URC galactic is positive
under the constraint
$\frac{1}{4}\log\left(r_{s}^{2}+r^{2}\right)+\frac{1}{2}\log(r_{s}+r)>\frac{1}{2}\tan^{-1}\left(\frac{r}{r_{s}}\right)$.
Again, for the NFW profile, we can obtain the active gravitational mass

$\mathcal{M}_{\mathcal{A}}=4\pi\rho_{s}r_{s}^{3}\left[\frac{r_{s}}{r_{s}+r}+\log(r_{s}+r)\right]_{r_{0}}^{R}.$
(53)

Note that the RHS of the Eq. (53) is a positive quantity for any values of the
parameters.
At last, for the SFDM profile, $\mathcal{M}_{\mathcal{A}}$ can be read as

$\mathcal{M}_{\mathcal{A}}=4\rho_{s}r_{s}\left[\frac{r_{s}^{2}\sin\left(\frac{\pi
r}{r_{s}}\right)}{\pi^{2}}-\frac{r_{s}r\cos\left(\frac{\pi
r}{r_{s}}\right)}{\pi}\right]_{r_{0}}^{R}.$ (54)

In this case, the active gravitational mass is positive under the constraint
$\frac{r_{s}^{2}\sin\left(\frac{\pi
r}{r_{s}}\right)}{\pi^{2}}>\frac{r_{s}r\cos\left(\frac{\pi
r}{r_{s}}\right)}{\pi}$.
Thus, we can conclude that the above DM models are physically acceptable under
some restrictions.

###  VI.3 Total gravitational energy

We have already noticed that the material forming the wormhole violates the
NEC and, therefore, must be exotic rather than normal matter. The total
gravitational energy of a structure composed of normal baryonic matter is
negative. Therefore, it is crucial to check the nature of the gravitational
energy in a wormhole background. Here, we follow the works of Lyndell-Bell et
al. [129], and Nandi et al. [130] and will try to find out the total
gravitational energy of the DM galactic wormholes. The total gravitational
energy $\mathcal{E}_{g}$ can be define as [130]

$\mathcal{E}_{g}=\mathcal{M}c^{2}-\mathcal{E}_{M},$ (55)

where $\mathcal{M}c^{2}$ represents the total energy, and it can be expressed
as

$\mathcal{M}c^{2}=\frac{1}{2}\int_{r_{0}}^{r}\mathcal{T}_{0}^{0}r^{2}dr+\frac{r_{0}}{2},$

where the quantity $\frac{r_{0}}{2}$ is linked to the effective mass [130].
$\mathcal{E}_{M}$ is the sum of other forms of energy like kinetic energy,
rest energy, internal energy, etc., defined by

$\mathcal{E}_{M}=\frac{1}{2}\int_{r_{0}}^{r}\mathcal{T}_{0}^{0}(g_{rr})^{\frac{1}{2}}r^{2}dr,\,\,\,\text{with}\quad
g_{rr}=\left(1-\frac{b(r)}{r}\right)^{-1}.$

It can be claimed that since $(g_{rr})^{\frac{1}{2}}>1$ (by definition), then
one can instantly deduce the criteria that $\mathcal{E}_{g}<0$ (attractive) if
$\mathcal{T}_{0}^{0}>0$ and $\mathcal{E}_{g}>0$ (repulsive) if
$\mathcal{T}_{0}^{0}<0$ [131]. Since it is very complicated to find the exact
solutions of the integral (55), we solve it numerically by setting the
integration range from the throat at $r_{0}$ to the embedded radial space of
the wormhole geometry (refer to Tables (2) and (3)). As depicted in Figs. 13
and 14, the results demonstrate that $\mathcal{E}_{g}>0$ signifies the
repulsive behavior in the proximity of the throat. Note that we obtain
$\mathcal{E}_{g}>0$ here despite $\mathcal{T}_{0}^{0}>0$. This is because the
matter distribution supporting the wormhole structure violates the NEC.
Similar behavior of $\mathcal{E}_{g}$ can be found in [132, 133, 134]. This
repulsive nature of $\mathcal{E}_{g}$ aligns with expectations for the
formation of a physically viable wormhole.

