Can a Higher Dimensional Model of the Universe Solve the Problem of Dark Matter?

1. Introduction

Dark matter/DM (see [1] for a detailed review) remains one of the greatest puzzles in our understanding of the cosmos, along with dark energy. The standard model of cosmology, called $\Lambda$CDM, which assumes that General Relativity(GR) is correct, is based on some yet undetected cold dark matter particles(CDM) and a small but positive cosmological constant $\Lambda$. Dark matter is considered to be weakly interacting massive particles (WIMPs). A lower bound on the mass of dark matter in a model-independent way is provided in [2]. The standard model of cosmology is successful in explaining a wide variety of observations, such as flat rotation curves of galaxies, CMB, and large-scale structure formation [3]. However, as of now, there is no direct detection of DM particles. Even the Standard Model of particle physics does not seem to contain good candidates for DM particles. Particles such as axions (see [4] for a review) and branons [5,6], which are massive brane fluctuations, are also proposed to be candidates of dark matter but have no direct evidence as of now. All this has led to alternative theoretical explorations for dark matter. This is usually done by modifying gravity (see [7] for a nice discussion on modifying gravity and [8,9] for detailed reviews). A simple proposal in this regard was given by Milgrom called MOND [10,11]. It argues that Newton’s law should be modified at large distances (or at small accelerations). Another theory was given by Moffat called Scalar-Tensor-Vector theory (STV) [12]. This also modifies gravitational dynamics at large distances. Dark matter and dark energy may also be explained using modified $f\left(R\right)$ gravity approaches [13].

A different approach in this direction was proposed by Boyle, Finn and Turok [14]. On the grounds of preserving CPT symmetry of the universe as a whole, the authors argued for the existence of an anti-universe, which provides an intriguing argument for dark matter. The existence of an anti-universe was also shown to explain cosmic acceleration without dark energy [15], thus providing a unified approach towards explaining dark matter and dark energy. Yet another approach is the spherical reduction of GR at large distances based on dilaton gravity (see [16] for a review). This approach modifies gravity at large distances and solves the problem of flat rotation curves of galaxies [17,18]. In [19,20], it was argued that logarithmic potential can explain phenomena usually attributed to dark matter.

In this work, we take a different perspective. We assume that GR is the correct theory of gravity on our 4D universe, but that phenomena usually attributed to dark matter may arise as “external” effects of higher dimensions. Specifically, we investigate a braneworld setup in which our 3-brane universe is embedded in a 6D Minkowski bulk with cascading gravity. Within this framework, we show that the on-brane propagator generates an effective logarithmic potential at large distances, corresponding to a $1/r$ force law. This force scaling naturally reproduces certain dark matter–like phenomena, such as flat galactic rotation curves, while avoiding the need to postulate new particle species. Our aim is not to provide a complete alternative to $\Lambda$CDM, but rather to highlight a mechanism by which higher-dimensional physics can mimic dark matter effects and motivate further phenomenological investigation.

The rest of this paper is organized as follows. In Section 2 we introduce the braneworld setup and describe how cascading gravity regulates the higher-dimensional theory. In Section 3, we derive the effective on-brane propagator, analyze its infrared behavior, and show how a logarithmic potential arises, leading to an emergent $1/r$ force law. We then discuss the implications for circular velocities and galactic rotation curves. Section 4 summarizes our findings and outlines open issues, while Appendix E and Appendix F provide detailed derivations of the logarithmic potential from the on-brane propagator and present a toy model illustrating infrared suppression of mediator modes.

2. The Setup

We consider a 6D Minkowski bulk with the 3-brane immersed in it. The brane is assumed to be tensionless. The assumption we make is that GR is the correct theory on the brane(our universe), with the modification being an “external” effect coming from the higher-dimensional bulk in which it is immersed. This leads to the action

$$S=M^4\int d^{6}X\sqrt{G}{R}_{\left(6\right)}+M_P^2\int d^{4}x\sqrt{\left|g\right|}R$$

(1)

where M is the 6D Planck mass, $M_P$ is the usual 4D Planck mass. $G\left(X\right)=G(x,y,z)$ denotes a 6D metric with $R_{\left(6\right)}$ as the 6D Ricci scalar. This is similar to the DGP model [21]. The subsequent analysis is also similar to this paper. The reader should note that, similar to the DGP model, the 6D action (1) suffers from ghost instabilities. In particular, the propagator on the brane is not well-defined. We, therefore, need to regulate this divergence, which we shall discuss later in this section. But before that, let us analyze the action (1) to understand what we are actually interested in doing. For simplicity, we now consider a 6D scalar field. Then, the action becomes

$$\begin{array}{cc}\hfill S& =M^4\int d^{4}xdydz{\partial }_A\Phi (x,y,x){\partial }^A\Phi (x,y,z)\hfill \\ \hfill & +M_P^2\int d^{4}xdydz\delta \left(y\right)\delta \left(z\right){\partial }_{\mu }\Phi (x,0,0){\partial }^{\mu }\Phi (x,0,0)\hfill \end{array}$$

