Abstract
This is a review with the ambitious goal of covering the recent progress in: (1) universal relations (in general relativity and alternative theories of gravity), and (2) neutron star models in alternative theories. We also aim to be complementary to recent reviews in the literature.
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Notes
- 1.
The perturbations, though, could be different and thus used for testing alternative theories of gravity.
- 2.
The newtonian limit is for \(\mathcal {C}\rightarrow 0\), while the relativistic limit is for \(\mathcal {C}\rightarrow 1/2\).
- 3.
There will be a further discussion on multipole moments when we talk about the 3-hair relations.
- 4.
The polynomial here is defined as \(Q^2_2(x)=\frac {3}{2} \left (x^2-1\right ) \ln \left (\frac {x+1}{x-1}\right )+\frac {5 x-3 x^3}{x^2-1}\)
- 5.
We should also note that the expansion in this case is not at infinity. It takes place at a buffer region outside the surface of the star and inside a radius given by the external field’s characteristic curvature radius.
- 6.
See also Lattimer and Lim (2013) where the results were extended to a wider range of equations of state. We remind here that these results apply to quark stars as well.
- 7.
The minus sign here is due to the fact that rotating neutron stars tend to be oblate due to rotation which gives a negative value for the quadrupole. On the other hand, an object that is prolate has a positive quadrupole.
- 8.
We should note that the numerical coefficients in these expressions depend on the specific configuration.
- 9.
In Chakrabarti et al. (2014) the authors arrive at the same result with χ values as low as 0.1, which get even closer to the slow rotation fit.
- 10.
The degeneracy is in the gravitational wave phase where a spin-spin coupling term has a contribution at the same order as the quadrupole term.
- 11.
Multipole moments in general are tensorial quantities. The rank of the tensor is given by the order of the corresponding multipole moment. In the case that we also have axisymmetry, the moments are multiples of the symmetric trace-free tensor product of the axis vector n a with itself. In this case therefore one can define scalar moments as \(\mathcal {P}_{\ell }=1/(\ell !) \mathcal {P}_{i^1\ldots i^{\ell }} n^{i^1}\ldots n^{i^{\ell }}\), where \(\mathcal {P}_{\ell }\) is the ℓ-th order moment. Since we will discuss about stationary and axisymmetric spacetimes we will only talk about the scalar moments.
- 12.
The Gegenbauer polynomials are given by the definition \(T_l^{1/2}(\mu )=\frac {(-1)^l \Gamma (l+2)}{2^{l+1/2} l! \Gamma (l+3/2)} (1-\mu ^2)^{-1/2} \frac {d^l}{d\mu ^l}(1-\mu ^2)^{l+1/2} \).
- 13.
A polytropic equation of state has the form P = kρ Γ, where the exponent can be written as Γ = 1 + 1∕n in terms of the polytropic index n.
- 14.
For n = 0 Lane–Emden admits an exact solution which is \(\vartheta =1-\frac {1}{6}\xi ^2\) with a surface at \(\xi _1=\sqrt {6}\), while for n = 1 it admits the solution \(\vartheta =\frac {\sin \xi }{\xi }\) with a surface at ξ 1 = π.
- 15.
One should always keep in mind that at the end of the day what matters most is not exactly the errors of the fitting function but instead the accuracy of solving the inverse problem, i.e. determining the stellar parameters from the observed gravitational wave frequencies and damping times.
- 16.
If we approximate the moment of inertia by MR 2 then we can see that \(\eta =M/R=\mathcal {C}\).
- 17.
- 18.
- 19.
The radius here is expressed in terms of Schwarzschild-like coordinates which reduce to being circumferential at the non-rotating limit. The radial coordinates in the Hartle and Thorne approach also have this property.
- 20.
The interested reader might want to also have a look at the review by Hartle (1978).
- 21.
For the interesting case of disformal coupling we refer the reader to (Minamitsuji and Silva 2016).
- 22.
As we will discuss below, this is true only in the static case. The rapid rotation magnifies the differences significantly and offers new possibilities for probing scalar-tensor theories of gravity.
- 23.
The neutron stars, though, can still have nonzero higher scalar multipoles and thus nontrivial scalar field.
- 24.
- 25.
For the I-C relations in EdGB, as we have noted earlier, this is not the case though. On the contrary we expect that the general relativistic relations to hold also in EdGB.
- 26.
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Acknowledgements
We would like thank Kent Yagi for providing us data for some of the plots as well as useful comments on the manuscript. We would also like to thank for their many useful comments, Hector O. Silva, Kostas Glampedakis, Jutta Kunz and Stoytcho Yazadjiev.
G.P. is supported by FCT contract IF/00797/2014/CP1214/CT0012 under the IF2014 Programme.
D.D. would like to thank the European Social Fund, the “Ministry of Science, Research and the Arts” Baden-Wurttemberg and the Bulgarian NSF Grant DFNI T02/6 for the support. DD is also indebted to the Baden-Württemberg Stiftung for the financial support of by the Eliteprogramme for Postdocs.
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Doneva, D.D., Pappas, G. (2018). Universal Relations and Alternative Gravity Theories. In: Rezzolla, L., Pizzochero, P., Jones, D., Rea, N., Vidaña, I. (eds) The Physics and Astrophysics of Neutron Stars. Astrophysics and Space Science Library, vol 457. Springer, Cham. https://doi.org/10.1007/978-3-319-97616-7_13
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