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Quadratic Gravity And Scalar Dark Matter In A Gravitational Wave Context.

Hontas Farmer and Shane Larson

We present, in the form of an interactive Mathematica notebook, a model of classical quadratic gravity with a massive scalar field in the context of extreme mass ratio in spirals which could be tested with the future Laser Interferometer Space Antenna. The Lagrangian is written, field equations derived, then solved exactly without approximation. Further computational investigation and evaluation was done to evaluate the consistency of the result with LIGO observations. The result is a model which does not disturb the classic solar system test of general relativity but which in extreme conditions models cosmological behavior, and which would have LISA observable consequences in the context of an extreme mass ratio in spiral. Quadratic gravity theories are some of the best candidates for alternative gravity models which have not been eliminated by LIGO observation. For example Starobinsky gravity, an inflationary model that fits available Planck data, while in the study of quantum gravity quadratic terms lead to a renormalization model. Given that data two parameters that appear in this model are precisely constrained to small but non-zero values. Investigation of the EMRI environment may result in this model being validated or refuted by experiment as this paper shows.

This Mathematica notebook is meant as a supplement to formal publication. It includes details which are not appropriate for, or not possible in, the format of a journal paper. For example some of the Mathematica code and results, if fully expanded make this document a few times longer, and there are interactive plots.

Introduction

Quadratic gravity theories are some of the best candidates for alternative gravity models which have not been eliminated by LIGO observation. For example Starobinsky gravity, an inflationary model that fits available Planck data, while in the study of quantum gravity quadratic terms lead to a renormalization model. For those reasons these models are very important and of great observational and theoretical interest.

The model examined in this notebook is a combination of quadratic gravity found in literature as Starobinsky inflation

1-2

with scalar field dark matter such as Fuzzy dark matter or axion dark matter very light scalar bosons

3-4

. This takes the form of an F(R) gravity model. The goal is to build a straight forward understanding of these models and to probe if and how LISA will test them.

These models have been investigated theoretically for static situations and application to cosmology or in the context of quantum field theory in curved space time. With the common assumption that the Ricci scalar would take on a small but non zero constant value

. Gravitational wave physics presents a unique challenge where the Ricci scalar is not zero or any other constant. While the space times of Black holes are of zero curvature the gravitational wave foreground that results from an in spiral of one black hole to another is not zero. Solutions to these situations have been achieved by way of various computational and approximations methods such as parameterized post Newtonian, parameterized post Einsteinian

, and EMRI Kluges

. These methods have all given good useful and computationally efficient results.

To definitively determine the physics beyond all reasonable doubt, I seek an exact mathematical model. In this day computational perturbative solutions are often thought of as enough, but if exact solutions could be found why not try and find them? Having them will allow better use of computational power.

The action which we will study is as follows.

S=

-g

f(R)-

μν

∇

ϕ-

mϕ





(

)

In which f(R) is given as the following.

f(R)=R+

βR

ξRϕ

Deriving the field equations

Deriving the field equations due to this action is a matter of taking functional/variational derivatives

. The immediate result of this process will be modified Einstein field equations, and also a modified equation for a scalar field in curved space time with a Starobinsky term. As demonstrated in CQ Geng how this will look given a particular f(R) is

Many people who study gravitational waves in the observational sense tend to ignore the fact that gravitational waves have a non zero Ricci curvature even though the background space time has zero curvature. Anyone who objects to this fact can read Sean Carolls textbook and write him an angry letter.

f'(R)

μν

f(R)

μν

∇

)f'(R)=

μν

This can then be traced as shown in CQ Geng into a form which is only in terms of invariants

f'(R)R-2f(R)+3f'(R)=

For the f(R) we are studying in this paper take a derivative with respect to R (d/dR as if it was X and this was a 1 D problem. Strange as that may feel since we know R is itself a function of four space time variables. That is the simple beauty of working with invariants.) Carrying out this problem as much as possible using the faultless math of Mathematica.

6β-1+ξ

R-3ξ

The process of taking the functional derivative of the action for the scalar field phi leads to the field equation for that field. The result is a set of two field equations which will need to be solved simultaneously to fully solve the problem.



-

6β

R-

2β



6β

(-

-ξR)ϕ=0



(

)

A solution to the above set of equations should not depend on the choice of coordinates. The problem as it stands is formulated in terms of invariant quantities and the solutions should themselves be invariants. When coordinates are chosen, and the correct limits are taken the amplitude of the fields should drop off as 1/r or

Solving the field equations up to an integral, analytically.