The values of $\mathcal{E}_{g}$ for different $r$.
---
$r$ |  URC |  NFW |  SFDM
1.5 |  0.498883 |  0.498341 |  0.498766
2.0 |  0.498456 |  0.497876 |  0.498566
2.5 |  0.498142 |  0.497577 |  0.498722
3.0 |  0.497897 |  0.49736 |  0.499071
3.5 |  0.497701 |  0.497193 |  0.499401
4.0 |  0.497541 |  0.497058 |  0.499534
4.5 |  0.497408 |  0.496948 |  0.499406
5.0 |  0.497296 |  0.496855 |  0.499104
Table 2: Table shows the values of $\mathcal{E}_{g}$ for different DM profiles
under the shape function (29). We use $\rho_{s}=0.004\,kpc^{-2}$,
$r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and $r_{0}=1$. The values of
$\mathcal{E}_{g}$ for different $r$.
---
$r$ |  URC |  NFW |  SFDM
1.5 |  0.499035 |  0.498545 |  0.49892
2.0 |  0.498829 |  0.49832 |  0.498818
2.5 |  0.498716 |  0.498212 |  0.498872
3.0 |  0.498639 |  0.498143 |  0.498982
3.5 |  0.498579 |  0.498092 |  0.499083
4.0 |  0.498531 |  0.498052 |  0.499122
4.5 |  0.498491 |  0.498019 |  0.499084
5.0 |  0.498457 |  0.497991 |  0.498992
Table 3: Table shows the values of $\mathcal{E}_{g}$ for different DM profiles
under the shape function (42). We use $\rho_{s}=0.004\,kpc^{-2}$,
$r_{s}=2\,kpc$, $\lambda=0.3$, $\eta=2$ and $r_{0}=1$. Figure 13: The
variation of $\mathcal{E}_{g}$ under the shape function (29). We use
$\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$, $\delta=0.9$, $k=0.2$ and
$r_{0}=1$. Figure 14: The variation of $\mathcal{E}_{g}$ under the shape
function (42). We use $\rho_{s}=0.004\,kpc^{-2}$, $r_{s}=2\,kpc$,
$\lambda=0.3$, $\eta=2$ and $r_{0}=1$.

###  VI.4 Embedding diagram

This subsection will discuss the embedding diagram that helps us visualize
wormhole space-time. As per the framework outlined by Morris and Thorne [19],
the embedding surface of the wormhole is represented by the function $z(r)$,
which satisfies the given differential equation:

$\frac{dz}{dr}=\pm\frac{1}{\sqrt{\frac{r}{b(r)}-1}}.$ (56)

In this expression, it is observed that $\frac{dz}{dr}$ diverges at the
wormhole’s throat, implying that the embedding surface assumes a vertical
orientation at the throat. Note that the above differential equation (56)
provides the following relation

$z(r)=\pm\int_{r_{0}}^{r}\frac{dr}{\sqrt{\frac{r}{b(r)}-1}}.$ (57)

Further, the radial distance of the wormhole can be read as

$l(r)=\pm\int_{r_{0}}^{r}\frac{dr}{\sqrt{1-\frac{r}{b(r)}}}.$ (58)

Note that the above integral is given in Eqs. (57) cannot be solved
analytically. Hence, we will solve it numerically by fixing some values of the
free parameters and by changing the upper limit $r$. The numerical plot for
embedding diagram $z(r)$ for the shape functions (29) and (42) is given in
Fig. 15. Moreover, one can check Fig. 16 for the full visualization of
wormholes.

Figure 15: $2D$ embedding diagram for the shape functions (29) (Red) and (42)
(Black). We use $\delta=0.9$, $k=0.2$, $\lambda=0.3$, $\eta=2$ and $r_{0}=1$.

Figure 16: Full visualization of the embedding diagram for the shape functions
(29) (left) and (42) (right). Left panel: we use $r_{0}=1$, $\delta=0.9$,
$k=0.2$. Right panel: we use $r_{0}=1$, $\lambda=0.3$, $\eta=2$.

