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Scalars in (d+1)-dimensions

Simulating scalar fields on lattices with \(d \neq 3\) spatial dimensions could be useful for two main reasons. First, one might wish to simulate a field theory intrinsically defined in \(d \neq 3\) dimensions, with scalar fields propagating, e.g. in a \((2+1)\)-dimensional expanding spacetime, \(ds^2 = a^2(\eta)(-d\eta^2 + \delta_{ij}dx_i dx_j)\), with \(i,j=1,2\). In this case, the corresponding EOM in \((2+1)\) dimensions can be discretized and solved using suitable evolution algorithms, in much the same way as for the theories in \((3+1)\) dimensions. Alternatively, one may wish to simulate a field theory defined in three spatial dimensions on a lower-dimensional lattice, thereby approximating its three-dimensional dynamics at a reduced computational cost. This approach is justified provided that the system is statistically isotropic, both in its initial conditions and throughout its evolution. This is typically the case for scalar-field interactions, as they can not create locally (neither globally) a preferred spatial direction, unless that was already imposed as an initial condition in the field gradients. Reducing the dimensionality from \(d=3\) to \(d=2\) can be therefore highly advantageous, accelerating simulations by a factor \(\sim N\), with \(N \approx 10^2-10^3\) in most applications. This enables investigations requiring very long evolution times, very large lattices, or extensive scans over model parameters, see e.g. Refs. 1, 2, 3. The ability of lower-dimensional simulations to accurately reproduce three-dimensional dynamics, must be assessed anyways on a case-by-case basis.

In \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) we have considered the second circumstance, namely simulating \((3+1)\)-dimensional physics on a reduced \((d+1)\)-dimensional lattice, with \(d=2\), or \(d=1\). For such purpose, the code allows to solve the singlet scalar field EOM in both \((1+1)\) and \((2+1)\) dimensions. More specifically, the code solves standard EOM (e.g. of the form given in Eq. (6) and Eq. (9) in Scalar-Scalar Interactions), but on a \(1-\) or \(2-\)dimensional spatial slice respectively. The discrete Laplacian operator in Eq. (6) from Scalar-Scalar Interactions, is then computed summing over the \(d\)-spatial directions, i.e. \(\nabla^2 \phi = \sum_{i=1}^{d} \partial^2 \phi / \partial x_i^2\), with \(d = 1\) or \(2\). A similar change applies to the gradient component of the scalar field's energy density, \({G}_{\phi} = \frac{1}{2 a^2} \sum_{i=1}^d (\partial\phi / \partial x_i)^2\), c.f. Eq. (10) from Scalar-Scalar Interactions. By modifying appropriately the initial spectrum of fluctuations, these simulations mimic the three-dimensional dynamics, see Sect. 7.1 of \(\mathtt{The~Art-II}\) (Ref. 4) for more extensive discussions.

Runnin in d=1,2 spatial dimensions

For running a scalar field model in either 1+1 or 2+1 dimensions, you just need to add in the model file (e.g. lphi4.h) the following:

static constexpr size_t NDim = 1;

or

static constexpr size_t NDim = 2;

This must be added together with the number of fields of each species and the number of potential terms.

Note: For now, simulations in 1+1 or 2+1 dimensions only work with PS_type = 2 spectra.

  1. S. Antusch, D. G. Figueroa, K. Marschall, and F. Torrenti. Energy distribution and equation of state of the early universe: matching the end of inflation and the onset of radiation domination. Phys. Lett. B, 811:135888, 2020. arXiv:2005.07563, doi:10.1016/j.physletb.2020.135888. ↩

  2. Stefan Antusch, Daniel G. Figueroa, Kenneth Marschall, and Francisco Torrenti. Characterizing the postinflationary reheating history: Single daughter field with quadratic-quadratic interaction. Phys. Rev. D, 105(4):043532, 2022. arXiv:2112.11280, doi:10.1103/PhysRevD.105.043532. ↩

  3. Stefan Antusch, Kenneth Marschall, and Francisco Torrenti. Equation of state during (p)reheating with trilinear interactions. JCAP, 11:002, 2025. arXiv:2507.13465, doi:10.1088/1475-7516/2025/11/002. ↩

  4. J. Baeza-Ballesteros, D. G. Figueroa, A. Florio, J. Lizarraga, N. Loayza, K. Marschall, T. Opferkuch, B. A. Stefanek, F. Torrentí, and A. Urio. The art of simulating the early universe. part ii. non-canonical cases & gravitational waves. JCAP, 06:087, 2026. arXiv:2512.15627, doi:10.1088/1475-7516/2026/06/087. ↩