To solve these analytically let us consider some extremal cases. Then using this simplified cases we can assemble an analytical solution.

Setting the Starobinsky coefficient to zero, β =0

This is not allowed due to the appearance of terms which contain beta in the denominator.



-

R-



(-

-ξR)ϕ=0



This would lead to expressions that are undefined mathematically.

Setting the Ricci scalar to zero, R =0 Case1

Setting R equal to zero causes the field equations to collapse to a simple form.



2β



6β

(-

)ϕ=0



This form of the equations is Ricci vacuum. The result is that ϕ is described by the Klein-Gordon equation in flat space time.

(-

)ϕ=0

Following the methods of Caroll and basic calculus. The equation can be simplified by transforming it into a first order equation using an appropriate Killing vector. Once this is done we get two complex valued equations one for the positive energy states and one for the negative energy states. The solutions to these give us exponentials which lead to wave functions. The solution works out to

ϕ=

-ω∫



ω∫



In which

is an appropriately chosen Killing vector.

There is a small modification to the stress energy invariant in this case.

T=-

3ξ



Setting the Starobinsky coefficient to zero, ϕ =0 Case2

In this case the scalar field does not exist the result is an equation which can be solved by the same methods as used for the Klein-Gordon equation in curved space time.

-

6β

The procedure and result are very similar. The difference is there is a recursive term that appears where in the value at one point depends on the value previously.

-ω∫



±

6β



ω∫



±

6β



The solution is then.

-ω∫



±

6β



ω∫



±

6β



Setting the Starobinsky coefficient to zero, ξ =0 Case 3

In this case the equations are decoupled with no explicit interaction between the fields in the field equations.



-

6β

(-

)ϕ=0



The solutions to the two cases above apply to this case.

ϕ=

-ω∫



ω∫



-ω∫



±

6β



ω∫



±

6β



The General Solution



-

6β

R-

2β



6β

(-

-ξR)ϕ=0



With the insight gained from the special cases the general solution can be written by combining these solutions. Furthermore as the computer algebra system points out in the section below an arbitrary function must be inserted to match the boundary condition at infinity.

In[]:=

σ(

)

Out[]=

σ[

]

large output

show less

show all

set size limit...

In which

σ(

)

is at simplest the interval between two points in the curved space time.

The general solution with the boundary condition that it goes to zero at infinity is given by.

ϕ=

σ(

)



∫

(-ω+ξR)

∫

(ω+ξR)



(

)

σ(

)



-ω-

6β

[R]±

6β

[R]





ω-

6β

[R]±

6β

[R]



(

)

Computer Assisted Evaluation Of Integrals.

The solution for the field equations found by the analytical method above contains a set of integrals that while doable would be tedious with many steps. In particular

∫



and

∫

(ω+ξR)

etc.

σ(

)



Exp-ω∫



+

Expω∫





(

)

ϕ=

σ(

)



Exp∫

(-ω+ξR)

+

Exp∫

(ω+ξR)



(

)

The above equations have the virtues of being stated in terms of invariants and in coordinate free form. Thus they satisfy all the requirements of General Relativity. The computer algebra system Mathematica does not care about this important fact. To encode this into the math we must choose a coordinate system and specify the normalized helical killing vector

in it. Naturally the problem has spherical symmetry so spherical coordinates` are used.

In[]:=

=-1-

2GM

,0,0,Ω

(Sin[θ])



To complete the solution we need the metric tensor to use it to write down the relativistic separation between the points.

For the computer.

Now we can write the full solution

Now for the scalar field.

Now we need to check the limits at the origin and at infinity. This needs to be finite at both.

This shows the gravitational wave Ricci goes to a constant the value of which will shrink as the amplitude of the EMRI orbit increases. Meanwhile the same is for Phifield. The boundary at infinity and the initial values being satisfied the solution for this system of equations, in Schwarzschild space time can be written as follows.

Computer Algebra Solution

Now to Solve for ϕ ...

With these findings the computer algebra solutions can be written down

This set of equations matches the major features of the solution found above and differs only in the simplifying assumptions needed to make the math doable by Mathematica. This gives me confidence that the math was done correctly by hand.