(2)

Here, A denotes 6D coordinates. We are interested in the effect of this 6D scalar field on the brane. For this, we consider the retarded classical Green’s function as

$$\begin{array}{cc}\hfill & (M^4{\partial }_A{\partial }^A+M_P^2\delta \left(y\right)\delta \left(z\right){\partial }_{\mu }{\partial }^{\mu })G_R(x,y,z;0,0,0)\hfill \\ \hfill & ={\delta }^{\left(4\right)}\left(x\right)\delta \left(y\right)\delta \left(z\right)\hfill \end{array}$$

(3)

Then, the potential mediated by scalar $\Phi$ on the brane is given by

$$V\left(r\right)=\int G_R(t,\overrightarrow{x},y=z=0;0,0,0,0)dt$$

(4)

The solution of (3) in Euclidean space is

$${\tilde{G}}_R(p,y,z)={\displaystyle \frac{1}{M_P^2{p}^2+M^{4}log(\mu /p)}}{e}^{-p\left|y\right|}{e}^{-p\left|z\right|}$$

(5)

where $p^2$ is the square of Euclidean 4-momentum. Using (4), we obtain the (normalized) potential as

$$\begin{array}{cc}\hfill V\left(r\right)& =-{\displaystyle \frac{1}{8{\pi }^2{M}_P^2}}{\displaystyle \frac{1}{r^2}}\{sin\left({\displaystyle \frac{r}{r_0}}\right)\mathrm{Ci}\left({\displaystyle \frac{r}{r_0}}\right)\hfill \\ \hfill & +{\displaystyle \frac{r}{2}}cos\left({\displaystyle \frac{r}{r_0}}\right)\left[\pi -2\mathrm{Si}\left({\displaystyle \frac{r}{r_0}}\right)\right]\}\hfill \end{array}$$

(6)

where $r_0={\displaystyle \frac{M_P^2}{2M^4}}$, $\mathrm{Ci}\left(z\right)=\gamma +ln\left(z\right)+{\int }_0^z(cos\left(t\right)-1)dt/t$, $\mathrm{Si}\left(z\right)={\int }_0^{z}sin\left(t\right)dt/t$ and $\gamma$ is the Euler-Masceroni constant. We are interested in the large distance behaviour of $V\left(r\right)$ since this is the effect exclusive to extra dimensions and reads

$$V(r>>r_0)=-{\displaystyle \frac{r_0}{8{\pi }^2{M}_P^2}}{\displaystyle \frac{1}{r^3}}$$

(7)

We see that the large distance behavior scales as $1/r^3$ in accordance with 6D theory. The small distance behavior of (6) gives the Newtonian potential. We now return to the problem of instability. This can be cured by using a second codimension-1 brane as a regulator. So, the setup is as follows: A (3+1)-brane is embedded on a (4+1)-brane in 6D bulk. We have a ${\mathcal{Z}}_2$ symmetry across both branes such that the coordinates of extra dimensions given by y and z along the 5th and 6th dimension, respectively, range from 0 to ∞. The codimension-1 brane is situated at $z=0$ while the codimension-2 brane is at $y=z=0$. This type of setup is called cascading gravity [22,23,24] and is ghost-free. The corresponding action is

$$\begin{array}{cc}\hfill S& =M_6^4\int d^{6}x\sqrt{-g_6}{R}_6+M_5^3\int d^{5}x\sqrt{-g_5}{R}_5\hfill \\ \hfill & +M_4^2\int d^{4}x\sqrt{-g_4}{R}_4\hfill \end{array}$$

(8)

In this case, we have a transition from $4D\to 5D\to 6D$ gravity, and the potential varies as $1/r\to 1/r^2\to 1/r^3$. The transitions depend on crossover scales. The 4D to 5D crossover scale is

$$m_5={\displaystyle \frac{M_5^3}{M_4^2}}$$

(9)

while 5D to 6D crossover scale is

$$m_6={\displaystyle \frac{M_6^4}{M_5^3}}$$

(10)

3. Derivation of the Effective $1/r$ Force

1. Cascading Set-Up and On-Brane Propagator

We use the standard “cascading gravity” toy model (a bulk scalar $\Phi$ with induced kinetic terms on the codimension-1 and codimension-2 branes), which faithfully captures the IR structure of the tensor sector. The action reads

$$\begin{array}{cc}\hfill & S={\displaystyle \frac{M_6^4}{2}}\int d^{6}X{({\partial }_A\Phi )}^2+{\displaystyle \frac{M_5^3}{2}}\int d^{5}x{({\partial }_a\Phi )}^2\hfill \\ & +{\displaystyle \frac{M_4^2}{2}}\int d^{4}x{({\partial }_{\mu }\Phi )}^2-\int d^{4}x\Phi J\left(x\right),\hfill \end{array}$$

(11)

where a static point source $J\left(x\right)=M{\delta }^{\left(3\right)}\left(\mathbf{x}\right)$ is placed on the codimension-2 brane at $y=z=0$.