##  VII Discussions and conclusions

Indeed, exploring wormhole geometry has recently sparked considerable
enthusiasm among theoretical researchers. Consequently, wormholes have been
identified in the galactic halo region, supported by various dark matter
density profiles. This paper searches for wormhole existence in the galactic
halo supported by three different DM profiles, such as URC, NFW, and SFDM
models within recently proposed $4D$ EGB gravity. Also, we employed the
Karmarkar condition to find the shape function for two different non-constant
redshift functions. The detailed outcomes of this work are as follows:
Firstly, we obtained the shape function using an embedding class- I approach
under the redshift function $\phi(r)=-\frac{k}{r}$. We noticed that the
obtained shape function follows the flare-out condition under asymptotically
flatness conditions within the range $0<\delta<r_{0}$. Later, we inserted the
shape function into the pressure components of the $4D$ EGB gravity. Further,
we have investigated the energy conditions in the galactic halo with DM
density profiles. We have observed that NEC is violated in the neighborhood of
the throat. It was noticed the influence of GB coefficient $\alpha$ in the
violation of NEC. Mathematically, one can check the RHS of Eqs. (33), (36),
and (37) by simply putting the values of the parameters and taking the GB
coefficient $\alpha>0$ provides a negative quantity. Also, we checked SEC for
each DM profile and found that SEC was violated. Moreover, as we increase the
value of $\alpha$, the contribution of violation becomes more. Additionally,
DEC for both pressures was found satisfying near the throat. Moreover, one can
check the summary of the energy conditions near the throat of the wormhole in
Table-1.
Similar to the previous case, we extracted the shape function under the
redshift function $\phi(r)=\frac{1}{2}\log(1+\frac{\eta^{2}}{r^{2}})$. We
investigated the necessary criteria for a traversable wormhole, i.e., the
flare-out condition, which is satisfied in the range $0<\lambda<r_{0}$. Later,
we checked the NEC, SEC, and DEC for obtained solutions in the galactic halo
regions. NEC is disrespected at the throat for $\alpha>0$. Moreover, one can
check numerically from the expression given in Eq. (46). DEC was investigated
for each DM profile, and it was found that DEC was satisfied with both
pressures in the galactic halos, except tangential DEC, which is violated in
the URC DM profile. The calculated energy conditions are shown in Table-1.
Note that the confirming violated behavior of energy conditions supports the
presence of the dark halos. We checked the behavior of energy conditions for
each DM halo profile and noticed the violations of energy conditions, which
means that traversable wormholes may exist in the galactic regions supported
by dark matter in the context of $4D$ EGB gravity.
Further, some physical features of wormholes, such as complexity factor,
active gravitational mass, total gravitational mass, and embedding diagrams,
have been explored in this paper. The complexity factor in the context of
galactic DM halo wormholes in EGB gravity has been calculated, and it was
noticed that the complexity factor converges to zero for high radial
coordinates and GB coefficient $\alpha>0$. Such a study has been done in Ref.
[128] for Casimir wormholes in higher dimensional EGB gravity. Moreover, the
active gravitational mass for each DM density profile has been performed, and
it has been observed that these DM models are physically acceptable under some
restrictions. Rahaman et al. [135] studied the active gravitational mass of
the NFW profile; however, in this paper, we have studied the active
gravitational mass of URC and SFDM models along with the NFW model. Further,
we have studied and numerically calculated the total gravitational energy for
each DM profile under obtained shape functions. it was observed that
$\mathcal{E}_{g}>0$, which signifies the presence of repulsion near the
throat. This characteristic nature of $\mathcal{E}_{g}$ aligns with
expectations for the formation of a physically viable wormhole.
In [136], wormhole geometry in the galactic halos has been explored with two
embedded wormhole-specific shape functions in the context of Einsteinian cubic
gravity. They used observational data within the signature of the M87 galaxy
and the Milky Way galaxy to check the effect of the dark matter halos.
Further, in [137], the author discussed wormhole solution with density profile
obtained from modified Newtonian dynamics (MOND) with or without a scalar
field in GR. Recently, wormhole solutions have been explored in $f(Q,T)$
gravity under different dark matter profiles such as URC and NFW profiles
[138]. In this paper, we have investigated wormhole solutions under URC, NFW,
and SFDM profiles in $4D$ EGB gravity with two different redshift functions.
Our study confirms that the obtained wormhole solutions might exist in the
galactic halos within $4D$ EGB gravity. Also, traversable wormholes in the
galactic halos with observational data sets in higher dimensional gravity
would be an interesting problem that is being actively considered.