Plotting The Fields

For plotting we need the constants of nature for our plots. For this a choice of units must be made. Given that the systems we are going to study are of broadly solar system scale the unit of time will be years, the unit of distance will be astronomical units and the unit of mass will be solar masses. This choice also facilitates computation by keeping the numbers manageable for any reasonable desktop computer. The ranges of values available are chosen to be certain that the graphs produced will show the important features. In this system of units the gravitational constant G will be given by

A constant κ appears multiple times in the equations it must also be defined.

A crucial consideration is the range of frequencies that it would even be reasonable to consider. Mathematically any arbitrary frequency can be input. Physically certain frequencies will result in physically impossible situations.

The range of frequencies will be from 320 cycles pear year to 320000 cycles per year. Upon careful consideration of all the factors the parameters used in all of these plots vary over the following ranges.

Take note of what the vertical axis does as you move the slider increasing the radial distance r. This shows that the inverse square law is being obeyed.

The following analyses will be done based on the solutions found by hand.

Tensor Analysis

Following with the textbook and in reference to CQ Gengs work using equation 7.7 in Caroll I was able to solve for the metric of the gravitational wave in terms of the Ricci curvature itself.

We will also use the trace reverse perturbation as shown in the book.

We will also work under the Lorenz gauge condition.

Using this gauge condition, substituting into equation 15 and simplifying results in the next equation.

To simplify equation 16 requires integration by parts then dropping terms that contain a derivative of the scalar field. So Ricci modified will be.

The following form allows for evaluating equation 16 more easily.

Carrying out integration by parts again, and dropping terms that have the derivative of the scalar field ϕ. In equations 18 and 19 R is the unmodified scalar.

Finding the strain scalar means multiplying by the Minkowski metric one more time.

Expanding this with series so the undoable integrals can be done. This is at the cost of exact precision. .

Now this integral can be done.

The scalar field works out to the following. The closed cells above this show the details of this computation.

For completeness the full tensor is found simply by application of the metric.

Thus we get the full gravitational wave

Plots of h

So we will plot the scalar as it is easy to plot in 1d and is directly comparable to data.

I do like how this compares to the available LIGO data at least in waveform and shape.

We will also plot the percentage difference between the modified and unmodified theories. Since the modifications to GR actually factor out this percentage difference takes on a simple form.

Plotting this percentage difference we get the following graph.

Now let us look at the percentage difference as a function of distance out to galactic cluster scale.

The percentage difference as a function of distance out to solar system scale.

Energy Considerations

Now that we have confidence in the solutions to the field equations, we will consider how the stress-energy invariant behaves in this model. This invariant will be a physically observable quantity that is proportional to and will give the same wave form as the graphs of strain vs time which are standard in gravitational wave physics.

The stress energy invariant is found directly in the field equations with a bit of algebraic manipulation.

This was done because the general solutions, shown again below have a great deal of complication. For R they contain terms that are quadratic in ϕ which itself depends on R and is linear in R.

Initially these were simplified to the following form.

Now R and ϕ as we need them to compute T can be done. Now for the Ricci scalar we have....

ϕ is very similar to Ricci

I need to take □R and □ϕ. First □R

The trace of the stress energy tensor for making these plots will be.

The result is that the output is so big Mathematica does not want to show all of it. The cell for this will be closed by default.

First let us do this with only the term that corresponds to pure, unaltered, General Relativity. A plot of the following term.

Now here is a similar graph with all the modifications available. With this interactive plot one can view how the wave will look from different points r, and angles ϕ (in the plane of the orbit) and θ the polar angle.

One can also vary the Starobinsky constant β and the coupling constant between gravity and the scalar field ξ. The mass of the Black hole M, amplitude of the gravitational wave, as well as the scalar field. Last but not least the frequency of the orbit Ω and of the fields ω can be manipulated. In reality only certain combinations may be possible. For example, we can presume that Ω=ω but with this as it is we can explore what if Ω is not equal to ω.

The percentage difference between the modified and unmodified stress energy invariants.

How different will the energy be from the pure GR case? To answer this question I will plot a manipulate for the absolute percentage difference between the stress energy invariant for pure GR with the stress energy invariant for the modified theory. This works out analytically as follows.

Now a plot of the percentage difference between the pure GR case and the modified GR case will be.