Fourier transforming along the brane’s spatial directions ($\mathbf{p}\equiv \left|\mathbf{p}\right|$) and solving the bulk equation with standard boundary conditions, the on-brane propagator is

$$G_{\mathrm{br}}\left(\mathbf{p}\right)={\displaystyle \frac{1}{M_4^2{\mathbf{p}}^2+M_5^3\mathbf{p}+M_6^4\mathcal{I}\left(\mathbf{p}\right)}},$$

(12)

where $\mathcal{I}\left(\mathbf{p}\right)$ encodes the codimension-2 logarithmic behavior. With thick-brane regularization one finds at small $\mathbf{p}$:

$$\mathcal{I}\left(\mathbf{p}\right)=log\left({\displaystyle \frac{\mu }{\mathbf{p}}}\right)+\mathcal{O}\left(1\right).$$

(13)

Defining the crossover scales

$$m_5\equiv {\displaystyle \frac{M_5^3}{M_4^2}},m_6\equiv {\displaystyle \frac{M_6^4}{M_5^3}},$$

(14)

we obtain

$$G_{\mathrm{br}}\left(\mathbf{p}\right)\simeq {\displaystyle \frac{1}{M_4^2}}{\displaystyle \frac{1}{{\mathbf{p}}^2+m_5\mathbf{p}+m_5{m}_{6}log(\mu /\mathbf{p})}}.$$

(15)

2. Long-Distance Potential on the Brane

The static potential is

$$\Phi \left(r\right)=M\int {\displaystyle \frac{d^3\mathbf{p}}{{\left(2\pi \right)}^3}}{e}^{i\mathbf{p}·\mathbf{r}}{G}_{\mathrm{br}}\left(\mathbf{p}\right).$$

(16)

At very large r (small $\mathbf{p}$), the logarithmic term dominates if $m_5>m_6$ near $\mathbf{p}\sim \sqrt{m_5{m}_6}$. In this regime

$$G_{\mathrm{br}}\left(\mathbf{p}\right)\approx {\displaystyle \frac{1}{M_4^2}}{\displaystyle \frac{1}{m_5{m}_{6}log(\mu /\mathbf{p})}}.$$

(17)

The effective 2D IR Fourier transform gives a logarithmic potential for the $1/p_{\perp }^2$ kernel (see Appendix E for details)

$$\Phi \left(r\right)={\displaystyle \frac{M}{2{\pi }^2{M}_4^2{m}_5{m}_6}}log\left({\displaystyle \frac{r}{r_0}}\right)+\cdots ,$$

(18)

so that the physical acceleration is

$$a_{\mathrm{eff}}\left(r\right)=-{\displaystyle \frac{d\Phi }{dr}}=-{\displaystyle \frac{M}{2{\pi }^2{M}_4^2{m}_5{m}_6}}{\displaystyle \frac{1}{r}}.$$

(19)

Thus, the codimension-2 IR $1/p_{\perp }^2$ kernel produces the desired $1/r$ scaling directly.

3. Circular Velocities and Flat Rotation Curves

For a test body in circular orbit,

$${\displaystyle \frac{v^2\left(r\right)}{r}}={\displaystyle \frac{G{M}_b\left(r\right)}{r^2}}+{\displaystyle \frac{M}{2{\pi }^2{M}_4^2{m}_5{m}_6}}{\displaystyle \frac{1}{r}}.$$

(20)

At $r\gg r_0$ the second term dominates, yielding

$$v\left(r\right)\to v_{\infty }=\sqrt{{\displaystyle \frac{M}{2{\pi }^2{M}_4^2{m}_5{m}_6}}},$$

(21)

which is constant for all sufficiently large radii. This reproduces flat galactic rotation curves without invoking particle dark matter.

4. Conclusion

In this work, we have shown, given the assumptions that within a braneworld setup, that the long-distance behavior of the on-brane propagator can generate an effective logarithmic potential, leading to a $1/r$ force law. This reproduces flat rotation curves without invoking particle dark matter, suggesting that what we attribute to dark matter might be viewed instead as an imprint of higher-dimensional structure.

We emphasize, however, that the present analysis should be regarded as a proof of concept. Several open issues remain: the precise origin of the anisotropic IR suppression and induced 2D kinetic terms, the consistency and stability of the setup at cosmological scales, and the quantitative confrontation with observational data (galaxy rotation curves, lensing, large-scale structure, and CMB). These aspects require careful study before such a framework can be considered a realistic alternative to $\Lambda$CDM.

Nevertheless, this work illustrates a novel pathway by which higher-dimensional physics can give rise to dark matter–like effects. Together with other approaches—such as CPT-symmetric cosmologies and variable brane tension scenarios—this contributes to the broader program of exploring whether phenomena attributed to dark matter and dark energy might originate from physics beyond the standard 4D picture.