## Data Availability

There are no new data associated with this article.

###### Acknowledgements.

ZH acknowledges the Department of Science and Technology (DST), Government of
India, New Delhi, for awarding a Senior Research Fellowship (File No.
DST/INSPIRE Fellowship/2019/IF190911). PKS acknowledges the National Board for
Higher Mathematics (NBHM) under the Department of Atomic Energy (DAE), Govt.
of India, for financial support to carry out the Research project No.:
02011/3/2022 NBHM(R.P.)/R&D II/2152 Dt.14.02.2022. We are very much grateful
to the honorable referees and to the editor for the illuminating suggestions
that have significantly improved our work in terms of research quality, and
presentation.

## References

  * [1] Leonard I. Schiff, Am. J. Phys. 28(4), 340-343 (1960).
  * [2] L. Flamm, Phys. Z. 17, 448 (1916).
  * [3] A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).
  * [4] J. A. Wheeler, Ann. Phys. 2, 604 (1957).
  * [5] R. W. Fuller and J. A. Wheeler, Phys. Rev. 128, 919 (1962).
  * [6] V. Khatsymovsky, Phys. Lett. B 320, 234 (1994).
  * [7] A. Ovgun, Eur. Phys. J. Plus 136, 987 (2021).
  * [8] W. Javed, S. Riaz, R. C. Pantig, and A. Ovgun, Eur. Phys. J. C 82, 1057 (2022).
  * [9] Y. Kumaran and A. Ovgun, Turk. J. Phys. 45, 247-267 (2021).
  * [10] A. Ovgun, Turk. J. Phys. 44, 465-471 (2020).
  * [11] W. Javed, R. Babar, and A. Ovgun, Phys. Rev. D 99, 084012 (2019).
  * [12] A. Ovgun, Phys. Rev. D 98, 044033 (2018).
  * [13] K. Jusufi and A. Ovgun, Phys. Rev. D 97, 024042 (2018).
  * [14] J. Maldacena, and L. Susskind, Fortschr. Phys. 61, 781-811 (2013).
  * [15] M. Visser, Nuclear Phys. B 328, 203-212 (1989).
  * [16] M. G. Richarte et al. Phys. Rev. D 96, 084022 (2017).
  * [17] A. Ovgun, Eur. Phys. J. Plus 131, 389 (2016).
  * [18] J. Maldacena, D. Stanford, Z. Yang, Fortschr. Phys. 65, 1700034 (2017).
  * [19] M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395-412 (1988).
  * [20] M. Visser, S. Kar, N. Dadhich, Phys. Rev. Lett. 90, 201102 (2003).
  * [21] N. M. Garcia, F. S. N. Lobo, M. Visser, Phys. Rev. D 86, 044026 (2012).
  * [22] R. Garattini and S. N. Francisco Lobo, Class. Quantum Gravity 24, 2401 (2007).
  * [23] F. Parsaei, S. Rastgoo, Phys. Rev. D 99, 104037 (2019).
  * [24] T. Karakasis, E. Papantonopoulos, and C. Vlachos, Phys. Rev. D 105, 024006 (2022).
  * [25] H. Golchin, M. R. Mehdizadeh, Eur. Phys. J. C 79, 777 (2019).
  * [26] A. Eid, Phys. Dark Univ. 30, 100705 (2020).
  * [27] D. J. Gogoi and U. D. Goswami, J. Cosmol. Astropart. Phys. 2023, 027 (2023).
  * [28] Z. Yousaf, M. Ilyas, and M. Z. Bhatti, Eur. Phys. J. Plus 132, 268 (2017).
  * [29] A. Chanda, S. Dey, and B. C. Paul, Gen. Relativ. Grav. 53, 78 (2021).