The question is will we be able to see these effects, even in principle, at the distances and masses of known nearby and distant Black holes? The following manipulate is created to do just that. This will be for masses from those of the smallest and nearest binary black hole candidates all the way up to distant and super massive ones.

Repeating the same computation with the distances at solar system scale.

Percent change in the integrated stress energies.

This section is concerned with integrating the stress energy invariant to see what we find. Then finding the percent change due to the modified theory VS pure GR.

I was directed to integrate this quantity, the stress energy invariant, with respect to time.

Mathematica can integrate the pure GR part with respect to time t easily.

Doing this for the full modified theory leads to a computation that seems to run without end. Even if left to run overnight it will not reach a result. This may in part be due to the limits of integration. Rather than putting a limit at zero... I will put it at the time at which the EMRI would contact the event horizon. Schwarzschild radius /c . Lets see if this makes a difference....

This has been left to run for up to 18 hours and it produces no result.

Trying something manually

This is why I feel it is better to proceed manually and integrate this not over time t but either the proper time τ, or some other invariant perhaps even R.

Upon integrating by parts the last term will vanish identically. Rearranging in order of escalating powers of R.

As for the integral over the killing vector that is easily done. Evaluating the definite integral gives

The result becomes even simpler.

The best way to interpret this integral is to compare it to the same integral over the unmodified theory by way of a percentage difference. In human readable terms.

Simplified

Every multiple of 10^14 AU is 1.58 billion ly, or 484.8 Mpc. Dark matter and dark energy type phenomena should do this and their effects on billion light year scales are not small.

Plotting this same thing on the scale of a solar system with a Sag A* mass black hole sitting in the center of it. On this scale how big of a percentage difference from GR and other known effects could be acceptable.

{r, 100, 1.974*^-4}

Plots In The Traditional Paper

These are the plots shown in the paper. These and infinite other plots can be produced using the above interactive graphs. The parameters used in creating these plots were as follows, with all other variables set to maximum.

Plots of the fields R and ϕ

This is the waveform for the scalar field ϕ. Here it is being measured in the same units as R. However note the time scale on this graph. The divisions are two orders of magnitude smaller.

This is a plot of the metric waveform, with modifications active. This is the quantity that LISA would directly observe. Compare this graph to the graph of R above. Note the time scales are the same but the amplitudes are very different. Large changes in the amplitude of h make little difference to the amplitude of R. Conversely large changes in the metric perturbation lead to comparatively small changes in curvature.

This is a plot of the metric waveform, with modifications active. This is the quantity that LISA would directly observe.

This is a plot of the percent difference in metric waveform from pure GR and the modified theory as a function of time. This shows that this modified model predicts the same behavior as GR far from the time of merger but will differ from it by a tiny percentage just before the merger. The ring down phase of the merger is where we would need to look for these differences.

The percent difference in the metric waves as a function of distance at intergalactic scale shows us that the variation from GR is not great and approaches GR as distance increases.

This is a plot of the percent difference in the metric strain with the modifications as a function of distance at solar system scale. Take special note of the size of the percent differences and the sharp peaks that appear. These sharp peaks are points where the ratio taken as part of the percent difference leads to points which get close to being undefined. They are not physical singular points but artefacts of this percent difference analysis.

This is a plot of the percent difference in the metric strain with the modifications as a function of distance at solar system scale.

Plots relating to the stress energy invariant T.

This graph is a plot of the stress energy invariant without any modifications.

This graph is a plot of the stress energy invariant with all modifications active.

Comparing these two graphs the most notable difference is that with the modifications active the vertical scale maxes out one order of magnitude higher. This would indicate that a large quantity of energy is being pumped into the scalar field and the interaction between the scalar field and the gravitational filed. Given that the scalar field is intended to be dark matter it would be sensible if the effect it had on the energy invariant was large.

The percentage differences in the energies are small at all distances. This means we can rule out dramatic differences in the trajectories of objects.

The same graph as the above but zooming in at solar system scale.

Next we look at a plot of the same quantity at solar system scale. The percentage difference is tiny, even near the singular points in this computation which are not physical.

Discussion.

From the perspective of detecting the effects of this model with LISA there is a good chance it will.

The values of the parameters.

LISA FP WG Views on Testing Such Theories.

To take its’ derivatives and integrals is not simple. That said, working with these exact solutions, which can be Taylor expanded to get series that allow approximation to any desired precision.

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