  * [30] J. L. Rosa, P. M. Kull, Eur. Phys. J. C 82, 1154 (2022).
  * [31] K. A. Bronnikov and S. -W. Kim, Phys. Rev. D 67, 064027 (2003).
  * [32] K. C. Wong, T. Harko, K. S. Cheng, Class. Quantum Grav. 28, 145023 (2011).
  * [33] Y. Tomikawa, T. Shiromizu, and K. Izumi, Phys. Rev. D 90, 126001 (2014).
  * [34] F. Javed, G. Mustafa, Ali Ovgun and M. F. Shamir, Eur. Phys. J. Plus 137, 61 (2022).
  * [35] I. P. Lobo, M. G. Richarte, J. P. Morais GraÃ§a, and H. Moradpour, Eur. Phys. J. Plus 135, 550 (2020).
  * [36] T. Tangphati, C.R. Muniz, A. Pradhan, and A. Banerjee, Phys. Dark Univ. 42, 101364 (2023).
  * [37] G. Mustafa et al. Phys. Lett. B 821, 136612 (2021).
  * [38] Z. Hassan et al. EPL 139, 39001 (2022).
  * [39] A. Banerjee, et al. Eur. Phys. J. C 81, 1031 (2021).
  * [40] Z. Hassan, S. Ghosh, P. K. Sahoo, and V. S. Hari Rao, Gen. Relativ. Gravit. 55, 90 (2023).
  * [41] M. Tayde, et al. Chinese Phys. C 46, 115101 (2022).
  * [42] M. Tayde, et al. Phys. Dark Univ. 42, 101288 (2023).
  * [43] C. C. Chalavadi, N. S. Kavya and V. Venkatesha, Eur. Phys. J. Plus 138, 885 (2023).
  * [44] B. Zwiebach, Phys. Lett. B 156, 315 (1985).
  * [45] D. J. Gross and J. H. Sloan, Nucl. Phys. B 291, 41 (1987).
  * [46] M.C. Bento and O. Bertolami, Phys. Lett. B 368, 198 (1996).
  * [47] C. Lanczos, Ann. Math. 39, 842 (1938).
  * [48] D. Lovelock, J. Math. Phys. 12, 498 (1971).
  * [49] D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985).
  * [50] D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020).
  * [51] K. Aoki, M.A. Gorji, and S. Mukohyama, Phys. Lett. B 810, 135843 (2020).
  * [52] K. Aoki, M.A. Gorji, and S. Mukohyama, J. Cosmol. Astropart. Phys. 2009, 014 (2020).
  * [53] R. A. Konoplya and A. Zhidenko, Phys. Dark Univ. 30, 100697 (2020).
  * [54] C.-Y. Zhang, P.-C. Li, and M. Guo, Eur. Phys. J. C 80, 874 (2020).
  * [55] H. Lu and Y. Pang, Phys. Lett. B 809, 135717 (2020).
  * [56] M. S. Churilova, Phys. Dark Univ. 31, 100748 (2021).
  * [57] D. V. Singh, S. G. Ghosh and S. D. Maharaj, Phys. Dark. Univ. 30, 100730 (2020).
  * [58] S. G. Ghosh and R. Kumar, Class. Quantum Grav. 37, 245008 (2020).
  * [59] D. D. Doneva and S. S. Yazadjiev, J. Cosmo. Astro. Phys. 2021, 024 (2021).
  * [60] R. A. Konoplya and A. Zhidenko, Phys. Rev. D 102, 064004 (2020).
  * [61] S.-W. Wei and Y.-X. Liu, Eur. Phys. J. Plus 136, 436 (2021).
  * [62] R. Kumar and S. G. Ghosh, J. Cosmo. Astro. Phys. 2020, 053 (2020).
  * [63] S. A. Hosseini Mansoori, Phys. Dark Univ. 31, 100776 (2021).
  * [64] M. Guo and P.-C. Li, Eur. Phys. J. C 80, 588 (2020).
  * [65] P. G. S. Fernandes, P. Carrilho, T. Clifton, and D. J. Mulryne, Class. Quantum Grav. 39, 063001 (2022).
  * [66] K. Jusufi, A. Banerjee, and S. G. Ghosh, Eur. Phys. J. C 80, 698 (2020).
  * [67] N. Godani, D. V. Singh, G. C. Samanta, Phys. Dark Univ. 35, 100952 (2022).
  * [68] Shweta, U. K. Sharma, and A. K. Mishra, Int. J. Geo. Mod. Phys. 20 2350140 (2023).
  * [69] F. Zwicky, Helv. Phys. Acta 6, 110 (1933)
  * [70] F. Zwicky, Astrophys. J. 86, 217 (1937).
  * [71] J. Oort, Bull. Astron. Ins. Nether. V 192 (1930).
  * [72] J. Oort, Bull. Astron. Ins. Nether. V 239 (1930).
  * [73] J. Diemand et al., Nature 454, 735 (2008).
  * [74] V. Springel et al., Mon. Not. R. Astron. Soc. 391, 1685 (2008).
  * [75] F. Rahaman, G. C. Shit, B. Sen, S. Ray, Astrophys. Space Sci. 361, 37 (2016).
  * [76] N. Sarkar, et al. Mod. Phys. Lett. A 34, 1950188 (2019).
  * [77] P. K. F. Kuhfittig, Eur. Phys. J. C 74, 2818 (2014).
  * [78] S.K. Maurya, B.S. Ratanpal, and M. Govender, Ann. Phys. 382, 36 (2017).
  * [79] S. Gedela, R.K. Bisht, and N. Pant, Neeraj, Mod. Phys. Lett. A 34, 1950157 (2019).
  * [80] P. Bhar, Eur. Phys. J. C 79, 138 (2019).
  * [81] S. Gedela, R. K. Bisht, and N. Pant, Eur. Phys. J. A 54, 207 (2018).
  * [82] S. Hansraj and L. Moodly, Eur. Phys. J. C 80, 496 (2020).
  * [83] N. F. Naidu, M. Govender, and S. D. Maharaj, Eur. Phys. J. C 78, 48 (2018).
  * [84] J. Ospino and L. A. Nunez, Eur. Phys. J. C 80, 166 (2020).
  * [85] S. C. Jaryal, Eur. Phys. J. C 80, 683 (2020).
  * [86] S. K. Maurya, et al. Eur. Phys. J. C 81, 729 (2021).
  * [87] R. A. Hennigar, et al. J. High Energ. Phys. 2020, 27 (2020).
  * [88] L. Ma and H. Lu, Eur. Phys. J. C 80, 1209 (2020).
  * [89] A. casalino, et al. Phys. Dark Univ. 31, 100770 (2021).
  * [90] J. Eiesland, Trans. Am. Math. Soc. 27, 213 (1925).
  * [91] L. P. Eisenhart, Riemannian Geometry, (Princeton University Press, Princeton, United States of America, 1966).
  * [92] K. R. Karmarkar, Proc. Indian Acad. Sci. A 27 56 (1948).
  * [93] S. N. Pandey and S. P. Sharma, Gen. Relativ. Gravit. 14, 113 (1981).
  * [94] P. K. F. Kuhfittig, Eur. Phys. J. C 74, 2818 (2014).
  * [95] F. Rahaman, P. Salucci, P. K. F. Kuhfittig, S. Ray, and M. Rahaman, Ann. Phys. 350, 561 (2014).
  * [96] J. H. Oort, Bull. Astro. Inst. Nether. 3, 275 (1927).
  * [97] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro. J., 462, 563 (1996).
  * [98] J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro. J., 490, 493 (1997).
  * [99] D. Merritt et al., Astro. J. 624, L85 (2005).
  * [100] D. Merritt et al., Astronomial J. 132, 2685 (2006).
  * [101] K. Jusufi, M. Jamil, and M. Rizwan, Gen. Rel. Grav. 51, 102 (2019).
  * [102] Z. Xu, M. Tang, G. Cao, and S.-N. Zhang, Eur. Phys. J. C 80, 70 (2020).
  * [103] G. Mustafa, S. K. Maurya, and S. Ray, Astro. J. 941, 170 (2022).
  * [104] A. Burkert, Astro. J 447, L25-L28, (1995).
  * [105] M. Persic, P. Salucci, F. Stel, Mon. Not. R. Astron. Soc. 281, 27 (1996).
  * [106] J. Dubinski and R. G. Carlberg, Astro. J. 378, 496 (1991).
  * [107] L. A. Urena-Lopez, T. Matos and R. Becerril, Class. Quant. Grav. 19, 6259 (2002).
  * [108] T. Harko, J. Cosmo. Astro. Phys. 05, 022 (2011).
  * [109] X. Hou et al., J. Cosmo. Astro. Phys. 2018, 015 (2018).
  * [110] Z. Xu et al., J. Cosmo. Astro. Phys. 2018, 038(2018).
  * [111] M. Cataldo, L. Liempi, and P. Rodriguez, Phys. Lett. B 757, 130-135 (2016).
  * [112] F. S. N. Lobo, Phys. Rev. D 71, 084011 (2005).
  * [113] A. S. Nezhada, M. R. Mehdizadehb, and H. Golchin, Eur. Phys. J. Plus 139, 110 (2024).
  * [114] Kar and Sahdev, Phys. Rev. D 52, 2030 (1995).
  * [115] L. A. Anchordoqui et al., Phys. Rev. D 57, 829 (1998).
  * [116] M. F. Shamir and I. Fayyaz, Eur. Phys. J. C 80, 1102 (2020).
  * [117] A. Malik et al. Eur. Phys. J. C 83, 522 (2023).
  * [118] S. Sarkar, N. Sarkar, and F. Rahaman Eur. Phys. J. C 80, 882 (2020).
  * [119] K. Jusufi, P. Channuie, and M. Jamil, Eur. Phys. J. C 80, 127 (2020).
  * [120] L. Herrera, Phys. Rev. D 97, 044010 (2018).
  * [121] Alfonso Garcia-Parrado Gomez-Lobo, Class. Quantum Grav. 25, 015006 (2008).
  * [122] L. Herrera et al., Phys. Rev. D 79, 064025 (2009).
  * [123] R. Casadio et al., Eur. Phys. J. C 79, 826 (2019).
  * [124] E. Contreras and E. Fuenmayor, Phys. Rev. D 103, 124065 (2021).
  * [125] C. Arias, E. Contreras, E. Fuenmayor, and A. Ramos, Ann. Phys. 436, 168671 (2022).
  * [126] M. Sharif and I. I. Butt, Eur. Phys. J. C 78, 688 (2018).
  * [127] Z. Yousaf, M. Yu. Khlopov, M. Z. Bhatti, and T. Naseer, MNRAS 495, 4334-4346 (2020).
  * [128] M. Zubair and M. Farooq, Eur. Phys. J. C 83, 507 (2023).
  * [129] D. Lynden-Bell, J. Katz, J. Bicak, Phys. Rev. D 75, 024040 (2007).
  * [130] K. K. Nandi, Y. Z. Zhang, R. G. Cai, and A. Panchenko, Phys. Rev. D 79, 024011 (2009).
  * [131] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973), pp. 467 and 603.
  * [132] P. Bhar, F. Rahaman, T. Manna, and A. Banerjee, Eur. Phys. J. C 76, 708 (2016).
  * [133] F. Rahaman, S. Karmakar, I. Karar, and S. Ray, Phys. Lett. B 746, 73-78 (2015).
  * [134] O. Sokoliuk, S. Mandal, P. K. Sahoo, and A. Baransky, Eur. Phys. J. C 82, 280 (2022).
  * [135] F. Rahaman, P. K. F. Kuhfitting, S. Ray, and N. Islam, Eur. Phys. J. C 74, 2750 (2014).
  * [136] G. Mustafa, F. Atamurotov, and Sushant G. Ghosh, Phys. Dark. Univ. 40, 101214 (2023).
  * [137] B. C. Paul, Class. Quantum Grav. 38 145022 (2021).
  * [138] M. Tayde, Z. Hassan, and P. K. Sahoo, Nuclear Phys. B 1000, 116478 (2024).